Abstract. tackle higher dimensional Burgers type equations [5-7].
May 25, 2019
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;
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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
In more compact form,
where and .
Subject to the initial
, , … , , 4)
the initial approximation of
The th-order approximate solution is defined by
the truncated series
, . 6)
Where is the well-defined th- residual function defined by
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
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
Proof: Since is
assumed to be analytic as well as for . It is obvious by the definition of residual function
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
the proof of Taylor’s Theorem 8, a number exists with
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
the technique discussed in Sections 2, we consider the ()-dimensional nonlinear Burgers’ equation 9-10
, , 11)
subject to the initial condition
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.
generalized residual power series mechanism to suggested problem, the initial
approximation is , and the th-order approximate solution has the form
, , 13)
For , we have and
Hence we get and therefore . Repeating this procedure for , we obtain that .
As , the solution takes the form
solution leads to the exact solution obtained by Taylor’s expansion.
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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