research article analytical solution of space-time...

Research ArticleAnalytical Solution of Space-Time FractionalFokker-Planck Equation by Homotopy PerturbationSumudu Transform Method

Ravi Shanker Dubey1 Badr Saad T Alkahtani2 and Abdon Atangana3

1Department of Mathematics Yagyavalkya Institute of Technology Jaipur 302022 India2Mathematics Department College of Science King Saud University PO Box 1142 Riyadh 11989 Saudi Arabia3Institute for Groundwater Studies Faculty of Natural and Agricultural Sciences University of the Free StateBloemfontein 9300 South Africa

Correspondence should be addressed to Ravi Shanker Dubey ravimath13gmailcom

Received 27 May 2014 Revised 11 September 2014 Accepted 21 September 2014

Academic Editor Samir B Belhaouari

Copyright copy 2015 Ravi Shanker Dubey et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An efficient approach based on homotopy perturbation method by using Sumudu transform is proposed to solve some linearand nonlinear space-time fractional Fokker-Planck equations (FPEs) in closed form The space and time fractional derivatives areconsidered in Caputo sense The homotopy perturbation Sumudu transform method (HPSTM) is a combined form of Sumudutransform homotopy perturbation method and Hersquos polynomials The nonlinear terms can be easily handled by the use of Hersquospolynomials Some examples show that the HPSTM is an effective tool for solving many space time fractional partial differentialequations

1 Introduction

Fokker-Planck equation (FPE) was introduced by AdriaanFokker and Max Planck to describe the time evolution ofthe probability density function of position and velocity of aparticle which is one of the classical widely used equations ofstatistical physics [1] FPE arises in a number of different fieldsin natural sciences Brownian motion [2] and the diffusionmodel of chemical reactions [3] are now largely employed invarious generalized forms in physics chemistry engineeringand biology [1] The FPE arises in kinetic theory [4] whereit describes the evolution of the one-particle distributionfunction of a dilute gas with long-range collisions such as aCoulomb gas Some applications of this type of equations canbe worked out in the works of He and Wu [5] Jumarie [6]Kamitani and Matsuba [7] Xu et al [8] and Zak [9]

The general FPE for the motion of a concentration fieldV(119909 119905) of one space variable 119909 at time 119905 has the form [1]

120597V (119909 119905)120597119905

= [minus120597119860 (119909)

120597119909+

1205972119861 (119909)

1205971199092] V (119909 119905) (1)

with initial condition

V (0 119909) = 119891 (119909) 119909 isin R (2)

where 119860(119909) and 119861(119909) gt 0 are called the drift and diffusioncoefficients This equation is also called the forward Kol-mogorov equation The drift and diffusion coefficients mayalso depend on time as

120597V (119909 119905)120597119905

= [minus120597119860 (119909 119905)

120597119909+

1205972119861 (119909 119905)

1205971199092] V (119909 119905) (3)

There is a more general form of Fokker-Planck equationwhich is called the nonlinear Fokker-Planck equation Thenonlinear Fokker-Planck equation has important applica-tions in various areas such as plasma physics surfacephysics population dynamics biophysics engineering neu-rosciences nonlinear hydrodynamics polymer physics laserphysics pattern formation psychology and so forth [10] Inthe one variable case the nonlinear FPE can be written as

120597V (119909 119905)120597119905

= [minus120597119860 (119909 119905 V)

120597119909+

1205972119861 (119909 119905 V)

1205971199092] V (119909 119905) (4)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 780929 7 pageshttpdxdoiorg1011552015780929

2 Mathematical Problems in Engineering

Due to vast range of applications of the FPE a lot of workhas been done to find numerical solution with this equationIn this context the works of Buet et al [11] Harrison [12]Palleschi et al [13] Vanaja [14] and Zorzano et al [15] areworth mentioning

For generalization of classical integer order of FPE (4)the following equation can be introducedThe Fokker-Planckequation with fractional space derivative is a particular caseof anomalous diffusion and Levy flights (see[16ndash19]) Thisequation is called nonlinear FPE with space-time fractionalderivatives [20]

120597120572V (119909 119905)120597119905120572

= [minus120597119860 (119909 119905 V)

120597119909+

1205972119861 (119909 119905 V)

1205971199092] V (119909 119905) (5)

where 119905 gt 0 119909 gt 0 and 0 lt 120572 le 1 It can be obtained fromthe general Fokker-Planck equation by replacing the spaceand time derivatives by fractional derivatives operator Thefunction V(119909 119905) is assumed to be a causal function of time andspace Particularly for 120572 = 1 the fractional FPE (5) reducesto the classical nonlinear FPE given by (4) in the case 119909 gt 0

In recent years researchers have studied the fractionalpartial differential equations and the fractional FPE toinvestigate various scientific models [21ndash24] In the presentpaper we obtain closed form solutions of a linear-nonlineartime fractional FPE using homotopy perturbation Sumudutransform method (HPSTM) see [25]

2 Some Mathematical Preliminariesand Definitions

21 Fundamental Properties of Fractional Calculus Firstly wemention some of the fundamental properties of the fractionalcalculus that is fractional derivatives and integrals

The Riemann-Liouville fractional integral operator oforder 120572 gt 0 of a function 119891(119905) isin 119862

120583 120583 ge minus1 is defined

by implementing the integration operator 119869120572 in the followingmanner

119869120572[119891 (119905)] =

1

Γ (120572)int119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591 120572 gt 0

1198690[119891 (119905)] = 119891 (119905)

119869120572119869120573[119891 (119905)] = 119869

120572+120573119891 (119905)

119869120573119869120572[119891 (119905)] = 119869

120572119869120573119891 (119905)

119869120572(119905120574) =

Γ (120574 + 1)

Γ (120572 + 120574 + 1)119905120572+120574

(6)

The fundamental properties of fractional integration andfractional differentiation have been introduced to the liter-ature by Podlubny [26]

The fractional derivative of 119891(119905) in the Caputo sense isdefined by the following relation [27]

119863120572

119905119891 (119905) = 119869

119898minus120572119863119899119891 (119905)

=1

Γ (119899 minus 120572)int119905

0

(119905 minus 120591)minus120572+119898minus1

119891(119898)

(120591) 119889120591

119898 minus 1 lt 120572 le 119898

(7)

where119898 isin 119873 and 119905 gt 0The relation between Riemann-Liouville fractional inte-

gral operator and the Caputo fractional derivative operatoris

119869120572

119905119863120572

119905119891 (119905) = 119891 (119905) minus

119898minus1

sum119896=0

119891119896 (0+) 119905119896

119896

119863minus120572

[119891 (119905)] =1

Γ (120572)int119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591

0 lt 120572 le 1

(8)

Further properties and applications can be found in [26 28ndash31]

22 The Sumudu Transform In early 90s Watugala [32]introduced a new integral transform named the Sumudutransform and applied it to the solution of ordinary differen-tial equation in control engineering problems The Sumudutransform is defined over the set of functions

119860 = 119891 (119905) | exist119872 1205911 1205912gt 0

1003816100381610038161003816119891 (119905)1003816100381610038161003816 lt 119872119890

|119905|120591119895

if 119905 isin (minus1)119895times [0infin)

(9)

the Sumudu transform is defined by

119866 (119906) = 119878 [119891 (119905)] = intinfin

0

119891 (119906119905) 119890minus119905119889119905 119906 isin (minus120591

1 1205912) (10)

Sumudu transformhasmany useful and important propertieslike linear property scale properties shifting propertiesduality with Laplace transforms and so forth Further detailand properties about this transform can be found in [25 33ndash37]

By using the Sumudu transform of multiple differentia-tion we obtain

119878 [119863120572

119905119891 (119905)] = 119906

119898minus120572[119866 (119906)

119906119898minus

119898minus1

sum119896=0

119891119896 (0)

119906119898minus119896]

= [119866 (119906)

119906120572minus

119898minus1

sum119896=0

119891119896 (0+)

119906120572minus119896] (119898 minus 1 lt 120572 le 119898)

(11)

where 119866(119906) = 119878[119891(119905)]

Mathematical Problems in Engineering 3

23 The Mittag-Leffler Function TheMittag-Leffler functionwhich is a generalization of exponential function (see [38]) isas follows

119864120572(119911) =

infin

sum119899=119900

119911119899

Γ (120572119899 + 1)(120572 isin C R (120572) gt 0) (12)

24 Adomian Decomposition Method The Adomian decom-position method (ADM) is a creative and effective methodfor exactly solving functional equations of various kindsThemethod was developed by Adomian [39] It is important tonote that a large amount of researchwork has been devoted tothe application of theADM in awide class of linear nonlinearordinary or partial differential equationsThe decompositionmethod provides the solution as an infinite series in whicheach term can be determined easily

3 Solution of Fractional DifferentialEquations by Homotopy PerturbationSumudu Transform Method (HPSTM)

The homotopy perturbation method (HPM) introduced byHe (see [40ndash43]) is a series expansion method used in thesolution of nonlinear partial differential equationsTheHPMuses a so-called convergence-control parameter to guaranteethe convergence of approximation series over a given intervalof physical parameters

We illustrate the basic idea of this method by consider-ing a general fractional nonlinear nonhomogeneous partialdifferential equation

119863120572

119905119880 (119909 119905) = 119871119880 (119909 119905) + 119873119880 (119909 119905) + 119891 (119909 119905) 120572 gt 0 (13)

with initial conditions

119863119896

0119880 (119909 0) = 119892

119896 (119896 = 0 1 119899 minus 1)

119863119899

0119880 (119909 0) = 0 119899 = [120572]

(14)

where 119863120572119905119880(119909 119905) is the Caputo fractional derivative of the

function 119880(119909 119905) 119871 is the linear differential operator 119873

represents the general nonlinear differential operator and119891(119909 119905) is a known function

Applying the Sumudu transform on both sides of (13) weget

119878 [119863120572

119905119880 (119909 119905)] = 119878 [119871119880 (119909 119905)] + 119878 [119873119880 (119909 119905)] + 119878 [119891 (119909 119905)]

(15)

Using the property of the Sumudu transform we have

119878 [119880 (119909 119905)] = 119906120572119878 [119871119880 (119909 119905) + 119873119880 (119909 119905)] + 119892 (119909 119905) (16)

Operating with the inverse Sumudu transform on both sidesof (16) we get

119880 (119909 119905) = 119866 (119909 119905) + 119878minus1

[119906120572119878 [119871119880 (119909 119905) + 119873119880 (119909 119905)]] (17)

where 119866(119909 119905) represents the term arising from the functionand the prescribed initial conditions

Now we use the homotopy perturbation method

119880 (119909 119905) =

infin

sum119899=0

119901119899119880119899(119909 119905) (18)

and the nonlinear term can be decomposed as

119873119880(119909 119905) =

infin

sum119899=0

119901119899119867119899(119880) (19)

on using Hersquos polynomials 119867119899(119880) (see [10 44]) which are

given by

119867119899(1198800 1198801 119880

119899) =

1

119899

120597119899

120597119901119899[119873(

infin

sum119894=0

119901119894119880119894)]

119901=0

(20)

Substituting (18) and (19) in (17) we getinfin

sum119899=0

119901119899119880119899(119909 119905)

= 119866 (119909 119905)

+ 119901(119878minus1

[119906120572119878 [119871

infin

sum119899=0

119901119899119880119899(119909 119905) +

infin

sum119899=0

119901119899119867119899(119880)]])

(21)

which shows the coupling of the Sumudu transform and theHPM by using Hersquos polynomials

On collecting the coefficients of powers of 119901 we obtain

1199010 1198800(119909 119905) = 119866 (119909 119905)

1199011 1198801(119909 119905) = 119878

minus1[119906120572119878 119871119880

0(119909 119905) + 119867

0(119880)]

1199012 1198802(119909 119905) = 119878

minus1[119906120572119878 119871119880

1(119909 119905) + 119867

1(119880)]

(22)

and similarly

119901119899 119880119899(119909 119905) = 119878

minus1[119906120572119878 119871119880

119899minus1(119909 119905) + 119867

119899minus1(119880)] (23)

Finally we approximate the analytical solution 119880(119909 119905) bytruncated series

119880 (119909 119905) = lim119873rarrinfin

119873

sum119899=0

119880119899(119909 119905) (24)

However as we have seen inmany cases the exact solution ina closed form may be obtained In addition the decomposi-tion series solutions generally converge very rapidly Abbaouiand Cherruault [45] had defined the classical approach ofconvergence of this type of series

4 Solution of Fokker-Planck FractionalDifferential Equation

Solution of the following nonlinear space-time fractionalFokker-Planck equations (FPE) is

120597120572V (119909 119905)120597119905120572

= [minus120597119860 (119909 119905 V)

120597119909+

1205972119861 (119909 119905 V)

1205971199092] V (119909 119905) (25)


with the initial condition

V (119909 0) = 119891 (119909) (26)

Applying the Sumudu transformon both sides of (25) subjectto initial condition (26) we have

119878 [V (119909 119905)] = 119891 (119909) + 119906120572119878

times [minus120597119860 (119909 119905 V)

120597119909V (119909 119905) +

1205972119861 (119909 119905 V)

1205971199092V (119909 119905)]

(27)

taking inverse Sumudu transform

V (119909 119905) = 119891 (119909) + 119878minus1

[119906120572119878minus

120597

120597119909

infin

sum119899=0

119860119899+

1205972

1205971199092

infin

sum119899=0

119861119899]

(28)

whereinfin

sum119899=0

119860119899= 119860 (119909 119905 V) sdot V (119909 119905)

infin

sum119899=0

119861119899= 119861 (119909 119905 V) sdot V (119909 119905)

(29)

Using HPMmethod we getinfin

sum119899=0

119901119899V119899(119909 119905)

= 119891 (119909) + 119901

times (119878minus1

[119906120572119878minus

infin

sum119899=0

119901119899119867119899(119909 119905 V) +

infin

sum119899=0

119901119899119867lowast

119899(119909 119905 V)])

(30)

where 119867119899(119909 119905 V) and 119867lowast

119899(119909 119905 V) are Hersquos polynomials that

represent the nonlinear termsTheHersquos polynomials are givenby

infin

sum119899=0

119867119899(119909 119905 V) =

120597

120597119909(

infin

sum119899=0

119860119899)

infin

sum119899=0

119867lowast

119899(119909 119905 V) =

1205972

1205971199092(

infin

sum119899=0

119861119899)

(31)

Using the above equation we can collect the coefficient ofpower of 119901

1199010 V0(119909 119905) = 119891 (119909)

1199011 V1(119909 119905) = 119878

minus1[119906120572119878 [minus119867

0+ 119867lowast

0]]

1199012 V2(119909 119905) = 119878

minus1[119906120572119878 [minus119867

1+ 119867lowast

1]]

(32)

and similarly

119901119899 V119899(119909 119905) = 119878

minus1[119906120572119878 [minus119867

119899minus1+ 119867lowast

119899minus1]] (33)

Finally we approximate the analytical solution V119899(119909 119905) by

truncated series [25]

V (119909 119905) = lim119873rarrinfin

119873

sum119899=0

V119899(119909 119905) (34)

the series solutions of the above equation converge veryrapidly [25 45]

5 Numerical Examples

In this section we will illustrate the HPSTM techniques byseveral examples These examples are somewhat artificial inthe sense that the exact answer for the special case 120572 = 1 isknown in advance and the initial and boundary conditionsare directly taken from this answer Nonetheless such anapproach is needed to evaluate the accuracy of the analyticaltechniques and to examine the effect of varying the order ofthe space- and time-fractional derivatives on the behaviorof the solution All the results are calculated by using thesymbolic calculus software MATLAB

Example 1 Consider the nonlinear time fractional FPE

119863120572

119905V (119909 119905) = [minus

120597

120597119909(3V minus

119909

2) +

1205972

1205971199092(119909V)] V (119909 119905) (35)

where 119905 gt 0 119909 gt 0 0 lt 120572 le 1 and 119863120572119905is the Caputo

fractional derivative defined by (7) and initial condition is

V (119909 0) = 119909 (36)

Then solution is given by V(119909 119905) = 119909119864120572(119905120572)

Solution 1 On using the method defined in Section 4 we getthe coefficient of powers of 119901

1199010 V0(119909 119905) = 119909

1199011 V1(119909 119905) =

119909119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

1199091199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

1199091199053120572

Γ (3120572 + 1)

(37)

hence V(119909 119905) is

V (119909 119905) = 119909(1 +119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 119909119864120572(119905120572)

(38)

Remark 2 Setting 120572 = 1 Example 1 reduces to nonlinear FPE

120597

120597119905V (119909 119905) = [minus

120597

120597119909(3V minus

119909

2) +

1205972

1205971199092(119909V)] V (119909 119905) (39)



V (119909 0) = 119909 (40)

and a solution as

V (119909 119905) = 119909119890119905 (41)


119863120572

119905V (119909 119905)

= [minus120597

120597119909(4V119909

minus119909

3) +

1205972

1205971199092(V)] V (119909 119905) 119909 gt 0 119905 gt 0

(42)

where 119905 gt 0 119909 gt 0 0 lt 120572 le 1 and 119863120572119905is Caputo fractional

derivative defined by (7) and initial condition is

V (119909 0) = 1199092 (43)


Solution 2 By using the method defined in Section 4 we getthe coefficient of power of 119901

1199010 V0(119909 119905) = 119909

2

1199011 V1(119909 119905) =

1199092119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

11990921199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

11990921199053120572

Γ (3120572 + 1)sdot sdot sdot

(44)

hence V(119909 119905) is

V (119909 119905)

= 1199092(1 +

119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 1199092119864120572(119905120572)

(45)

Remark 4 Setting120572 = 1 Example 3 reduces to nonlinear FPE

120597

120597119905V (119909 119905)

= [minus120597

120597119909(4V119909

minus119909

3) +

1205972

1205971199092(V)] V (119909 119905) 119909 gt 0 119905 gt 0

(46)


V (119909 0) = 1199092 (47)

and a solution as

V (119909 119905) = 1199092119890119905 (48)


119863120572

119905V (119909 119905)

= [minus120597

120597119909(119909) +

1205972

1205971199092(1199092

2)] V (119909 119905) 119909 gt 0 119905 gt 0

(49)

where 119905 gt 0 119909 gt 0 0 lt 120572 le 1 and 119863120572

119905is Caputo fractional


V (119909 0) = 119909 (50)

Solution 3 By using the method defined in Section 4 we getthe coefficient of 119901

1199010 V0(119909 119905) = 119909

1199011 V1(119909 119905) =

119909119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

1199091199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

1199091199053120572

Γ (3120572 + 1)

(51)

hence V(119909 119905) is

V (119909 119905) = 119909(1 +119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 119909119864120572(119905120572)

(52)

Remark 6 Setting 120572 = 1 in problem Example 5 reduces tononlinear FPE

120597

120597119905V (119909 119905) = [minus

120597

120597119909(119909) +

1205972

1205971199092(1199092

2)] V (119909 119905)

119909 gt 0 119905 gt 0

(53)


V (119909 0) = 119909 (54)

and a solution as

V (119909 119905) = 119909119890119905 (55)

The approximate solutions for (42) and (49) shown in Tables1 and 2 respectively were obtained for different values of 119909and 119905 by using the homotopy perturbation Sumudu transformmethod The exact solutions of (42) and (49) are V(119909 119905) =

1199092119890119905 and V(119909 119905) = 119909119890119905 respectively The accuracy of our

approximate solutions can be improved by computing moreterms of the approximate solutions Both the methods vizADM and HPSTM are used adroitly to obtain the exactvalues and get comparisons From the numerical resultsin Tables 1 and 2 it is easy to conclude that the solutioncontinuously depends on the space-fractional derivative andthe approximate solutions of (42) and (49) obtained by using


Table 1 Comparison study for solution of (42) for 120572 = 1 when119905 = 02 04 06 and 119909 = 025 050 075 10 by various methods

119905 119909 HPSTM ADM (see [20]) EXACT (see [20])02 025 0076333 0076333 007633802 050 0305333 0305333 030535102 075 0687000 0687000 068703902 100 1221333 1221333 122140304 025 0093167 0093167 009323904 050 0372667 0372667 037295604 075 0838500 0838500 083915104 100 1490667 1490667 149182506 025 0113500 0113500 011388206 050 0454000 0454000 045553006 075 1021500 1021500 102494206 100 1816000 1816000 1822119



the Adomian decomposition method are the same as thoseobtained by the homotopy perturbation Sumudu transformmethod It is to be noted that only the fourth-order term ofthe series of the abovesaid methods was used in evaluatingthe approximate solutions for Tables 1 and 2

6 Conclusion

Within the scope of theHPSTM we derive the solution of thenonlinear time fractional Fokker-Planck equationsWemadeuse of the Caputo derivative because it allows usual initialconditionsThe numerical result shows that the method usedis very simple and straightforward to implement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to the anonymous referee for hishercareful reading andmany correctionssuggestions to improvethis paper This project was supported by Deanship ofScientific Research College of Science Research Center KingSaud University

References

[1] H Risken The FokkerPlanck Equation Method of Solution andApplications Springer Berlin Germany 1989

[2] S Chandresekhar ldquoStochastic problems in physics and astron-omyrdquo Reviews of Modern Physics vol 15 pp 1ndash89 1943

[3] H A Kramers ldquoBrownian motion in a field of force and thediffusion model of chemical reactionsrdquo Physica vol 7 pp 284ndash304 1940

[4] A Fokker ldquoThe median energy of rotating electrical dipoles inradiation fieldsrdquo Annalen Der Physik vol 43 pp 810ndash820 1914

[5] J H He and X H Wu ldquoConstruction of solitary solutionand compaction-like solution by variational iteration methodrdquoChaos Solitons amp Fractals vol 29 pp 108ndash113 2006

[6] G Jumarie ldquoFractional Brownian motions via random walk inthe complex plane and via fractional derivative comparisonand further results on their Fokker-Planck equationsrdquo ChaosSolitons amp Fractals vol 22 pp 907ndash925 2004

[7] Y Kamitani and I Matsuba ldquoSelf-similar characteristics ofneural networks based on Fokker-Planck equationrdquo ChaosSolitons amp Fractals vol 20 no 2 pp 329ndash335 2004

[8] Y Xu F Y Ren J R Liang and W Y Qiu ldquoStretched Gaussianasymptotic behavior for fractional Fokker-Planck equation onfractal structure in external force fieldsrdquo Chaos Solitons ampFractals vol 20 no 3 pp 581ndash586 2004

[9] M Zak ldquoExpectation-based intelligent controlrdquo Chaos Solitonsamp Fractals vol 28 pp 616ndash626 2006

[10] S T Mohyud-Din M A Noor and K I Noor ldquoTraveling wavesolutions of seventh-order generalized KdV equations usingHersquos polynomialsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 10 no 2 pp 227ndash233 2009

[11] C Buet S Dellacherie and R Sentis ldquoNumerical solution ofan ionic Fokker-Planck equation with electronic temperaturerdquoSIAM Journal on Numerical Analysis vol 39 no 4 pp 1219ndash1253 2001

[12] G W Harrison ldquoNumerical solution of the Fokker-Planckequation using moving finite elementsrdquo Numerical Methods forPartial Differential Equations vol 4 no 3 pp 219ndash232 1988

[13] V Palleschi F Sarri G Marcozzi and M R Torquati ldquoNumer-ical solution of the Fokker-Planck equation a fast and accuratealgorithmrdquo Physics Letters A vol 146 no 7-8 pp 378ndash386 1990

[14] V Vanaja ldquoNumerical solution of a simple Fokker-PlanckequationrdquoAppliedNumericalMathematics vol 9 no 6 pp 533ndash540 1992

[15] M P Zorzano H Mais and L Vazquez ldquoNumerical solutionof two-dimensional Fokker-Planck equationsrdquo Applied Mathe-matics and Computation vol 98 no 2-3 pp 109ndash117 1999

[16] A A Dubkov B Spagnolo and V V Uchaikin ldquoLevy flightsuperdiffusion an introductionrdquo International Journal of Bifur-cation and Chaos in Applied Sciences and Engineering vol 18no 9 pp 2649ndash2672 2008


[17] A A Dubkov and B Spagnolo ldquoAcceleration of diffusion inrandomly switching potential with supersymmetryrdquo PhysicalReview E vol 72 no 4 part 1 Article ID 041104 8 pages 2005

[18] A Dubkov and B Spagnolo ldquoGeneralized Wiener process andKolmogorovrsquos equation for diffusion induced by non-Gaussiannoise sourcerdquo Fluctuation and Noise Letters vol 5 no 2 ppL267ndashL274 2005

[19] A Dubkov and B Spagnolo ldquoLangevin approach to Levyflights in fixed potentials exact results for stationary probabilitydistributionsrdquo Acta Physica Polonica B vol 38 no 5 pp 1745ndash1758 2007

[20] Z Odibat and S Momani ldquoNumerical solution of Fokker-Planck equation with space- and time-fractional derivativesrdquoPhysics Letters A vol 369 pp 349ndash358 2007

[21] Q Yang F Liu and I Turner ldquoComputationally efficientnumerical methods for time and space-fractional Fokker-Planck equationsrdquo Physica Scripta vol 2009 no T136 ArticleID 014026 7 pages 2009

[22] Q Yang F Liu and I Turner ldquoStability and convergence ofan effective numerical method for the time-space fractionalFokker-Planck equation with a nonlinear source termrdquo Inter-national Journal of Differential Equations vol 2010 Article ID464321 22 pages 2010

[23] A Yildirim ldquoAnalytical approach to Fokker-Planck equationwith space- and timefractional derivatives by means of thehomotopy perturbation methodrdquo Journal of King Saud Univer-sity (Science) vol 22 pp 257ndash264 2010

[24] P Zhuang F Liu V Anh and I Turner ldquoNumerical treatmentfor the fractional Fokker-Planck equationrdquoAnziam Journal vol48 pp C759ndashC774 2007

[25] A Atangana and A Kılıcman ldquoThe use of Sumudu transformfor solving certain nonlinear fractional heat-like equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 737481 12pages 2013

[26] F Mainardi ldquoFractional calculus some basic problems in con-tinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds vol 378 pp 291ndash348 Springer New York NYUSA 1997

[27] M Caputo Elasticitae Dissipazione Zanichelli Bologna Italy1969 [Italian]

[28] G Jumarie ldquoLaplacersquos transform of fractional order via theMittag-Leffler function andmodified Riemann-Liouville deriv-ativerdquoAppliedMathematics Letters vol 22 no 11 pp 1659ndash16642009

[29] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press New YorkNY USA 1999

[30] F Mainardi ldquoThe fundamental solutions for the fractional dif-fusion-wave equationrdquo Applied Mathematics Letters vol 9 no6 pp 23ndash28 1996

[31] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus amp Applied Analysis vol 4 no 2 pp 153ndash192 2001

[32] G K Watugala ldquoSumudu transform a new integral trans-form to solve differential equations and control engineeringproblemsrdquo International Journal of Mathematical Education inScience and Technology vol 24 no 1 pp 35ndash43 1993

[33] F B M Belgacem ldquoApplications of the sumudu transformto indefinite periodic parabolic equationsrdquo in Proceedings ofthe 6th International Conference on Mathematical Problems

amp Aerospace Sciences (ICNPAA rsquo06) chapter 6 pp 51ndash60Cambridge Scientific Cambridge UK 2007

[34] F B BelgacemA A Karaballi and S L Kalla ldquoAnalytical inves-tigations of the Sumudu transform and applications to integralproduction equationsrdquo Mathematical Problems in Engineeringvol 2003 no 3 pp 103ndash118 2003

[35] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[36] V B L Chaurasia R S Dubey and F B M Belgacem ldquoFrac-tional radial diffusion equation analytical solution via Hankeland Sumudu transformsrdquo Mathematics in Engineering Scienceand Aerospace vol 3 no 2 pp 1ndash10 2012

[37] Q D Katatbeh and F B M Belgacem ldquoApplications of theSumudu transform to fractional differential equationsrdquoNonlin-ear Studies vol 18 no 1 pp 99ndash112 2011

[38] G M Mittag-Leffler Sur la Nouvelle Fonction Ea(x) vol 137 CR Academy of Science Paris France 1903

[39] G Adomian Solving Frontier Problems of Physics The DecompOsition Method Kluwer Academic Boston Mass USA 1994

[40] J-H He ldquoThe homotopy perturbation method nonlinear oscil-lators with discontinuitiesrdquoAppliedMathematics and Computa-tion vol 151 no 1 pp 287ndash292 2004

[41] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[42] A A Hemeda ldquoHomotopy perturbation method for solvingsystems of nonlinear coupled equationsrdquo Applied MathematicalSciences vol 6 no 93ndash96 pp 4787ndash4800 2012

[43] J Saberi-Nadjafi and A Ghorbani ldquoHersquos homotopy perturba-tionmethod an effective tool for solving nonlinear integral andintegro-differential equationsrdquo Computers amp Mathematics withApplications vol 58 no 11-12 pp 2379ndash2390 2009

[44] A Ghorbani ldquoBeyond Adomian polynomials he polynomialsrdquoChaos Solitons and Fractals vol 39 no 3 pp 1486ndash1492 2009

[45] K Abbaoui and Y Cherruault ldquoNew ideas for proving conver-gence of decomposition methodsrdquo Computers amp Mathematicswith Applications vol 29 no 7 pp 103ndash108 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Due to vast range of applications of the FPE a lot of workhas been done to find numerical solution with this equationIn this context the works of Buet et al [11] Harrison [12]Palleschi et al [13] Vanaja [14] and Zorzano et al [15] areworth mentioning

For generalization of classical integer order of FPE (4)the following equation can be introducedThe Fokker-Planckequation with fractional space derivative is a particular caseof anomalous diffusion and Levy flights (see[16ndash19]) Thisequation is called nonlinear FPE with space-time fractionalderivatives [20]

120597120572V (119909 119905)120597119905120572

= [minus120597119860 (119909 119905 V)

120597119909+

1205972119861 (119909 119905 V)

1205971199092] V (119909 119905) (5)

where 119905 gt 0 119909 gt 0 and 0 lt 120572 le 1 It can be obtained fromthe general Fokker-Planck equation by replacing the spaceand time derivatives by fractional derivatives operator Thefunction V(119909 119905) is assumed to be a causal function of time andspace Particularly for 120572 = 1 the fractional FPE (5) reducesto the classical nonlinear FPE given by (4) in the case 119909 gt 0

In recent years researchers have studied the fractionalpartial differential equations and the fractional FPE toinvestigate various scientific models [21ndash24] In the presentpaper we obtain closed form solutions of a linear-nonlineartime fractional FPE using homotopy perturbation Sumudutransform method (HPSTM) see [25]

2 Some Mathematical Preliminariesand Definitions

21 Fundamental Properties of Fractional Calculus Firstly wemention some of the fundamental properties of the fractionalcalculus that is fractional derivatives and integrals

The Riemann-Liouville fractional integral operator oforder 120572 gt 0 of a function 119891(119905) isin 119862

120583 120583 ge minus1 is defined

by implementing the integration operator 119869120572 in the followingmanner

119869120572[119891 (119905)] =

1

Γ (120572)int119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591 120572 gt 0

1198690[119891 (119905)] = 119891 (119905)

119869120572119869120573[119891 (119905)] = 119869

120572+120573119891 (119905)

119869120573119869120572[119891 (119905)] = 119869

120572119869120573119891 (119905)

119869120572(119905120574) =

Γ (120574 + 1)

Γ (120572 + 120574 + 1)119905120572+120574

(6)

The fundamental properties of fractional integration andfractional differentiation have been introduced to the liter-ature by Podlubny [26]

The fractional derivative of 119891(119905) in the Caputo sense isdefined by the following relation [27]

119863120572

119905119891 (119905) = 119869

119898minus120572119863119899119891 (119905)

=1

Γ (119899 minus 120572)int119905

0

(119905 minus 120591)minus120572+119898minus1

119891(119898)

(120591) 119889120591

119898 minus 1 lt 120572 le 119898

(7)

where119898 isin 119873 and 119905 gt 0The relation between Riemann-Liouville fractional inte-

gral operator and the Caputo fractional derivative operatoris

119869120572

119905119863120572

119905119891 (119905) = 119891 (119905) minus

119898minus1

sum119896=0

119891119896 (0+) 119905119896

119896

119863minus120572

[119891 (119905)] =1

Γ (120572)int119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591

0 lt 120572 le 1

(8)

Further properties and applications can be found in [26 28ndash31]

22 The Sumudu Transform In early 90s Watugala [32]introduced a new integral transform named the Sumudutransform and applied it to the solution of ordinary differen-tial equation in control engineering problems The Sumudutransform is defined over the set of functions

119860 = 119891 (119905) | exist119872 1205911 1205912gt 0

1003816100381610038161003816119891 (119905)1003816100381610038161003816 lt 119872119890

|119905|120591119895

if 119905 isin (minus1)119895times [0infin)

(9)

the Sumudu transform is defined by

119866 (119906) = 119878 [119891 (119905)] = intinfin

0

119891 (119906119905) 119890minus119905119889119905 119906 isin (minus120591

1 1205912) (10)

Sumudu transformhasmany useful and important propertieslike linear property scale properties shifting propertiesduality with Laplace transforms and so forth Further detailand properties about this transform can be found in [25 33ndash37]

By using the Sumudu transform of multiple differentia-tion we obtain

119878 [119863120572

119905119891 (119905)] = 119906

119898minus120572[119866 (119906)

119906119898minus

119898minus1

sum119896=0

119891119896 (0)

119906119898minus119896]

= [119866 (119906)

119906120572minus

119898minus1

sum119896=0

119891119896 (0+)

119906120572minus119896] (119898 minus 1 lt 120572 le 119898)

(11)

where 119866(119906) = 119878[119891(119905)]



119864120572(119911) =

infin

sum119899=119900

119911119899

Γ (120572119899 + 1)(120572 isin C R (120572) gt 0) (12)





119863120572

119905119880 (119909 119905) = 119871119880 (119909 119905) + 119873119880 (119909 119905) + 119891 (119909 119905) 120572 gt 0 (13)


119863119896

0119880 (119909 0) = 119892

119896 (119896 = 0 1 119899 minus 1)

119863119899

0119880 (119909 0) = 0 119899 = [120572]

(14)





119878 [119863120572

119905119880 (119909 119905)] = 119878 [119871119880 (119909 119905)] + 119878 [119873119880 (119909 119905)] + 119878 [119891 (119909 119905)]

(15)


119878 [119880 (119909 119905)] = 119906120572119878 [119871119880 (119909 119905) + 119873119880 (119909 119905)] + 119892 (119909 119905) (16)


119880 (119909 119905) = 119866 (119909 119905) + 119878minus1

[119906120572119878 [119871119880 (119909 119905) + 119873119880 (119909 119905)]] (17)



119880 (119909 119905) =

infin

sum119899=0

119901119899119880119899(119909 119905) (18)


119873119880(119909 119905) =

infin

sum119899=0

119901119899119867119899(119880) (19)


given by

119867119899(1198800 1198801 119880

119899) =

1

119899

120597119899

120597119901119899[119873(

infin

sum119894=0

119901119894119880119894)]

119901=0

(20)


sum119899=0

119901119899119880119899(119909 119905)

= 119866 (119909 119905)

+ 119901(119878minus1

[119906120572119878 [119871

infin

sum119899=0

119901119899119880119899(119909 119905) +

infin

sum119899=0

119901119899119867119899(119880)]])

(21)



1199010 1198800(119909 119905) = 119866 (119909 119905)

1199011 1198801(119909 119905) = 119878

minus1[119906120572119878 119871119880

0(119909 119905) + 119867

0(119880)]

1199012 1198802(119909 119905) = 119878

minus1[119906120572119878 119871119880

1(119909 119905) + 119867

1(119880)]

(22)

and similarly

119901119899 119880119899(119909 119905) = 119878

minus1[119906120572119878 119871119880

119899minus1(119909 119905) + 119867

119899minus1(119880)] (23)


119880 (119909 119905) = lim119873rarrinfin

119873

sum119899=0

119880119899(119909 119905) (24)




120597120572V (119909 119905)120597119905120572

= [minus120597119860 (119909 119905 V)

120597119909+

1205972119861 (119909 119905 V)

1205971199092] V (119909 119905) (25)



V (119909 0) = 119891 (119909) (26)


119878 [V (119909 119905)] = 119891 (119909) + 119906120572119878

times [minus120597119860 (119909 119905 V)

120597119909V (119909 119905) +

1205972119861 (119909 119905 V)

1205971199092V (119909 119905)]

(27)


V (119909 119905) = 119891 (119909) + 119878minus1

[119906120572119878minus

120597

120597119909

infin

sum119899=0

119860119899+

1205972

1205971199092

infin

sum119899=0

119861119899]

(28)

whereinfin

sum119899=0

119860119899= 119860 (119909 119905 V) sdot V (119909 119905)

infin

sum119899=0

119861119899= 119861 (119909 119905 V) sdot V (119909 119905)

(29)


sum119899=0

119901119899V119899(119909 119905)

= 119891 (119909) + 119901

times (119878minus1

[119906120572119878minus

infin

sum119899=0

119901119899119867119899(119909 119905 V) +

infin

sum119899=0

119901119899119867lowast

119899(119909 119905 V)])

(30)




infin

sum119899=0

119867119899(119909 119905 V) =

120597

120597119909(

infin

sum119899=0

119860119899)

infin

sum119899=0

119867lowast

119899(119909 119905 V) =

1205972

1205971199092(

infin

sum119899=0

119861119899)

(31)


1199010 V0(119909 119905) = 119891 (119909)

1199011 V1(119909 119905) = 119878

minus1[119906120572119878 [minus119867

0+ 119867lowast

0]]

1199012 V2(119909 119905) = 119878

minus1[119906120572119878 [minus119867

1+ 119867lowast

1]]

(32)

and similarly

119901119899 V119899(119909 119905) = 119878

minus1[119906120572119878 [minus119867


119899minus1]] (33)



V (119909 119905) = lim119873rarrinfin

119873

sum119899=0

V119899(119909 119905) (34)





119863120572

119905V (119909 119905) = [minus

120597

120597119909(3V minus

119909

2) +

1205972

1205971199092(119909V)] V (119909 119905) (35)



V (119909 0) = 119909 (36)



1199010 V0(119909 119905) = 119909

1199011 V1(119909 119905) =

119909119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

1199091199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

1199091199053120572

Γ (3120572 + 1)

(37)

hence V(119909 119905) is

V (119909 119905) = 119909(1 +119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 119909119864120572(119905120572)

(38)


120597

120597119905V (119909 119905) = [minus

120597

120597119909(3V minus

119909

2) +

1205972

1205971199092(119909V)] V (119909 119905) (39)



V (119909 0) = 119909 (40)

and a solution as

V (119909 119905) = 119909119890119905 (41)


119863120572

119905V (119909 119905)

= [minus120597

120597119909(4V119909

minus119909

3) +

1205972

1205971199092(V)] V (119909 119905) 119909 gt 0 119905 gt 0

(42)



V (119909 0) = 1199092 (43)



1199010 V0(119909 119905) = 119909

2

1199011 V1(119909 119905) =

1199092119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

11990921199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

11990921199053120572


(44)

hence V(119909 119905) is

V (119909 119905)

= 1199092(1 +

119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 1199092119864120572(119905120572)

(45)


120597

120597119905V (119909 119905)

= [minus120597

120597119909(4V119909

minus119909

3) +

1205972

1205971199092(V)] V (119909 119905) 119909 gt 0 119905 gt 0

(46)


V (119909 0) = 1199092 (47)

and a solution as

V (119909 119905) = 1199092119890119905 (48)


119863120572

119905V (119909 119905)

= [minus120597

120597119909(119909) +

1205972

1205971199092(1199092

2)] V (119909 119905) 119909 gt 0 119905 gt 0

(49)




V (119909 0) = 119909 (50)


1199010 V0(119909 119905) = 119909

1199011 V1(119909 119905) =

119909119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

1199091199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

1199091199053120572

Γ (3120572 + 1)

(51)

hence V(119909 119905) is

V (119909 119905) = 119909(1 +119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 119909119864120572(119905120572)

(52)


120597

120597119905V (119909 119905) = [minus

120597

120597119909(119909) +

1205972

1205971199092(1199092

2)] V (119909 119905)

119909 gt 0 119905 gt 0

(53)


V (119909 0) = 119909 (54)

and a solution as

V (119909 119905) = 119909119890119905 (55)










6 Conclusion




Acknowledgment


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119864120572(119911) =

infin

sum119899=119900

119911119899

Γ (120572119899 + 1)(120572 isin C R (120572) gt 0) (12)





119863120572

119905119880 (119909 119905) = 119871119880 (119909 119905) + 119873119880 (119909 119905) + 119891 (119909 119905) 120572 gt 0 (13)


119863119896

0119880 (119909 0) = 119892

119896 (119896 = 0 1 119899 minus 1)

119863119899

0119880 (119909 0) = 0 119899 = [120572]

(14)





119878 [119863120572

119905119880 (119909 119905)] = 119878 [119871119880 (119909 119905)] + 119878 [119873119880 (119909 119905)] + 119878 [119891 (119909 119905)]

(15)


119878 [119880 (119909 119905)] = 119906120572119878 [119871119880 (119909 119905) + 119873119880 (119909 119905)] + 119892 (119909 119905) (16)


119880 (119909 119905) = 119866 (119909 119905) + 119878minus1

[119906120572119878 [119871119880 (119909 119905) + 119873119880 (119909 119905)]] (17)



119880 (119909 119905) =

infin

sum119899=0

119901119899119880119899(119909 119905) (18)


119873119880(119909 119905) =

infin

sum119899=0

119901119899119867119899(119880) (19)


given by

119867119899(1198800 1198801 119880

119899) =

1

119899

120597119899

120597119901119899[119873(

infin

sum119894=0

119901119894119880119894)]

119901=0

(20)


sum119899=0

119901119899119880119899(119909 119905)

= 119866 (119909 119905)

+ 119901(119878minus1

[119906120572119878 [119871

infin

sum119899=0

119901119899119880119899(119909 119905) +

infin

sum119899=0

119901119899119867119899(119880)]])

(21)



1199010 1198800(119909 119905) = 119866 (119909 119905)

1199011 1198801(119909 119905) = 119878

minus1[119906120572119878 119871119880

0(119909 119905) + 119867

0(119880)]

1199012 1198802(119909 119905) = 119878

minus1[119906120572119878 119871119880

1(119909 119905) + 119867

1(119880)]

(22)

and similarly

119901119899 119880119899(119909 119905) = 119878

minus1[119906120572119878 119871119880

119899minus1(119909 119905) + 119867

119899minus1(119880)] (23)


119880 (119909 119905) = lim119873rarrinfin

119873

sum119899=0

119880119899(119909 119905) (24)




120597120572V (119909 119905)120597119905120572

= [minus120597119860 (119909 119905 V)

120597119909+

1205972119861 (119909 119905 V)

1205971199092] V (119909 119905) (25)



V (119909 0) = 119891 (119909) (26)


119878 [V (119909 119905)] = 119891 (119909) + 119906120572119878

times [minus120597119860 (119909 119905 V)

120597119909V (119909 119905) +

1205972119861 (119909 119905 V)

1205971199092V (119909 119905)]

(27)


V (119909 119905) = 119891 (119909) + 119878minus1

[119906120572119878minus

120597

120597119909

infin

sum119899=0

119860119899+

1205972

1205971199092

infin

sum119899=0

119861119899]

(28)

whereinfin

sum119899=0

119860119899= 119860 (119909 119905 V) sdot V (119909 119905)

infin

sum119899=0

119861119899= 119861 (119909 119905 V) sdot V (119909 119905)

(29)


sum119899=0

119901119899V119899(119909 119905)

= 119891 (119909) + 119901

times (119878minus1

[119906120572119878minus

infin

sum119899=0

119901119899119867119899(119909 119905 V) +

infin

sum119899=0

119901119899119867lowast

119899(119909 119905 V)])

(30)




infin

sum119899=0

119867119899(119909 119905 V) =

120597

120597119909(

infin

sum119899=0

119860119899)

infin

sum119899=0

119867lowast

119899(119909 119905 V) =

1205972

1205971199092(

infin

sum119899=0

119861119899)

(31)


1199010 V0(119909 119905) = 119891 (119909)

1199011 V1(119909 119905) = 119878

minus1[119906120572119878 [minus119867

0+ 119867lowast

0]]

1199012 V2(119909 119905) = 119878

minus1[119906120572119878 [minus119867

1+ 119867lowast

1]]

(32)

and similarly

119901119899 V119899(119909 119905) = 119878

minus1[119906120572119878 [minus119867


119899minus1]] (33)



V (119909 119905) = lim119873rarrinfin

119873

sum119899=0

V119899(119909 119905) (34)





119863120572

119905V (119909 119905) = [minus

120597

120597119909(3V minus

119909

2) +

1205972

1205971199092(119909V)] V (119909 119905) (35)



V (119909 0) = 119909 (36)



1199010 V0(119909 119905) = 119909

1199011 V1(119909 119905) =

119909119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

1199091199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

1199091199053120572

Γ (3120572 + 1)

(37)

hence V(119909 119905) is

V (119909 119905) = 119909(1 +119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 119909119864120572(119905120572)

(38)


120597

120597119905V (119909 119905) = [minus

120597

120597119909(3V minus

119909

2) +

1205972

1205971199092(119909V)] V (119909 119905) (39)



V (119909 0) = 119909 (40)

and a solution as

V (119909 119905) = 119909119890119905 (41)


119863120572

119905V (119909 119905)

= [minus120597

120597119909(4V119909

minus119909

3) +

1205972

1205971199092(V)] V (119909 119905) 119909 gt 0 119905 gt 0

(42)



V (119909 0) = 1199092 (43)



1199010 V0(119909 119905) = 119909

2

1199011 V1(119909 119905) =

1199092119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

11990921199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

11990921199053120572


(44)

hence V(119909 119905) is

V (119909 119905)

= 1199092(1 +

119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 1199092119864120572(119905120572)

(45)


120597

120597119905V (119909 119905)

= [minus120597

120597119909(4V119909

minus119909

3) +

1205972

1205971199092(V)] V (119909 119905) 119909 gt 0 119905 gt 0

(46)


V (119909 0) = 1199092 (47)

and a solution as

V (119909 119905) = 1199092119890119905 (48)


119863120572

119905V (119909 119905)

= [minus120597

120597119909(119909) +

1205972

1205971199092(1199092

2)] V (119909 119905) 119909 gt 0 119905 gt 0

(49)




V (119909 0) = 119909 (50)


1199010 V0(119909 119905) = 119909

1199011 V1(119909 119905) =

119909119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

1199091199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

1199091199053120572

Γ (3120572 + 1)

(51)

hence V(119909 119905) is

V (119909 119905) = 119909(1 +119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 119909119864120572(119905120572)

(52)


120597

120597119905V (119909 119905) = [minus

120597

120597119909(119909) +

1205972

1205971199092(1199092

2)] V (119909 119905)

119909 gt 0 119905 gt 0

(53)


V (119909 0) = 119909 (54)

and a solution as

V (119909 119905) = 119909119890119905 (55)










6 Conclusion




Acknowledgment


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











V (119909 0) = 119891 (119909) (26)


119878 [V (119909 119905)] = 119891 (119909) + 119906120572119878

times [minus120597119860 (119909 119905 V)

120597119909V (119909 119905) +

1205972119861 (119909 119905 V)

1205971199092V (119909 119905)]

(27)


V (119909 119905) = 119891 (119909) + 119878minus1

[119906120572119878minus

120597

120597119909

infin

sum119899=0

119860119899+

1205972

1205971199092

infin

sum119899=0

119861119899]

(28)

whereinfin

sum119899=0

119860119899= 119860 (119909 119905 V) sdot V (119909 119905)

infin

sum119899=0

119861119899= 119861 (119909 119905 V) sdot V (119909 119905)

(29)


sum119899=0

119901119899V119899(119909 119905)

= 119891 (119909) + 119901

times (119878minus1

[119906120572119878minus

infin

sum119899=0

119901119899119867119899(119909 119905 V) +

infin

sum119899=0

119901119899119867lowast

119899(119909 119905 V)])

(30)




infin

sum119899=0

119867119899(119909 119905 V) =

120597

120597119909(

infin

sum119899=0

119860119899)

infin

sum119899=0

119867lowast

119899(119909 119905 V) =

1205972

1205971199092(

infin

sum119899=0

119861119899)

(31)


1199010 V0(119909 119905) = 119891 (119909)

1199011 V1(119909 119905) = 119878

minus1[119906120572119878 [minus119867

0+ 119867lowast

0]]

1199012 V2(119909 119905) = 119878

minus1[119906120572119878 [minus119867

1+ 119867lowast

1]]

(32)

and similarly

119901119899 V119899(119909 119905) = 119878

minus1[119906120572119878 [minus119867


119899minus1]] (33)



V (119909 119905) = lim119873rarrinfin

119873

sum119899=0

V119899(119909 119905) (34)





119863120572

119905V (119909 119905) = [minus

120597

120597119909(3V minus

119909

2) +

1205972

1205971199092(119909V)] V (119909 119905) (35)



V (119909 0) = 119909 (36)



1199010 V0(119909 119905) = 119909

1199011 V1(119909 119905) =

119909119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

1199091199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

1199091199053120572

Γ (3120572 + 1)

(37)

hence V(119909 119905) is

V (119909 119905) = 119909(1 +119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 119909119864120572(119905120572)

(38)


120597

120597119905V (119909 119905) = [minus

120597

120597119909(3V minus

119909

2) +

1205972

1205971199092(119909V)] V (119909 119905) (39)



V (119909 0) = 119909 (40)

and a solution as

V (119909 119905) = 119909119890119905 (41)


119863120572

119905V (119909 119905)

= [minus120597

120597119909(4V119909

minus119909

3) +

1205972

1205971199092(V)] V (119909 119905) 119909 gt 0 119905 gt 0

(42)



V (119909 0) = 1199092 (43)



1199010 V0(119909 119905) = 119909

2

1199011 V1(119909 119905) =

1199092119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

11990921199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

11990921199053120572


(44)

hence V(119909 119905) is

V (119909 119905)

= 1199092(1 +

119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 1199092119864120572(119905120572)

(45)


120597

120597119905V (119909 119905)

= [minus120597

120597119909(4V119909

minus119909

3) +

1205972

1205971199092(V)] V (119909 119905) 119909 gt 0 119905 gt 0

(46)


V (119909 0) = 1199092 (47)

and a solution as

V (119909 119905) = 1199092119890119905 (48)


119863120572

119905V (119909 119905)

= [minus120597

120597119909(119909) +

1205972

1205971199092(1199092

2)] V (119909 119905) 119909 gt 0 119905 gt 0

(49)




V (119909 0) = 119909 (50)


1199010 V0(119909 119905) = 119909

1199011 V1(119909 119905) =

119909119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

1199091199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

1199091199053120572

Γ (3120572 + 1)

(51)

hence V(119909 119905) is

V (119909 119905) = 119909(1 +119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 119909119864120572(119905120572)

(52)


120597

120597119905V (119909 119905) = [minus

120597

120597119909(119909) +

1205972

1205971199092(1199092

2)] V (119909 119905)

119909 gt 0 119905 gt 0

(53)


V (119909 0) = 119909 (54)

and a solution as

V (119909 119905) = 119909119890119905 (55)










6 Conclusion




Acknowledgment


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











V (119909 0) = 119909 (40)

and a solution as

V (119909 119905) = 119909119890119905 (41)


119863120572

119905V (119909 119905)

= [minus120597

120597119909(4V119909

minus119909

3) +

1205972

1205971199092(V)] V (119909 119905) 119909 gt 0 119905 gt 0

(42)



V (119909 0) = 1199092 (43)



1199010 V0(119909 119905) = 119909

2

1199011 V1(119909 119905) =

1199092119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

11990921199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

11990921199053120572


(44)

hence V(119909 119905) is

V (119909 119905)

= 1199092(1 +

119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 1199092119864120572(119905120572)

(45)


120597

120597119905V (119909 119905)

= [minus120597

120597119909(4V119909

minus119909

3) +

1205972

1205971199092(V)] V (119909 119905) 119909 gt 0 119905 gt 0

(46)


V (119909 0) = 1199092 (47)

and a solution as

V (119909 119905) = 1199092119890119905 (48)


119863120572

119905V (119909 119905)

= [minus120597

120597119909(119909) +

1205972

1205971199092(1199092

2)] V (119909 119905) 119909 gt 0 119905 gt 0

(49)




V (119909 0) = 119909 (50)


1199010 V0(119909 119905) = 119909

1199011 V1(119909 119905) =

119909119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) =

1199091199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) =

1199091199053120572

Γ (3120572 + 1)

(51)

hence V(119909 119905) is

V (119909 119905) = 119909(1 +119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)+

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot )

V (119909 119905) = 119909119864120572(119905120572)

(52)


120597

120597119905V (119909 119905) = [minus

120597

120597119909(119909) +

1205972

1205971199092(1199092

2)] V (119909 119905)

119909 gt 0 119905 gt 0

(53)


V (119909 0) = 119909 (54)

and a solution as

V (119909 119905) = 119909119890119905 (55)










6 Conclusion




Acknowledgment


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















6 Conclusion




Acknowledgment


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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