Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 769102 10 pageshttpdxdoiorg1011552013769102

Research ArticleA Note on the Triple Laplace Transform and Its Applications toSome Kind of Third-Order Differential Equation

Abdon Atangana

Institute for Groundwater Studies Faculty of Natural and Agricultural Sciences University of the Free StateBloemfontein 9300 South Africa

Correspondence should be addressed to Abdon Atangana abdonatanganayahoofr

Received 25 March 2013 Accepted 20 May 2013

Academic Editor R K Bera

Copyright copy 2013 Abdon Atangana This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We introduced a relatively new operator called the triple Laplace transformWe presented some properties and theorems about therelatively new operator We examine the triple Laplace transform of some function of three variables We make use of the operatorto solve some kind of third-order differential equation called ldquoMboctara equationsrdquo

1 Introduction

The topic of partial differential equations is one of the mostimportant subjects in mathematics and other sciences Thebehaviour of the solution very much depends essentiallyon the classification of PDEs therefore the problem ofclassification for partial differential equations is very naturaland well known since the classification governs the sufficientnumber and the type of the conditions in order to determinewhether the problem is well posed and has a unique solutionThe Laplace transform has been intensively used to solvenonlinear and linear equations [1ndash7] The Laplace transformis used frequently in engineering and physics the output of alinear time invariant system can be calculated by convolvingits unit impulse response with the input signal Performingthis calculation in Laplace space turns the convolution intoa multiplication the latter is easier to solve because of itsalgebraic form The Laplace transform can also be usedto solve differential equations and is used extensively inelectrical engineering [1ndash7] The Laplace transform reducesa linear differential equation to an algebraic equation whichcan then be solved by the formal rules of algebraThe originaldifferential equation can then be solved by applying theinverse Laplace transform The English electrical engineerOliver Heaviside first proposed a similar scheme althoughwithout using the Laplace transform and the resulting oper-ational calculus is credited as theHeaviside calculus RecentlyKılıcman et al [8ndash11] extended the Laplace transform to

the concept of double Laplace transform This new operatorhas been intensively used to solve some kind of differentialequation [11] and fractional differential equations The aimof this work is to extend the Laplace transform to the tripleLaplace transform We will start with the definition of thetriple Laplace transform

2 Definitions and Theorems

Definition 1 Let 119891 be a continuous function of three vari-ables then the triple Laplace transformof119891(119909 119910 119905) is definedby

119871119909119910119905

[119891 (119909 119910 119905)]

= 119865 (119901 119904 119896) ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

(1)

where 119909 119910 119905 gt 0 and 119901 119904 119896 are Laplace variables and

119891 (119909 119910 119905)

=1

2120587119894int120572+119894infin

120572minus119894infin

119890119901119909

times [1

2120587119894int120573+119894infin

120573minus119894infin

119890119904119910

2 Abstract and Applied Analysis

times [1

2120587119894int120583+119894infin

120583minus119894infin

119890119896119905

times 119865(119901 119904 119896) 119889119896] 119889119904] 119889119901

(2)

is the inverse triple Laplace transform

Property 2 Assuming that the continuous function 119891(119909 119910 119905)is triple Laplace transformable then

119871119905119910119909

[1205973119891 (119909 119910 119905)

120597119909120597119910120597119905]

= 119901119904119896119865 (119901 119904 119896) minus 119901119904119865 (119901 119904 0) minus 119901119904119865 (119901 0 119896)

+ 119901119865 (119901 0 0) minus 119904119896119865 (0 119904 119896) + 119904119865 (0 119904 0)

+ 119896119865 (0 0 119896) minus 119865 (0 0 0)

119871119909119909119905

[1205973119891 (119909 119910 119905)

1205971199051205971199092]

= 1198961199012119865 (119901 119910 119896) minus 119901119896119865 (0 119910 119896) minus120597119865 (0 119910 119896)

120597119909

minus 1199012119865 (119901 119910 0) + 119901119865 (0 119910 0) +120597119865 (0 119910 0)

120597119909

119871119909119909119909

[1205973119891 (119909 119910 119905)

1205971199093]

= 1199013119865 (119901 119910 119905) minus 1199012119865 (0 119910 119905)

minus 119901120597119865 (0 119910 119905)

120597119909minus

1205972119865 (0 119910 119905)

1205971199092

(3)

3 Uniqueness and Existence of the TripleLaplace Transform

In this section we will study the uniqueness and existenceof triple Laplace transform First of all let 119891(119909 119910 119905) bea continuous function on the interval [0infin) which is ofexponential order that is for some 119886 119887 119888 isin 119877 Consider

sup119909119910119905gt0

100381610038161003816100381610038161003816100381610038161003816

119891 (119909 119910 119905)

exp [119886119909 + 119887119910 + 119888119905]

100381610038161003816100381610038161003816100381610038161003816lt 0 (4)

Under the previous condition the triple Laplace transform

119865 (119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

(5)

exists for all 119901 gt 119886 119904 gt 119887 and 119896 gt 119888 and is in actualityinfinitely differentiable with respect to 119901 gt 119886 119904 gt 119887 and 119896 gt 119888All functions in this study are assumed to be of exponentialorder The following theorem shows that 119891(119909 119910 119905) can beuniquely obtained from 119865(119901 119904 119905)

Theorem 3 Let 119891(119909 119910 119905) and 119892(119909 119910 119905) be continuous func-tions defined for 119909 119910 119905 ge 0 and having Laplace transforms119865(119901 119904 119896) and 119866(119901 119904 119896) respectively If 119865(119901 119904 119896) = 119866(119901 119904 119896)then 119891(119909 119910 119905) = 119892(119909 119910 119905)

Proof From the definition of the inverse Laplace transform if120572 120573 and 120583 are sufficiently large then the integral expressionby

119891 (119909 119910 119905)

=1

2120587119894int120572+119894infin

120572minus119894infin

119890119901119909

times [1

2120587119894int120573+119894infin

120573minus119894infin

119890119904119910

times [1

2120587119894int120583+119894infin

120583minus119894infin

119890119896119905

times 119865(119901 119904 119896) 119889119896] 119889119904] 119889119901

(6)

for the triple inverse Laplace transform can be used to obtain

119891 (119909 119910 119905)

=1

2120587119894int120572+119894infin

120572minus119894infin

119890119901119909

times [1

2120587119894int120573+119894infin

120573minus119894infin

119890119904119910

times [1

2120587119894int120583+119894infin

120583minus119894infin

119890119896119905

times 119865(119901 119904 119896) 119889119896] 119889119904] 119889119901

(7)

By hypothesis we have that 119865(119901 119904 119896) = 119866(119901 119904 119896) thenreplacing this in the previous expression we have the follow-ing

119891 (119909 119910 119905)

=1

2120587119894int120572+119894infin

120572minus119894infin

119890119901119909

times [1

2120587119894int120573+119894infin

120573minus119894infin

119890119904119910

times [1

2120587119894int120583+119894infin

120583minus119894infin

119890119896119905

times 119866(119901 119904 119896) 119889119896] 119889119904] 119889119901

(8)

Abstract and Applied Analysis 3

which boil down to119891 (119909 119910 119905)

=1

2120587119894int120572+119894infin

120572minus119894infin

119890119901119909

times [1

2120587119894int120573+119894infin

120573minus119894infin

119890119904119910

times [1

2120587119894int120583+119894infin

120583minus119894infin

119890119896119905

times 119866(119901 119904 119896) 119889119896] 119889119904] 119889119901

= 119892 (119909 119910 119905)

(9)

and this proves the uniqueness of the triple Laplace trans-form

Theorem 4 If at the point (119901 119904 119896) the integrals

1198651

(119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 1198911

(119909 119910 119905) 119889119909 119889119910 119889119905

1198652

(119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 1198912

(119909 119910 119905) 119889119909 119889119910 119889119905

(10)

are convergent and in addition if

1198653

(119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 1198913

(119909 119910 119905) 119889119909 119889119910 119889119905

(11)

is absolutely convergent then the following expression

119865 (119901 119904 119896) = 1198651

(119901 119904 119896) 1198652

(119901 119904 119896) 1198653

(119901 119904 119896) (12)

is the Laplace transform of the function

119891 (119909 119910 119905)

= int119905

0

int119910

0

int119909

0

1198913

(119909 minus (1199091

+ 120588) 119910 minus (1199101

+ 120590)

119905 minus (1199051

+ 120591)) 1198912

(1199091

minus 120588 1199101

minus 120590 1199051

minus 120591)

times 1198911

(120588 120590 120591) 119889120588 119889120590 119889120591

(13)

and the integral

119865 (119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

(14)

is convergent at the point (119901 119904 119896) for the readers who areinterested they can see the proof in [11 12]

Theorem 5 A function 119891(119909 119910 119905) which is continuous on[0 infin) and satisfies the growth condition (4) can be recoveredfrom only 119865(119901 119904 119896) as

119891 (119909 119910 119905) = lim1198991rarrinfin

1198992rarrinfin

1198993rarrinfin

(minus1)1198991+1198992+1198993

119899111989921198993

(1198991

119909)1198991+1

(1198992

119910)1198992+1

times (1198993

119905)1198993+1

Χ1198991+1198992+1198993 [1198991

1199091198992

1199101198993

119905]

(15)

Evidently the main difficulty in using Theorem 5 for com-puting the inverse Laplace transform is the repeated symbolicdifferentiation of 119865(119901 119904 119896)

Let us see how Theorem 5 can be applicable Let usconsider the following functions

119891 (119909 119910 119905) = exp [minus119886119909 minus 119887119910 minus 119888119905] (16)

Naturally the triple Laplace transform of the previous func-tion is given later as

119865 (119901 119904 119896) =1

(119901 minus 119886) (119904 minus 119887) (119896 minus 119888) (17)

Nowapplying the high-ordermixedderivative to the previousexpression we obtain the following

1205971198991+1198992+1198993 [119865 (119901 119904 119896)]

120597119901119899112059711990411989921205971198961198993= 119899111989921198993(minus1)1198991+1198992+1198993

times (119886+119875)minus1minus1198991(119904+119887)

minus1minus1198992(119888+119896)

minus1minus1198993

(18)

ApplyingTheorem 5 in the previous expression we obtain thefollowing result

119891 (119909 119910 119905) = lim1198991rarrinfin

1198992rarrinfin

1198993rarrinfin

1198991

1+11989911198992

1+11989921198993

1+1198993

1199091198991+11199101198992+11199051198993+1(119886 +

1198991

119909)minus1198991minus1

times (119887 +1198992

119910)minus1198992minus1

(119888 +1198993

119905)minus1198993minus1

(19)

Making a change of variable in the previous expression weobtain the following simplified result

119891 (119909 119910 119905) = lim1198991rarrinfin

1198992rarrinfin

1198993rarrinfin

(1 +1198861198991

119909)minus1198991minus1

(1 +1198871198992

119910)minus1198992minus1

times (1 +1198881198993

119905)minus1198993minus1

(20)

Using together the application of logarithm and theLrsquoHopitalrsquos rule on the previous expression we arrive at thefollowing result

ln (119891 (119909 119910 119905)) = minus119886119909 minus 119887119910 minus 119888119905 997904rArr 119891 (119909 119910 119905)

= exp [minus119886119909 minus 119887119910 minus 119888119905] (21)

4 Abstract and Applied Analysis

4 Some Properties of TripleLaplace Transform

In this section we present some properties of the tripleLaplace transform Note that these properties follow fromthose of the double Laplace transform introduced byKılıcman and Eltayeb [8]The properties of the triple Laplacetransform will enable us to find further transform pairs119891(119909 119910 119905) 119865(119901 119904 119896)

(i) 119865 (119901 + 119886 119904 + 119887 119896 + 119889)

= 119871119909119910119905

[119890minus119886119909minus119910119887minus119888119905119891 (119909 119910 119905)] (119901 119904 119896) (22)

We will present the proof

119871119909119910119905

[119890minus119886119909minus119910119887minus119888119905119891 (119909 119910 119905)] (119901 119904 119896)

= ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905] exp [minus119886119909]

times exp [minus119887119910] exp [minus119888119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

intinfin

0

exp [minus119901119909] exp [minus119886119909]

times (∬infin

0

exp [minus119904119910] exp [minus119896119905] exp [minus119887119910]

times exp [minus119888119905] 119891 (119909 119910 119905) 119889119905 119889119910) 119889119905

(23)

Note that the integral inside the bracket satisfies the proper-ties of the double Laplace transform and is given as [11]

(∬infin

0

exp [minus119904119910] exp [minus119896119905] exp [minus119887119910] exp [minus119888119905]

times 119891 (119909 119910 119905) 119889119905 119889119910) = 119865 (119909 119904 + 119887 119896 + 119889)

(24)

Thus

intinfin

0

exp [minus119901119909] exp [minus119886119909] 119865 (119909 119904 + 119887 119896 + 119889) 119889119905

= 119865 (119901 + 119886 119904 + 119887 119896 + 119889)

(25)

and this completes the proof(ii) The following can also be observed

1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) = 119871

119909119910119905[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896) (26)

We will present the proof

119871119909119910119905

[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896)

= ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905]

times 119891 (120572119909 120573119910 120574119905) 119889119909 119889119910 119889119905

intinfin

0

exp [minus119901119909] (∬infin

0

exp [minus119904119910] exp [minus119896119905]

times 119891 (120572119909 120573119910 120574119905) 119889119910 119889119905) 119889119909

(27)

Note that the double integral inside the bracket satisfies theproperty of the double Laplace transform as [11]

(∬infin

0

exp [minus119904119910] exp [minus119896119905] 119891 (120572119909 120573119910 120574119905) 119889119910 119889119905)

=1

120573120574119865 (120572119909

119904

120573119896

120574)

(28)

Thus

119871119909119910119905

[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896)

= intinfin

0

exp [minus119901119909]1

120573120574119865 (120572119909

119904

120573119896

120574) 119889119909

=1

120572120573120574119865 (

119901

120572

119904

120573119896

120574)

(29)

and this completes the proof(iii) The following property can also be observed

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= 119871119909119910119905

[(minus1)119899+119898+V119909119899119910119898119905V119891 (119909 119910 119905)] (119901 119904 119896)

(30)

We will present the proof

119865 (119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905]

times 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

(31)

Then

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899+119898+V

120597119901119899120597119904119899120597119896V(∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905)

(32)

Now making use of the convergence properties of theimproper integral involved we can interchange the opera-tion of differentiation and integration and differentiate with

Abstract and Applied Analysis 5

respect to 119901 119904 and 119896 under the integral sign Thus we arriveat the following expression

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899

120597119901119899intinfin

0

exp [minus119901119909]

times (120597119898+V

120597119904119899120597119896V∬infin

0

exp [minus119904119910] exp [minus119896119905]

times 119891 (119909 119910 119905) 119889119910 119889119905) 119889119909

(33)

Note that the expression in the bracket satisfies the propertyof the double Laplace transform as [11]

120597119898+V

120597119904119899120597119896V∬infin

0

exp [minus119904119910] exp [minus119896119905] 119891 (119909 119910 119905) 119889119910 119889119905

= 119871119910119905

[(minus1)119898+V119910119898119905V119891 (119909 119910 119905)] (119904 119896)

(34)

Thus

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899

120597119901119899intinfin

0

exp [minus119901119909]

times (119871119910119905

[(minus1)119898+V119910119898119905V119891 (119909 119910 119905)] (119904 119896)) 119889119909

(35)

And finally we obtain

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= 119871119909119910119905

[(minus1)119899+119898+V119909119899119910119898119905V119891 (119909 119910 119905)] (119901 119904 119896)

(36)

and this completes the proofNow using the previous three properties we will show the

proof of Theorem 5

Proof of Theorem 5 Let us define the set of functions depend-ing on parameters 119898 119899 and V as

ℎ119898119899V (119909 119910 119905) =

119898119898+1119899119899+1VV+1

119898119899V119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905 (37)

It worth noting that the previous function is a kind of three-dimensional density of probability and it therefore followsthat

∭infin

0

ℎ119898119899V (119909 119910 119905) 119889119909 119889119910 119889119905 = 1 (38)

In addition of this we will have that

lim119898rarrinfin

119899rarrinfin

Vrarrinfin

∭infin

0

ℎ119898119899V (119909 119910 119905) 120595 (119909 119910 119905) 119889119909 119889119910 119889119905 = 120595 (1 1 1)

(39)

where 120595(119909 119910 119905) is any continuous function Let Ψ(119901 119904 119896)denote the triple Laplace transform of the continuousfunction 120595(119909 119910 119905) However if one defines the function119872(119909 119910 119905) = 119891(119909120572 119910120573 119905120574) making use of the second pro-perty established in (29) we arrive at the following

1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) = 119871

119909119910119905[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896) (40)

Here if one applies the third property in particular by rep-lacing 119901 = 119898119909 119904 = 119899119910 119896 = V119905 as follows

119871119909119910119905

(119872 (119909 119910 119905)) =1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) (41)

120597119899+119898+V [119871119909119910119905

(119872 (119909 119910 119905))]

120597119901119899120597119904119899120597119896V

=120597119899+119898+V [(1120572120573120574) 119865 (119901120572 119904120573 119896120574)]

120597119901119899120597119904119899120597119896V

=1

120572119898+1120573119899+1120574V+1

times120597119899+119898+V [119865 (119901120572 119904120573 119896120574)]

120597119901119899120597119904119899120597119896V

(42)

Now let us put 120595(119909 119910 119905) = 119890minus119901119909minus119904119910minus119896119905119872(119909 119910 119905) Now if wemake use of (38) we obtain the following

120595 (1 1 1) = 119890minus119901minus119904minus119896119872 (1 1 1) = 119890minus119901minus119904minus119896119891 (120572 120573 120574)

= lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V∭infin

0

119909119898119910119899119905V119890minus119901119909minus119904119910minus119896119905

times 119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905) 119889119909 119889119910 119889119905

= lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V119871119909119910119905

[119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905)]

(43)

Now taking into account properties (i) and (ii) (42) togetherwith the function 119872(119909 119910 119905) we arrive at the following

119871119909119910119905

[119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905)]

= (minus1)119898+119899+V

120597119899+119898+V [119871119909119910119905

(119890minus119898119909minus119899119910minus119896119905Ψ (119909 119910 119905)) (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= (minus1)119898+119899+V 1

120572119898120573119899120574V

times120597119899+119898+V [119871

119909119910119905(Ψ (119909 119910 119905)) (119901 + 119898 119904 + 119899 119896 + V)]

120597119901119899120597119904119899120597119896V

= (minus1)119898+119899+V 1

120572119898120573119899120574V

times((120597119899+119898+V [119871119909119910119905

(119891 (120572119909 120573119910 120574119905))

times(119901+119898

120572119904+119899

120573119896+V

120574)]) (120597119901119899120597119904119899120597119896V)

minus1

)

6 Abstract and Applied Analysis

= (minus1)119898+119899+V

times1

120572119898120573119899120574V120597119899+119898+V [119865 ((119901+119898) 120572 (119904+119899) 120573 (119896+V) 120574)]

120597119901119899120597119904119899120597119896V

(44)

Now observe that from (44) with the fact that 119891(120572 120573 119905) =

120595(1 1 1)119890119901+119904+119896 we arrive at the following

119891 (120572 120573 119905)

= 119890119901+119904+119896 lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V

times (119898

120572)119898+1

(119899

120573)119899+1

(V

120574)V+1

times120597119899+119898+V [119865 ((119901 + 119898) 120572 (119904 + 119899) 120573 (119896 + V) 120574)]

120597119901119899120597119904119899120597119896V

(45)

The previously mentioned is true for any 119901 119904 119896 in thecomplete space in particular for 119901 = 0 119904 = 0 119896 = 0 andin this case Theorem 5 is covered

5 Application to Third-Order PartialDifferential Equation

In this section we present the application of this operatorfor solving some kind of third-order partial differentialequations

Example 1 consider the following third-order partial differ-ential equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = 0 (46)

The previous equation is called the Mboctara equation and issubjected to the following boundaries and initial conditions

119906 (119909 119910 0) = 119890119909+119910 119906 (119909 0 119905) = 119890119909minus119905

119906 (0 119910 119905) = 119890119910minus119905 119906 (119909 119910 1) = 119890119909+119910minus1(47)

Now applying the triple Laplace transform on both sides of(46) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896) = 119866 (119901 119904 119896) (48)

Here119866 (119901 119904 119896) = 119901119904119880 (119901 119904 0) + 119901119904119880 (119901 0 119896) minus 119901119880 (119901 0 0)

+ 119904119896119880 (0 119904 119896) minus 119904119880 (0 119904 0) minus 119896119880 (0 0 119896)

+ 119880 (0 0 0)

(49)

Factorising the right side of equation (49) we obtain thefollowing

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896 (50)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896] = 119890119909+119910minus119905 (51)

This is the exact solution for Mboctara equation

Example 2 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = minus119890119909minus2119910+119905 (52)

subjected to the following initial and boundaries conditions

119906 (119909 0 0) = 119890119909 120597119905119906 (119909 0 119905) = 119890119909+119905 120597

119909119906 (119909 0 119905) = 119890119909+119905

119906 (0 0 0) = 1 119906 (119909 05 119905) = 119890119909+119905minus1

(53)

Now applying the triple Laplace transform on both sidesof (52) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) minus1

(1 + 119901) (2 + 119904) (1 + 119896)

(54)

Factorising the right side of (54) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896

(55)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896]

= 119890119909minus2119910+119905

(56)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 3 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = cos (119909) cos (119910) cos (minus119905)

minus sin (119909) sin (119910) sin (minus119905)

(57)

subjected to the following initial and boundaries conditions

119906 (119909 119910 0) = cos (119909) cos (119910)

120597119905119906 (119909 119910 0) = 120597

119909119906 (0 119910 119905) = 120597

119910119906 (119909 0 119905) = 0

119906 (119909120587

2 119905) = 119906 (119909 119910

120587

2) = 119906 (

120587

2 119910 119905) = 0

(58)

Abstract and Applied Analysis 7

Table 1 Table of triple Laplace transform for some function of three variables

Functions119891(119909 119910 119905) Triple laplace transform 119865(119901 119904 119896)

119886119887119888119886119887119888

119901119904119896

1199091199101199051

119901211990421198962

119909119899119910119898119905V 119899 119898 V are natural numbers 119896minus1minusV119904minus1minus119898119901minus119899minus1Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119909119899119910119898119905V119890minus119886119909minus119887119910minus119888119905 (119896 + 119888)minus1minusV(119904 + 119887)minus1minus119898(119901 + 119886)minus119899minus1

Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119890minus119886119909minus119887119910minus1198881199051

(119886 + 119901) (119887 + 119904) (119888 + 119896)

cos(119909) cos(119910) cos(119905)119896119904119901

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin(119909) sin(119910) sin(119905)1

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin (119909 + 119910 + 119905)minus1 + 119901119904 + 119896 (119901 + 119904)

(1 + 1199012) (1 + 1199042) (1 + 1198962)

cos(119909 + 119910 + 119905) minus119896 + 119901 + 119904 minus 119896119901119904

(1 + 1199012) (1 + 1199042) (1 + 1198962)

radic119909119910119905120587radic120587

83radic119896119904119901

119890119886119909+119910119887+119888119905 sinh(119886119909)sinh(119887119910) sinh(119888119905)(119887) (119888) (119886)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

119890119886119909+119910119887+119888119905 cosh(119886119909) cosh(119887119910) cosh(119888119905)(119887 minus 119904) (119888 minus 119896) (119886 minus 119901)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

Erf [119886

2radic119909]Erf[

119887

2radic119910]Erf [

119888

2radic119905]

119890minusradic1198882119896minusradic1198872119904

119896119901119904(minus1 + 119890minus

radic1198882119896) (1 minus 119890minus

radic1198862119901) (minus1 + 119890minus

radic1198872119904)

sin(119886119909)

119909

sin(119887119910)

119910

sin(119888119905)

119905arctan(

radic1198862

119901) arctan(

radic1198872

119904) arctan(

radic1198882

119896)

cos(119886119909)

119909119899cos(119887119910)

119910119898cos(119888119905)

119905V

119896minus1+V(1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898 cos [119888119905] cos((minus1 + 119898) arctan[radic1198872

119904]) Γ (1 minus 119898) Γ (1 minus V)

times (1 +1198862

1199012)

12(minus1+119899)

119901minus1+119899 cos((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

sin(119886119909)

119909119899sin(119887119910)

119910119898sin(119888119905)

119905V

119896minus1+V (1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898Γ (1 minus 119898) Γ (1 minus V) (1 +1198862

1199012)

12(minus1+119899)

times 119901minus1+119899Γ (1 minus 119899) sign (119886) sin ((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

119869119899

(119909) 119869119899

(119910) 119869119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 minus

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199042)

[Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199012)

119868119899

(119909) 119868119899

(119910) 119868119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 1 + 119899

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199042)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199012)

8 Abstract and Applied Analysis

500

400

300

200

100

06

4

2

0Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (41)

x

(a)

500

400

300

200

100

06

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

y x

(b)

64

2

0

0

20

40

60

Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

x

(c)

6

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (411)

minus1

minus05

0

05

1

xy

(d)

Figure 1 Numerical simulation of the exact solutions of the Homogeneous and non-homogeneous Mboctara equations

Now applying the triple Laplace transform on both sides of(57) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) +119896119904119901

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

minus1

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

(59)

Factorising the right side of (59) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962 )

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042 ) (minus1 + 1198962)

1 + 119901119904119896

(60)

Abstract and Applied Analysis 9

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905)

= 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896]

= cos (119909) cos (119910) cos (minus119905)

(61)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 4 consider the following nonlinear nonhomoge-neous with variable coefficient Mboctara equation

119890119909+119910+119905120597119909119910119905

119906 (119909 119910 119905) minus 31199062 (119909 119910 119905) + 119890119909+119910+119905119906 (119909 119910 119905)

= 1198902119909+2119910+2119905

119906119909

(119909 119910 0) = 119890119909+119910 119906 (0 0 0) = 1

119906 (1 0 0) = 119890 120597119909119910119905

119906 (0 0 0) = 1

(62)

Now applying the triple Laplace transform on both sidesof (62) and then using the properties of the triple Laplacetransform and after factorising as in the previous examplesand taking the inverse triple Laplace transform we obtainthe following as an exact solution of this type of Mboctaraequation

119906 (119909 119910 119905) = 119890119909+119910+119905 (63)

Thenumerical simulations of the exact solutions of theMboc-tara equation are depicted in Figure 1(a) (41) Figure 1(b)(46) Figure 1(c) (46) and Figure 1(d) (411) respectively

6 Triple Laplace Transform of SomeFunctions of Three Variables

In this section we examine the triple Laplace transform ofsome functions in Table 1

119871119909119910119905

(119884119899

(119909) 119884119899

(119910) 119884119899

(119905))

= 8minus119899(119896119904119901)minus1minus119899

1198621199041198883 [119899120587]

times ( minus (41198962)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1198962)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1198962))

times ( minus (41199042)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199042)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199042))

times ( minus (41199012)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199012)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199012))

(64)

7 Conclusion

This work presents the definition of the triple Laplace trans-form Some triple Laplace transform is presented in Table 1Some theorems and properties of this new relatively newoperator are presented Applications of the new operator forsolving some kind of third-order partial differential equationscalledMboctara equation are presentedNumerical solutionsof the Mboctara equation are given

References

[1] G L Lamb Jr Introductory Applications of Partial DifferentialEquations with Emphasis on Wave Propagation and DiffusionJohn Wiley amp Sons New York NY USA 1995

[2] U T Myint Differential Equations of Mathematical PhysicsAmerican Elsevier New York NY USA 1980

[3] C Constanda Solution Techniques for Elementary Partial Differ-ential Equations Chapman amp HallCRC New York NY USA2002

[4] D G Duffy Transform Methods for Solving Partial DifferentialEquations CRC Press New York NY USA 2004

[5] A Babakhani and R S Dahiya ldquoSystems of multi-dimensionalLaplace transforms and a heat equationrdquo in Proceedings of the16th Conference onAppliedMathematics vol 7 ofElectronicJour-nal of Differential Equations pp 25ndash36 University of CentralOklahoma Edmond Okla USA 2001

[6] Y A Brychkov H -J Glaeske A P Prudnikov and V K TuanMultidimensional Integral Transformations Gordon and BreachScience Publishers Philadelphia Pa USA 1992

[7] A Atangana and A Kilicman ldquoA possible generalization ofacoustic wave equation using the concept of perturbed deriva-tive orderrdquo Mathematical Problems in Engineering vol 2013Article ID 696597 6 pages 2013

[8] A Kılıcman and H Eltayeb ldquoA note on the classifications ofhyperbolic and elliptic equations with polynomial coefficientsrdquoApplied Mathematics Letters vol 21 no 11 pp 1124ndash1128 2008

10 Abstract and Applied Analysis

[9] H Eltayeb A Kılıcman and P Ravi Agarwal ldquoAn analysis onclassifications of hyperbolic and elliptic PDEsrdquo MathematicalSciences vol 6 article 47 2012

[10] H Eltayeb andAKılıcman ldquoAnote ondouble laplace transformand telegraphic equationsrdquo Abstract and Applied Analysis vol2013 Article ID 932578 6 pages 2013

[11] A Kılıcman and H E Gadain ldquoOn the applications of Laplaceand Sumudu transformsrdquo Journal of the Franklin Institute vol347 no 5 pp 848ndash862 2010

[12] R P Kanwal Generalized Functions Theory and ApplicationsBirkhaauser Boston Mass USA 2004

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Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

times [1

2120587119894int120583+119894infin

120583minus119894infin

119890119896119905

times 119865(119901 119904 119896) 119889119896] 119889119904] 119889119901

(2)

is the inverse triple Laplace transform

Property 2 Assuming that the continuous function 119891(119909 119910 119905)is triple Laplace transformable then

119871119905119910119909

[1205973119891 (119909 119910 119905)

120597119909120597119910120597119905]

= 119901119904119896119865 (119901 119904 119896) minus 119901119904119865 (119901 119904 0) minus 119901119904119865 (119901 0 119896)

+ 119901119865 (119901 0 0) minus 119904119896119865 (0 119904 119896) + 119904119865 (0 119904 0)

+ 119896119865 (0 0 119896) minus 119865 (0 0 0)

119871119909119909119905

[1205973119891 (119909 119910 119905)

1205971199051205971199092]

= 1198961199012119865 (119901 119910 119896) minus 119901119896119865 (0 119910 119896) minus120597119865 (0 119910 119896)

120597119909

minus 1199012119865 (119901 119910 0) + 119901119865 (0 119910 0) +120597119865 (0 119910 0)

120597119909

119871119909119909119909

[1205973119891 (119909 119910 119905)

1205971199093]

= 1199013119865 (119901 119910 119905) minus 1199012119865 (0 119910 119905)

minus 119901120597119865 (0 119910 119905)

120597119909minus

1205972119865 (0 119910 119905)

1205971199092

(3)

3 Uniqueness and Existence of the TripleLaplace Transform

In this section we will study the uniqueness and existenceof triple Laplace transform First of all let 119891(119909 119910 119905) bea continuous function on the interval [0infin) which is ofexponential order that is for some 119886 119887 119888 isin 119877 Consider

sup119909119910119905gt0

100381610038161003816100381610038161003816100381610038161003816

119891 (119909 119910 119905)

exp [119886119909 + 119887119910 + 119888119905]

100381610038161003816100381610038161003816100381610038161003816lt 0 (4)

Under the previous condition the triple Laplace transform

119865 (119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

(5)

exists for all 119901 gt 119886 119904 gt 119887 and 119896 gt 119888 and is in actualityinfinitely differentiable with respect to 119901 gt 119886 119904 gt 119887 and 119896 gt 119888All functions in this study are assumed to be of exponentialorder The following theorem shows that 119891(119909 119910 119905) can beuniquely obtained from 119865(119901 119904 119905)

Theorem 3 Let 119891(119909 119910 119905) and 119892(119909 119910 119905) be continuous func-tions defined for 119909 119910 119905 ge 0 and having Laplace transforms119865(119901 119904 119896) and 119866(119901 119904 119896) respectively If 119865(119901 119904 119896) = 119866(119901 119904 119896)then 119891(119909 119910 119905) = 119892(119909 119910 119905)

Proof From the definition of the inverse Laplace transform if120572 120573 and 120583 are sufficiently large then the integral expressionby

119891 (119909 119910 119905)

=1

2120587119894int120572+119894infin

120572minus119894infin

119890119901119909

times [1

2120587119894int120573+119894infin

120573minus119894infin

119890119904119910

times [1

2120587119894int120583+119894infin

120583minus119894infin

119890119896119905

times 119865(119901 119904 119896) 119889119896] 119889119904] 119889119901

(6)

for the triple inverse Laplace transform can be used to obtain

119891 (119909 119910 119905)

=1

2120587119894int120572+119894infin

120572minus119894infin

119890119901119909

times [1

2120587119894int120573+119894infin

120573minus119894infin

119890119904119910

times [1

2120587119894int120583+119894infin

120583minus119894infin

119890119896119905

times 119865(119901 119904 119896) 119889119896] 119889119904] 119889119901

(7)

By hypothesis we have that 119865(119901 119904 119896) = 119866(119901 119904 119896) thenreplacing this in the previous expression we have the follow-ing

119891 (119909 119910 119905)

=1

2120587119894int120572+119894infin

120572minus119894infin

119890119901119909

times [1

2120587119894int120573+119894infin

120573minus119894infin

119890119904119910

times [1

2120587119894int120583+119894infin

120583minus119894infin

119890119896119905

times 119866(119901 119904 119896) 119889119896] 119889119904] 119889119901

(8)

Abstract and Applied Analysis 3

which boil down to119891 (119909 119910 119905)

=1

2120587119894int120572+119894infin

120572minus119894infin

119890119901119909

times [1

2120587119894int120573+119894infin

120573minus119894infin

119890119904119910

times [1

2120587119894int120583+119894infin

120583minus119894infin

119890119896119905

times 119866(119901 119904 119896) 119889119896] 119889119904] 119889119901

= 119892 (119909 119910 119905)

(9)

and this proves the uniqueness of the triple Laplace trans-form

Theorem 4 If at the point (119901 119904 119896) the integrals

1198651

(119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 1198911

(119909 119910 119905) 119889119909 119889119910 119889119905

1198652

(119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 1198912

(119909 119910 119905) 119889119909 119889119910 119889119905

(10)

are convergent and in addition if

1198653

(119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 1198913

(119909 119910 119905) 119889119909 119889119910 119889119905

(11)

is absolutely convergent then the following expression

119865 (119901 119904 119896) = 1198651

(119901 119904 119896) 1198652

(119901 119904 119896) 1198653

(119901 119904 119896) (12)

is the Laplace transform of the function

119891 (119909 119910 119905)

= int119905

0

int119910

0

int119909

0

1198913

(119909 minus (1199091

+ 120588) 119910 minus (1199101

+ 120590)

119905 minus (1199051

+ 120591)) 1198912

(1199091

minus 120588 1199101

minus 120590 1199051

minus 120591)

times 1198911

(120588 120590 120591) 119889120588 119889120590 119889120591

(13)

and the integral

119865 (119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

(14)

is convergent at the point (119901 119904 119896) for the readers who areinterested they can see the proof in [11 12]

Theorem 5 A function 119891(119909 119910 119905) which is continuous on[0 infin) and satisfies the growth condition (4) can be recoveredfrom only 119865(119901 119904 119896) as

119891 (119909 119910 119905) = lim1198991rarrinfin

1198992rarrinfin

1198993rarrinfin

(minus1)1198991+1198992+1198993

119899111989921198993

(1198991

119909)1198991+1

(1198992

119910)1198992+1

times (1198993

119905)1198993+1

Χ1198991+1198992+1198993 [1198991

1199091198992

1199101198993

119905]

(15)

Evidently the main difficulty in using Theorem 5 for com-puting the inverse Laplace transform is the repeated symbolicdifferentiation of 119865(119901 119904 119896)

Let us see how Theorem 5 can be applicable Let usconsider the following functions

119891 (119909 119910 119905) = exp [minus119886119909 minus 119887119910 minus 119888119905] (16)

Naturally the triple Laplace transform of the previous func-tion is given later as

119865 (119901 119904 119896) =1

(119901 minus 119886) (119904 minus 119887) (119896 minus 119888) (17)

Nowapplying the high-ordermixedderivative to the previousexpression we obtain the following

1205971198991+1198992+1198993 [119865 (119901 119904 119896)]

120597119901119899112059711990411989921205971198961198993= 119899111989921198993(minus1)1198991+1198992+1198993

times (119886+119875)minus1minus1198991(119904+119887)

minus1minus1198992(119888+119896)

minus1minus1198993

(18)

ApplyingTheorem 5 in the previous expression we obtain thefollowing result

119891 (119909 119910 119905) = lim1198991rarrinfin

1198992rarrinfin

1198993rarrinfin

1198991

1+11989911198992

1+11989921198993

1+1198993

1199091198991+11199101198992+11199051198993+1(119886 +

1198991

119909)minus1198991minus1

times (119887 +1198992

119910)minus1198992minus1

(119888 +1198993

119905)minus1198993minus1

(19)

Making a change of variable in the previous expression weobtain the following simplified result

119891 (119909 119910 119905) = lim1198991rarrinfin

1198992rarrinfin

1198993rarrinfin

(1 +1198861198991

119909)minus1198991minus1

(1 +1198871198992

119910)minus1198992minus1

times (1 +1198881198993

119905)minus1198993minus1

(20)

Using together the application of logarithm and theLrsquoHopitalrsquos rule on the previous expression we arrive at thefollowing result

ln (119891 (119909 119910 119905)) = minus119886119909 minus 119887119910 minus 119888119905 997904rArr 119891 (119909 119910 119905)

= exp [minus119886119909 minus 119887119910 minus 119888119905] (21)

4 Abstract and Applied Analysis

4 Some Properties of TripleLaplace Transform

In this section we present some properties of the tripleLaplace transform Note that these properties follow fromthose of the double Laplace transform introduced byKılıcman and Eltayeb [8]The properties of the triple Laplacetransform will enable us to find further transform pairs119891(119909 119910 119905) 119865(119901 119904 119896)

(i) 119865 (119901 + 119886 119904 + 119887 119896 + 119889)

= 119871119909119910119905

[119890minus119886119909minus119910119887minus119888119905119891 (119909 119910 119905)] (119901 119904 119896) (22)

We will present the proof

119871119909119910119905

[119890minus119886119909minus119910119887minus119888119905119891 (119909 119910 119905)] (119901 119904 119896)

= ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905] exp [minus119886119909]

times exp [minus119887119910] exp [minus119888119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

intinfin

0

exp [minus119901119909] exp [minus119886119909]

times (∬infin

0

exp [minus119904119910] exp [minus119896119905] exp [minus119887119910]

times exp [minus119888119905] 119891 (119909 119910 119905) 119889119905 119889119910) 119889119905

(23)

Note that the integral inside the bracket satisfies the proper-ties of the double Laplace transform and is given as [11]

(∬infin

0

exp [minus119904119910] exp [minus119896119905] exp [minus119887119910] exp [minus119888119905]

times 119891 (119909 119910 119905) 119889119905 119889119910) = 119865 (119909 119904 + 119887 119896 + 119889)

(24)

Thus

intinfin

0

exp [minus119901119909] exp [minus119886119909] 119865 (119909 119904 + 119887 119896 + 119889) 119889119905

= 119865 (119901 + 119886 119904 + 119887 119896 + 119889)

(25)

and this completes the proof(ii) The following can also be observed

1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) = 119871

119909119910119905[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896) (26)

We will present the proof

119871119909119910119905

[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896)

= ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905]

times 119891 (120572119909 120573119910 120574119905) 119889119909 119889119910 119889119905

intinfin

0

exp [minus119901119909] (∬infin

0

exp [minus119904119910] exp [minus119896119905]

times 119891 (120572119909 120573119910 120574119905) 119889119910 119889119905) 119889119909

(27)

Note that the double integral inside the bracket satisfies theproperty of the double Laplace transform as [11]

(∬infin

0

exp [minus119904119910] exp [minus119896119905] 119891 (120572119909 120573119910 120574119905) 119889119910 119889119905)

=1

120573120574119865 (120572119909

119904

120573119896

120574)

(28)

Thus

119871119909119910119905

[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896)

= intinfin

0

exp [minus119901119909]1

120573120574119865 (120572119909

119904

120573119896

120574) 119889119909

=1

120572120573120574119865 (

119901

120572

119904

120573119896

120574)

(29)

and this completes the proof(iii) The following property can also be observed

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= 119871119909119910119905

[(minus1)119899+119898+V119909119899119910119898119905V119891 (119909 119910 119905)] (119901 119904 119896)

(30)

We will present the proof

119865 (119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905]

times 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

(31)

Then

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899+119898+V

120597119901119899120597119904119899120597119896V(∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905)

(32)

Now making use of the convergence properties of theimproper integral involved we can interchange the opera-tion of differentiation and integration and differentiate with

Abstract and Applied Analysis 5

respect to 119901 119904 and 119896 under the integral sign Thus we arriveat the following expression

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899

120597119901119899intinfin

0

exp [minus119901119909]

times (120597119898+V

120597119904119899120597119896V∬infin

0

exp [minus119904119910] exp [minus119896119905]

times 119891 (119909 119910 119905) 119889119910 119889119905) 119889119909

(33)

Note that the expression in the bracket satisfies the propertyof the double Laplace transform as [11]

120597119898+V

120597119904119899120597119896V∬infin

0

exp [minus119904119910] exp [minus119896119905] 119891 (119909 119910 119905) 119889119910 119889119905

= 119871119910119905

[(minus1)119898+V119910119898119905V119891 (119909 119910 119905)] (119904 119896)

(34)

Thus

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899

120597119901119899intinfin

0

exp [minus119901119909]

times (119871119910119905

[(minus1)119898+V119910119898119905V119891 (119909 119910 119905)] (119904 119896)) 119889119909

(35)

And finally we obtain

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= 119871119909119910119905

[(minus1)119899+119898+V119909119899119910119898119905V119891 (119909 119910 119905)] (119901 119904 119896)

(36)

and this completes the proofNow using the previous three properties we will show the

proof of Theorem 5

Proof of Theorem 5 Let us define the set of functions depend-ing on parameters 119898 119899 and V as

ℎ119898119899V (119909 119910 119905) =

119898119898+1119899119899+1VV+1

119898119899V119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905 (37)

It worth noting that the previous function is a kind of three-dimensional density of probability and it therefore followsthat

∭infin

0

ℎ119898119899V (119909 119910 119905) 119889119909 119889119910 119889119905 = 1 (38)

In addition of this we will have that

lim119898rarrinfin

119899rarrinfin

Vrarrinfin

∭infin

0

ℎ119898119899V (119909 119910 119905) 120595 (119909 119910 119905) 119889119909 119889119910 119889119905 = 120595 (1 1 1)

(39)

where 120595(119909 119910 119905) is any continuous function Let Ψ(119901 119904 119896)denote the triple Laplace transform of the continuousfunction 120595(119909 119910 119905) However if one defines the function119872(119909 119910 119905) = 119891(119909120572 119910120573 119905120574) making use of the second pro-perty established in (29) we arrive at the following

1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) = 119871

119909119910119905[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896) (40)

Here if one applies the third property in particular by rep-lacing 119901 = 119898119909 119904 = 119899119910 119896 = V119905 as follows

119871119909119910119905

(119872 (119909 119910 119905)) =1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) (41)

120597119899+119898+V [119871119909119910119905

(119872 (119909 119910 119905))]

120597119901119899120597119904119899120597119896V

=120597119899+119898+V [(1120572120573120574) 119865 (119901120572 119904120573 119896120574)]

120597119901119899120597119904119899120597119896V

=1

120572119898+1120573119899+1120574V+1

times120597119899+119898+V [119865 (119901120572 119904120573 119896120574)]

120597119901119899120597119904119899120597119896V

(42)

Now let us put 120595(119909 119910 119905) = 119890minus119901119909minus119904119910minus119896119905119872(119909 119910 119905) Now if wemake use of (38) we obtain the following

120595 (1 1 1) = 119890minus119901minus119904minus119896119872 (1 1 1) = 119890minus119901minus119904minus119896119891 (120572 120573 120574)

= lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V∭infin

0

119909119898119910119899119905V119890minus119901119909minus119904119910minus119896119905

times 119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905) 119889119909 119889119910 119889119905

= lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V119871119909119910119905

[119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905)]

(43)

Now taking into account properties (i) and (ii) (42) togetherwith the function 119872(119909 119910 119905) we arrive at the following

119871119909119910119905

[119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905)]

= (minus1)119898+119899+V

120597119899+119898+V [119871119909119910119905

(119890minus119898119909minus119899119910minus119896119905Ψ (119909 119910 119905)) (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= (minus1)119898+119899+V 1

120572119898120573119899120574V

times120597119899+119898+V [119871

119909119910119905(Ψ (119909 119910 119905)) (119901 + 119898 119904 + 119899 119896 + V)]

120597119901119899120597119904119899120597119896V

= (minus1)119898+119899+V 1

120572119898120573119899120574V

times((120597119899+119898+V [119871119909119910119905

(119891 (120572119909 120573119910 120574119905))

times(119901+119898

120572119904+119899

120573119896+V

120574)]) (120597119901119899120597119904119899120597119896V)

minus1

)

6 Abstract and Applied Analysis

= (minus1)119898+119899+V

times1

120572119898120573119899120574V120597119899+119898+V [119865 ((119901+119898) 120572 (119904+119899) 120573 (119896+V) 120574)]

120597119901119899120597119904119899120597119896V

(44)

Now observe that from (44) with the fact that 119891(120572 120573 119905) =

120595(1 1 1)119890119901+119904+119896 we arrive at the following

119891 (120572 120573 119905)

= 119890119901+119904+119896 lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V

times (119898

120572)119898+1

(119899

120573)119899+1

(V

120574)V+1

times120597119899+119898+V [119865 ((119901 + 119898) 120572 (119904 + 119899) 120573 (119896 + V) 120574)]

120597119901119899120597119904119899120597119896V

(45)

The previously mentioned is true for any 119901 119904 119896 in thecomplete space in particular for 119901 = 0 119904 = 0 119896 = 0 andin this case Theorem 5 is covered

5 Application to Third-Order PartialDifferential Equation

In this section we present the application of this operatorfor solving some kind of third-order partial differentialequations

Example 1 consider the following third-order partial differ-ential equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = 0 (46)

The previous equation is called the Mboctara equation and issubjected to the following boundaries and initial conditions

119906 (119909 119910 0) = 119890119909+119910 119906 (119909 0 119905) = 119890119909minus119905

119906 (0 119910 119905) = 119890119910minus119905 119906 (119909 119910 1) = 119890119909+119910minus1(47)

Now applying the triple Laplace transform on both sides of(46) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896) = 119866 (119901 119904 119896) (48)

Here119866 (119901 119904 119896) = 119901119904119880 (119901 119904 0) + 119901119904119880 (119901 0 119896) minus 119901119880 (119901 0 0)

+ 119904119896119880 (0 119904 119896) minus 119904119880 (0 119904 0) minus 119896119880 (0 0 119896)

+ 119880 (0 0 0)

(49)

Factorising the right side of equation (49) we obtain thefollowing

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896 (50)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896] = 119890119909+119910minus119905 (51)

This is the exact solution for Mboctara equation

Example 2 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = minus119890119909minus2119910+119905 (52)

subjected to the following initial and boundaries conditions

119906 (119909 0 0) = 119890119909 120597119905119906 (119909 0 119905) = 119890119909+119905 120597

119909119906 (119909 0 119905) = 119890119909+119905

119906 (0 0 0) = 1 119906 (119909 05 119905) = 119890119909+119905minus1

(53)

Now applying the triple Laplace transform on both sidesof (52) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) minus1

(1 + 119901) (2 + 119904) (1 + 119896)

(54)

Factorising the right side of (54) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896

(55)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896]

= 119890119909minus2119910+119905

(56)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 3 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = cos (119909) cos (119910) cos (minus119905)

minus sin (119909) sin (119910) sin (minus119905)

(57)

subjected to the following initial and boundaries conditions

119906 (119909 119910 0) = cos (119909) cos (119910)

120597119905119906 (119909 119910 0) = 120597

119909119906 (0 119910 119905) = 120597

119910119906 (119909 0 119905) = 0

119906 (119909120587

2 119905) = 119906 (119909 119910

120587

2) = 119906 (

120587

2 119910 119905) = 0

(58)

Abstract and Applied Analysis 7

Table 1 Table of triple Laplace transform for some function of three variables

Functions119891(119909 119910 119905) Triple laplace transform 119865(119901 119904 119896)

119886119887119888119886119887119888

119901119904119896

1199091199101199051

119901211990421198962

119909119899119910119898119905V 119899 119898 V are natural numbers 119896minus1minusV119904minus1minus119898119901minus119899minus1Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119909119899119910119898119905V119890minus119886119909minus119887119910minus119888119905 (119896 + 119888)minus1minusV(119904 + 119887)minus1minus119898(119901 + 119886)minus119899minus1

Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119890minus119886119909minus119887119910minus1198881199051

(119886 + 119901) (119887 + 119904) (119888 + 119896)

cos(119909) cos(119910) cos(119905)119896119904119901

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin(119909) sin(119910) sin(119905)1

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin (119909 + 119910 + 119905)minus1 + 119901119904 + 119896 (119901 + 119904)

(1 + 1199012) (1 + 1199042) (1 + 1198962)

cos(119909 + 119910 + 119905) minus119896 + 119901 + 119904 minus 119896119901119904

(1 + 1199012) (1 + 1199042) (1 + 1198962)

radic119909119910119905120587radic120587

83radic119896119904119901

119890119886119909+119910119887+119888119905 sinh(119886119909)sinh(119887119910) sinh(119888119905)(119887) (119888) (119886)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

119890119886119909+119910119887+119888119905 cosh(119886119909) cosh(119887119910) cosh(119888119905)(119887 minus 119904) (119888 minus 119896) (119886 minus 119901)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

Erf [119886

2radic119909]Erf[

119887

2radic119910]Erf [

119888

2radic119905]

119890minusradic1198882119896minusradic1198872119904

119896119901119904(minus1 + 119890minus

radic1198882119896) (1 minus 119890minus

radic1198862119901) (minus1 + 119890minus

radic1198872119904)

sin(119886119909)

119909

sin(119887119910)

119910

sin(119888119905)

119905arctan(

radic1198862

119901) arctan(

radic1198872

119904) arctan(

radic1198882

119896)

cos(119886119909)

119909119899cos(119887119910)

119910119898cos(119888119905)

119905V

119896minus1+V(1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898 cos [119888119905] cos((minus1 + 119898) arctan[radic1198872

119904]) Γ (1 minus 119898) Γ (1 minus V)

times (1 +1198862

1199012)

12(minus1+119899)

119901minus1+119899 cos((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

sin(119886119909)

119909119899sin(119887119910)

119910119898sin(119888119905)

119905V

119896minus1+V (1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898Γ (1 minus 119898) Γ (1 minus V) (1 +1198862

1199012)

12(minus1+119899)

times 119901minus1+119899Γ (1 minus 119899) sign (119886) sin ((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

119869119899

(119909) 119869119899

(119910) 119869119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 minus

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199042)

[Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199012)

119868119899

(119909) 119868119899

(119910) 119868119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 1 + 119899

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199042)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199012)

8 Abstract and Applied Analysis

500

400

300

200

100

06

4

2

0Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (41)

x

(a)

500

400

300

200

100

06

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

y x

(b)

64

2

0

0

20

40

60

Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

x

(c)

6

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (411)

minus1

minus05

0

05

1

xy

(d)

Figure 1 Numerical simulation of the exact solutions of the Homogeneous and non-homogeneous Mboctara equations

Now applying the triple Laplace transform on both sides of(57) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) +119896119904119901

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

minus1

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

(59)

Factorising the right side of (59) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962 )

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042 ) (minus1 + 1198962)

1 + 119901119904119896

(60)

Abstract and Applied Analysis 9

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905)

= 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896]

= cos (119909) cos (119910) cos (minus119905)

(61)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 4 consider the following nonlinear nonhomoge-neous with variable coefficient Mboctara equation

119890119909+119910+119905120597119909119910119905

119906 (119909 119910 119905) minus 31199062 (119909 119910 119905) + 119890119909+119910+119905119906 (119909 119910 119905)

= 1198902119909+2119910+2119905

119906119909

(119909 119910 0) = 119890119909+119910 119906 (0 0 0) = 1

119906 (1 0 0) = 119890 120597119909119910119905

119906 (0 0 0) = 1

(62)

Now applying the triple Laplace transform on both sidesof (62) and then using the properties of the triple Laplacetransform and after factorising as in the previous examplesand taking the inverse triple Laplace transform we obtainthe following as an exact solution of this type of Mboctaraequation

119906 (119909 119910 119905) = 119890119909+119910+119905 (63)

Thenumerical simulations of the exact solutions of theMboc-tara equation are depicted in Figure 1(a) (41) Figure 1(b)(46) Figure 1(c) (46) and Figure 1(d) (411) respectively

6 Triple Laplace Transform of SomeFunctions of Three Variables

In this section we examine the triple Laplace transform ofsome functions in Table 1

119871119909119910119905

(119884119899

(119909) 119884119899

(119910) 119884119899

(119905))

= 8minus119899(119896119904119901)minus1minus119899

1198621199041198883 [119899120587]

times ( minus (41198962)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1198962)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1198962))

times ( minus (41199042)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199042)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199042))

times ( minus (41199012)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199012)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199012))

(64)

7 Conclusion

This work presents the definition of the triple Laplace trans-form Some triple Laplace transform is presented in Table 1Some theorems and properties of this new relatively newoperator are presented Applications of the new operator forsolving some kind of third-order partial differential equationscalledMboctara equation are presentedNumerical solutionsof the Mboctara equation are given

References

[1] G L Lamb Jr Introductory Applications of Partial DifferentialEquations with Emphasis on Wave Propagation and DiffusionJohn Wiley amp Sons New York NY USA 1995

[2] U T Myint Differential Equations of Mathematical PhysicsAmerican Elsevier New York NY USA 1980

[3] C Constanda Solution Techniques for Elementary Partial Differ-ential Equations Chapman amp HallCRC New York NY USA2002

[4] D G Duffy Transform Methods for Solving Partial DifferentialEquations CRC Press New York NY USA 2004

[5] A Babakhani and R S Dahiya ldquoSystems of multi-dimensionalLaplace transforms and a heat equationrdquo in Proceedings of the16th Conference onAppliedMathematics vol 7 ofElectronicJour-nal of Differential Equations pp 25ndash36 University of CentralOklahoma Edmond Okla USA 2001

[6] Y A Brychkov H -J Glaeske A P Prudnikov and V K TuanMultidimensional Integral Transformations Gordon and BreachScience Publishers Philadelphia Pa USA 1992

[7] A Atangana and A Kilicman ldquoA possible generalization ofacoustic wave equation using the concept of perturbed deriva-tive orderrdquo Mathematical Problems in Engineering vol 2013Article ID 696597 6 pages 2013

[8] A Kılıcman and H Eltayeb ldquoA note on the classifications ofhyperbolic and elliptic equations with polynomial coefficientsrdquoApplied Mathematics Letters vol 21 no 11 pp 1124ndash1128 2008

10 Abstract and Applied Analysis

[9] H Eltayeb A Kılıcman and P Ravi Agarwal ldquoAn analysis onclassifications of hyperbolic and elliptic PDEsrdquo MathematicalSciences vol 6 article 47 2012

[10] H Eltayeb andAKılıcman ldquoAnote ondouble laplace transformand telegraphic equationsrdquo Abstract and Applied Analysis vol2013 Article ID 932578 6 pages 2013

[11] A Kılıcman and H E Gadain ldquoOn the applications of Laplaceand Sumudu transformsrdquo Journal of the Franklin Institute vol347 no 5 pp 848ndash862 2010

[12] R P Kanwal Generalized Functions Theory and ApplicationsBirkhaauser Boston Mass USA 2004

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

which boil down to119891 (119909 119910 119905)

=1

2120587119894int120572+119894infin

120572minus119894infin

119890119901119909

times [1

2120587119894int120573+119894infin

120573minus119894infin

119890119904119910

times [1

2120587119894int120583+119894infin

120583minus119894infin

119890119896119905

times 119866(119901 119904 119896) 119889119896] 119889119904] 119889119901

= 119892 (119909 119910 119905)

(9)

and this proves the uniqueness of the triple Laplace trans-form

Theorem 4 If at the point (119901 119904 119896) the integrals

1198651

(119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 1198911

(119909 119910 119905) 119889119909 119889119910 119889119905

1198652

(119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 1198912

(119909 119910 119905) 119889119909 119889119910 119889119905

(10)

are convergent and in addition if

1198653

(119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 1198913

(119909 119910 119905) 119889119909 119889119910 119889119905

(11)

is absolutely convergent then the following expression

119865 (119901 119904 119896) = 1198651

(119901 119904 119896) 1198652

(119901 119904 119896) 1198653

(119901 119904 119896) (12)

is the Laplace transform of the function

119891 (119909 119910 119905)

= int119905

0

int119910

0

int119909

0

1198913

(119909 minus (1199091

+ 120588) 119910 minus (1199101

+ 120590)

119905 minus (1199051

+ 120591)) 1198912

(1199091

minus 120588 1199101

minus 120590 1199051

minus 120591)

times 1198911

(120588 120590 120591) 119889120588 119889120590 119889120591

(13)

and the integral

119865 (119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

(14)

is convergent at the point (119901 119904 119896) for the readers who areinterested they can see the proof in [11 12]

Theorem 5 A function 119891(119909 119910 119905) which is continuous on[0 infin) and satisfies the growth condition (4) can be recoveredfrom only 119865(119901 119904 119896) as

119891 (119909 119910 119905) = lim1198991rarrinfin

1198992rarrinfin

1198993rarrinfin

(minus1)1198991+1198992+1198993

119899111989921198993

(1198991

119909)1198991+1

(1198992

119910)1198992+1

times (1198993

119905)1198993+1

Χ1198991+1198992+1198993 [1198991

1199091198992

1199101198993

119905]

(15)

Evidently the main difficulty in using Theorem 5 for com-puting the inverse Laplace transform is the repeated symbolicdifferentiation of 119865(119901 119904 119896)

Let us see how Theorem 5 can be applicable Let usconsider the following functions

119891 (119909 119910 119905) = exp [minus119886119909 minus 119887119910 minus 119888119905] (16)

Naturally the triple Laplace transform of the previous func-tion is given later as

119865 (119901 119904 119896) =1

(119901 minus 119886) (119904 minus 119887) (119896 minus 119888) (17)

Nowapplying the high-ordermixedderivative to the previousexpression we obtain the following

1205971198991+1198992+1198993 [119865 (119901 119904 119896)]

120597119901119899112059711990411989921205971198961198993= 119899111989921198993(minus1)1198991+1198992+1198993

times (119886+119875)minus1minus1198991(119904+119887)

minus1minus1198992(119888+119896)

minus1minus1198993

(18)

ApplyingTheorem 5 in the previous expression we obtain thefollowing result

119891 (119909 119910 119905) = lim1198991rarrinfin

1198992rarrinfin

1198993rarrinfin

1198991

1+11989911198992

1+11989921198993

1+1198993

1199091198991+11199101198992+11199051198993+1(119886 +

1198991

119909)minus1198991minus1

times (119887 +1198992

119910)minus1198992minus1

(119888 +1198993

119905)minus1198993minus1

(19)

Making a change of variable in the previous expression weobtain the following simplified result

119891 (119909 119910 119905) = lim1198991rarrinfin

1198992rarrinfin

1198993rarrinfin

(1 +1198861198991

119909)minus1198991minus1

(1 +1198871198992

119910)minus1198992minus1

times (1 +1198881198993

119905)minus1198993minus1

(20)

Using together the application of logarithm and theLrsquoHopitalrsquos rule on the previous expression we arrive at thefollowing result

ln (119891 (119909 119910 119905)) = minus119886119909 minus 119887119910 minus 119888119905 997904rArr 119891 (119909 119910 119905)

= exp [minus119886119909 minus 119887119910 minus 119888119905] (21)

4 Abstract and Applied Analysis

4 Some Properties of TripleLaplace Transform

In this section we present some properties of the tripleLaplace transform Note that these properties follow fromthose of the double Laplace transform introduced byKılıcman and Eltayeb [8]The properties of the triple Laplacetransform will enable us to find further transform pairs119891(119909 119910 119905) 119865(119901 119904 119896)

(i) 119865 (119901 + 119886 119904 + 119887 119896 + 119889)

= 119871119909119910119905

[119890minus119886119909minus119910119887minus119888119905119891 (119909 119910 119905)] (119901 119904 119896) (22)

We will present the proof

119871119909119910119905

[119890minus119886119909minus119910119887minus119888119905119891 (119909 119910 119905)] (119901 119904 119896)

= ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905] exp [minus119886119909]

times exp [minus119887119910] exp [minus119888119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

intinfin

0

exp [minus119901119909] exp [minus119886119909]

times (∬infin

0

exp [minus119904119910] exp [minus119896119905] exp [minus119887119910]

times exp [minus119888119905] 119891 (119909 119910 119905) 119889119905 119889119910) 119889119905

(23)

Note that the integral inside the bracket satisfies the proper-ties of the double Laplace transform and is given as [11]

(∬infin

0

exp [minus119904119910] exp [minus119896119905] exp [minus119887119910] exp [minus119888119905]

times 119891 (119909 119910 119905) 119889119905 119889119910) = 119865 (119909 119904 + 119887 119896 + 119889)

(24)

Thus

intinfin

0

exp [minus119901119909] exp [minus119886119909] 119865 (119909 119904 + 119887 119896 + 119889) 119889119905

= 119865 (119901 + 119886 119904 + 119887 119896 + 119889)

(25)

and this completes the proof(ii) The following can also be observed

1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) = 119871

119909119910119905[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896) (26)

We will present the proof

119871119909119910119905

[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896)

= ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905]

times 119891 (120572119909 120573119910 120574119905) 119889119909 119889119910 119889119905

intinfin

0

exp [minus119901119909] (∬infin

0

exp [minus119904119910] exp [minus119896119905]

times 119891 (120572119909 120573119910 120574119905) 119889119910 119889119905) 119889119909

(27)

Note that the double integral inside the bracket satisfies theproperty of the double Laplace transform as [11]

(∬infin

0

exp [minus119904119910] exp [minus119896119905] 119891 (120572119909 120573119910 120574119905) 119889119910 119889119905)

=1

120573120574119865 (120572119909

119904

120573119896

120574)

(28)

Thus

119871119909119910119905

[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896)

= intinfin

0

exp [minus119901119909]1

120573120574119865 (120572119909

119904

120573119896

120574) 119889119909

=1

120572120573120574119865 (

119901

120572

119904

120573119896

120574)

(29)

and this completes the proof(iii) The following property can also be observed

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= 119871119909119910119905

[(minus1)119899+119898+V119909119899119910119898119905V119891 (119909 119910 119905)] (119901 119904 119896)

(30)

We will present the proof

119865 (119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905]

times 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

(31)

Then

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899+119898+V

120597119901119899120597119904119899120597119896V(∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905)

(32)

Now making use of the convergence properties of theimproper integral involved we can interchange the opera-tion of differentiation and integration and differentiate with

Abstract and Applied Analysis 5

respect to 119901 119904 and 119896 under the integral sign Thus we arriveat the following expression

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899

120597119901119899intinfin

0

exp [minus119901119909]

times (120597119898+V

120597119904119899120597119896V∬infin

0

exp [minus119904119910] exp [minus119896119905]

times 119891 (119909 119910 119905) 119889119910 119889119905) 119889119909

(33)

Note that the expression in the bracket satisfies the propertyof the double Laplace transform as [11]

120597119898+V

120597119904119899120597119896V∬infin

0

exp [minus119904119910] exp [minus119896119905] 119891 (119909 119910 119905) 119889119910 119889119905

= 119871119910119905

[(minus1)119898+V119910119898119905V119891 (119909 119910 119905)] (119904 119896)

(34)

Thus

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899

120597119901119899intinfin

0

exp [minus119901119909]

times (119871119910119905

[(minus1)119898+V119910119898119905V119891 (119909 119910 119905)] (119904 119896)) 119889119909

(35)

And finally we obtain

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= 119871119909119910119905

[(minus1)119899+119898+V119909119899119910119898119905V119891 (119909 119910 119905)] (119901 119904 119896)

(36)

and this completes the proofNow using the previous three properties we will show the

proof of Theorem 5

Proof of Theorem 5 Let us define the set of functions depend-ing on parameters 119898 119899 and V as

ℎ119898119899V (119909 119910 119905) =

119898119898+1119899119899+1VV+1

119898119899V119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905 (37)

It worth noting that the previous function is a kind of three-dimensional density of probability and it therefore followsthat

∭infin

0

ℎ119898119899V (119909 119910 119905) 119889119909 119889119910 119889119905 = 1 (38)

In addition of this we will have that

lim119898rarrinfin

119899rarrinfin

Vrarrinfin

∭infin

0

ℎ119898119899V (119909 119910 119905) 120595 (119909 119910 119905) 119889119909 119889119910 119889119905 = 120595 (1 1 1)

(39)

where 120595(119909 119910 119905) is any continuous function Let Ψ(119901 119904 119896)denote the triple Laplace transform of the continuousfunction 120595(119909 119910 119905) However if one defines the function119872(119909 119910 119905) = 119891(119909120572 119910120573 119905120574) making use of the second pro-perty established in (29) we arrive at the following

1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) = 119871

119909119910119905[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896) (40)

Here if one applies the third property in particular by rep-lacing 119901 = 119898119909 119904 = 119899119910 119896 = V119905 as follows

119871119909119910119905

(119872 (119909 119910 119905)) =1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) (41)

120597119899+119898+V [119871119909119910119905

(119872 (119909 119910 119905))]

120597119901119899120597119904119899120597119896V

=120597119899+119898+V [(1120572120573120574) 119865 (119901120572 119904120573 119896120574)]

120597119901119899120597119904119899120597119896V

=1

120572119898+1120573119899+1120574V+1

times120597119899+119898+V [119865 (119901120572 119904120573 119896120574)]

120597119901119899120597119904119899120597119896V

(42)

Now let us put 120595(119909 119910 119905) = 119890minus119901119909minus119904119910minus119896119905119872(119909 119910 119905) Now if wemake use of (38) we obtain the following

120595 (1 1 1) = 119890minus119901minus119904minus119896119872 (1 1 1) = 119890minus119901minus119904minus119896119891 (120572 120573 120574)

= lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V∭infin

0

119909119898119910119899119905V119890minus119901119909minus119904119910minus119896119905

times 119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905) 119889119909 119889119910 119889119905

= lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V119871119909119910119905

[119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905)]

(43)

Now taking into account properties (i) and (ii) (42) togetherwith the function 119872(119909 119910 119905) we arrive at the following

119871119909119910119905

[119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905)]

= (minus1)119898+119899+V

120597119899+119898+V [119871119909119910119905

(119890minus119898119909minus119899119910minus119896119905Ψ (119909 119910 119905)) (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= (minus1)119898+119899+V 1

120572119898120573119899120574V

times120597119899+119898+V [119871

119909119910119905(Ψ (119909 119910 119905)) (119901 + 119898 119904 + 119899 119896 + V)]

120597119901119899120597119904119899120597119896V

= (minus1)119898+119899+V 1

120572119898120573119899120574V

times((120597119899+119898+V [119871119909119910119905

(119891 (120572119909 120573119910 120574119905))

times(119901+119898

120572119904+119899

120573119896+V

120574)]) (120597119901119899120597119904119899120597119896V)

minus1

)

6 Abstract and Applied Analysis

= (minus1)119898+119899+V

times1

120572119898120573119899120574V120597119899+119898+V [119865 ((119901+119898) 120572 (119904+119899) 120573 (119896+V) 120574)]

120597119901119899120597119904119899120597119896V

(44)

Now observe that from (44) with the fact that 119891(120572 120573 119905) =

120595(1 1 1)119890119901+119904+119896 we arrive at the following

119891 (120572 120573 119905)

= 119890119901+119904+119896 lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V

times (119898

120572)119898+1

(119899

120573)119899+1

(V

120574)V+1

times120597119899+119898+V [119865 ((119901 + 119898) 120572 (119904 + 119899) 120573 (119896 + V) 120574)]

120597119901119899120597119904119899120597119896V

(45)

The previously mentioned is true for any 119901 119904 119896 in thecomplete space in particular for 119901 = 0 119904 = 0 119896 = 0 andin this case Theorem 5 is covered

5 Application to Third-Order PartialDifferential Equation

In this section we present the application of this operatorfor solving some kind of third-order partial differentialequations

Example 1 consider the following third-order partial differ-ential equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = 0 (46)

The previous equation is called the Mboctara equation and issubjected to the following boundaries and initial conditions

119906 (119909 119910 0) = 119890119909+119910 119906 (119909 0 119905) = 119890119909minus119905

119906 (0 119910 119905) = 119890119910minus119905 119906 (119909 119910 1) = 119890119909+119910minus1(47)

Now applying the triple Laplace transform on both sides of(46) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896) = 119866 (119901 119904 119896) (48)

Here119866 (119901 119904 119896) = 119901119904119880 (119901 119904 0) + 119901119904119880 (119901 0 119896) minus 119901119880 (119901 0 0)

+ 119904119896119880 (0 119904 119896) minus 119904119880 (0 119904 0) minus 119896119880 (0 0 119896)

+ 119880 (0 0 0)

(49)

Factorising the right side of equation (49) we obtain thefollowing

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896 (50)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896] = 119890119909+119910minus119905 (51)

This is the exact solution for Mboctara equation

Example 2 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = minus119890119909minus2119910+119905 (52)

subjected to the following initial and boundaries conditions

119906 (119909 0 0) = 119890119909 120597119905119906 (119909 0 119905) = 119890119909+119905 120597

119909119906 (119909 0 119905) = 119890119909+119905

119906 (0 0 0) = 1 119906 (119909 05 119905) = 119890119909+119905minus1

(53)

Now applying the triple Laplace transform on both sidesof (52) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) minus1

(1 + 119901) (2 + 119904) (1 + 119896)

(54)

Factorising the right side of (54) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896

(55)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896]

= 119890119909minus2119910+119905

(56)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 3 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = cos (119909) cos (119910) cos (minus119905)

minus sin (119909) sin (119910) sin (minus119905)

(57)

subjected to the following initial and boundaries conditions

119906 (119909 119910 0) = cos (119909) cos (119910)

120597119905119906 (119909 119910 0) = 120597

119909119906 (0 119910 119905) = 120597

119910119906 (119909 0 119905) = 0

119906 (119909120587

2 119905) = 119906 (119909 119910

120587

2) = 119906 (

120587

2 119910 119905) = 0

(58)

Abstract and Applied Analysis 7

Table 1 Table of triple Laplace transform for some function of three variables

Functions119891(119909 119910 119905) Triple laplace transform 119865(119901 119904 119896)

119886119887119888119886119887119888

119901119904119896

1199091199101199051

119901211990421198962

119909119899119910119898119905V 119899 119898 V are natural numbers 119896minus1minusV119904minus1minus119898119901minus119899minus1Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119909119899119910119898119905V119890minus119886119909minus119887119910minus119888119905 (119896 + 119888)minus1minusV(119904 + 119887)minus1minus119898(119901 + 119886)minus119899minus1

Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119890minus119886119909minus119887119910minus1198881199051

(119886 + 119901) (119887 + 119904) (119888 + 119896)

cos(119909) cos(119910) cos(119905)119896119904119901

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin(119909) sin(119910) sin(119905)1

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin (119909 + 119910 + 119905)minus1 + 119901119904 + 119896 (119901 + 119904)

(1 + 1199012) (1 + 1199042) (1 + 1198962)

cos(119909 + 119910 + 119905) minus119896 + 119901 + 119904 minus 119896119901119904

(1 + 1199012) (1 + 1199042) (1 + 1198962)

radic119909119910119905120587radic120587

83radic119896119904119901

119890119886119909+119910119887+119888119905 sinh(119886119909)sinh(119887119910) sinh(119888119905)(119887) (119888) (119886)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

119890119886119909+119910119887+119888119905 cosh(119886119909) cosh(119887119910) cosh(119888119905)(119887 minus 119904) (119888 minus 119896) (119886 minus 119901)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

Erf [119886

2radic119909]Erf[

119887

2radic119910]Erf [

119888

2radic119905]

119890minusradic1198882119896minusradic1198872119904

119896119901119904(minus1 + 119890minus

radic1198882119896) (1 minus 119890minus

radic1198862119901) (minus1 + 119890minus

radic1198872119904)

sin(119886119909)

119909

sin(119887119910)

119910

sin(119888119905)

119905arctan(

radic1198862

119901) arctan(

radic1198872

119904) arctan(

radic1198882

119896)

cos(119886119909)

119909119899cos(119887119910)

119910119898cos(119888119905)

119905V

119896minus1+V(1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898 cos [119888119905] cos((minus1 + 119898) arctan[radic1198872

119904]) Γ (1 minus 119898) Γ (1 minus V)

times (1 +1198862

1199012)

12(minus1+119899)

119901minus1+119899 cos((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

sin(119886119909)

119909119899sin(119887119910)

119910119898sin(119888119905)

119905V

119896minus1+V (1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898Γ (1 minus 119898) Γ (1 minus V) (1 +1198862

1199012)

12(minus1+119899)

times 119901minus1+119899Γ (1 minus 119899) sign (119886) sin ((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

119869119899

(119909) 119869119899

(119910) 119869119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 minus

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199042)

[Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199012)

119868119899

(119909) 119868119899

(119910) 119868119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 1 + 119899

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199042)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199012)

8 Abstract and Applied Analysis

500

400

300

200

100

06

4

2

0Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (41)

x

(a)

500

400

300

200

100

06

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

y x

(b)

64

2

0

0

20

40

60

Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

x

(c)

6

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (411)

minus1

minus05

0

05

1

xy

(d)

Figure 1 Numerical simulation of the exact solutions of the Homogeneous and non-homogeneous Mboctara equations

Now applying the triple Laplace transform on both sides of(57) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) +119896119904119901

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

minus1

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

(59)

Factorising the right side of (59) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962 )

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042 ) (minus1 + 1198962)

1 + 119901119904119896

(60)

Abstract and Applied Analysis 9

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905)

= 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896]

= cos (119909) cos (119910) cos (minus119905)

(61)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 4 consider the following nonlinear nonhomoge-neous with variable coefficient Mboctara equation

119890119909+119910+119905120597119909119910119905

119906 (119909 119910 119905) minus 31199062 (119909 119910 119905) + 119890119909+119910+119905119906 (119909 119910 119905)

= 1198902119909+2119910+2119905

119906119909

(119909 119910 0) = 119890119909+119910 119906 (0 0 0) = 1

119906 (1 0 0) = 119890 120597119909119910119905

119906 (0 0 0) = 1

(62)

Now applying the triple Laplace transform on both sidesof (62) and then using the properties of the triple Laplacetransform and after factorising as in the previous examplesand taking the inverse triple Laplace transform we obtainthe following as an exact solution of this type of Mboctaraequation

119906 (119909 119910 119905) = 119890119909+119910+119905 (63)

Thenumerical simulations of the exact solutions of theMboc-tara equation are depicted in Figure 1(a) (41) Figure 1(b)(46) Figure 1(c) (46) and Figure 1(d) (411) respectively

6 Triple Laplace Transform of SomeFunctions of Three Variables

In this section we examine the triple Laplace transform ofsome functions in Table 1

119871119909119910119905

(119884119899

(119909) 119884119899

(119910) 119884119899

(119905))

= 8minus119899(119896119904119901)minus1minus119899

1198621199041198883 [119899120587]

times ( minus (41198962)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1198962)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1198962))

times ( minus (41199042)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199042)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199042))

times ( minus (41199012)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199012)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199012))

(64)

7 Conclusion

This work presents the definition of the triple Laplace trans-form Some triple Laplace transform is presented in Table 1Some theorems and properties of this new relatively newoperator are presented Applications of the new operator forsolving some kind of third-order partial differential equationscalledMboctara equation are presentedNumerical solutionsof the Mboctara equation are given

References

[1] G L Lamb Jr Introductory Applications of Partial DifferentialEquations with Emphasis on Wave Propagation and DiffusionJohn Wiley amp Sons New York NY USA 1995

[2] U T Myint Differential Equations of Mathematical PhysicsAmerican Elsevier New York NY USA 1980

[3] C Constanda Solution Techniques for Elementary Partial Differ-ential Equations Chapman amp HallCRC New York NY USA2002

[4] D G Duffy Transform Methods for Solving Partial DifferentialEquations CRC Press New York NY USA 2004

[5] A Babakhani and R S Dahiya ldquoSystems of multi-dimensionalLaplace transforms and a heat equationrdquo in Proceedings of the16th Conference onAppliedMathematics vol 7 ofElectronicJour-nal of Differential Equations pp 25ndash36 University of CentralOklahoma Edmond Okla USA 2001

[6] Y A Brychkov H -J Glaeske A P Prudnikov and V K TuanMultidimensional Integral Transformations Gordon and BreachScience Publishers Philadelphia Pa USA 1992

[7] A Atangana and A Kilicman ldquoA possible generalization ofacoustic wave equation using the concept of perturbed deriva-tive orderrdquo Mathematical Problems in Engineering vol 2013Article ID 696597 6 pages 2013

[8] A Kılıcman and H Eltayeb ldquoA note on the classifications ofhyperbolic and elliptic equations with polynomial coefficientsrdquoApplied Mathematics Letters vol 21 no 11 pp 1124ndash1128 2008

10 Abstract and Applied Analysis

[9] H Eltayeb A Kılıcman and P Ravi Agarwal ldquoAn analysis onclassifications of hyperbolic and elliptic PDEsrdquo MathematicalSciences vol 6 article 47 2012

[10] H Eltayeb andAKılıcman ldquoAnote ondouble laplace transformand telegraphic equationsrdquo Abstract and Applied Analysis vol2013 Article ID 932578 6 pages 2013

[11] A Kılıcman and H E Gadain ldquoOn the applications of Laplaceand Sumudu transformsrdquo Journal of the Franklin Institute vol347 no 5 pp 848ndash862 2010

[12] R P Kanwal Generalized Functions Theory and ApplicationsBirkhaauser Boston Mass USA 2004

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Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

4 Some Properties of TripleLaplace Transform

In this section we present some properties of the tripleLaplace transform Note that these properties follow fromthose of the double Laplace transform introduced byKılıcman and Eltayeb [8]The properties of the triple Laplacetransform will enable us to find further transform pairs119891(119909 119910 119905) 119865(119901 119904 119896)

(i) 119865 (119901 + 119886 119904 + 119887 119896 + 119889)

= 119871119909119910119905

[119890minus119886119909minus119910119887minus119888119905119891 (119909 119910 119905)] (119901 119904 119896) (22)

We will present the proof

119871119909119910119905

[119890minus119886119909minus119910119887minus119888119905119891 (119909 119910 119905)] (119901 119904 119896)

= ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905] exp [minus119886119909]

times exp [minus119887119910] exp [minus119888119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

intinfin

0

exp [minus119901119909] exp [minus119886119909]

times (∬infin

0

exp [minus119904119910] exp [minus119896119905] exp [minus119887119910]

times exp [minus119888119905] 119891 (119909 119910 119905) 119889119905 119889119910) 119889119905

(23)

Note that the integral inside the bracket satisfies the proper-ties of the double Laplace transform and is given as [11]

(∬infin

0

exp [minus119904119910] exp [minus119896119905] exp [minus119887119910] exp [minus119888119905]

times 119891 (119909 119910 119905) 119889119905 119889119910) = 119865 (119909 119904 + 119887 119896 + 119889)

(24)

Thus

intinfin

0

exp [minus119901119909] exp [minus119886119909] 119865 (119909 119904 + 119887 119896 + 119889) 119889119905

= 119865 (119901 + 119886 119904 + 119887 119896 + 119889)

(25)

and this completes the proof(ii) The following can also be observed

1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) = 119871

119909119910119905[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896) (26)

We will present the proof

119871119909119910119905

[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896)

= ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905]

times 119891 (120572119909 120573119910 120574119905) 119889119909 119889119910 119889119905

intinfin

0

exp [minus119901119909] (∬infin

0

exp [minus119904119910] exp [minus119896119905]

times 119891 (120572119909 120573119910 120574119905) 119889119910 119889119905) 119889119909

(27)

Note that the double integral inside the bracket satisfies theproperty of the double Laplace transform as [11]

(∬infin

0

exp [minus119904119910] exp [minus119896119905] 119891 (120572119909 120573119910 120574119905) 119889119910 119889119905)

=1

120573120574119865 (120572119909

119904

120573119896

120574)

(28)

Thus

119871119909119910119905

[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896)

= intinfin

0

exp [minus119901119909]1

120573120574119865 (120572119909

119904

120573119896

120574) 119889119909

=1

120572120573120574119865 (

119901

120572

119904

120573119896

120574)

(29)

and this completes the proof(iii) The following property can also be observed

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= 119871119909119910119905

[(minus1)119899+119898+V119909119899119910119898119905V119891 (119909 119910 119905)] (119901 119904 119896)

(30)

We will present the proof

119865 (119901 119904 119896) = ∭infin

0

exp [minus119901119909] exp [minus119904119910] exp [minus119896119905]

times 119891 (119909 119910 119905) 119889119909 119889119910 119889119905

(31)

Then

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899+119898+V

120597119901119899120597119904119899120597119896V(∭infin

0

exp [minus119901119909] exp [minus119904119910]

times exp [minus119896119905] 119891 (119909 119910 119905) 119889119909 119889119910 119889119905)

(32)

Now making use of the convergence properties of theimproper integral involved we can interchange the opera-tion of differentiation and integration and differentiate with

Abstract and Applied Analysis 5

respect to 119901 119904 and 119896 under the integral sign Thus we arriveat the following expression

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899

120597119901119899intinfin

0

exp [minus119901119909]

times (120597119898+V

120597119904119899120597119896V∬infin

0

exp [minus119904119910] exp [minus119896119905]

times 119891 (119909 119910 119905) 119889119910 119889119905) 119889119909

(33)

Note that the expression in the bracket satisfies the propertyof the double Laplace transform as [11]

120597119898+V

120597119904119899120597119896V∬infin

0

exp [minus119904119910] exp [minus119896119905] 119891 (119909 119910 119905) 119889119910 119889119905

= 119871119910119905

[(minus1)119898+V119910119898119905V119891 (119909 119910 119905)] (119904 119896)

(34)

Thus

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899

120597119901119899intinfin

0

exp [minus119901119909]

times (119871119910119905

[(minus1)119898+V119910119898119905V119891 (119909 119910 119905)] (119904 119896)) 119889119909

(35)

And finally we obtain

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= 119871119909119910119905

[(minus1)119899+119898+V119909119899119910119898119905V119891 (119909 119910 119905)] (119901 119904 119896)

(36)

and this completes the proofNow using the previous three properties we will show the

proof of Theorem 5

Proof of Theorem 5 Let us define the set of functions depend-ing on parameters 119898 119899 and V as

ℎ119898119899V (119909 119910 119905) =

119898119898+1119899119899+1VV+1

119898119899V119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905 (37)

It worth noting that the previous function is a kind of three-dimensional density of probability and it therefore followsthat

∭infin

0

ℎ119898119899V (119909 119910 119905) 119889119909 119889119910 119889119905 = 1 (38)

In addition of this we will have that

lim119898rarrinfin

119899rarrinfin

Vrarrinfin

∭infin

0

ℎ119898119899V (119909 119910 119905) 120595 (119909 119910 119905) 119889119909 119889119910 119889119905 = 120595 (1 1 1)

(39)

where 120595(119909 119910 119905) is any continuous function Let Ψ(119901 119904 119896)denote the triple Laplace transform of the continuousfunction 120595(119909 119910 119905) However if one defines the function119872(119909 119910 119905) = 119891(119909120572 119910120573 119905120574) making use of the second pro-perty established in (29) we arrive at the following

1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) = 119871

119909119910119905[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896) (40)

Here if one applies the third property in particular by rep-lacing 119901 = 119898119909 119904 = 119899119910 119896 = V119905 as follows

119871119909119910119905

(119872 (119909 119910 119905)) =1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) (41)

120597119899+119898+V [119871119909119910119905

(119872 (119909 119910 119905))]

120597119901119899120597119904119899120597119896V

=120597119899+119898+V [(1120572120573120574) 119865 (119901120572 119904120573 119896120574)]

120597119901119899120597119904119899120597119896V

=1

120572119898+1120573119899+1120574V+1

times120597119899+119898+V [119865 (119901120572 119904120573 119896120574)]

120597119901119899120597119904119899120597119896V

(42)

Now let us put 120595(119909 119910 119905) = 119890minus119901119909minus119904119910minus119896119905119872(119909 119910 119905) Now if wemake use of (38) we obtain the following

120595 (1 1 1) = 119890minus119901minus119904minus119896119872 (1 1 1) = 119890minus119901minus119904minus119896119891 (120572 120573 120574)

= lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V∭infin

0

119909119898119910119899119905V119890minus119901119909minus119904119910minus119896119905

times 119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905) 119889119909 119889119910 119889119905

= lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V119871119909119910119905

[119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905)]

(43)

Now taking into account properties (i) and (ii) (42) togetherwith the function 119872(119909 119910 119905) we arrive at the following

119871119909119910119905

[119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905)]

= (minus1)119898+119899+V

120597119899+119898+V [119871119909119910119905

(119890minus119898119909minus119899119910minus119896119905Ψ (119909 119910 119905)) (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= (minus1)119898+119899+V 1

120572119898120573119899120574V

times120597119899+119898+V [119871

119909119910119905(Ψ (119909 119910 119905)) (119901 + 119898 119904 + 119899 119896 + V)]

120597119901119899120597119904119899120597119896V

= (minus1)119898+119899+V 1

120572119898120573119899120574V

times((120597119899+119898+V [119871119909119910119905

(119891 (120572119909 120573119910 120574119905))

times(119901+119898

120572119904+119899

120573119896+V

120574)]) (120597119901119899120597119904119899120597119896V)

minus1

)

6 Abstract and Applied Analysis

= (minus1)119898+119899+V

times1

120572119898120573119899120574V120597119899+119898+V [119865 ((119901+119898) 120572 (119904+119899) 120573 (119896+V) 120574)]

120597119901119899120597119904119899120597119896V

(44)

Now observe that from (44) with the fact that 119891(120572 120573 119905) =

120595(1 1 1)119890119901+119904+119896 we arrive at the following

119891 (120572 120573 119905)

= 119890119901+119904+119896 lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V

times (119898

120572)119898+1

(119899

120573)119899+1

(V

120574)V+1

times120597119899+119898+V [119865 ((119901 + 119898) 120572 (119904 + 119899) 120573 (119896 + V) 120574)]

120597119901119899120597119904119899120597119896V

(45)

The previously mentioned is true for any 119901 119904 119896 in thecomplete space in particular for 119901 = 0 119904 = 0 119896 = 0 andin this case Theorem 5 is covered

5 Application to Third-Order PartialDifferential Equation

In this section we present the application of this operatorfor solving some kind of third-order partial differentialequations

Example 1 consider the following third-order partial differ-ential equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = 0 (46)

The previous equation is called the Mboctara equation and issubjected to the following boundaries and initial conditions

119906 (119909 119910 0) = 119890119909+119910 119906 (119909 0 119905) = 119890119909minus119905

119906 (0 119910 119905) = 119890119910minus119905 119906 (119909 119910 1) = 119890119909+119910minus1(47)

Now applying the triple Laplace transform on both sides of(46) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896) = 119866 (119901 119904 119896) (48)

Here119866 (119901 119904 119896) = 119901119904119880 (119901 119904 0) + 119901119904119880 (119901 0 119896) minus 119901119880 (119901 0 0)

+ 119904119896119880 (0 119904 119896) minus 119904119880 (0 119904 0) minus 119896119880 (0 0 119896)

+ 119880 (0 0 0)

(49)

Factorising the right side of equation (49) we obtain thefollowing

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896 (50)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896] = 119890119909+119910minus119905 (51)

This is the exact solution for Mboctara equation

Example 2 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = minus119890119909minus2119910+119905 (52)

subjected to the following initial and boundaries conditions

119906 (119909 0 0) = 119890119909 120597119905119906 (119909 0 119905) = 119890119909+119905 120597

119909119906 (119909 0 119905) = 119890119909+119905

119906 (0 0 0) = 1 119906 (119909 05 119905) = 119890119909+119905minus1

(53)

Now applying the triple Laplace transform on both sidesof (52) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) minus1

(1 + 119901) (2 + 119904) (1 + 119896)

(54)

Factorising the right side of (54) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896

(55)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896]

= 119890119909minus2119910+119905

(56)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 3 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = cos (119909) cos (119910) cos (minus119905)

minus sin (119909) sin (119910) sin (minus119905)

(57)

subjected to the following initial and boundaries conditions

119906 (119909 119910 0) = cos (119909) cos (119910)

120597119905119906 (119909 119910 0) = 120597

119909119906 (0 119910 119905) = 120597

119910119906 (119909 0 119905) = 0

119906 (119909120587

2 119905) = 119906 (119909 119910

120587

2) = 119906 (

120587

2 119910 119905) = 0

(58)

Abstract and Applied Analysis 7

Table 1 Table of triple Laplace transform for some function of three variables

Functions119891(119909 119910 119905) Triple laplace transform 119865(119901 119904 119896)

119886119887119888119886119887119888

119901119904119896

1199091199101199051

119901211990421198962

119909119899119910119898119905V 119899 119898 V are natural numbers 119896minus1minusV119904minus1minus119898119901minus119899minus1Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119909119899119910119898119905V119890minus119886119909minus119887119910minus119888119905 (119896 + 119888)minus1minusV(119904 + 119887)minus1minus119898(119901 + 119886)minus119899minus1

Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119890minus119886119909minus119887119910minus1198881199051

(119886 + 119901) (119887 + 119904) (119888 + 119896)

cos(119909) cos(119910) cos(119905)119896119904119901

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin(119909) sin(119910) sin(119905)1

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin (119909 + 119910 + 119905)minus1 + 119901119904 + 119896 (119901 + 119904)

(1 + 1199012) (1 + 1199042) (1 + 1198962)

cos(119909 + 119910 + 119905) minus119896 + 119901 + 119904 minus 119896119901119904

(1 + 1199012) (1 + 1199042) (1 + 1198962)

radic119909119910119905120587radic120587

83radic119896119904119901

119890119886119909+119910119887+119888119905 sinh(119886119909)sinh(119887119910) sinh(119888119905)(119887) (119888) (119886)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

119890119886119909+119910119887+119888119905 cosh(119886119909) cosh(119887119910) cosh(119888119905)(119887 minus 119904) (119888 minus 119896) (119886 minus 119901)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

Erf [119886

2radic119909]Erf[

119887

2radic119910]Erf [

119888

2radic119905]

119890minusradic1198882119896minusradic1198872119904

119896119901119904(minus1 + 119890minus

radic1198882119896) (1 minus 119890minus

radic1198862119901) (minus1 + 119890minus

radic1198872119904)

sin(119886119909)

119909

sin(119887119910)

119910

sin(119888119905)

119905arctan(

radic1198862

119901) arctan(

radic1198872

119904) arctan(

radic1198882

119896)

cos(119886119909)

119909119899cos(119887119910)

119910119898cos(119888119905)

119905V

119896minus1+V(1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898 cos [119888119905] cos((minus1 + 119898) arctan[radic1198872

119904]) Γ (1 minus 119898) Γ (1 minus V)

times (1 +1198862

1199012)

12(minus1+119899)

119901minus1+119899 cos((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

sin(119886119909)

119909119899sin(119887119910)

119910119898sin(119888119905)

119905V

119896minus1+V (1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898Γ (1 minus 119898) Γ (1 minus V) (1 +1198862

1199012)

12(minus1+119899)

times 119901minus1+119899Γ (1 minus 119899) sign (119886) sin ((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

119869119899

(119909) 119869119899

(119910) 119869119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 minus

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199042)

[Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199012)

119868119899

(119909) 119868119899

(119910) 119868119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 1 + 119899

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199042)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199012)

8 Abstract and Applied Analysis

500

400

300

200

100

06

4

2

0Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (41)

x

(a)

500

400

300

200

100

06

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

y x

(b)

64

2

0

0

20

40

60

Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

x

(c)

6

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (411)

minus1

minus05

0

05

1

xy

(d)

Figure 1 Numerical simulation of the exact solutions of the Homogeneous and non-homogeneous Mboctara equations

Now applying the triple Laplace transform on both sides of(57) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) +119896119904119901

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

minus1

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

(59)

Factorising the right side of (59) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962 )

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042 ) (minus1 + 1198962)

1 + 119901119904119896

(60)

Abstract and Applied Analysis 9

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905)

= 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896]

= cos (119909) cos (119910) cos (minus119905)

(61)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 4 consider the following nonlinear nonhomoge-neous with variable coefficient Mboctara equation

119890119909+119910+119905120597119909119910119905

119906 (119909 119910 119905) minus 31199062 (119909 119910 119905) + 119890119909+119910+119905119906 (119909 119910 119905)

= 1198902119909+2119910+2119905

119906119909

(119909 119910 0) = 119890119909+119910 119906 (0 0 0) = 1

119906 (1 0 0) = 119890 120597119909119910119905

119906 (0 0 0) = 1

(62)

Now applying the triple Laplace transform on both sidesof (62) and then using the properties of the triple Laplacetransform and after factorising as in the previous examplesand taking the inverse triple Laplace transform we obtainthe following as an exact solution of this type of Mboctaraequation

119906 (119909 119910 119905) = 119890119909+119910+119905 (63)

Thenumerical simulations of the exact solutions of theMboc-tara equation are depicted in Figure 1(a) (41) Figure 1(b)(46) Figure 1(c) (46) and Figure 1(d) (411) respectively

6 Triple Laplace Transform of SomeFunctions of Three Variables

In this section we examine the triple Laplace transform ofsome functions in Table 1

119871119909119910119905

(119884119899

(119909) 119884119899

(119910) 119884119899

(119905))

= 8minus119899(119896119904119901)minus1minus119899

1198621199041198883 [119899120587]

times ( minus (41198962)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1198962)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1198962))

times ( minus (41199042)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199042)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199042))

times ( minus (41199012)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199012)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199012))

(64)

7 Conclusion

This work presents the definition of the triple Laplace trans-form Some triple Laplace transform is presented in Table 1Some theorems and properties of this new relatively newoperator are presented Applications of the new operator forsolving some kind of third-order partial differential equationscalledMboctara equation are presentedNumerical solutionsof the Mboctara equation are given

References

[1] G L Lamb Jr Introductory Applications of Partial DifferentialEquations with Emphasis on Wave Propagation and DiffusionJohn Wiley amp Sons New York NY USA 1995

[2] U T Myint Differential Equations of Mathematical PhysicsAmerican Elsevier New York NY USA 1980

[3] C Constanda Solution Techniques for Elementary Partial Differ-ential Equations Chapman amp HallCRC New York NY USA2002

[4] D G Duffy Transform Methods for Solving Partial DifferentialEquations CRC Press New York NY USA 2004

[5] A Babakhani and R S Dahiya ldquoSystems of multi-dimensionalLaplace transforms and a heat equationrdquo in Proceedings of the16th Conference onAppliedMathematics vol 7 ofElectronicJour-nal of Differential Equations pp 25ndash36 University of CentralOklahoma Edmond Okla USA 2001

[6] Y A Brychkov H -J Glaeske A P Prudnikov and V K TuanMultidimensional Integral Transformations Gordon and BreachScience Publishers Philadelphia Pa USA 1992

[7] A Atangana and A Kilicman ldquoA possible generalization ofacoustic wave equation using the concept of perturbed deriva-tive orderrdquo Mathematical Problems in Engineering vol 2013Article ID 696597 6 pages 2013

[8] A Kılıcman and H Eltayeb ldquoA note on the classifications ofhyperbolic and elliptic equations with polynomial coefficientsrdquoApplied Mathematics Letters vol 21 no 11 pp 1124ndash1128 2008

10 Abstract and Applied Analysis

[9] H Eltayeb A Kılıcman and P Ravi Agarwal ldquoAn analysis onclassifications of hyperbolic and elliptic PDEsrdquo MathematicalSciences vol 6 article 47 2012

[10] H Eltayeb andAKılıcman ldquoAnote ondouble laplace transformand telegraphic equationsrdquo Abstract and Applied Analysis vol2013 Article ID 932578 6 pages 2013

[11] A Kılıcman and H E Gadain ldquoOn the applications of Laplaceand Sumudu transformsrdquo Journal of the Franklin Institute vol347 no 5 pp 848ndash862 2010

[12] R P Kanwal Generalized Functions Theory and ApplicationsBirkhaauser Boston Mass USA 2004

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Abstract and Applied Analysis 5

respect to 119901 119904 and 119896 under the integral sign Thus we arriveat the following expression

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899

120597119901119899intinfin

0

exp [minus119901119909]

times (120597119898+V

120597119904119899120597119896V∬infin

0

exp [minus119904119910] exp [minus119896119905]

times 119891 (119909 119910 119905) 119889119910 119889119905) 119889119909

(33)

Note that the expression in the bracket satisfies the propertyof the double Laplace transform as [11]

120597119898+V

120597119904119899120597119896V∬infin

0

exp [minus119904119910] exp [minus119896119905] 119891 (119909 119910 119905) 119889119910 119889119905

= 119871119910119905

[(minus1)119898+V119910119898119905V119891 (119909 119910 119905)] (119904 119896)

(34)

Thus

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

=120597119899

120597119901119899intinfin

0

exp [minus119901119909]

times (119871119910119905

[(minus1)119898+V119910119898119905V119891 (119909 119910 119905)] (119904 119896)) 119889119909

(35)

And finally we obtain

120597119899+119898+V [119865 (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= 119871119909119910119905

[(minus1)119899+119898+V119909119899119910119898119905V119891 (119909 119910 119905)] (119901 119904 119896)

(36)

and this completes the proofNow using the previous three properties we will show the

proof of Theorem 5

Proof of Theorem 5 Let us define the set of functions depend-ing on parameters 119898 119899 and V as

ℎ119898119899V (119909 119910 119905) =

119898119898+1119899119899+1VV+1

119898119899V119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905 (37)

It worth noting that the previous function is a kind of three-dimensional density of probability and it therefore followsthat

∭infin

0

ℎ119898119899V (119909 119910 119905) 119889119909 119889119910 119889119905 = 1 (38)

In addition of this we will have that

lim119898rarrinfin

119899rarrinfin

Vrarrinfin

∭infin

0

ℎ119898119899V (119909 119910 119905) 120595 (119909 119910 119905) 119889119909 119889119910 119889119905 = 120595 (1 1 1)

(39)

where 120595(119909 119910 119905) is any continuous function Let Ψ(119901 119904 119896)denote the triple Laplace transform of the continuousfunction 120595(119909 119910 119905) However if one defines the function119872(119909 119910 119905) = 119891(119909120572 119910120573 119905120574) making use of the second pro-perty established in (29) we arrive at the following

1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) = 119871

119909119910119905[119891 (120572119909 120573119910 120574119905)] (119901 119904 119896) (40)

Here if one applies the third property in particular by rep-lacing 119901 = 119898119909 119904 = 119899119910 119896 = V119905 as follows

119871119909119910119905

(119872 (119909 119910 119905)) =1

120572120573120574119865 (

119901

120572

119904

120573119896

120574) (41)

120597119899+119898+V [119871119909119910119905

(119872 (119909 119910 119905))]

120597119901119899120597119904119899120597119896V

=120597119899+119898+V [(1120572120573120574) 119865 (119901120572 119904120573 119896120574)]

120597119901119899120597119904119899120597119896V

=1

120572119898+1120573119899+1120574V+1

times120597119899+119898+V [119865 (119901120572 119904120573 119896120574)]

120597119901119899120597119904119899120597119896V

(42)

Now let us put 120595(119909 119910 119905) = 119890minus119901119909minus119904119910minus119896119905119872(119909 119910 119905) Now if wemake use of (38) we obtain the following

120595 (1 1 1) = 119890minus119901minus119904minus119896119872 (1 1 1) = 119890minus119901minus119904minus119896119891 (120572 120573 120574)

= lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V∭infin

0

119909119898119910119899119905V119890minus119901119909minus119904119910minus119896119905

times 119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905) 119889119909 119889119910 119889119905

= lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V119871119909119910119905

[119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905)]

(43)

Now taking into account properties (i) and (ii) (42) togetherwith the function 119872(119909 119910 119905) we arrive at the following

119871119909119910119905

[119909119898119910119899119905V119890minus119898119909minus119899119910minusV119905Ψ (119909 119910 119905)]

= (minus1)119898+119899+V

120597119899+119898+V [119871119909119910119905

(119890minus119898119909minus119899119910minus119896119905Ψ (119909 119910 119905)) (119901 119904 119896)]

120597119901119899120597119904119899120597119896V

= (minus1)119898+119899+V 1

120572119898120573119899120574V

times120597119899+119898+V [119871

119909119910119905(Ψ (119909 119910 119905)) (119901 + 119898 119904 + 119899 119896 + V)]

120597119901119899120597119904119899120597119896V

= (minus1)119898+119899+V 1

120572119898120573119899120574V

times((120597119899+119898+V [119871119909119910119905

(119891 (120572119909 120573119910 120574119905))

times(119901+119898

120572119904+119899

120573119896+V

120574)]) (120597119901119899120597119904119899120597119896V)

minus1

)

6 Abstract and Applied Analysis

= (minus1)119898+119899+V

times1

120572119898120573119899120574V120597119899+119898+V [119865 ((119901+119898) 120572 (119904+119899) 120573 (119896+V) 120574)]

120597119901119899120597119904119899120597119896V

(44)

Now observe that from (44) with the fact that 119891(120572 120573 119905) =

120595(1 1 1)119890119901+119904+119896 we arrive at the following

119891 (120572 120573 119905)

= 119890119901+119904+119896 lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V

times (119898

120572)119898+1

(119899

120573)119899+1

(V

120574)V+1

times120597119899+119898+V [119865 ((119901 + 119898) 120572 (119904 + 119899) 120573 (119896 + V) 120574)]

120597119901119899120597119904119899120597119896V

(45)

The previously mentioned is true for any 119901 119904 119896 in thecomplete space in particular for 119901 = 0 119904 = 0 119896 = 0 andin this case Theorem 5 is covered

5 Application to Third-Order PartialDifferential Equation

In this section we present the application of this operatorfor solving some kind of third-order partial differentialequations

Example 1 consider the following third-order partial differ-ential equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = 0 (46)

The previous equation is called the Mboctara equation and issubjected to the following boundaries and initial conditions

119906 (119909 119910 0) = 119890119909+119910 119906 (119909 0 119905) = 119890119909minus119905

119906 (0 119910 119905) = 119890119910minus119905 119906 (119909 119910 1) = 119890119909+119910minus1(47)

Now applying the triple Laplace transform on both sides of(46) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896) = 119866 (119901 119904 119896) (48)

Here119866 (119901 119904 119896) = 119901119904119880 (119901 119904 0) + 119901119904119880 (119901 0 119896) minus 119901119880 (119901 0 0)

+ 119904119896119880 (0 119904 119896) minus 119904119880 (0 119904 0) minus 119896119880 (0 0 119896)

+ 119880 (0 0 0)

(49)

Factorising the right side of equation (49) we obtain thefollowing

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896 (50)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896] = 119890119909+119910minus119905 (51)

This is the exact solution for Mboctara equation

Example 2 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = minus119890119909minus2119910+119905 (52)

subjected to the following initial and boundaries conditions

119906 (119909 0 0) = 119890119909 120597119905119906 (119909 0 119905) = 119890119909+119905 120597

119909119906 (119909 0 119905) = 119890119909+119905

119906 (0 0 0) = 1 119906 (119909 05 119905) = 119890119909+119905minus1

(53)

Now applying the triple Laplace transform on both sidesof (52) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) minus1

(1 + 119901) (2 + 119904) (1 + 119896)

(54)

Factorising the right side of (54) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896

(55)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896]

= 119890119909minus2119910+119905

(56)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 3 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = cos (119909) cos (119910) cos (minus119905)

minus sin (119909) sin (119910) sin (minus119905)

(57)

subjected to the following initial and boundaries conditions

119906 (119909 119910 0) = cos (119909) cos (119910)

120597119905119906 (119909 119910 0) = 120597

119909119906 (0 119910 119905) = 120597

119910119906 (119909 0 119905) = 0

119906 (119909120587

2 119905) = 119906 (119909 119910

120587

2) = 119906 (

120587

2 119910 119905) = 0

(58)

Abstract and Applied Analysis 7

Table 1 Table of triple Laplace transform for some function of three variables

Functions119891(119909 119910 119905) Triple laplace transform 119865(119901 119904 119896)

119886119887119888119886119887119888

119901119904119896

1199091199101199051

119901211990421198962

119909119899119910119898119905V 119899 119898 V are natural numbers 119896minus1minusV119904minus1minus119898119901minus119899minus1Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119909119899119910119898119905V119890minus119886119909minus119887119910minus119888119905 (119896 + 119888)minus1minusV(119904 + 119887)minus1minus119898(119901 + 119886)minus119899minus1

Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119890minus119886119909minus119887119910minus1198881199051

(119886 + 119901) (119887 + 119904) (119888 + 119896)

cos(119909) cos(119910) cos(119905)119896119904119901

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin(119909) sin(119910) sin(119905)1

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin (119909 + 119910 + 119905)minus1 + 119901119904 + 119896 (119901 + 119904)

(1 + 1199012) (1 + 1199042) (1 + 1198962)

cos(119909 + 119910 + 119905) minus119896 + 119901 + 119904 minus 119896119901119904

(1 + 1199012) (1 + 1199042) (1 + 1198962)

radic119909119910119905120587radic120587

83radic119896119904119901

119890119886119909+119910119887+119888119905 sinh(119886119909)sinh(119887119910) sinh(119888119905)(119887) (119888) (119886)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

119890119886119909+119910119887+119888119905 cosh(119886119909) cosh(119887119910) cosh(119888119905)(119887 minus 119904) (119888 minus 119896) (119886 minus 119901)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

Erf [119886

2radic119909]Erf[

119887

2radic119910]Erf [

119888

2radic119905]

119890minusradic1198882119896minusradic1198872119904

119896119901119904(minus1 + 119890minus

radic1198882119896) (1 minus 119890minus

radic1198862119901) (minus1 + 119890minus

radic1198872119904)

sin(119886119909)

119909

sin(119887119910)

119910

sin(119888119905)

119905arctan(

radic1198862

119901) arctan(

radic1198872

119904) arctan(

radic1198882

119896)

cos(119886119909)

119909119899cos(119887119910)

119910119898cos(119888119905)

119905V

119896minus1+V(1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898 cos [119888119905] cos((minus1 + 119898) arctan[radic1198872

119904]) Γ (1 minus 119898) Γ (1 minus V)

times (1 +1198862

1199012)

12(minus1+119899)

119901minus1+119899 cos((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

sin(119886119909)

119909119899sin(119887119910)

119910119898sin(119888119905)

119905V

119896minus1+V (1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898Γ (1 minus 119898) Γ (1 minus V) (1 +1198862

1199012)

12(minus1+119899)

times 119901minus1+119899Γ (1 minus 119899) sign (119886) sin ((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

119869119899

(119909) 119869119899

(119910) 119869119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 minus

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199042)

[Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199012)

119868119899

(119909) 119868119899

(119910) 119868119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 1 + 119899

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199042)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199012)

8 Abstract and Applied Analysis

500

400

300

200

100

06

4

2

0Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (41)

x

(a)

500

400

300

200

100

06

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

y x

(b)

64

2

0

0

20

40

60

Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

x

(c)

6

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (411)

minus1

minus05

0

05

1

xy

(d)

Figure 1 Numerical simulation of the exact solutions of the Homogeneous and non-homogeneous Mboctara equations

Now applying the triple Laplace transform on both sides of(57) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) +119896119904119901

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

minus1

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

(59)

Factorising the right side of (59) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962 )

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042 ) (minus1 + 1198962)

1 + 119901119904119896

(60)

Abstract and Applied Analysis 9

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905)

= 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896]

= cos (119909) cos (119910) cos (minus119905)

(61)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 4 consider the following nonlinear nonhomoge-neous with variable coefficient Mboctara equation

119890119909+119910+119905120597119909119910119905

119906 (119909 119910 119905) minus 31199062 (119909 119910 119905) + 119890119909+119910+119905119906 (119909 119910 119905)

= 1198902119909+2119910+2119905

119906119909

(119909 119910 0) = 119890119909+119910 119906 (0 0 0) = 1

119906 (1 0 0) = 119890 120597119909119910119905

119906 (0 0 0) = 1

(62)

Now applying the triple Laplace transform on both sidesof (62) and then using the properties of the triple Laplacetransform and after factorising as in the previous examplesand taking the inverse triple Laplace transform we obtainthe following as an exact solution of this type of Mboctaraequation

119906 (119909 119910 119905) = 119890119909+119910+119905 (63)

Thenumerical simulations of the exact solutions of theMboc-tara equation are depicted in Figure 1(a) (41) Figure 1(b)(46) Figure 1(c) (46) and Figure 1(d) (411) respectively

6 Triple Laplace Transform of SomeFunctions of Three Variables

In this section we examine the triple Laplace transform ofsome functions in Table 1

119871119909119910119905

(119884119899

(119909) 119884119899

(119910) 119884119899

(119905))

= 8minus119899(119896119904119901)minus1minus119899

1198621199041198883 [119899120587]

times ( minus (41198962)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1198962)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1198962))

times ( minus (41199042)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199042)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199042))

times ( minus (41199012)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199012)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199012))

(64)

7 Conclusion

This work presents the definition of the triple Laplace trans-form Some triple Laplace transform is presented in Table 1Some theorems and properties of this new relatively newoperator are presented Applications of the new operator forsolving some kind of third-order partial differential equationscalledMboctara equation are presentedNumerical solutionsof the Mboctara equation are given

References

[1] G L Lamb Jr Introductory Applications of Partial DifferentialEquations with Emphasis on Wave Propagation and DiffusionJohn Wiley amp Sons New York NY USA 1995

[2] U T Myint Differential Equations of Mathematical PhysicsAmerican Elsevier New York NY USA 1980

[3] C Constanda Solution Techniques for Elementary Partial Differ-ential Equations Chapman amp HallCRC New York NY USA2002

[4] D G Duffy Transform Methods for Solving Partial DifferentialEquations CRC Press New York NY USA 2004

[5] A Babakhani and R S Dahiya ldquoSystems of multi-dimensionalLaplace transforms and a heat equationrdquo in Proceedings of the16th Conference onAppliedMathematics vol 7 ofElectronicJour-nal of Differential Equations pp 25ndash36 University of CentralOklahoma Edmond Okla USA 2001

[6] Y A Brychkov H -J Glaeske A P Prudnikov and V K TuanMultidimensional Integral Transformations Gordon and BreachScience Publishers Philadelphia Pa USA 1992

[7] A Atangana and A Kilicman ldquoA possible generalization ofacoustic wave equation using the concept of perturbed deriva-tive orderrdquo Mathematical Problems in Engineering vol 2013Article ID 696597 6 pages 2013

[8] A Kılıcman and H Eltayeb ldquoA note on the classifications ofhyperbolic and elliptic equations with polynomial coefficientsrdquoApplied Mathematics Letters vol 21 no 11 pp 1124ndash1128 2008

10 Abstract and Applied Analysis

[9] H Eltayeb A Kılıcman and P Ravi Agarwal ldquoAn analysis onclassifications of hyperbolic and elliptic PDEsrdquo MathematicalSciences vol 6 article 47 2012

[10] H Eltayeb andAKılıcman ldquoAnote ondouble laplace transformand telegraphic equationsrdquo Abstract and Applied Analysis vol2013 Article ID 932578 6 pages 2013

[11] A Kılıcman and H E Gadain ldquoOn the applications of Laplaceand Sumudu transformsrdquo Journal of the Franklin Institute vol347 no 5 pp 848ndash862 2010

[12] R P Kanwal Generalized Functions Theory and ApplicationsBirkhaauser Boston Mass USA 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

= (minus1)119898+119899+V

times1

120572119898120573119899120574V120597119899+119898+V [119865 ((119901+119898) 120572 (119904+119899) 120573 (119896+V) 120574)]

120597119901119899120597119904119899120597119896V

(44)

Now observe that from (44) with the fact that 119891(120572 120573 119905) =

120595(1 1 1)119890119901+119904+119896 we arrive at the following

119891 (120572 120573 119905)

= 119890119901+119904+119896 lim119898rarrinfin

119899rarrinfin

Vrarrinfin

119898119898+1119899119899+1VV+1

119898119899V

times (119898

120572)119898+1

(119899

120573)119899+1

(V

120574)V+1

times120597119899+119898+V [119865 ((119901 + 119898) 120572 (119904 + 119899) 120573 (119896 + V) 120574)]

120597119901119899120597119904119899120597119896V

(45)

The previously mentioned is true for any 119901 119904 119896 in thecomplete space in particular for 119901 = 0 119904 = 0 119896 = 0 andin this case Theorem 5 is covered

5 Application to Third-Order PartialDifferential Equation

In this section we present the application of this operatorfor solving some kind of third-order partial differentialequations

Example 1 consider the following third-order partial differ-ential equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = 0 (46)

The previous equation is called the Mboctara equation and issubjected to the following boundaries and initial conditions

119906 (119909 119910 0) = 119890119909+119910 119906 (119909 0 119905) = 119890119909minus119905

119906 (0 119910 119905) = 119890119910minus119905 119906 (119909 119910 1) = 119890119909+119910minus1(47)

Now applying the triple Laplace transform on both sides of(46) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896) = 119866 (119901 119904 119896) (48)

Here119866 (119901 119904 119896) = 119901119904119880 (119901 119904 0) + 119901119904119880 (119901 0 119896) minus 119901119880 (119901 0 0)

+ 119904119896119880 (0 119904 119896) minus 119904119880 (0 119904 0) minus 119896119880 (0 0 119896)

+ 119880 (0 0 0)

(49)

Factorising the right side of equation (49) we obtain thefollowing

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896 (50)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896] = 119890119909+119910minus119905 (51)

This is the exact solution for Mboctara equation

Example 2 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = minus119890119909minus2119910+119905 (52)

subjected to the following initial and boundaries conditions

119906 (119909 0 0) = 119890119909 120597119905119906 (119909 0 119905) = 119890119909+119905 120597

119909119906 (119909 0 119905) = 119890119909+119905

119906 (0 0 0) = 1 119906 (119909 05 119905) = 119890119909+119905minus1

(53)

Now applying the triple Laplace transform on both sidesof (52) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) minus1

(1 + 119901) (2 + 119904) (1 + 119896)

(54)

Factorising the right side of (54) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896

(55)

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905) = 119871minus1119909119910119905

[119866 (119901 119904 119896) minus 1 (1 + 119901) (2 + 119904) (1 + 119896)

1 + 119901119904119896]

= 119890119909minus2119910+119905

(56)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 3 Let us consider the following nonhomogeneousMboctara equation

120597119909119910119905

119906 (119909 119910 119905) + 119906 (119909 119910 119905) = cos (119909) cos (119910) cos (minus119905)

minus sin (119909) sin (119910) sin (minus119905)

(57)

subjected to the following initial and boundaries conditions

119906 (119909 119910 0) = cos (119909) cos (119910)

120597119905119906 (119909 119910 0) = 120597

119909119906 (0 119910 119905) = 120597

119910119906 (119909 0 119905) = 0

119906 (119909120587

2 119905) = 119906 (119909 119910

120587

2) = 119906 (

120587

2 119910 119905) = 0

(58)

Abstract and Applied Analysis 7

Table 1 Table of triple Laplace transform for some function of three variables

Functions119891(119909 119910 119905) Triple laplace transform 119865(119901 119904 119896)

119886119887119888119886119887119888

119901119904119896

1199091199101199051

119901211990421198962

119909119899119910119898119905V 119899 119898 V are natural numbers 119896minus1minusV119904minus1minus119898119901minus119899minus1Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119909119899119910119898119905V119890minus119886119909minus119887119910minus119888119905 (119896 + 119888)minus1minusV(119904 + 119887)minus1minus119898(119901 + 119886)minus119899minus1

Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119890minus119886119909minus119887119910minus1198881199051

(119886 + 119901) (119887 + 119904) (119888 + 119896)

cos(119909) cos(119910) cos(119905)119896119904119901

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin(119909) sin(119910) sin(119905)1

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin (119909 + 119910 + 119905)minus1 + 119901119904 + 119896 (119901 + 119904)

(1 + 1199012) (1 + 1199042) (1 + 1198962)

cos(119909 + 119910 + 119905) minus119896 + 119901 + 119904 minus 119896119901119904

(1 + 1199012) (1 + 1199042) (1 + 1198962)

radic119909119910119905120587radic120587

83radic119896119904119901

119890119886119909+119910119887+119888119905 sinh(119886119909)sinh(119887119910) sinh(119888119905)(119887) (119888) (119886)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

119890119886119909+119910119887+119888119905 cosh(119886119909) cosh(119887119910) cosh(119888119905)(119887 minus 119904) (119888 minus 119896) (119886 minus 119901)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

Erf [119886

2radic119909]Erf[

119887

2radic119910]Erf [

119888

2radic119905]

119890minusradic1198882119896minusradic1198872119904

119896119901119904(minus1 + 119890minus

radic1198882119896) (1 minus 119890minus

radic1198862119901) (minus1 + 119890minus

radic1198872119904)

sin(119886119909)

119909

sin(119887119910)

119910

sin(119888119905)

119905arctan(

radic1198862

119901) arctan(

radic1198872

119904) arctan(

radic1198882

119896)

cos(119886119909)

119909119899cos(119887119910)

119910119898cos(119888119905)

119905V

119896minus1+V(1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898 cos [119888119905] cos((minus1 + 119898) arctan[radic1198872

119904]) Γ (1 minus 119898) Γ (1 minus V)

times (1 +1198862

1199012)

12(minus1+119899)

119901minus1+119899 cos((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

sin(119886119909)

119909119899sin(119887119910)

119910119898sin(119888119905)

119905V

119896minus1+V (1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898Γ (1 minus 119898) Γ (1 minus V) (1 +1198862

1199012)

12(minus1+119899)

times 119901minus1+119899Γ (1 minus 119899) sign (119886) sin ((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

119869119899

(119909) 119869119899

(119910) 119869119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 minus

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199042)

[Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199012)

119868119899

(119909) 119868119899

(119910) 119868119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 1 + 119899

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199042)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199012)

8 Abstract and Applied Analysis

500

400

300

200

100

06

4

2

0Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (41)

x

(a)

500

400

300

200

100

06

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

y x

(b)

64

2

0

0

20

40

60

Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

x

(c)

6

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (411)

minus1

minus05

0

05

1

xy

(d)

Figure 1 Numerical simulation of the exact solutions of the Homogeneous and non-homogeneous Mboctara equations

Now applying the triple Laplace transform on both sides of(57) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) +119896119904119901

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

minus1

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

(59)

Factorising the right side of (59) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962 )

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042 ) (minus1 + 1198962)

1 + 119901119904119896

(60)

Abstract and Applied Analysis 9

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905)

= 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896]

= cos (119909) cos (119910) cos (minus119905)

(61)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 4 consider the following nonlinear nonhomoge-neous with variable coefficient Mboctara equation

119890119909+119910+119905120597119909119910119905

119906 (119909 119910 119905) minus 31199062 (119909 119910 119905) + 119890119909+119910+119905119906 (119909 119910 119905)

= 1198902119909+2119910+2119905

119906119909

(119909 119910 0) = 119890119909+119910 119906 (0 0 0) = 1

119906 (1 0 0) = 119890 120597119909119910119905

119906 (0 0 0) = 1

(62)

Now applying the triple Laplace transform on both sidesof (62) and then using the properties of the triple Laplacetransform and after factorising as in the previous examplesand taking the inverse triple Laplace transform we obtainthe following as an exact solution of this type of Mboctaraequation

119906 (119909 119910 119905) = 119890119909+119910+119905 (63)

Thenumerical simulations of the exact solutions of theMboc-tara equation are depicted in Figure 1(a) (41) Figure 1(b)(46) Figure 1(c) (46) and Figure 1(d) (411) respectively

6 Triple Laplace Transform of SomeFunctions of Three Variables

In this section we examine the triple Laplace transform ofsome functions in Table 1

119871119909119910119905

(119884119899

(119909) 119884119899

(119910) 119884119899

(119905))

= 8minus119899(119896119904119901)minus1minus119899

1198621199041198883 [119899120587]

times ( minus (41198962)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1198962)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1198962))

times ( minus (41199042)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199042)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199042))

times ( minus (41199012)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199012)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199012))

(64)

7 Conclusion

This work presents the definition of the triple Laplace trans-form Some triple Laplace transform is presented in Table 1Some theorems and properties of this new relatively newoperator are presented Applications of the new operator forsolving some kind of third-order partial differential equationscalledMboctara equation are presentedNumerical solutionsof the Mboctara equation are given

References

[1] G L Lamb Jr Introductory Applications of Partial DifferentialEquations with Emphasis on Wave Propagation and DiffusionJohn Wiley amp Sons New York NY USA 1995

[2] U T Myint Differential Equations of Mathematical PhysicsAmerican Elsevier New York NY USA 1980

[3] C Constanda Solution Techniques for Elementary Partial Differ-ential Equations Chapman amp HallCRC New York NY USA2002

[4] D G Duffy Transform Methods for Solving Partial DifferentialEquations CRC Press New York NY USA 2004

[5] A Babakhani and R S Dahiya ldquoSystems of multi-dimensionalLaplace transforms and a heat equationrdquo in Proceedings of the16th Conference onAppliedMathematics vol 7 ofElectronicJour-nal of Differential Equations pp 25ndash36 University of CentralOklahoma Edmond Okla USA 2001

[6] Y A Brychkov H -J Glaeske A P Prudnikov and V K TuanMultidimensional Integral Transformations Gordon and BreachScience Publishers Philadelphia Pa USA 1992

[7] A Atangana and A Kilicman ldquoA possible generalization ofacoustic wave equation using the concept of perturbed deriva-tive orderrdquo Mathematical Problems in Engineering vol 2013Article ID 696597 6 pages 2013

[8] A Kılıcman and H Eltayeb ldquoA note on the classifications ofhyperbolic and elliptic equations with polynomial coefficientsrdquoApplied Mathematics Letters vol 21 no 11 pp 1124ndash1128 2008

10 Abstract and Applied Analysis

[9] H Eltayeb A Kılıcman and P Ravi Agarwal ldquoAn analysis onclassifications of hyperbolic and elliptic PDEsrdquo MathematicalSciences vol 6 article 47 2012

[10] H Eltayeb andAKılıcman ldquoAnote ondouble laplace transformand telegraphic equationsrdquo Abstract and Applied Analysis vol2013 Article ID 932578 6 pages 2013

[11] A Kılıcman and H E Gadain ldquoOn the applications of Laplaceand Sumudu transformsrdquo Journal of the Franklin Institute vol347 no 5 pp 848ndash862 2010

[12] R P Kanwal Generalized Functions Theory and ApplicationsBirkhaauser Boston Mass USA 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Table 1 Table of triple Laplace transform for some function of three variables

Functions119891(119909 119910 119905) Triple laplace transform 119865(119901 119904 119896)

119886119887119888119886119887119888

119901119904119896

1199091199101199051

119901211990421198962

119909119899119910119898119905V 119899 119898 V are natural numbers 119896minus1minusV119904minus1minus119898119901minus119899minus1Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119909119899119910119898119905V119890minus119886119909minus119887119910minus119888119905 (119896 + 119888)minus1minusV(119904 + 119887)minus1minus119898(119901 + 119886)minus119899minus1

Γ (1 + 119899) Γ (1 + 119898) Γ (1 + V)

119890minus119886119909minus119887119910minus1198881199051

(119886 + 119901) (119887 + 119904) (119888 + 119896)

cos(119909) cos(119910) cos(119905)119896119904119901

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin(119909) sin(119910) sin(119905)1

(1 + 1199012) (1 + 1199042) (1 + 1198962)

sin (119909 + 119910 + 119905)minus1 + 119901119904 + 119896 (119901 + 119904)

(1 + 1199012) (1 + 1199042) (1 + 1198962)

cos(119909 + 119910 + 119905) minus119896 + 119901 + 119904 minus 119896119901119904

(1 + 1199012) (1 + 1199042) (1 + 1198962)

radic119909119910119905120587radic120587

83radic119896119904119901

119890119886119909+119910119887+119888119905 sinh(119886119909)sinh(119887119910) sinh(119888119905)(119887) (119888) (119886)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

119890119886119909+119910119887+119888119905 cosh(119886119909) cosh(119887119910) cosh(119888119905)(119887 minus 119904) (119888 minus 119896) (119886 minus 119901)

(minus2119886119901 + 1199012) (minus2119887119904 + 1199042) (minus2119888119896 + 1198962)

Erf [119886

2radic119909]Erf[

119887

2radic119910]Erf [

119888

2radic119905]

119890minusradic1198882119896minusradic1198872119904

119896119901119904(minus1 + 119890minus

radic1198882119896) (1 minus 119890minus

radic1198862119901) (minus1 + 119890minus

radic1198872119904)

sin(119886119909)

119909

sin(119887119910)

119910

sin(119888119905)

119905arctan(

radic1198862

119901) arctan(

radic1198872

119904) arctan(

radic1198882

119896)

cos(119886119909)

119909119899cos(119887119910)

119910119898cos(119888119905)

119905V

119896minus1+V(1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898 cos [119888119905] cos((minus1 + 119898) arctan[radic1198872

119904]) Γ (1 minus 119898) Γ (1 minus V)

times (1 +1198862

1199012)

12(minus1+119899)

119901minus1+119899 cos((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

sin(119886119909)

119909119899sin(119887119910)

119910119898sin(119888119905)

119905V

119896minus1+V (1 +1198872

1199042)

12(minus1+119898)

119904minus1+119898Γ (1 minus 119898) Γ (1 minus V) (1 +1198862

1199012)

12(minus1+119899)

times 119901minus1+119899Γ (1 minus 119899) sign (119886) sin ((119899 minus 1) arctan(|119886|

119901)) Γ (1 minus 119899)

119869119899

(119909) 119869119899

(119910) 119869119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 minus

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199042)

[Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 minus

1

1199012)

119868119899

(119909) 119868119899

(119910) 119868119899(119905)

8minus119899(119896119904119901)minus1minus119899Hypergeometric21198651 (

1 + 119899

2

2 + 119899

2 1 + 119899

1

1198962)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199042)

Hypergeometric21198651 (1 + 119899

2

2 + 119899

2 1 + 119899

1

1199012)

8 Abstract and Applied Analysis

500

400

300

200

100

06

4

2

0Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (41)

x

(a)

500

400

300

200

100

06

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

y x

(b)

64

2

0

0

20

40

60

Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

x

(c)

6

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (411)

minus1

minus05

0

05

1

xy

(d)

Figure 1 Numerical simulation of the exact solutions of the Homogeneous and non-homogeneous Mboctara equations

Now applying the triple Laplace transform on both sides of(57) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) +119896119904119901

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

minus1

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

(59)

Factorising the right side of (59) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962 )

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042 ) (minus1 + 1198962)

1 + 119901119904119896

(60)

Abstract and Applied Analysis 9

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905)

= 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896]

= cos (119909) cos (119910) cos (minus119905)

(61)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 4 consider the following nonlinear nonhomoge-neous with variable coefficient Mboctara equation

119890119909+119910+119905120597119909119910119905

119906 (119909 119910 119905) minus 31199062 (119909 119910 119905) + 119890119909+119910+119905119906 (119909 119910 119905)

= 1198902119909+2119910+2119905

119906119909

(119909 119910 0) = 119890119909+119910 119906 (0 0 0) = 1

119906 (1 0 0) = 119890 120597119909119910119905

119906 (0 0 0) = 1

(62)

Now applying the triple Laplace transform on both sidesof (62) and then using the properties of the triple Laplacetransform and after factorising as in the previous examplesand taking the inverse triple Laplace transform we obtainthe following as an exact solution of this type of Mboctaraequation

119906 (119909 119910 119905) = 119890119909+119910+119905 (63)

Thenumerical simulations of the exact solutions of theMboc-tara equation are depicted in Figure 1(a) (41) Figure 1(b)(46) Figure 1(c) (46) and Figure 1(d) (411) respectively

6 Triple Laplace Transform of SomeFunctions of Three Variables

In this section we examine the triple Laplace transform ofsome functions in Table 1

119871119909119910119905

(119884119899

(119909) 119884119899

(119910) 119884119899

(119905))

= 8minus119899(119896119904119901)minus1minus119899

1198621199041198883 [119899120587]

times ( minus (41198962)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1198962)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1198962))

times ( minus (41199042)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199042)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199042))

times ( minus (41199012)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199012)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199012))

(64)

7 Conclusion

This work presents the definition of the triple Laplace trans-form Some triple Laplace transform is presented in Table 1Some theorems and properties of this new relatively newoperator are presented Applications of the new operator forsolving some kind of third-order partial differential equationscalledMboctara equation are presentedNumerical solutionsof the Mboctara equation are given

References

[1] G L Lamb Jr Introductory Applications of Partial DifferentialEquations with Emphasis on Wave Propagation and DiffusionJohn Wiley amp Sons New York NY USA 1995

[2] U T Myint Differential Equations of Mathematical PhysicsAmerican Elsevier New York NY USA 1980

[3] C Constanda Solution Techniques for Elementary Partial Differ-ential Equations Chapman amp HallCRC New York NY USA2002

[4] D G Duffy Transform Methods for Solving Partial DifferentialEquations CRC Press New York NY USA 2004

[5] A Babakhani and R S Dahiya ldquoSystems of multi-dimensionalLaplace transforms and a heat equationrdquo in Proceedings of the16th Conference onAppliedMathematics vol 7 ofElectronicJour-nal of Differential Equations pp 25ndash36 University of CentralOklahoma Edmond Okla USA 2001

[6] Y A Brychkov H -J Glaeske A P Prudnikov and V K TuanMultidimensional Integral Transformations Gordon and BreachScience Publishers Philadelphia Pa USA 1992

[7] A Atangana and A Kilicman ldquoA possible generalization ofacoustic wave equation using the concept of perturbed deriva-tive orderrdquo Mathematical Problems in Engineering vol 2013Article ID 696597 6 pages 2013

[8] A Kılıcman and H Eltayeb ldquoA note on the classifications ofhyperbolic and elliptic equations with polynomial coefficientsrdquoApplied Mathematics Letters vol 21 no 11 pp 1124ndash1128 2008

10 Abstract and Applied Analysis

[9] H Eltayeb A Kılıcman and P Ravi Agarwal ldquoAn analysis onclassifications of hyperbolic and elliptic PDEsrdquo MathematicalSciences vol 6 article 47 2012

[10] H Eltayeb andAKılıcman ldquoAnote ondouble laplace transformand telegraphic equationsrdquo Abstract and Applied Analysis vol2013 Article ID 932578 6 pages 2013

[11] A Kılıcman and H E Gadain ldquoOn the applications of Laplaceand Sumudu transformsrdquo Journal of the Franklin Institute vol347 no 5 pp 848ndash862 2010

[12] R P Kanwal Generalized Functions Theory and ApplicationsBirkhaauser Boston Mass USA 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

500

400

300

200

100

06

4

2

0Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (41)

x

(a)

500

400

300

200

100

06

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

y x

(b)

64

2

0

0

20

40

60

Time

64

20

Exact solution of the nonhomogeneous Mboctara equation (46)

x

(c)

6

4

2

0

64

20

Exact solution of the nonhomogeneous Mboctara equation (411)

minus1

minus05

0

05

1

xy

(d)

Figure 1 Numerical simulation of the exact solutions of the Homogeneous and non-homogeneous Mboctara equations

Now applying the triple Laplace transform on both sides of(57) we obtain the following

119901119904119896119880 (119901 119904 119896) + 119880 (119901 119904 119896)

= 119866 (119901 119904 119896) +119896119904119901

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

minus1

(1 + 1199012) (1 + 1199042) (minus1 + 1198962)

(59)

Factorising the right side of (59) we obtain the following

119880 (119901 119904 119896) =119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962 )

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042 ) (minus1 + 1198962)

1 + 119901119904119896

(60)

Abstract and Applied Analysis 9

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905)

= 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896]

= cos (119909) cos (119910) cos (minus119905)

(61)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 4 consider the following nonlinear nonhomoge-neous with variable coefficient Mboctara equation

119890119909+119910+119905120597119909119910119905

119906 (119909 119910 119905) minus 31199062 (119909 119910 119905) + 119890119909+119910+119905119906 (119909 119910 119905)

= 1198902119909+2119910+2119905

119906119909

(119909 119910 0) = 119890119909+119910 119906 (0 0 0) = 1

119906 (1 0 0) = 119890 120597119909119910119905

119906 (0 0 0) = 1

(62)

Now applying the triple Laplace transform on both sidesof (62) and then using the properties of the triple Laplacetransform and after factorising as in the previous examplesand taking the inverse triple Laplace transform we obtainthe following as an exact solution of this type of Mboctaraequation

119906 (119909 119910 119905) = 119890119909+119910+119905 (63)

Thenumerical simulations of the exact solutions of theMboc-tara equation are depicted in Figure 1(a) (41) Figure 1(b)(46) Figure 1(c) (46) and Figure 1(d) (411) respectively

6 Triple Laplace Transform of SomeFunctions of Three Variables

In this section we examine the triple Laplace transform ofsome functions in Table 1

119871119909119910119905

(119884119899

(119909) 119884119899

(119910) 119884119899

(119905))

= 8minus119899(119896119904119901)minus1minus119899

1198621199041198883 [119899120587]

times ( minus (41198962)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1198962)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1198962))

times ( minus (41199042)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199042)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199042))

times ( minus (41199012)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199012)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199012))

(64)

7 Conclusion

This work presents the definition of the triple Laplace trans-form Some triple Laplace transform is presented in Table 1Some theorems and properties of this new relatively newoperator are presented Applications of the new operator forsolving some kind of third-order partial differential equationscalledMboctara equation are presentedNumerical solutionsof the Mboctara equation are given

References

[1] G L Lamb Jr Introductory Applications of Partial DifferentialEquations with Emphasis on Wave Propagation and DiffusionJohn Wiley amp Sons New York NY USA 1995

[2] U T Myint Differential Equations of Mathematical PhysicsAmerican Elsevier New York NY USA 1980

[3] C Constanda Solution Techniques for Elementary Partial Differ-ential Equations Chapman amp HallCRC New York NY USA2002

[4] D G Duffy Transform Methods for Solving Partial DifferentialEquations CRC Press New York NY USA 2004

[5] A Babakhani and R S Dahiya ldquoSystems of multi-dimensionalLaplace transforms and a heat equationrdquo in Proceedings of the16th Conference onAppliedMathematics vol 7 ofElectronicJour-nal of Differential Equations pp 25ndash36 University of CentralOklahoma Edmond Okla USA 2001

[6] Y A Brychkov H -J Glaeske A P Prudnikov and V K TuanMultidimensional Integral Transformations Gordon and BreachScience Publishers Philadelphia Pa USA 1992

[7] A Atangana and A Kilicman ldquoA possible generalization ofacoustic wave equation using the concept of perturbed deriva-tive orderrdquo Mathematical Problems in Engineering vol 2013Article ID 696597 6 pages 2013

[8] A Kılıcman and H Eltayeb ldquoA note on the classifications ofhyperbolic and elliptic equations with polynomial coefficientsrdquoApplied Mathematics Letters vol 21 no 11 pp 1124ndash1128 2008

10 Abstract and Applied Analysis

[9] H Eltayeb A Kılıcman and P Ravi Agarwal ldquoAn analysis onclassifications of hyperbolic and elliptic PDEsrdquo MathematicalSciences vol 6 article 47 2012

[10] H Eltayeb andAKılıcman ldquoAnote ondouble laplace transformand telegraphic equationsrdquo Abstract and Applied Analysis vol2013 Article ID 932578 6 pages 2013

[11] A Kılıcman and H E Gadain ldquoOn the applications of Laplaceand Sumudu transformsrdquo Journal of the Franklin Institute vol347 no 5 pp 848ndash862 2010

[12] R P Kanwal Generalized Functions Theory and ApplicationsBirkhaauser Boston Mass USA 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

Now applying the inverse triple Laplace transform on theprevious equation we obtain the following solution

119906 (119909 119910 119905)

= 119871minus1119909119910119905

[119866 (119901 119904 119896)

1 + 119901119904119896+

119896119904119901 (1 + 1199012) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896

minus1 (1 + 1199012 ) (1 + 1199042) (minus1 + 1198962)

1 + 119901119904119896]

= cos (119909) cos (119910) cos (minus119905)

(61)

This is the exact solution for nonhomogeneous Mboctaraequation

Example 4 consider the following nonlinear nonhomoge-neous with variable coefficient Mboctara equation

119890119909+119910+119905120597119909119910119905

119906 (119909 119910 119905) minus 31199062 (119909 119910 119905) + 119890119909+119910+119905119906 (119909 119910 119905)

= 1198902119909+2119910+2119905

119906119909

(119909 119910 0) = 119890119909+119910 119906 (0 0 0) = 1

119906 (1 0 0) = 119890 120597119909119910119905

119906 (0 0 0) = 1

(62)

Now applying the triple Laplace transform on both sidesof (62) and then using the properties of the triple Laplacetransform and after factorising as in the previous examplesand taking the inverse triple Laplace transform we obtainthe following as an exact solution of this type of Mboctaraequation

119906 (119909 119910 119905) = 119890119909+119910+119905 (63)

Thenumerical simulations of the exact solutions of theMboc-tara equation are depicted in Figure 1(a) (41) Figure 1(b)(46) Figure 1(c) (46) and Figure 1(d) (411) respectively

6 Triple Laplace Transform of SomeFunctions of Three Variables

In this section we examine the triple Laplace transform ofsome functions in Table 1

119871119909119910119905

(119884119899

(119909) 119884119899

(119910) 119884119899

(119905))

= 8minus119899(119896119904119901)minus1minus119899

1198621199041198883 [119899120587]

times ( minus (41198962)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1198962)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1198962))

times ( minus (41199042)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199042)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199042))

times ( minus (41199012)119899

Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 minus 119899 minus

1

1199012)

+ cos (119899120587)Hypergeometric21198651

times (1 + 119899

22 + 119899

2 1 + 119899 minus

1

1199012))

(64)

7 Conclusion

This work presents the definition of the triple Laplace trans-form Some triple Laplace transform is presented in Table 1Some theorems and properties of this new relatively newoperator are presented Applications of the new operator forsolving some kind of third-order partial differential equationscalledMboctara equation are presentedNumerical solutionsof the Mboctara equation are given

References

[1] G L Lamb Jr Introductory Applications of Partial DifferentialEquations with Emphasis on Wave Propagation and DiffusionJohn Wiley amp Sons New York NY USA 1995

[2] U T Myint Differential Equations of Mathematical PhysicsAmerican Elsevier New York NY USA 1980

[3] C Constanda Solution Techniques for Elementary Partial Differ-ential Equations Chapman amp HallCRC New York NY USA2002

[4] D G Duffy Transform Methods for Solving Partial DifferentialEquations CRC Press New York NY USA 2004

[5] A Babakhani and R S Dahiya ldquoSystems of multi-dimensionalLaplace transforms and a heat equationrdquo in Proceedings of the16th Conference onAppliedMathematics vol 7 ofElectronicJour-nal of Differential Equations pp 25ndash36 University of CentralOklahoma Edmond Okla USA 2001

[6] Y A Brychkov H -J Glaeske A P Prudnikov and V K TuanMultidimensional Integral Transformations Gordon and BreachScience Publishers Philadelphia Pa USA 1992

[7] A Atangana and A Kilicman ldquoA possible generalization ofacoustic wave equation using the concept of perturbed deriva-tive orderrdquo Mathematical Problems in Engineering vol 2013Article ID 696597 6 pages 2013

[8] A Kılıcman and H Eltayeb ldquoA note on the classifications ofhyperbolic and elliptic equations with polynomial coefficientsrdquoApplied Mathematics Letters vol 21 no 11 pp 1124ndash1128 2008

10 Abstract and Applied Analysis

[9] H Eltayeb A Kılıcman and P Ravi Agarwal ldquoAn analysis onclassifications of hyperbolic and elliptic PDEsrdquo MathematicalSciences vol 6 article 47 2012

[10] H Eltayeb andAKılıcman ldquoAnote ondouble laplace transformand telegraphic equationsrdquo Abstract and Applied Analysis vol2013 Article ID 932578 6 pages 2013

[11] A Kılıcman and H E Gadain ldquoOn the applications of Laplaceand Sumudu transformsrdquo Journal of the Franklin Institute vol347 no 5 pp 848ndash862 2010

[12] R P Kanwal Generalized Functions Theory and ApplicationsBirkhaauser Boston Mass USA 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

[9] H Eltayeb A Kılıcman and P Ravi Agarwal ldquoAn analysis onclassifications of hyperbolic and elliptic PDEsrdquo MathematicalSciences vol 6 article 47 2012

[10] H Eltayeb andAKılıcman ldquoAnote ondouble laplace transformand telegraphic equationsrdquo Abstract and Applied Analysis vol2013 Article ID 932578 6 pages 2013

[11] A Kılıcman and H E Gadain ldquoOn the applications of Laplaceand Sumudu transformsrdquo Journal of the Franklin Institute vol347 no 5 pp 848ndash862 2010

[12] R P Kanwal Generalized Functions Theory and ApplicationsBirkhaauser Boston Mass USA 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of