Abstract. In this work, a

recently developed semi-analytic technique, so called the residual

power series method, is generalized to process higher-dimensional linear and

nonlinear partial differential equations. The obtained solution takes a form of

an infinite power series which can, in turn, be expressed in a closed exact

form. The results reveal that the proposed generalization is very effective,

convenient and simple. This is

achieved by handling the -dimensional Burgers equation.

Key words. -dimensional nonlinear partial differential equations;

Generalized residual power series method; Convergence analysis; Exact solution;

Burgers equation.

1. Introduction

Over the last four years, a

recent developed technique, namely the residual power series method (RPSM), for

solving linear and nonlinear ordinary differential equations of integer and

fractional orders have been proposed 1-4. In the current work, we improve

this method to process linear and nonlinear partial differential equations

(NPDEs) of higher dimensional and orders.

Due to their broad variety

of relevance, comparative studies, using the Adomian decomposition method,

homotopy perturbation method, variational iteration method, differential

transform method and its reduction, were discussed deeply to tackle higher

dimensional Burgers type equations 5-7.

2. The GPSM

Consider the -dimensional,th – order NPDE in general form

, , . 1)

Where is assumed to be

sufficiently smooth on the indicated domain contains the starting values. represents the th derivative of the analytic function with respect to

independent variable , and in the same way for other independent variables.

The generalized RPSM

(shortly GRPSM) assumes the solution , of Eq.(1), in a form of power series

. 2)

In more compact form,

, 3)

where and .

Subject to the initial

conditions

, , … , , 4)

the initial approximation of

will be

. 5)

The th-order approximate solution is defined by

the truncated series

, . 6)

subject to

. 7)

Where is the well-defined th- residual function defined by

, 8)

which represents the basic

idea of the GRPSM. The exact analytic solution of the initial-value problem,

Eq.(1) and Eq.(4), is given by

. 9)

Provided that the series has

exact closed form.

3. Convergence Analysis

In this part, convergence

analysis and error estimating for using GRPSM are studied. The presented

scheme, under considerations mentioned in the previous section, approaches the

exact analytic solution as more and more terms found.

Theorem 1. If is an analytic

operator on an open interval containing , then the residual function vanishes as approaches the

infinity.

Proof: Since is

assumed to be analytic as well as for . It is obvious by the definition of residual function

Eq.(7).

Lemma 1. Suppose that , then

, . 10)

Proof: The th-derivative of with respect to is continuous at . Therefore

,

.

Theorem 2. The approximate truncated series solution defined in Eq.(6) and

obtained by applying the GRPSM for solving the

Eqs.(1) and (4) is the th Taylor polynomial of about . In general, as , the series solution in Eq.(9) concise the Taylor series

expansion of centered at .

Proof: For , it is clear from the initial approximation of in Eq.(5). For , it suffices to prove that

.

Applying Eq.(7) to the th-order approximate solution given in Eq.(6) ,

and using the result of Theorem 1, we get

As a result of Lemma 1,

.

Which completes the proof.

Corollary 1. Suppose that the truncated series defined in Eq.(6) is used as an

approximation to the solution of problem Eqs.(1)- (4) on

,

then numbers , satisfies , and exist with

.

Proof: Theorem 1 implies that

.

Following

the proof of Taylor’s Theorem 8, a number exists with

.

Since

the st-derivative of the analytic function with respect to is bounded on , a number also exists with for all . Hence the result.

Corollary 2. The GRPSM results the exact analytic solution if it is a polynomial of .

4. Numerical Illustration

To illustrate

the technique discussed in Sections 2, we consider the ()-dimensional nonlinear Burgers’ equation 9-10

, , 11)

subject to the initial condition

. 12)

Eq. (11) is also

known as Richard’s equation, which is used in the study of cellular automata,

and interacting particle systems. It describes the flow pattern of the particle

in a lattice fluid past an impenetrable obstacle; it can be also used as a

model to describe the water flow in soils.

Applying the

generalized residual power series mechanism to suggested problem, the initial

approximation is , and the th-order approximate solution has the form

, , 13)

which

satisfies

. 14)

For , we have and

.

Hence we get and therefore . Repeating this procedure for , we obtain that .

As , the solution takes the form

.

The series

solution leads to the exact solution obtained by Taylor’s expansion.

5. Conclusions

In this work, we have improved an analytic

solution procedure, called the generalized residual power series method, for

solving higher dimensional partial differential equations. The results

validate the efficiency and reliability of the aforesaid technique that

are achieved by handling the (m+1)-dimensional Burgers equation. The method is a powerful mathematical tool for

solving a wide range of problems arising in engineering and sciences.

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