# Variational Homotopy Perturbation Method for Solving Higher Dimensional Initial Boundary Value Problems

Authors: Muhammad Aslam Noor; Syed Tauseef Mohyud-Din
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

We suggest and analyze a  technique by combining the variational iteration method
and the homotopy perturbation method. This method is called the variational homotopy
perturbation method (VHPM). We use this method for solving higher dimensional initial boundary
value problems with variable coefficients. The developed algorithm is quite efficient and is
practically well suited for use in these problems. The proposed scheme finds the solution without any
discritization, transformation, or restrictive assumptions and avoids the round-off errors. Several
examples are given to check the reliability and efficiency of the proposed technique.


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## Body

## 1. Introduction

The numerical
and analytical solutions of higher dimensional initial boundary value problems
of variable coefficients, linear and nonlinear, are of considerable
significance for applied sciences. Examples of linear models are Euler-Darboux equation
[1], Lambropoubs’ equation [2] and Tricomi equation [3] given by
(1.1)(x−y)uxy+(αux−βuy)=0,uxy+axux+byuy+cxyu+ut=0,uyy=yuxx,respectively. Examples
of nonlinear models are introduced in Kdv equation [4–7] of variable coefficients
and Clairaut’s equation [5] given by
(1.2)ut+αtnuux+βtmuxxx=0,u=xux+yuy+f(ux,uy),respectively; see
[1–7]. Several numerical and analytical techniques
including the spectral methods, characteristics method, and Adomian’s
decomposition method have been developed for solving these problems; see [1–7] and the
references therein. For implementation of the Adomian decomposition
method, one has to find the so-called the Adomian polynomial, which is itself a
difficult problem. To overcome these
difficulties and drawbacks, He [8–18] developed variational iteration method
for solving linear and nonlinear problems, which arise in various branches of
pure and applied sciences. It is worth mentioning that the origin of
variational iteration method can be traced back to Inokuti et al. [19]. It has been shown that the variational iteration method is user friendly.
Furthermore, He [8–14] also introduced the homotopy perturbation method, which
is developed by combining the standard homotopy and perturbation method. In
these methods the solution is given in an infinite series usually converging to
an accurate solution, see [8–31]. We  would like
to  mention that  Noor [32]
used the  homotopy  perturbation method  for suggesting a  number of iterative methods  solving
nonlinear equations of the type f(x)=0. This is another application of the homotopy
perturbation method.Motivated and
inspired by the on-going research in these areas, we consider a new method, which is called the variational
homotopy perturbation method (VHPM). This method is suggested by combining the
variational iteration technique and the homotopy perturbation method. The
suggested VHPM provides the solution in a rapid convergent series which may
lead the solution in a closed form and is in full agreement with [7], where similar problems were solved by using the
decomposition method. The fact that the proposed technique solves nonlinear
problems without using the so-called Adomian’s polynomials is a clear advantage
of this algorithm over the decomposition method. In
this algorithm, the correct functional is developed [8, 15–19, 21, 22, 23, 24, 25]
and the Lagrange multipliers are calculated optimally via variational theory.
Finally, the homotopy perturbation is implemented on the correct functional and
the comparison of like powers of p gives solutions of various orders. The
developed algorithm takes full advantage of variational iteration and the
homotopy perturbation methods. It is worth mentioning that the VHPM is applied
without any discretization, restrictive assumption, or transformation and is
free from round-off errors. Unlike the method of separation of variables that
require initial and boundary conditions, the VHPM provides an analytical
solution by using the initial conditions only. The boundary conditions can be
used only to justify the obtained result. The proposed method work efficiently
and the results so far are very encouraging and reliable. We would like to emphasize that the VHPM may be considered
as an important and significant refinement of the previously developed
techniques and can be viewed as an alternative to the recently developed methods such as Adomian’s
decomposition, variational iterations, and homotopy perturbation methods. Several examples
are given to verify the reliability and efficiency of the variational homotopy perturbation method (VHPM).


## 2. Variational Iteration Method

To illustrate
the basic concept of the technique, we consider the following general
differential equation:(2.1)Lu+Nu=g(x),where L is a
linear operator, N a nonlinear operator, and g(x) the  forcing term. According to variational
iteration method [8, 15–19, 21, 22, 23, 24, 25], we can construct a correct functional as follows:
(2.2)un+1(x)=un(x)+∫0xλ(Lun(s)+Nu˜n(s)−g(s))ds,where λ is a Lagrange multiplier [8, 15–19], which can
be identified optimally via a variational iteration method. The subscripts n
denote the nth approximation, u˜n is considered as a restricted variation. That
is, δu˜n=0; (2.2) is called a correct functional. The
solution of the linear problems can be solved in a single iteration step due to
the exact identification of the Lagrange multiplier. The principles of the variational
iteration method and its applicability for various kinds of differential
equations are given in [8, 15–19]. In this method, it is required first to
determine the Lagrange multiplier λ optimally. The successive approximation un+1,n≥0 of the solution u will be readily obtained
upon using the determined Lagrange multiplier and any selective function u0, consequently, the solution is given by u=lim⁡n→∞un.


## 3. Homotopy Perturbation Method

To explain the
homotopy perturbation method, we consider a general equation of the type,(3.1)L(u)=0,where L is any integral or differential operator. We
define a convex homotopy H(u,p) by
(3.2)H(u,p)=(1−p)F(u)+pL(u),where F(u) is a
functional operator with known solutions v0, which can be obtained
easily. It is clear that, for
(3.3)H(u,p)=0,we have
(3.4)H(u,0)=F(u),H(u,1)=L(u).This shows that H(u,p) continuously traces an implicitly defined
curve from a starting point H(v0,0) to a solution function H(f,1).
The embedding parameter monotonically increases from zero to unit as the
trivial problem F(u)=0 is continuously deforms the original problem L(u)=0.
The embedding parameter p∈(0,1] can be considered as an expanding
parameter [8–14, 26, 27, 28, 29, 30, 31]. The homotopy
perturbation method uses the homotopy parameter p as an expanding parameter [8–14] to obtain
(3.5)u=∑i=0∞piui=u0+pu1+p2u2+p3u3+….If p→1, then (3.5) corresponds to (3.2) and becomes the
approximate solution of the form
(3.6)f=lim⁡p→1u=∑i=0∞ui.It is well known that
series (3.5) is convergent for most of the cases and also the rate of convergence
is dependent on L(u);
see [8–14]. We assume that (3.6) has a unique solution. The comparisons of like powers
of p give solutions of various
orders.


## 4. Variational Homotopy Perturbation Method (VHPM)

To convey the
basic idea of the variational homotopy perturbation method, we consider the
following general differential equation:(4.1)Lu+Nu=g(x),where L is a linear operator, N a
nonlinear operator, and g(x) the  forcing
term. According to variational iteration method [8, 15–19], we can construct a
correct functional as follows:
(4.2)un+1(x)=un(x)+∫0xλ(ξ)(Lun(ξ)+Nu˜n(ξ)−g(ξ))dξ,where λ is a Lagrange multiplier [8, 15–19], which
can be identified optimally via variational iteration method. The subscripts n
denote the nth approximation, u˜n is considered as a restricted variation. That
is, δu˜n=0; (4.2) is called as a correct functional. Now,
we apply the homotopy perturbation method,
(4.3)∑n=0∞p(n)un=u0(x)+p∫0xλ(ξ)(∑n=0∞p(n)L(un)+∑n=0∞p(n)N(u˜n))dξ−∫0xλ(ξ)g(ξ)dξ,which is the
variational homotopy perturbation method and is formulated by the coupling of
variational iteration method and Adomian’s polynomials. A comparison of like
powers of p gives solutions of various orders.


## 5. Numerical Applications

In this section,
we apply the VHPM developed in Section4 for solving higher dimensional initial
boundary value problems with variable coefficient. We develop the correct
functional and calculate the Lagrange multipliers optimally via variational
theory. The homotopy perturbation method is implemented on the correct
functional and finally, the comparison of like powers of p gives solutions of
various orders. Numerical results reveal that the VHPM is easy to implement and
reduces the computational work to a tangible level while still maintaining a
very higher level of accuracy. For the sake of comparison, we take the same
examples as used in [7, 20].Example 5.1.
Consider the two-dimensional
initial boundary value problem:(5.1)utt=12y2uxx+12x2uyy,0<x,y<1,t>0,with boundary conditions
(5.2)u(0,y,t)=y2e−t,u(1,y,t)=(1+y2)e−t,u(x,0,t)=y2e−t,u(x,1,t)=(1+x2)e−t,and the initial conditions
(5.3)u(x,y,0)=x2+y2,ut(x,y,0)=−(x2+y2).The correct
functional is given as
(5.4)un+1(x,y,z,t)=(x2+y2)−(x2+y2)t+∫0tλ(ξ)(∂2un∂t2−12(y2(u˜n)xx+x2(u˜n)yy))dξ,where u˜n is considered as a restricted variation.
Making the above functional stationary, the Lagrange multiplier can be
determined as λ=ξ−t, which yields the following iteration formula:
(5.5)un+1(x,y,z,t)=(x2+y2)−(x2+y2)t+∫0t(ξ−t)(∂2un∂t2−12(y2(un)xx+x2(un)yy))dξ.Applying the variational homotopy perturbation method, we have
(5.6)u0+pu1+p2u2+…=(x2+y2)−(x2+y2)t+p∫0t(ξ−t)(∂2u0∂t2+p∂2u1∂t2+p2∂2u2∂t2+…)dξ−12p∫0t(ξ−t)((y2(∂2u˜0∂x2+p∂2u˜1∂x2+p2∂2u˜2∂x2+…))+(x2(∂2u˜0∂y2+p∂2u˜1∂y2+p2∂2u˜2∂y2+…)))dξ.Comparing the coefficient
of like powers of p, we have
(5.7)p(0):u0(x,y,t)=(x2+y2)−(x2+y2)t,p(1):u1(x,y,t)=(x2+y2)t22!−(x2+y2)t33!,p(2):u2(x,y,t)=(x2+y2)t44!−(x2+y2)t55!,p(3):u3(x,y,t)=(x2+y2)t55!−(x2+y2)t77!,p(4):u4(x,y,t)=(x2+y2)t88!−(x2+y2)t99!,p(5):u5(x,y,t)=(x2+y2)t1010!−(x2+y2)t1111!,⋮The series solution is given by
(5.8)u(x,y,t)=(x2+y2)(1−t+t22!−t33!+t44!−t55!+t66!−t77!+t88!−…),and in a closed form by u(x,y,t)=(x2+y2)e−t, which is in full agreement with [7].Example 5.2.
Consider the three-dimensional initial boundary value problem(5.9)utt=145x2uxx+145y2uyy+145z2uzz−u,0<x,y<1,t<0subject to the Neumann boundary conditions
(5.10)ux(0,y,z,t)=0,ux(1,y,z,t)=6y6z6sinh⁡t,uy(x,0,z,t)=0,uy(x,1,z,t)=6x6z6sinh⁡t,uz(x,y,0,t)=0,uz(x,y,1,t)=6x6y6sinh⁡t,and the initial conditions
(5.11)u(x,y,z,0)=0,ut(x,y,z,0)=x6y6z6.The correct functional is given by
(5.12)un+1(x,y,z,t)=(x6y6z6)t+∫0tλ(ξ)(∂2un∂t2−145(x2(u˜n)xx+y2(u˜n)yy+z2(u˜n)zz)+u˜n)dξ,
where u˜n is considered as a restricted variation. Making the above functional stationary, the Lagrange multiplier can be determined as λ=ξ−t, which yields the following iteration formula:
(5.13)un+1(x,y,z,t)=(x6y6z6)t+∫0t(ξ−t)(∂2un∂t2−145(x2(un)xx+y2(un)yy+z2(un)zz)+un)dξ.
Applying the variational homotopy perturbation method,
(5.14)u0+pu1+p2u2+…=(x2+y2)−(x2+y2)t+p∫0t(ξ−t)(∂2u0∂t2+p∂2u1∂t2+p2∂2u2∂t2+…)dξ−145p∫0t(ξ−t)((x2(∂2u˜0∂x2+p∂2u˜1∂x2+p2∂2u˜2∂x2+…))+(y2(∂2u˜0∂y2+p∂2u˜1∂y2+p2∂2u˜2∂y2+…)))dξ−145p∫0t(ξ−t)z2(∂2u˜0∂z2+p∂2u˜1∂z2+p2∂2u˜2∂z2+…)dξ+p∫0t(ξ−t)(u0+pu1+p2u2+…)dξ.
Comparing the coefficient of like powers of p, we have
(5.15)p(0):u0(x,y,z,t)=x6y6z6t,p(1):u1(x,y,z,t)=x6y6z6t33!,p(2):u2(x,y,z,t)=x6y6z6t55!,p(3):u3(x,y,z,t)=x6y6z6t77!,p(4):u4(x,y,z,t)=x6y6z6t99!,⋮
The series solution is given by
(5.16)u(x,y,z,t)=x6y6z6(t+t33!+t55!+t77!+t99!+…)=x6y6z6sinh⁡t,
which is in full agreement with [7].Example 5.3.
Consider the two-dimensional nonlinear inhomogeneous initial boundary value problem(5.17)utt=2x2+2y2+152(xuxx2+yuyy2),0<x,y<1,t>0
with boundary conditions
(5.18)u(0,y,t)=y2t2+yt6,u(1,y,t)=(1+y2)t2+(1+y)t6,u(x,0,t)=x2t2+xt6,u(x,1,t)=(1+x2)t2+(1+x)t6,
and the initial conditions
(5.19)u(x,y,0)=0,ut(x,y,0)=0,
The correct functional is given as
(5.20)un+1(x,y,z,t)=∫0tλ(ξ)(∂2un∂t2−152(x(u˜n2)xx+y2(u˜n2)yy)−2(x2+y2))dξ,
where u˜n is considered as a restricted variation. Making the above functional stationary, the Lagrange multiplier can be determined as λ=ξ−t, which yields the following iteration formula
(5.21)un+1(x,y,z,t)=∫0t(ξ−t)(∂2un∂t2−152(x(u˜n2)xx+y2(u˜n2)yy)−2(x2+y2))dξ.
Applying the variational homotopy perturbation method, we have
(5.22)u0+pu1+p2u2+…=p∫0t(ξ−t)(∂2u0∂t2+p∂2u1∂t2+p2∂2u2∂t2+…)dξ−152p(ξ−t)(x(∂2u˜0∂x2+p∂2u˜1∂x2+p2∂2u˜2∂x2+…)xx2+∫0ty(∂2u˜0∂y2+p∂2u˜1∂y2+p2∂2u˜2∂y2+…)yy2−2(x2+y2))dξ.
Comparing the coefficient of like powers of p, we have
(5.23)p(0):u0(x,y,t)=0,p(1):u1(x,y,t)=(x2+y2)t2,p(2):u2(x,y,t)=(x+y)t6,p(3):u3(x,y,t)=0,⋮
The solution is obtained as u(x,y,t)=(x2+y2)t2+(x+y)t6, which is in full agreement with [7].Example 5.4.
Consider the three-dimensional nonlinear initial boundary value problem(5.24)utt=(2−t2)+u−(e−xuxx2+e−yuyy2+e−zuzz2),0<x,y<1,t<0
subject to the Neumann boundary conditions
(5.25)ux(0,y,z,t)=1,ux(1,y,z,t)=e,uy(x,0,z,t)=0,uy(x,1,z,t)=e,uz(x,y,0,t)=1,uz(x,y,1,t)=e,
and the initial conditions(5.26)u(x,y,z,0)=ex+ey+ez,ut(x,y,z,0)=0.
The correct functional is given as
(5.27)un+1(x,y,z,t)=(ex+ey+ez)+∫0tλ(ξ)(∂2un∂t2+(e−x(u˜n)xx2+e−y(u˜n)yy2+e−z(u˜n)zz2)−u˜n)dξ−∫0tλ(ξ)(2−t2)dξ,
where u˜n is considered as a restricted variation. Making the above functional stationary, the Lagrange multiplier can be determined as λ=ξ−t, which yields the following iteration formula:
(5.28)un+1(x,y,z,t)=(ex+ey+ez)+∫0t(ξ−t)(∂2un∂t2+(e−x(u˜n)xx2+e−y(u˜n)yy2+e−z(u˜n)zz2)−u˜n)dξ−∫0tλ(ξ)(2−t2)dξ,
Applying the variational homotopy perturbation method, we have
(5.29)u0+pu1+…=(ex+ey+ez)+p∫0t(ξ−t)((∂2u0∂t2+p∂2u1∂t2+p2∂2u2∂t2+…)+e−x(∂2u˜0∂x2+p∂2u˜1∂x2+p2∂2u˜2∂x2+…)xx2)dξ+p∫0t(ξ−t)(e−y(∂2u˜0∂y2+p∂2u˜1∂y2+p2∂2u˜2∂y2+…)xx2+∫0te−z(∂2u˜0∂z2+p∂2u˜1∂z2+p2∂2u˜2∂z2+…)yy2)dξ−p∫0t(ξ−t)((u0+pu1+p2u2+…)+(2−t2))dξ.
Comparing the coefficient of like powers of p, we have
(5.30)p(0):u0(x,y,z,t)=(ex+ey+ez)+t2−t412,p(1):u1(x,y,z,t)=t412−t6360,p(2):u2(x,y,z,t)=t6360−t820160,⋮
The solution is obtained as u(x,y,z,t)=(ex+ey+ez)+t2, which is in full agreement with [7].Remark 5.5.
We would like to point out that Noor [32] used the homotopy perturbation method for suggesting some iterative-type methods for solving nonlinear equations f(x)=0 coupled with system of equations. Also it has been shown [32] that the homotopy perturbation method and Adomian decomposition method are equivalent. This application of the homotopy method is quite different in nature. It is an interesting problem to consider such type of applications of the variational homotopy method in solving nonlinear equations.


## 6. Conclusions

In this paper,
we develop the variational homotopy perturbation method (VHPM) for solving
nonlinear problems. We used the variational homotopy perturbation method for
solving the higher dimensional initial boundary value problems with variable
coefficient. The proposed method is successfully implemented by using the
initial conditions only. There are two important points to make here. First,
unlike the implicit and explicit finite difference methods, the solution here
is given in a closed form and by using the initial conditions only. Second, the
VHPM avoids the cumbersome of the computational methods while still maintaining
the higher level of accuracy. The fact that the variational homotopy
perturbation method solves nonlinear problems without using the Adomian’s
polynomials can be considered as a clear advantage of this technique over the
decomposition method. It is observed that the proposed scheme exploits full
advantage of variational iteration method and the homotopy perturbation method.
Finally, we conclude that the VHPM may be considered as a nice refinement in
existing numerical techniques.



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