a study of general second-order partial differential ... · partial differential equations, [2]...

Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2471–2492© Research India Publicationshttp://www.ripublication.com/gjpam.htm

A Study of General Second-Order Partial DifferentialEquations Using Homotopy Perturbation Method

Mohammed S. Mechee1, Adil M. Al-Rammahi, and Ghassan A. Al-Juaifri

Department of Mathematics,Faculty of Computer Science and Mathematics,

Kufa university, Najaf, Iraq.

Abstract

In this work, we have studied a general class of linear second-order partial differen-tial equations which is used as mathematical models in many physically significantfields and applied science. The homotopy perturbation method (HPM) has beenused for solving generalized linear second-order partial differential equation. Also,we have tested the HPM on the solving of different implementations which showthe efficiency and accuracy of the method. The approximated solutions are agreewell with analytical solutions for the tested problems Moreover, the approximatedsolutions proved that the proposed method to be efficient and high accurate.

AMS subject classification:Keywords: PDE, Second-order, Homotopy, HPM, Partial Differential Equation.

1. Introduction

The most important mathematical models for physical phenomena is the differentialequation. Motion of objects, Fluid and heat flow, bending and cracking of materials,vibrations, chemical reactions and nuclear reactions are all modeled by systems of differ-ential equations. Moreover, Numerous mathematical models in science and engineeringare expressed in terms of unknown quantities and their derivatives. Many applicationsof differential equations (DEs), particularly ODEs of different orders, can be found inthe mathematical modeling of real life problems ([22]).

1Corresponding author.

2472 Mohammed S. Mechee, et al.

The homotopy perturbation method (HPM), which is a well-known, is efficient tech-nique to find the approximate solutions for ordinary and partial differential equationswhich describe different fields of science, physical phenomena, engineering, mechanics,and so on. HPM was proposed by Ji-Huan He in 1999 for solving linear and nonlinear dif-ferential equations and integral equations. Many researchers used HPM to approximatethe solutions of differential equations and integral equations ([36], [15] & [20]).

Many researchers published some papers in solving some classes differential equa-tions using HPM. For example, [8] used HPM for solving a linear and nonlinear second-order two-point boundary value problems while [10] was solved the Black-Scholes equa-tion for a simple European option in this method to obtain a new efficient recurrent re-lation to solve Black-Scholes equation. Moreover, numerous researches used HPM forsolving nonlinear differential equations, [32] was solved nonlinear DEs, which yields theMaclaurin series of the exact solution, [7] developed a third-order explicit approximationto find the roots of the dispersion relation for water waves that propagate over dissipativemedia, [39] solved the nonlinear PB equation describing spherical and planar colloidalparticles immersed in an arbitrary valence and mixed electrolyte solution, [28] solvedcertain non-linear, non-smooth oscillators, [34] solved nonlinear vibration analysis offunctionally graded plate while [13] applied HPM for solving nonlinear oscillators withdiscontinuities, nonlinear Duffing equation and some nonlinear ODEs. For class of linearpartial differential equations, [2] solved the Korteweg-de Vries (KdV) equation and con-vergence study of HPM, [4] used HPM to solve time-dependent differential equations,[3] solved advection Problem, vibrating beam equation linear and nonlinear PDEs andthe system of nonlinear PDEs and [4] used the homotopy perturbation method to solvetime-dependent differential. Also, many researchers used HPM for solving the class ofnon-linear PDEs, [34] was approximated solution for free nonlinear vibration of thinrectangular laminated FGM plates, [19] has solved nonlinear PDEs, [35] was used toimplement the nonlinear Korteweg-deVries equation, [21] combined HPM-Padé approx-imant to acquire the approximate analytical solution of the KdV equation, [31] solvedParabolic equations and Periodic equation linear and nonlinear PDEs, [16] obtained thesolution of a second-order non-linear wave equation, [9] utilized to derive approximateexplicit analytical solution for the nonlinear foam drainage equation, [23] modified thealgorithm which provides approximate solutions in the form of convergent series witheasily computable components, [4] solved time-dependent differential equations while[12], solved non-linear problems using the homotopy technique. However, for the sys-tem of DEs, [5] solved systems of second-order BVPs, [17] solved SEIR model, [33]solution of a model for HIV infection of CD4 + T cells, [29] solution of the system ofnonlinear ordinary differential equations governing on the problem, [26] solved the sys-tem of linear equations. [25] solution of linear and non-linear sixth-order boundary valueproblems and system of differential equations, [17] solved system of linear Fredholmintegral equations (LFIEs). [38] has solved a linear Fredholm type integro-differentialequations with separable kernel. [17] solved non-linear Fredholm integral equations,[30] used modified HPM for solving the system linear and nonlinear of Volterra integralequations, [18] solved generalized Abel integral equation. Lastly, for the differential

A Study of General Second-Order Partial Differential Equations 2473

equations of fractional type, [27] solved nonlinear differential equations of fractionalorder, [14] solved nonlinear problems of fractional Riccati differential equation & [37]solved the space-time fractional advection-dispersion equation.

Recently, we have studied a wide class of linear second-order partial differentialequations which are used as mathematical models in many physically significant fieldsand applied science. The approximated solutions of this class of partial differential equa-tions have studied using homotopy perturbation method (HPM). The proposed methodapplied for solving different examples for this class of partial differential equations. Theapproximated solutions for different tested problems show that the HPM is more effi-cient in the iterations complexity and high accurate in the absolute errors. It has beenhighlighted that the use of HPM is more suitable to approximate the solutions of generalpartial differential equations.

2. Preliminary

2.1. Homotopy Perturbation Method(HPM)

In this section, we present a brief description of the HPM, to illustrate the basic ideasof the homotopy perturbation method, we consider the following differential equation([24], [8], [6] & [1]):

A(u) − f (τ) = 0, τ ∈ � (2.1)

with boundary conditions:

B(u,∂u

∂τ) = 0, τ ∈ ∂� (2.2)

where A is general differential operator, B is a boundary operator, f (τ) a known analyticfunction and ∂� is the boundary of the domain �. The operator A can be generallydivided into two parts of L and N where L is linear part, while N is the nonlinear partin the DE, Therefore Eq. (2.1) can be rewritten as follows ([11]):

L(u) + N(u) − f (τ) = 0 (2.3)

By using homotopy technique, One can construct a homotopy

V (τ , p) : � × [0, 1] �→ R

which satisfies:

H(v, p) = (1 − p)[L(v) − L(u0)] + p[L(v) + N(v) − f (τ)] = 0 (2.4)

or

H(v, p) = L(v) − L(u0) + pL(u0 + p[N(v) − f (τ)]) = 0 (2.5)


where p ∈ [0, 1], τ ∈ � & p is called homotopy parameter and uo is an initialapproximation for the solution of equation (2.1) which satisfies the boundary conditionsobviously, Using equation (2.4) or (2.5), we have the following equation:

H(v, 0) = L(v) − L(u0) = 0 (2.6)

and

H(v, 1) = L(v) + N(v) − f (τ) = 0 (2.7)

Assume that the solution of (2.4) or (2.5) can be expressed as a series in p as follows:

V = v0 + pv1 + p2v2 + p3v3 + · · · =∞∑i=0

pivi (2.8)

set p → 1 results in the approximate solution of (2.1).Consequently,

u(τ) = limp→1

V = v0 + v2 + v3 + · · · =∞∑i=0

vi (2.9)

It is worth to note that the major advantage of He’s homotopy perturbation method is thatthe perturbation equation can be freely constructed in many ways and approximation canalso be freely selected.

3. Analysis of HPM for Solving Second-Order PartialDifferential Equation

In this section, we have study the general second-order partial differential equation andintroduce a review of the homotopy perturbation method for solving first-order partialdifferential equation.

The general second-order partial differential equation in �n domain has the followingform:

n∑i,j=1

Ai,j (Tn)∂2u(Tn)

∂ti∂tj+

n∑i=1

Bi(Tn)∂u(Tn)

∂ti+ C(Tn)u(Tn) = G(Tn) (3.1)

subject to the initial condition

u(Tn−1, 0) = f (Tn−1)

∂u(Tn−1, 0)

∂tn= h(Tn−1),


such thatTn = (t1, t2, t3, . . . , tn), Tn−1 = (t1, t2, t3, . . . , tn−1)

andAi,j (Tn) = Aj,i(Tn)

whereAi,j (Tn), Bi(Tn), C(Tn), G(Tn) & f (Tn−1),

are the given functions of n independent variables.We describe a general technique for solving second-order partial differential equa-

tions in which the solution u : �n → � is a function of n variables according thefollowing algorithms:

3.1. The Proposed method

Consider Equation (3.1) and the following cases:

1. Case1: An,n(Tn) �= 0Firstly, we start with the initial approximation u(Tn−1, 0) = f (Tn) + th(Tn),Secondly, we can construct a homotopy for PDE (3.1) as follow:

H(u, p) = (1 − p)

(∂2u(Tn)

∂t2n

− ∂2u0(Tn)

∂t2n

)+ p

⎛⎝ n∑

i,j=1

Ai,j (Tn

⎞⎠ ∂2u(Tn)

∂ti∂tj

+n∑

i=1

Bi(Tn)∂u(Tn)

∂ti+ C(Tn)u(Tn) − G(Tn)) = 0 (3.2)

Thirdly, suppose that the solution of the Equation (3.1) is in the form

x(t) = x0 + px1 + p2x2 + p3x3 + . . . (3.3)

Therefore,

H(u, p) = (1 − p)

( ∞∑k=0

pk ∂2uk(Tn)

∂t2n

− ∂2u0(Tn)

∂t2n

)

+p

⎛⎝ ∞∑

k=0

n∑i,j=1

Ai,j (Tn)pk ∂2uk(Tn)

∂ti∂tj

+∞∑

k=0

n∑i=1

Bi(Tn)pk ∂uk(Tn)

∂ti+ C(Tn)

∞∑k=0

pkuk(Tn) − G(Tn)

)= 0

(3.4)


Fourthly, collecting terms of the same power of p gives, as show in the followingequations: By collecting the terms of the same powers of p we obtain the followingequation:

p0 : ∂2u0(Tn)

∂2tn− ∂2u0(Tn)

∂2tn= 0

p1 : ∂2u1(Tn)

∂2tn+

n∑i,j=1

Ai,j (Tn)∂2u0(Tn)

∂ti∂tj+

n∑i=1

Bi(Tn)∂u0(Tn)

∂ti

+ C(Tn)u0(Tn) − G(Tn) = 0

p2 : ∂2u2(Tn)

∂2tn− ∂2u1(Tn)

∂2tn+

n∑i,j=1

Ai,j (Tn)∂2u1(Tn)

∂ti∂tj+

n∑i=1

Bi(Tn)∂u1(Tn)

∂ti

+ C(Tn)u1(Tn) = 0

p3 : ∂2u3(Tn)

∂2tn− ∂2u2(Tn)

∂2tn+

n∑i,j=1

Ai,j (Tn)∂2u2(Tn)

∂ti∂tj+

n∑i=1

Bi(Tn)∂u2(Tn)

∂ti

+ C(Tn)u2(Tn) = 0

p4 : ∂2u4(Tn)

∂2tn− ∂2u3(Tn)

∂2tn+

n∑i,j=1

Ai,j (Tn)∂2u3(Tn)

∂ti∂tj+

n∑i=1

Bi(Tn)∂u3(Tn)

∂ti

+ C(Tn)u3(Tn) = 0

p5 : ∂2u5(Tn)

∂2tn− ∂2u4(Tn)

∂2tn+

n∑i,j=1

Ai,j (Tn)∂2u4(Tn)

∂ti∂tj+

n∑i=1

Bi(Tn)∂u4(Tn)

∂ti

+ C(Tn)u4(Tn) = 0

. . .

Hence, for n = 2, 3, 4, . . . we have,

pm : ∂2um(Tn)

∂2tn− ∂2um−1(Tn)

∂2tn+

n∑i,j=1

Ai,j (Tn)∂2um−1(Tn)

∂ti∂tj

+n∑

i=1

Bi(Tn)∂um−1(Tn)

∂ti+ C(Tn)um−1(Tn) = 0

. . .

Finally, using the Equations (3.5) with some simplifications, then the we get the


following sequence of the solutions:

u0(Tn) = f (Tn) + th(Tn)

u1(Tn) = −∫ ∫

Tn

⎛⎝∂2u1(Sn)

∂2sn+

n∑i,j=1

Ai,j (Sn)∂2u0(Tn)

∂ti∂tj+

n∑i=1

Bi(Tn)∂u0(Tn)

∂ti

+ C(Sn)u0(Sn) − G(Tn)

)dsndtn

u2(Tn) = −∫ ∫

Tn

⎛⎝∂2u2(Sn)

∂2sn− ∂2u1(Sn)

∂2sn+

n∑i,j=1

Ai,j (Sn)∂2u1(Sn)

∂si∂sj

+n∑

i=1

Bi(Sn)∂u1(Sn)

∂si+ C(Sn)u1(Sn)

)dsndtn

u3(Tn) = −∫ ∫

Tn

⎛⎝∂2u3(Sn)

∂2sn− ∂2u2(Sn)

∂2sn+

n∑i,j=1

Ai,j (Sn)∂2u2(Sn)

∂si∂sj

+n∑

i=1

Bi(Sn)∂u2(Sn)

∂si+ C(Sn)u2(Sn)

)dsndtn

u4(Tn) = −∫ ∫

Tn

⎛⎝∂2u4(Sn)

∂2sn− ∂2u3(Sn)

∂2sn+

n∑i,j=1

Ai,j (Sn)∂2u3(Sn)

∂si∂sj

+n∑

i=1

Bi(Sn)∂u3(Sn)

∂si+ C(Sn)u3(Sn)

)dsndtn (3.5)

and

u5(Tn) = −∫ ∫

Tn

⎛⎝∂2u5(Tn)

∂2sn− ∂2u4(Sn)

∂2sn+

n∑i,j=1

Ai,j (Sn)∂2u4(Sn)

∂si∂tj

+n∑

i=1

Bi(Sn)∂u4(Sn)

∂si+ C(Sn)u4(Sn)

)dsndtn

. . .

Hence, the general term has the following form:

un(Tn) =∫ ∫

Tn

⎛⎝∂2um(Sn)

∂2sn− ∂2um−1(Sn)

∂2sn+

n∑i,j=1

Ai,j (Sn)∂2um−1(Sn)

∂si∂tj

+n∑

i=1

Bi(Sn)∂um−1(Sn)

∂si+ C(Sn)um−1(Sn)

)dsndtn, n = 2, 3, 4, . . . .

. . .


Where Sn = (s1, s2, s3, . . . , sn), Then the solution of equation (3.1) is

u(Tn) = u0(Tn) + u1(Tn) + u2(Tn) + u3(Tn) + u4(Tn) + u5(Tn) + . . .(3.6)

2. Case2: An,n(Tn) = 0 and Bn(Tn) �= 0.

Firstly, we start with the initial approximation

u(Tn−1, 0) = f (Tn−1).

We start with the initial approximation u(Tn−1, 0) = f (Tn).

Secondly, we can construct a homotopy for PDE (3.1) as follow:

H(u, p) = (1 − p)

(∂u(Tn)

∂tn− ∂u0(Tn)

∂tn

)

+p

⎛⎝ n∑

i,j=1

Ai,j (Tn)∂2u(Tn)

∂ti∂tj+

n∑i=1

Bi(Tn)∂u(Tn)

∂ti

+ C(Tn)u(Tn) − G(Tn)

)= 0 (3.7)

Thirdly, suppose that the solution of the Equation (3.1) is in the form

x(t) = x0 + px1 + p2x2 + p3x3 + . . . (3.8)

Therefore,

H(u, p) = (1 − p)

( ∞∑k=0

pk ∂uk(Tn)

∂tn− ∂u0(Tn)

∂tn

)

+p

⎛⎝ ∞∑

k=0

n∑i,j=1

Ai,j (Tn)pk ∂2uk(Tn)

∂ti∂tj

+∞∑

k=0

n∑i=1

Bi(Tn)pk ∂uk(Tn)

∂ti+ C(Tn)

∞∑k=0

pkuk(Tn) − G(Tn)

)= 0

(3.9)

Fourthly, by collecting the terms of the same powers of p, we obtain the following


equation:

p0 : ∂u0(Tn)

∂tn− ∂u0(Tn)

∂tn= 0

p1 : ∂u1(Tn)

∂tn+

n∑i,j=1

Ai,j (Tn)∂2u0(Tn)

∂ti∂tj+

n−1∑i=1

Bi(Tn)∂u0(Tn)

∂ti

+ C(Tn)u0(Tn) − G(Tn) = 0

p2 : ∂u2(Tn)

∂tn+

n∑i,j=1

Ai,j (Tn)∂2u1(Tn)

∂ti∂tj+

n−1∑i=1

Bi(Tn)∂u1(Tn)

∂ti

+ C(Tn)u1(Tn) = 0

p3 : ∂u3(Tn)

∂tn+

n∑i,j=1

Ai,j (Tn)∂2u2(Tn)

∂ti∂tj+

n−1∑i=1

Bi(Tn)∂u2(Tn)

∂ti

+ C(Tn)u2(Tn) = 0

p4 : ∂u4(Tn)

∂tn+

n∑i,j=1

Ai,j (Tn)∂2u3(Tn)

∂ti∂tj+

n−1∑i=1

Bi(Tn)∂u3(Tn)

∂ti

+ C(Tn)u3(Tn) = 0

p5 : ∂u5(Tn)

∂tn+

n∑i,j=1

Ai,j (Tn)∂2u4(Tn)

∂ti∂tj+

n−1∑i=1

Bi(Tn)∂u4(Tn)

∂ti

+ C(Tn)u4(Tn) = 0

. . .

Hence, for n = 2, 3, 4, . . . we have,

pm : ∂um(Tn)

∂tn+

n−1∑i,j=1


∂ti∂tj+

n−1∑i=1

Bi(Tn)∂um−1(Tn)

∂ti

+ C(Tn)um−1(Tn) = 0

. . .

Finally, using the Equation (3.10) with some simplifications, then the we get the


following sequence of the solutions:

u0(Tn) = f (Tn)

u1(Tn) = −∫ ⎛

⎝ n∑i,j=1

Ai,j (Tn)∂2u0(Tn)

∂ti∂tj+

n−1∑i=1

Bi(Tn)∂u0(Tn)

∂ti

+ C(Tn)u0(Tn) − G(Tn)

)dtn

u2(Tn) = −∫ ⎛

⎝ n∑i,j=1

Ai,j (Tn)∂2u1(Tn)

∂ti∂tj+

n−1∑i=1

Bi(Tn)∂u1(Tn)

∂ti

+ C(Tn)u1(Tn)

)dtn

u3(Tn) = −∫ ⎛

⎝ n∑i,j=1

Ai,j (Tn)∂2u2(Tn)

∂ti∂tj+

n−1∑i=1

Bi(Tn)∂u2(Tn)

∂ti

+ C(Tn)u2(Tn)

)dtn

u4(Tn) = −∫ ⎛

⎝ n∑i,j=1

Ai,j (Tn)∂2u3(Tn)

∂ti∂tj+

n−1∑i=1

Bi(Tn)∂u3(Tn)

∂ti

+ C(Tn)u3(Tn)

)dtn

and

u5(Tn) = −∫ ⎛

⎝ n∑i,j=1

Ai,j (Tn)∂2u4(Tn)

∂ti∂tj+

n−1∑i=1

Bi(Tn)∂u4(Tn)

∂ti

+ C(Tn)u4(Tn)

)dtn

. . .

Hence, the general term has the following form:

un(Tn) =∫ ⎛

⎝ n∑i,j=1


∂ti∂tj+

n−1∑i=1

Bi(Tn)∂um−1(Tn)

∂ti

+ C(Tn)um−1(Tn)

)dtn, n = 2, 3, 4, . . . .

. . .

Then, Equation (3.1) has the following solution

u(Tn) = u0(Tn) + u1(Tn) + u2(Tn) + u3(Tn) + u4(Tn) + u5(Tn) + . . .(3.10)


4. Implementations

In order to assess the accuracy of the solving second-order PDE using homotopy per-turbation method (HPM) of some problems, we have introduced different examples tocompare the approximated solutions with the exact solutions for tested problems, wewill consider the following problems.

4.1. Problem 1

Consider the following partial differential equation:

uxx(x, y) + uyy(x, y) = 0, x, y ∈ R (4.1)

subject to the initial conditionsu(x, 0) = 0

uy(x, 0) = e−x.

Comparing Equation(4.1) with general Equation (3.1) we have

n = 2, A1,1 = 1, A2,2 = 1, B1 = B2 = 0 G = 0 and C = 0

The initial approximation has the form u0(x, y) = ye−x substituting (3.5) into (4.1), wehave

u1(x, y) = −∫ ∫

y

(∂2u0(x, s)

∂2x+ ∂2u0(x, s)

∂2s

)dsdy

= −y3

3! e−x

u2(x, y) = −∫ ∫

y

(∂2u1(x, s)

∂2x

)dsdy

= y5

5! e−x

u3(x, y) = −∫ ∫

y

(∂2u2(x, s)

∂2x

)dsdy

= −y7

7! e−x

u4(x, y) = −∫ ∫

y

(∂2u3(x, s)

∂2x

)dsdy

= y9

9! e−x

u5(x, y) = −∫ ∫

y

(∂2u4(x, s)

∂2x

)dsdy

= −y11

11! e−x


Figure 1: Approximated solution of Problem1 using HPM

and,

um(x, y)) = −∫ ∫

y

(∂2um−1(x, s)

∂2x

)dsdy

= (−1)my2m+1

(2m + 1)!e−x

Then, the general solution of equation (4.1) is written as follow:

u(x, y) = u0(x, y) + u1(x, y) + u2(x, y) + u3(x, y) + u4(x, y) + u5(x, y) + . . .

= sin(y)e−x

4.2. Problem2

Consider the following Helmholtz equation:

uxx(x, y) + uyy(x, y) + 8u(x, y) = 0, x, y ∈ R (4.2)

subject to the initial conditions

u(0, y) = sin(2y)

uy(0, y) = 0

comparing Equation (4.2) we have

n = 2, A1,1 = 1, A2,2 = 1, G = 0 and C = 8


The initial approximation has the form u0(x, y) = sin(2y) substituting (3.5) into (4.2),we have

u1(x, y) = −∫ ∫

y

(∂2u0(x, s)

∂2x+ ∂2u0(x, s)

∂2s+ 8u0(x, s)

)ds dy

= −(2x)2

2! sin(2y)

u2(x, y) = −∫ ∫

y

(∂2u1(x, s)

∂2x+ 8u1(x, s)

)ds dy

= (2x)4

4! sin(2y)

u3(x, y) = −∫ ∫

y

(∂2u2(x, s)

∂2x+ 8u2(x, s)

)ds dy

= −(2x)6

6! sin(2y)

u4(x, y) = −∫ ∫

y

(∂2u3(x, s)

∂2x+ 8u3(x, s)

)ds dy

= (2x)8

8! sin(2y)

u5(x, y) = −∫ ∫

y

(∂2u4(x, s)

∂2x+ 8u4(x, s)

)ds dy

= −(2x)10

10! sin(2y)

and,

um(x, y)) = −∫ ∫

y

(∂2um−1(x, s)

∂2x+ 8um−1(x, s)

)ds dy

= (−1)my2m+1

(2m + 1)!e−x

Then, the general solution of Equation (4.2) is written as follow:

u(x, y) = u0(x, y) + u1(x, y) + u2(x, y) + u3(x, y) + u4(x, y) + u5(x, y) + . . .

= sin(2y)

(1 − (2x)2

2! + (2x)4

4! − (2x)6

6! + (2x)8

8! − (2x)10

10! + . . .

)= sin(2y) cos(2y)


Figure 2: Approximated solution of Problem2 using HPM

4.3. Problem3

Consider the following Heat equation in 3-dimension :

ut(x, y, z, t) = λ(uxx(x, y, z, t)+uyy(x, y, z, t)+uzz(x, y, z, t)), x, y, z ∈ R, t > 0(4.3)


u(x, y, z, 0) = sin(x) sin(y) sin(z)

Comparing Equation (4.3) we have

n = 4, A1,1 = A2,2 = A3,3 = −λ,

A4,4 = B1 = B2 = B3 = G = C = 0 & B4 = 1

The initial approximation has the form

u0(x, y, z, t) = sin(x) sin(y) sin(z)

Substituting Equation (3.10) into the Equation (4.3), we have

u1(x, y, z, t) = −∫ (

λ

(∂2u0(x, y, z, t)

∂2x+ ∂2u0(x, y, z, t)

∂2y

+∂2u0(x, y, z, t)

∂2y− ∂u0(x, y, z, t)

∂t

))dt

= −3λt sin(x) sin(y) sin(z)


u2(x, y, z, t) = −∫ (

λ

(∂2u1(x, y, z, t)

∂2x+ ∂2u1(x, y, z, t)

∂2y+ ∂2u1(x, y, z, t)

∂2y

))dt

= (3λ)2 t2

2! sin(x) sin(y) sin(z)

u3(x, y, z, t) = −∫ (

λ

(∂2u2(x, y, z, t)

∂2x+ ∂2u2(x, y, z, t)

∂2y+ ∂2u2(x, y, z, t)

∂2y

))dt

= −(3λ)3 t3


u4(x, y, z, t) = −∫ (

λ

(∂2u3(x, y, z, t)

∂2x+ ∂2u3(x, y, z, t)

∂2y+ ∂2u3(x, y, z, t)

∂2y

))dt

= (3λ)4 t4


u5(x, y, z, t) = −∫ (

λ

(∂2u4(x, y, z, t)

∂2x+ ∂2u4(x, y, z, t)

∂2y+ ∂2u4(x, y, z, t)

∂2y

))dt

= −(3λ)5 t5


and

um(x, y, z, t) = −∫ (

λ

(∂2um−1(x, y, z, t)

∂2x+ ∂2um−1(x, y, z, t)

∂2y

+∂2um−1(x, y, z, t)

∂2y

))dt

= (−3λ)mtm

m! sin(x) sin(y) sin(z)

Then, the general solution of Equation (4.3) is written as follow:

u(x, y, z, t) = u0(x, y, z, t) + u1(x, y, z, t) + u2(x, y, z, t) + u3(x, y, z, t)

+u4(x, y, z, t) + u5(x, y, z, t) + · · ·=

(1 − 3λt + (3λ)2 t2

2! − (3λ)3 t3

3! + (3λ)4 t4

4! − (3λ)5 t5

5! + . . .

+(−3λ)mtm

m!)

sin(x) sin(y) sin(z)

= e−3λt sin(x) sin(y) sin(z) (4.4)


Figure 3: Approximated solution of Problem3 using HPM at λ = x = y = 1.

Figure 4: Approximated solution of Problem3 using HPM at λ = 1&x = y = 0.

4.4. Problem4

Consider the following 3-dimension parabolic-like equation:

ut(x, y, z, t) = 1

36(x2uxx(x, y, z, t) + y2uyy(x, y, z, t)

+z2uzz(x, y, z, t)) + (xyz)4,

0 ≤ x, y, z ≤ 1, t > 0 (4.5)


Figure 5: Approximated solution of Problem 3 using HPM at λ = x = t = 1.


u(x, y, z, 0) = 0

comparing Equation (4.5) we have

n = 4, A1,1 = −x2

36, A2,2 = −y2

36, A3,3 = −z2

36, A4,4 = 0,

B1 = B2 = B3 = 0, B4 = 1 G = C = 0

The initial approximation has the form u0(x, y, z, t)) = 0 substituting Equation (3.10)into Equation (4.5), we have

u1(x, y, z, t) = 1

36

∫ (x2 ∂2u0(x, y, z, t)

∂2x+ y2 ∂2u0(x, y, z, t)

∂2y

+z2 ∂2u0(x, y, z, t)

∂2y− (xyz)4

)dt

= (xyz)4t

u2(x, y, z, t) = 1

36

∫ (x2 ∂2u1(x, y, z, t)

∂2x+ y2 ∂2u1(x, y, z, t)

∂2y

+z2 ∂2u1(x, y, z, t)

∂2y

)dt

= (xyz)4 t2

2!


u3(x, y, z, t) = 1

36

∫ (x2 ∂2u2(x, y, z, t)

∂2x+ y2 ∂2u2(x, y, z, t)

∂2y

+z2 ∂2u2(x, y, z, t)

∂2y

)dt

= (xyz)4 t3

3!u4(x, y, z, t) = 1

36

∫ (x2 ∂2u3(x, y, z, t)

∂2x+ y2 ∂2u3(x, y, z, t)

∂2y

+z2 ∂2u3(x, y, z, t)

∂2y

)dt

= (xyz)4 t4

4!u5(x, y, z, t) = 1

36

∫ (x2 ∂2u4(x, y, z, t)

∂2x+ y2 ∂2u4(x, y, z, t)

∂2y

+z2 ∂2u4(x, y, z, t)

∂2y

)dt

= (xyz)4 t5

5!and,

um(x, y, z, t) = 1

36

∫ (x2 ∂2um−1(x, y, z, t)

∂2x+ y2 ∂2um−1(x, y, z, t)

∂2y

+z2 ∂2um−1(x, y, z, t)

∂2y

)dt

= (xyz)4 tm

m!Then, the general solution of Equation (4.5) is written as follow:

u(x, y, z, t) = u0(x, y, z, t) + u1(x, y, z, t) + u2(x, y, z, t) + u3(x, y, z, t)

+u4(x, y, z, t) + u5(x, y, z, t) + · · ·= (xyz)4(t + t2

2! + t3

3! + t4

4! + t5

5! + · · · + tm

m!)= (xyz)4(et − 1)

5. Discussion and Conclusion

In this paper, the homotopy perturbation method (HPM) has been studied for solvinggeneralized linear second-order partial differential equations. The approximated solu-tions of this class of PDEs have been studied. Also, we have tested the HPM on the


Figure 6: Approximated solution of Problem4 using HPM at x = y = 1.

Figure 7: Approximated solution of Problem4 using HPM at t = y = 1.

solving of different implementations which are show the efficiency and accuracy of theproposed method. The approximated solutions are agree well with analytical solutionsfor the tested problems Moreover, the approximated solutions proved that the proposedmethod to be efficient and high accurate.

Acknowledgements

The authors would like to thank university of Kufa for supporting this research project.


Figure 8: Approximated solution of Problem4 using HPM at t = 0, y = 1.

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