Numer AlgorDOI 10.1007/s11075-011-9448-7

ORIGINAL PAPER

Hypercomplex mathematics and HPMfor the time-delayed Burgers equationwith convergence analysis

Davood Rostamy · K. Karimi

Received: 22 July 2010 / Accepted: 25 January 2011© Springer Science+Business Media, LLC 2011

Abstract We investigate the analytical and numerical solutions of the time-delayed Burgers equation, by applying the idea of commutative hypercomplexmathematics and the homotopy perturbation method. Moreover, we discuss atgreat length the convergence conditions of the homotopy perturbation Method(HPM) by using the Banach fixed point theory , which could provide a gooditeration algorithm. Finally, we also give some numerical illustrations to theobtained results.

Keywords Time-delayed Burgers equation · Commutative hypercomplexmathematics · Homotopy perturbation method

1 Introduction

The HPM method is an easy technique for finding a power series as a solutionof nonlinear differential equations [3, 12–15]. At present this method is verypopular and we can find a lot of publications with applications of this methodin journals. We can observe a number of ecstatic words about possibilities ofthis method. It is also shown that the method, with the help of series, providesa powerful mathematical tool for solving other nonlinear evolution equations

D. Rostamy (B) · K. KarimiDepartment of Mathematics, Imam Khomeini International University, Qazvin, Irane-mail: rostamy@khayam.ut.ac.ir

K. Karimie-mail: Kobra.karimi@yahoo.com

D. RostamyDepartment of Mathematics, Islamic Azad University, South Tehran Branch, Tehran, Iran

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arising in mathematical physics. But we observe that HPM does not tell usanything about the possibility of convergence analysis. This paper emphasizethe need for convergence conditions.

On the other hand, many phenomena in physics and other fields such asbiology, chemistry, mechanics, etc., are described by time delay differentialequations [9, 16]. They arise when the rate of change of a time-dependentprocess in its mathematical modeling is not only determined by its present statebut also by a certain past state. Recent studies such as dynamics [10], chemical[6], economical and biological processes [11] have shown that delay differentialequation plays an important role in explaining many different phenomena.One of the well know delay differential equations which is an important featurein reaction-diffusion and convection-diffusion systems [1, 7, 8] are time delayedBurgers equation. Moreover, the generalized time-delayed Burgers equation ispresented in [8, 16]. It takes the following form:

As(u) = f (u), (1)

where

As(u) = τutt + [1 − τ fu]ut − uxx + pusux,

and

f (u) = qu(1 − u),

such that τ, p, q ∈ R and s ∈ N (see [7]).In this paper, there’s an emphasis on assessing the convergence analysis for

HPM on the above problem. Hence, we try to obtain the classical solutions by anew idea that is the commutative hypercomplex mathematics [4, 5]. Therefore,this paper is organized as follows:

In Section 2, we introduce hypercomplex mathematics and in Section 3,we use hypercomplex mathematics to obtain solutions of the time-delayedBurgers equation. In Section 4, we recall idea of homotopy method andin Section 5, we verify the contraction mapping in homotopy method forthis problem. The numerical solutions of the problem are obtained by usinghomotopy perturbation method in Section 6, also, we compare the numericaland exact results in this section. Finally, Section 7 is devoted to the conclusionremarks.

2 Review of commutative hypercomplex mathematics

Systems of hypercomplex numbers, which had been studied and developedat the end of the 19th century, are nowadays quite unknown to the scientificcommunity. It is believed that study of their applications ended just beforeone of the fundamental discoveries of the 20th century, Einstein’s equivalencebetween space and time. Owing to this equivalence, not-defined quadraticforms have got concrete physical meaning and have been recently recognizedto be in strong relationship with a system of bi-dimensional hypercomplex

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numbers. The commutative hypercomplex mathematics is an extension ofcomplex numbers that obeys the axioms of the classical complex variables. It is4-D independent variable, so we will use the notation Z = 1x + iy + jz + kct,where x, y, z, ct are real and Z belong to an element of the commutativehypercomplex algebra. In the fourth component, t represents time, and c isa scale factor [4, 5].

Analytic function is defined as following:

u(Z) = u(ξ)e1 + u(η)e2,

ξ = (x − ct) + i(y + z),

η = (x + ct) + i(y − z),

e1 =(

1 − k2

), e2 =

(1 + k

2

).

The 4-D function u(Z) is analytic if both u(ξ) and u(η) are analytic in the clas-sical complex variable sense. Now, we introduce operators such as derivativeand integral for functions of a 4-D variable. They obey the function definitionthat we already have. Therefore, they are as following:

oper(Z) = oper(ξ)e1 + oper(η)e2.

The result is that we can apply all of the powerful tools of complex analysis tofour-space problems.

The 4-D Cauchy-Riemann(C-R) conditions which have a number of inter-esting is:

dudZ

= 1∂u∂x

= −i∂u∂y

= −j∂u∂z

= k∂u∂ct

. (2)

C-R conditions say that the derivative of a 4-D analytic function is the samewithin a sign in all four coordinate directions. The first two equalities arethe same as for complex variables. These equations can be used to reducea partial differential equation in several real, independent variables to anordinary differential equation in one 4-D variable. By doing so, we would beimposing continuity conditions on the PDE, because the C-R conditions area statement of continuity. PDEs are typically derived with the assumption ofcontinuity, but without its explicit inclusion because convenient means havenot been available. Note carefully that we are not constraining any potentialsolution, because the C-R conditions hold for any and all analytic functions.

3 Exact solutions of the time-delayed Burgers equation

It is also shown that the hypercomplex mathematics, with the help of symboliccomputation, provides a powerful mathematical tool for solving other non-linear evolution equations arising in mathematical physics. In this section we

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impose a proposition that says how to get the analytic solutions of the time-delayed Burgers equation.

Theorem 3.1 The general analytical solution of (1) in 4-D space is as follows:

u(Z) = −√3β(u(Z))

3q(s + 1)(

12 − kc

) tan h

[√3β(u(Z))(Z + B)

3(s + 1)(τc2 − 1)

].

For q = 0 and s = 1 we have:

u(Z) = −kcp

+√

2Ap

− c2

p2tan

⎡⎣

√2Ap − c2

p2

− 2p (τc2 − 1)

(Z + B)

⎤⎦ .

Where A, B are the arbitrary 4-D constants of integration, k is a 4-D algebraicbasis element , c is a scale factor and

β(u(Z))

=√(

3pus+1(Z) + qu3(Z)s + qu3(Z) − 3kcu(Z)s − 3kcu(Z) − 3As − 3A)

q(

1

2− kc

)(s + 1).

Proof As mentioned previous section, our basic approach to solution is to firstconvert the (nonlinear) time-delayed Burgers equation to an ODE, then solveit by means of classical methods. To convert partial differentials to ordinaryderivatives, we shall use the 4-D Cauchy–Riemann equations (2), where Z =1x + iy + jz + kct. Making the partial derivative conversions, (1) converts to

τc2 d2u(Z)

dZ2+ kc

du(Z)

dZ(1 − q(1 − 2u(Z))) − d2u(Z)

dZ2+ pus(Z)

du(Z)

dZ= qu(Z)(1 − u(Z)).

This equation is nonlinear ordinary differential equation, but is solvable bydirect methods. A first integration yields:

(τc2 − 1

) du(Z)

dZ+ kcu(Z)(1 − q + u(Z)) + p

s + 1us+1(Z)

= q(

1

2u2(Z) − 1

3u3(Z)

)+ A,

where A is an arbitrary 4-D constant of integration. This is again integrable.We complete the square on the left, move like terms to separate sides of theequation, then integrate to get:

dZτc2 − 1

= du(Z)−ps+1 us+1(Z) − q

3 u3(Z) + q(

12 − kc

)u2(Z) + kcu(Z) + A

, (3)

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and we have

Z + Bτc2 − 1

=(s + 1)

√3 arctan h

⎛⎝ −√

3q(

12 −kc

)(s+1)u(Z)√

(3pus+1(Z)+qu3(Z)s+qu3(Z)−3kcu(Z)s−3kcu(Z)−3As−3A)q(

12 −kc

)(s+1)

⎞⎠

√(3pus+1(Z) + qu3(Z)s + qu3(Z) − 3kcu(Z)s − 3kcu(Z) − 3As − 3A

)q(

12 − kc

)(s + 1)

(4)

where B is another arbitrary 4-D constant of integration. Therefore, we have:

u(Z) = −√3β(u(Z))

3q(s + 1)(

12 − kc

) tan h

[√3β(u(Z))(Z + B)

3(s + 1)(τc2 − 1)

]. (5)

Equation (5) can be converted to the special case when we put s = 1 and q = 0:

u(Z) = −kcp

+√

2Ap

− c2

p2tan

⎡⎣

√2Ap − c2

p2

− 2p

(τc2 − 1

) (Z + B)

⎤⎦ . (6)

Equation (6) is dependent upon the condition 2Ap − c2

p2 > (0, 0, 0, 0). ��

Above theorem gives us the general analytical solution for the time-delayedBurgers equation in terms of the 4-D commutative hypercomplex variable Z.It is the complete solution for the ODE form in as much as we have integratedtwice and have a solution including two arbitrary constants of integration.

We have obtained a solution u(Z) in terms of one variable having threespace dimensions and time. One question that must be answered is, “Doesit reduce to a solution of the original, one-dimensional time-delayed Burgersequation when the y, z components are set to zero?”, this is enough to check.Setting y = z = 0 in (5) and (6), we get:

u(x, ct) = −√3β(u(x, ct))

3q(s + 1)(

12 − kc

) tan h

[√3β(u(x, ct))(x + kct + B)

3(s + 1)(τc2 − 1)

], (7)

and

u(x, ct) = −kcp

+√

2Ap

− c2

p2tan

⎡⎣

√2Ap − c2

p2

− 2p (τc2 − 1)

(x + kct + B)

⎤⎦ , (8)

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where

β(u(x, ct))

=√(

3pus+1(x, ct)+qu3(x, ct)s+qu3(x, ct)−3kcu(x, ct)s − 3kcu(x, ct) − 3As − 3A)

q(

1

2− kc

)(s + 1).

For (8), taking the requisite partial derivatives of u(x, ct) in the usual way,we have:

ut = kc2Ap − c2

p2

−2p (τc2 − 1)

sec2

⎡⎣

√2Ap − c2

p2

−2p (τc2 − 1)

(x + kct + B)

⎤⎦ ,

τutt = kc

√2Ap − c2

p2

−2p (τc2 − 1)

tan

⎡⎣

√2Ap − c2

p2

− 2p (τc2 − 1)

(x + kct + B)

⎤⎦ kcτ

2Ap − c2

p2

−1p (τc2 − 1)

× sec2

⎡⎣

√2Ap − c2

p2

− 2p (τc2 − 1)

(x + kct + B)

⎤⎦ ,

−puux =⎛⎝−kc

p+

√2Ap

− c2

p2tan

⎡⎣

√2Ap − c2

p2

−2p (τc2 − 1)

(x + kct + B)

⎤⎦⎞⎠

×p

⎛⎝ 2A

p − c2

p2

2p (τc2 − 1)

sec2

⎡⎣

√2Ap − c2

p2

− 2p (τc2 − 1)

(x + kct + B)

⎤⎦⎞⎠ ,

uxx =2Ap − c2

p2

2p (τc2 − 1)

tan

⎡⎣

√2Ap − c2

p2

− 2p (τc2 − 1)

(x + kct + B)

⎤⎦

×√

2Ap − c2

p2

− 1p (τc2 − 1)

sec2

⎡⎣

√2Ap − c2

p2

− 2p (τc2 − 1)

(x + kct + B)

⎤⎦ .

If we combine these equations by addition, then we get the following one-dimensional time-delayed Burgers equation:

A(u) = 0, (9)

where A(u) = A1(u) = τutt + [1 − τ fu]ut − uxx + pu1ux. We can obtain (1) ifwe repeat the above process for (7).

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4 The Homotopy Perturbation Method (HPM)

In this section, we introduce the homotopy perturbation method and discussthe convergence analysis. To illustrate the basic ideas of the homotopy pertur-bation method, we consider the following nonlinear differential equation:

T(u) − F(r) = 0, r ∈ �, (10)

with the boundary conditions

B(

u,∂u∂n

)= 0, r ∈ �, (11)

where T is a general differential operator, B is a boundary operator, F(r) is aknown analytical function and � is the boundary of the domain �. Generallyspeaking, the operator T can be divided into two parts which are L and N,where L is linear, but N is nonlinear. Therefore (10) can be rewritten asfollows:

L(u) − N(u) − F(r) = 0. (12)

By the homotopy perturbation technique, we construct a homotopyv(r, θ) : � × [0, 1] −→ R with satisfies

H(v, θ)=(1−θ)[L(v)−L(u0)]+θ [T(v)−F(r)]=0, θ ∈ [0, 1], r ∈ �, (13)

or

H(v, θ) = L(v) − L(u0) + θ L(u0) + θ [N(v) − F(r)] = 0, (14)

where θ ∈ [0, 1] is an embedding parameter and u0 is an initial approximationof (10). Obviously, from these definitions we will have

H(v, 0) = L(v) − L(u0) = 0, H(v, 1) = T(v) − F(r) = 0.

The process of changing of θ from zero to unity is just that of v(r, θ) fromu0(r) to u(r). In topology, this is called deformation, and L(v) − L(u0) andT(v) − F(r) are called homotopies. According to the HPM, we can first usethe embedding parameter θ as a “small paramter”, and assume that thesolution of (13) and (14) can be written as a power series in θ :

v = v0 + θv1 + θ2v2 + . . . . (15)

Setting θ = 1 results in the approximate solution of (10):

u = limθ→1

v = v0 + v1 + v2 + . . . .

5 Contraction mapping in HPM

The simplicity of contraction mapping in HPM for (1) is the fact that itis uncomplicated and can be understood easily if we can prove it for (9).Therefore, in this section, we apply the homotopy perturbation method for

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(9), after that we present an example to show the efficiency and high accuracyof the described method for solving (9).

In order to solve (9) by means of homotopy perturbation method, accordingto (13), we can construct a convex homotopy such that

H(v, θ)=(1−θ)

[τ

∂2v

∂t2−τ

∂2u0

∂t2

]+θ

[τ

∂2v

∂t2+ ∂v

∂t− ∂2v

∂x2+ pv

∂v

∂x

]= 0, (16)

substituting (15) in (16) and equating the coefficient of like powers of θ yield:

θ0 : τ∂2v0

∂t2− τ

∂2u0

∂t2= 0,

θ1 : τ∂2v1

∂t2+ ∂v0

∂t+ pv0

∂v0

∂x− ∂2v0

∂x2= 0,

θ2 : τ∂2v2

∂t2+ ∂v1

∂t+ pv1

∂v0

∂x+ pv0

∂v1

∂x− ∂2v1

∂x2= 0,

...

θn : τ∂2vn

∂t2+ ∂vn−1

∂t+ p

n−1∑i=0

vi∂v(n−1)−i

∂x− ∂2vn−1

∂x2= 0. (17)

Then starting with an initial approximation and solving the above equations,we get the nth approximation of the exact solution as un = v0 + v1 + v2 +· · · + vn. On the other hand for investigating of convergence analysis, werearrangement (17) to the following iterative method:

v j = −1

τ

∮�

(∂v j−1

∂t+ p

n−1∑i=0

vi∂v( j−1)−i

∂x− ∂2v j−1

∂x2

)d� = A j−1(v j−1), (18)

where j = 1, 2, 3, ..., n. We show that the above iterative method is a contrac-tion.

The following definition [2] is imposed, then convergence of the homotopyperturbation method is discussed.

Definition 5.1 Let k be a non-negative integer, r ∈ [1, ∞). Then Sobolev spaceWk,r(�) is the set of all the functions v ∈ Lr(�) such that for each multi-indexα with |α| ≤ k, the αth weak derivative ∂αv exists and ∂αv ∈ Lr(�). The normin the space Wk,r(�) is defined as

‖v‖Wk,r(�) =⎧⎨⎩[∑

|α|≤k ‖∂αv‖rLr(�)

]1/r, 1 ≤ r < ∞,

max|α|≤k ‖∂αv‖L∞(�), p = ∞.

Theorem 5.2 In (18) we def ine, A j : Wk,r → Wk,r and if A′j , A′′

j are boundedin some neighborhood. Then (i) A j is a contraction mapping, that is

∀v, w ∈ Wk,r; ‖A j(v) − A j(w)‖Wk,r ≤ γ ‖v − w‖Wk,r , 0 < γ < 1.

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On the other hand, according to Banach’s f ixed point theorem, having the f ixedpoint u, that is u = A j(u). Assume that the sequence generated by homotopyperturbation method can be written as

Vn = A j(Vn−1), Vn−1 =n−1∑i=0

ui, n = 1, 2, 3, . . . ,

and suppose that V0 = v0 = u0 ,then,(ii) The sequence of {Vn}∞n=1 is convergence, i.e.,

∃u ∈ Wk,r, limn→∞ Vn = u.

(iii) u in (ii) is the exact solution.

Proof

(i) By using Definition 5.3, Lemmas 5.4 and 5.5 and Theorem 5.6 we canconclude this claim.

(ii) It is enough to show that, {Vn}∞n=1 is a Cauchy sequence in the Sobolevspace. For this reason, consider, for every n, m ∈ N, n ≥ m, we have

‖Vn − Vm‖Wk,r = ‖(Vn − Vn−1) + (Vn−1 − Vn−2)

+ · · · + (Vm+1 − Vm)‖Wk,r

≤ ‖Vn − Vn−1‖Wk,r + ‖Vn−1 − Vn−2‖Wk,r

+ · · · + ‖Vm+1 − Vm‖Wk,r

= ‖A j(Vn−1) − A j(Vn−2)‖Wk,r + ‖A j(Vn−2)

−A j(Vn−3)‖Wk,r + · · · + ‖A j(Vm) − A j(Vm−1)‖Wk,r

≤ γ n−1‖V1 − v0‖Wk,r + γ n−2‖V1 − v0‖Wk,r

+ · · · + γ m‖V1 − v0‖Wk,r

≤ (γ m + γ m+1 + . . . )‖V1−v0‖Wk,r = γ m

1 − γ‖V1 − v0‖Wk,r ,

Since ‖V1 − v0‖Wk,r < ∞, hence, limn.m→+∞ ‖Vn − Vm‖Wk,r = 0, i.e.,{Vn}∞n=1 is a Cauchy sequence in the Sobolev space Wk,r and it impliesthat

∃u ∈ Wk,r, limn→+∞ Vn = u.

(iii) Using (ii), we have

A j(u) = A j

(lim

n→+∞ Vn

)= lim

n→+∞ A j(Vn) = limn→+∞ Vn+1 = u,

i.e., u is a solution of (9).��

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We can split the operator of Aj to the following linear and nonlinear operators:

A j(u) = AL, j(u) + AN, j(u),

where AL, j , AN, j are linear and nonlinear operators respectively. Moreover,We use the idea of the derivative of an operator by generalizing the definitionfor the derivative of a function of one variable. Hence, this generalization canbe made in different ways. We are led to several possible definitions for thederivative of an operator but in this paper we will use the definition of Frechetderivative. In the following we recall it:

Definition 5.3 Let A j : Wk,r → Wk,r and u ∈ Wk,r and there exists a boundedlinear operator A′

j(u) such that for all ‖�u‖Wk,r −→ 0

lim‖�u‖Wk,r −→0

‖A j(u + �u) − A j(u) − A′j(u)�u‖Wk,r

‖�u‖Wk,r= 0

then we say that A j is strongly differentiable at u . The operator A′j(u) is called

the strong or Frechet derivative.

We can also define the higher order of differentiate it (see [2]). In elemen-tary numerical analysis Taylor’s theorem is frequently used in analyzing algo-rithms. For the operator of A j there exists an extension of Taylor’s Theoremwhich is equally used. In the following we highlight some lemmas about it.

Lemma 5.4 The second Frechet derivative of AL, j is zero that is for every u ∈Wk,r we have A′

L, j(u) = AL, j.

Proof It is obvious from the above definition. Of course, we note that this doesnot say that A′

L, j and AL, j are identical, but that A′L, j has the same value AL at

all points u ∈ Wk,r. The observation is analogous to the fact that the derivativeof linear operator is a constant and we have A′′

L, j = 0. ��

Lemma 5.5 The third Frechet derivative of AN, j is zero.

Proof Consider the operator of AN,0(u) = p∮�

uuxd�, we can write the fol-lowing operator from the above definition:

A′N,0(u)( ) = p

∮�

((ux)( ) + (u)( )x)d�,

A′′N,0(u)( )( ) = 2p

∮�

( )x( )d�,

A′′′N,0(u) = 0,

as is easily verified. We can repeat the above proof for every AN, j, j =1, 2, ..., n − 1 by induction. ��

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Theorem 5.4 The operator of A j in (18) is a contraction mapping if γ < 1 , A′j

and A′′j are bounded in some neighborhood.

Proof we can write:

‖A j(u) − A j(u)‖Wk,r = ‖A j(u) − A j(u) − A′j(u)(u − u) + A′

j(u)(u − u)‖Wk,r

≤ ‖A j(u) − A j(u) − A′j(u)(u − u)‖Wk,r

+‖A′j(u)(u − u)‖Wk,r .

According to the theorem of generalized Taylor (see [11]) and the abovelemmas, we can write the following inequality, where l j(u, u) is the linesegment between u, u :

‖A j(u) − A j(u)‖Wk,r ≤ supu∈l j(u,u)

‖A′′N, j(u)‖Wk,r

‖u − u‖2Wk,r

2

+ ‖A′j(u)‖Wk,r ‖u − u‖Wk,r ≤ γ ‖u − u‖Wk,r ,

where γ = supu∈l j(u,u) ‖A′′N, j(u)‖Wk,r

‖u−u‖Wk,r

2 + ‖A′j(u)‖Wk,r . ��

6 Numerical experimental

Example 1 We consider (9) with τ = 0.5, p = 0.1 and the initial conditionu(x, 0) = 1 − tanh(0.05025x). According to (18) we have:

v0(x, t) = 1 − tanh(0.05025x),

v1(x, t) = 1

2

(201

20000− 201

20000tanh

(201

4000x)2

+ 201

4000000tanh

(201

4000x)

− 201

4000000tanh

(201

4000x)3

)t2,

v2(x, t) = − 1

500000000000000

1

cosh(

2014000 x

)5

(1

12

(14 sinh

(201

4000x)

cosh

(201

4000x)4

− 50502512533 sinh

(201

4000x)

cosh

(201

4000x)2

+ 50 cosh

(201

4000x)5

+ 256293844 sinh

(201

4000x)

+ 757518800 cosh

(201

4000x)

− 505012600 cosh

(201

4000x)3)

t4 + 1

6

(10050000000000 cosh

(201

4000x)3

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Table 1 For Example 1, some values of exact and numerical solutions for t = 0.1, s = 1 andp = 0.1

xi −60 −40 −20 0 20 40 60 80 100

uE(x, t) 1.99523 1.9649 1.76474 1.00251 0.237353 0.0354437 0.482284e-2 0.647525e-3 0.86781e-4uH PM 1.99520 1.9647 1.76376 1.00014 0.236359 0.0352796 0.480038e-1 0.644537e-3 0.86385e-4

+ 50250000000 sinh

(201

4000x)

cosh

(201

4000x)2

t3

)

+ 1

2

(− 39999

4000000tanh

(201

4000x)

+ 39999

4000000tanh

(201

4000x)3

− 201

10000tanh

(201

4000x)2

+ 201

10000

)t2

...

and so on. In this manner the other components can be easily obtained. InTable 1 we represent the values of the exact uE(x, t) and numerical solutionuH PM for t = 0.1. In the latter two figures we use the notation |uE(x, t) − uH PM|as error. It is clear from this table and following Fig. 1 that the differencebetween the exact and the numerical solutions is very small.

We repeat one’s deed for p = 10−i, i = 2, 3, ..., 6 and we get Fig. 2. In theseexperimental results we observe that near p = 0, in the method of HPM wehave not stability same as the classical solution.

Fig. 1 Error function for theintervals −100 ≤ x ≤ 100 ,0 ≤ t ≤ 0.4 and p = 0.1

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Fig. 2 Error function for theintervals −100 ≤ x ≤ 100 and10−6 ≤ p ≤ 0.1

Example 2 We consider (1) with τ = 0.0, p = 0.1, s = 2 and the exact is asfollows:

u(x, t) = −3

6400000000

1√6 − 6 tanh

((1

10

)x)

×(−1 + tanh

((1

10

)x))2

(27 tanh

((1

10

)x)7

+ 558 tanh

((1

10

)x)6

+ 21261 tanh

((1

10

)x)5

+ 90138 tanh

((1

10

)x)4

− 61323 tanh

((1

10

)x)3

− 79110 tanh

((1

10

)x)2

+ 26835 tanh

((1

10

)x)

+ 7214

)(tanh

((1

10

)x)

+1

)t3

)

+ 9

3200000

1√6 − 6 tanh

((1

10

)x)

×(((

−1 + tanh

(1

10x))2 (

21

(tanh

(1

10x))3

+ 147

(tanh

(1

10x))2

− 81 tanh

(1

10x)

− 47

)(tanh

(1

10x)

+ 1

)t2

)

− 3

4000

t(

3 tanh((

110

)x)4+30 tanh

((1

10

)x)3−16 tanh

((1

10

)x)2−30 tanh

((1

10

)x)+13

)√

6 − 6 tanh((

110

)x)

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+ √1.5 − 1.5 tanh(0.1x)

+⎛⎝ 1

720000000000000000

1√6 − 6 tanh

((110

)x) (−1 + tanh

((1

10

)x))

×(

1

4

(−6031715625 + 38156315624 tanh

((1

10

)x)2

+ 23475318750 tanh

((1

10

)x)

− 590417859380 tanh

((1

10

)x)6

+ 693973828180 tanh

((1

10

)x)8

− 457963875000 tanh

((1

10

)x)7

+ 84222281200 tanh

((1

10

)x)9

− 249290662500 tanh

((1

10

)x)3

+ 131008134380 tanh

((1

10

)x)4

+ 571741031250 tanh

((1

10

)x)5

− 295174378120 tanh

((1

10

)x)10

+ 28342153124 tanh

((1

10

)x)12

+ 24084337500 tanh

((1

10

)x)11

+ 143521875 tanh

((1

10

)x)14

+ 3731568750 tanh

((1

10

)x)13

))t4

+(

1/3

(−27337500000 tanh

((1

10

)x)11

+ 262480500000000 tanh

((1

10

)x)4

+ 112716562500000 tanh

((1

10

)x)3

− 134439750000000 tanh

((1

10

)x)2

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− 48265875000000 tanh

((1

10

)x)8

+ 243516375000000 tanh

((1

10

)x)7

− 86568750000000 tanh

((1

10

)x)6

− 348370875000000 tanh

((1

10

)x)5

+ 12562087500000 tanh

((1

10

)x)

− 20396812500000 tanh

((1

10

)x)9

− 510300000000 tanh

((1

10

)x)1

0 + 7304175000000

))t3

)⎞⎟⎠.

According to HPM we have:

v0(x, t) = √1.5 − 1.5 tanh(.1x),

v1(x, t) = −3

4000

t(

3 tanh((

110

)x)4+30 tanh

((1

10

)x)3−16 tanh

((1

10

)x)2−30 tanh

((1

10

)x)+13

)√

6 − 6 tanh((

110

)x) ,

v2(x, t) = 9

3200000

1√6 − 6 tanh

((1

10

)x)((

−1 + tanh

((1

10

)x))2

×(

21 tanh

((1

10x)3

+147 tanh

((1

10

)x)2

−81 tanh

((1

10

)x)

−47

)

×(

tanh

((1

10

)x)

+ 1

)t2

)

and so on. In this manner the other components can be easily obtained. InTable 2 we represent the values of the exact uE(x, t) and numerical solutionuH PM for t = 1.0.

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Table 2 For Example 2, some values of exact and numerical solutions for t = 1.0, s = 2 andp = 0.1

xi −40 −30 −20 20 30 40 50

uE(x, t) 1.73177 1.72995 1.71671 .234582 0.869905e-1 0.32037e-1 0.0117875uH PM 1.731749077 1.729825205 1.715815611 0.2344242928 0.08697856065 0.03203392997 0.01178215219

7 Conclusion remarks

We have found the most general characteristic function for the time-delayedBurgers equation. The equation was solved numerically by the homotopyperturbation method. In our opinion, in some sense that the research resultswill be of theoretical significance and practical value for constructing the exactas well as approximate solution of nonlinear evolution equations. We observe,that HPM does not tell us anything about the possibility of convergenceanalysis for HPM but in this paper, we need to suitable conditions. Therefore,one of the shortcoming in this method is removed.

Acknowledgements The authors would like to thank the referees for some useful comments.The support of Islamic Azad University of South Tehran Branch is gratefully acknowledged.

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