Critical Thinking

Abstract. tackle higher dimensional Burgers type equations [5-7].

 

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.

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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.

References

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