International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 603280 | 8 pages | https://doi.org/10.5402/2012/603280

A New Efficient Method for Solving Two-Dimensional Burgers' Equation

Academic Editor: V. Rai
Received14 May 2012
Accepted18 Jun 2012
Published13 Aug 2012

Abstract

We introduce a new hybrid of the Laplace transform method and new homotopy perturbation method (LTNHPM) that efficiently solves nonlinear two-dimensional Burgersโ€™ equation. Three examples are given to demonstrate the efficiency of the new method.

1. Introduction

The system of partial differential equation of the following form ๐œ•๐‘ข๐œ•๐‘ก+๐‘ข๐œ•๐‘ฃ๐œ•๐‘ฅ+๐‘ฃ๐œ•๐‘ข=1๐œ•๐‘ฆ๐‘…๎‚ต๐œ•2๐‘ข๐œ•๐‘ฅ2+๐œ•2๐‘ข๐œ•๐‘ฆ2๎‚ถ,๐œ•๐‘ข๐œ•๐‘ก+๐‘ข๐œ•๐‘ข๐œ•๐‘ฅ+๐‘ฃ๐œ•๐‘ฃ=1๐œ•๐‘ฆ๐‘…๎‚ต๐œ•2๐‘ข๐œ•๐‘ฅ2+๐œ•2๐‘ข๐œ•๐‘ฆ2๎‚ถ,(๐‘ฅ,๐‘ฆ)โˆˆฮฉ,๐‘ก>0,(1) subject to the initial conditions: ๐‘ฃ๐‘ข(๐‘ฅ,๐‘ฆ,0)=๐‘“(๐‘ฅ,๐‘ฆ),(๐‘ฅ,๐‘ฆ)โˆˆฮฉ,(๐‘ฅ,๐‘ฆ,0)=๐‘”(๐‘ฅ,๐‘ฆ),(๐‘ฅ,๐‘ฆ)โˆˆฮฉ,(2) is called the system of two-dimensional Burgersโ€™ equation, where ฮฉ={(๐‘ฅ,๐‘ฆ)|๐‘Žโ‰ค๐‘ฅโ‰ค๐‘,๐‘Žโ‰ค๐‘ฆโ‰ค๐‘}, ๐‘ข(๐‘ฅ,๐‘ฆ,๐‘ก) and ๐‘ฃ(๐‘ฅ,๐‘ฆ,๐‘ก) are the velocity components to be determined, ๐‘“ and ๐‘” are known functions, and ๐‘… is the Reynolds number. The Burgers model of turbulence is a very important fluid dynamic model, and the study of this model and the theory of shock waves have been considered by many authors, both to obtain a conceptual understanding of a class of physical flows and for testing various numerical methods. The mathematical properties of Burgersโ€™ equation have been studied by Burgers [1]. Nonlinear phenomena play a crucial role in applied mathematics and physics. The importance of obtaining the exact or approximate solutions of PDEs in physics and mathematics is still a hot topic as regards seeking new methods for obtaining new exact or approximate solutions [2โ€“5]. For that purpose, different methods have been put forward for seeking various exact solutions of multifarious physical models described using nonlinear PDEs. A well-known model was first introduced by Bateman [6], who found its steady solutions, descriptive of certain viscous flows. It was later proposed by Burgers [1] as one of a class of equations describing mathematical models of turbulence. In the context of gas dynamics, it was discussed by Hopf [7] and Cole [8]. They also illustrated independently that the Burgers equation can be solved exactly for an arbitrary initial condition. Benton and Platzman [9] have surveyed the analytical solutions of the one-dimensional Burgers equation. It can be considered as a simplified form of the Navier-Stokes equation [10] due to the form of the nonlinear convection term and the occurrence of the viscosity term. The numerical solution of the Burgers equation is of great importance due to the application of the equation in the approximate theory of flow through a shock wave, travelling in a viscous fluid [8] and in the Burgers model of turbulence [11]. It can be solved analytically for arbitrary initial conditions [7]. Numerical methods such as finite difference, finite element, and classical ones like Fourier series, Fourier integral, and Laplace transformation commonly used for solving these methods either need a lot of computations and have less convergence speed and accuracy or solve only certain types of problems. Therefore, science and engineering researchers attempt to propose new methods for solving functional equations.

In this paper, we propose a new hybrid of Laplace transform method and new homotopy perturbation method [12] to obtain exact and numerical solutions of the system of two-dimensional Burgersโ€™ equation. Finally, three examples are given to illustrate the proposed approach.

2. Analysis of the Method

For solving system of two-dimensional Burgersโ€™ equation by LTNHPM, we construct the following homotopy: ๐‘ˆ๐‘กโˆ’๐‘ข0๎‚€๐‘ข(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ๐‘ˆ๐‘ฅ+๐‘‰๐‘ˆ๐‘ฆโˆ’1๐‘…๎€ท๐‘ˆ๐‘ฅ๐‘ฅ+๐‘ˆ๐‘ฆ๐‘ฆ๎€ธ๎‚๐‘‰=0,๐‘กโˆ’๐‘ฃ0๎‚€๐‘ฃ(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ๐‘‰๐‘ฅ+๐‘‰๐‘‰๐‘ฆโˆ’1๐‘…๎€ท๐‘‰๐‘ฅ๐‘ฅ+๐‘‰๐‘ฆ๐‘ฆ๎€ธ๎‚=0,(3) where ๐‘โˆˆ[0,1] is an embedding parameter and ๐‘ข0(๐‘ฅ,๐‘ฆ,๐‘ก) and ๐‘ฃ0(๐‘ฅ,๐‘ฆ,๐‘ก) are initial approximation of solution of (1).

By applying Laplace transform on both sides of (3), we have L๎‚†๐‘ˆ๐‘กโˆ’๐‘ข0๎‚€๐‘ข(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ๐‘ˆ๐‘ฅ+๐‘‰๐‘ˆ๐‘ฆโˆ’1๐‘…๎€ท๐‘ˆ๐‘ฅ๐‘ฅ+๐‘ˆ๐‘ฆ๐‘ฆ๎€ธL๎‚†๐‘‰๎‚๎‚‡=0,๐‘กโˆ’๐‘ฃ0๎‚€๐‘ฃ(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ๐‘‰๐‘ฅ+๐‘‰๐‘‰๐‘ฆโˆ’1๐‘…๎€ท๐‘‰๐‘ฅ๐‘ฅ+๐‘‰๐‘ฆ๐‘ฆ๎€ธ๎‚๎‚‡=0.(4)

Using the differential property of Laplace transform we have๎‚†๐‘ข๐‘ L{๐‘ˆ}โˆ’๐‘ˆ(๐‘ฅ,๐‘ฆ,0)=L0๎‚€๐‘ข(๐‘ฅ,๐‘ฆ,๐‘ก)โˆ’๐‘0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ๐‘ˆ๐‘ฅ+๐‘‰๐‘ˆ๐‘ฆโˆ’1๐‘…๎€ท๐‘ˆ๐‘ฅ๐‘ฅ+๐‘ˆ๐‘ฆ๐‘ฆ๎€ธ,๎‚†๐‘ฃ๎‚๎‚‡๐‘ L{๐‘‰}โˆ’๐‘‰(๐‘ฅ,๐‘ฆ,0)=L0๎‚€๐‘ฃ(๐‘ฅ,๐‘ฆ,๐‘ก)โˆ’๐‘0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ๐‘‰๐‘ฅ+๐‘‰๐‘‰๐‘ฆโˆ’1๐‘…๎€ท๐‘‰๐‘ฅ๐‘ฅ+๐‘‰๐‘ฆ๐‘ฆ๎€ธ,๎‚๎‚‡(5)

or1L{๐‘ˆ}=๐‘ ๎‚€๎‚†๐‘ข๐‘ˆ(๐‘ฅ,๐‘ฆ,0)+L0๎‚€๐‘ข(๐‘ฅ,๐‘ฆ,๐‘ก)โˆ’๐‘0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ๐‘ˆ๐‘ฅ+๐‘‰๐‘ˆ๐‘ฆโˆ’1๐‘…๎€ท๐‘ˆ๐‘ฅ๐‘ฅ+๐‘ˆ๐‘ฆ๐‘ฆ๎€ธ,1๎‚๎‚‡๎‚L{๐‘‰}=๐‘ ๎‚€๎‚†๐‘ฃ๐‘‰(๐‘ฅ,๐‘ฆ,0)+L0๎‚€๐‘ฃ(๐‘ฅ,๐‘ฆ,๐‘ก)โˆ’๐‘0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ๐‘‰๐‘ฅ+๐‘‰๐‘‰๐‘ฆโˆ’1๐‘…๎€ท๐‘‰๐‘ฅ๐‘ฅ+๐‘‰๐‘ฆ๐‘ฆ๎€ธ.๎‚๎‚‡๎‚(6)

By applying inverse Laplace transform on both sides of (6), we have๐‘ˆ(๐‘ฅ,๐‘ฆ,๐‘ก)=Lโˆ’1๎‚†1๐‘ ๎‚€๎‚†๐‘ข๐‘ˆ(๐‘ฅ,๐‘ฆ,0)+L0๎‚€๐‘ข(๐‘ฅ,๐‘ฆ,๐‘ก)โˆ’๐‘0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ๐‘ˆ๐‘ฅ+๐‘‰๐‘ˆ๐‘ฆโˆ’1๐‘…๎€ท๐‘ˆ๐‘ฅ๐‘ฅ+๐‘ˆ๐‘ฆ๐‘ฆ๎€ธ,๎‚๎‚‡๎‚๎‚‡๐‘‰(๐‘ฅ,๐‘ฆ,๐‘ก)=Lโˆ’1๎‚†1๐‘ ๎‚€๎‚†๐‘ฃ๐‘‰(๐‘ฅ,๐‘ฆ,0)+L0๎‚€๐‘ฃ(๐‘ฅ,๐‘ฆ,๐‘ก)โˆ’๐‘0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ๐‘‰๐‘ฅ+๐‘‰๐‘‰๐‘ฆโˆ’1๐‘…๎€ท๐‘‰๐‘ฅ๐‘ฅ+๐‘‰๐‘ฆ๐‘ฆ๎€ธ.๎‚๎‚‡๎‚๎‚‡(7)

According to the HPM, we use the embedding parameter ๐‘ as a small parameter and assume that the solutions of (7) can be represented as a power series in ๐‘ as ๐‘ˆ=๐‘ˆ0+๐‘๐‘ˆ1+๐‘2๐‘ˆ2+โ‹ฏ,๐‘‰=๐‘‰0+๐‘๐‘‰1+๐‘2๐‘‰2+โ‹ฏ.(8) Substituting (8) into (7) and equating the terms with the identical powers of ๐‘ lead to๐‘0โˆถโŽงโŽชโŽจโŽชโŽฉ๐‘ˆ0=Lโˆ’1๎‚†1๐‘ ๎€ท๎€ฝ๐‘ข๐‘ˆ(๐‘ฅ,๐‘ฆ,0)+L0๎‚‡,๐‘‰(๐‘ฅ,๐‘ฆ,๐‘ก)๎€พ๎€ธ0=Lโˆ’1๎‚†1๐‘ ๎€ท๎€ฝ๐‘ฃ๐‘‰(๐‘ฅ,๐‘ฆ,0)+L0๎‚‡,๐‘(๐‘ฅ,๐‘ฆ,๐‘ก)๎€พ๎€ธ1โˆถโŽงโŽชโŽจโŽชโŽฉ๐‘ˆ1=Lโˆ’1๎‚†โˆ’1๐‘ ๎‚€L๎‚†๐‘ข0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ0๎€ท๐‘ˆ0๎€ธ๐‘ฅ+๐‘‰0๎€ท๐‘ˆ0๎€ธ๐‘ฆโˆ’1๐‘…๎‚€๎€ท๐‘ˆ0๎€ธ๐‘ฅ๐‘ฅ+๎€ท๐‘ˆ0๎€ธ๐‘ฆ๐‘ฆ,๐‘‰๎‚๎‚‡๎‚๎‚‡1=Lโˆ’1๎‚†โˆ’1๐‘ ๎‚€L๎‚†๐‘ฃ0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ0๎€ท๐‘‰0๎€ธ๐‘ฅ+๐‘‰0๎€ท๐‘‰0๎€ธ๐‘ฆโˆ’1๐‘…๎‚€๎€ท๐‘‰0๎€ธ๐‘ฅ๐‘ฅ+๎€ท๐‘‰0๎€ธ๐‘ฆ๐‘ฆ,๐‘๎‚๎‚‡๎‚๎‚‡2โˆถโŽงโŽชโŽจโŽชโŽฉ๐‘ˆ2=Lโˆ’1๎‚†โˆ’1๐‘ ๎‚€L๎‚†๐‘ˆ0๎€ท๐‘ˆ1๎€ธ๐‘ฅ+๐‘ˆ1๎€ท๐‘ˆ0๎€ธ๐‘ฅ+๐‘‰0๎€ท๐‘ˆ1๎€ธ๐‘ฆ+๐‘‰1๎€ท๐‘ˆ0๎€ธ๐‘ฆโˆ’1๐‘…๎‚€๎€ท๐‘ˆ1๎€ธ๐‘ฅ๐‘ฅ+๎€ท๐‘ˆ1๎€ธ๐‘ฆ๐‘ฆ,๐‘‰๎‚๎‚‡๎‚๎‚‡2=Lโˆ’1๎‚†โˆ’1๐‘ ๎‚€L๎‚†๐‘ˆ0๎€ท๐‘‰1๎€ธ๐‘ฅ+๐‘ˆ1๎€ท๐‘‰0๎€ธ๐‘ฅ+๐‘‰0๎€ท๐‘‰1๎€ธ๐‘ฆ+๐‘‰1๎€ท๐‘‰0๎€ธ๐‘ฆโˆ’1๐‘…๎‚€๎€ท๐‘‰1๎€ธ๐‘ฅ๐‘ฅ+๎€ท๐‘‰1๎€ธ๐‘ฆ๐‘ฆ,โ‹ฎ๐‘๎‚๎‚‡๎‚๎‚‡๐‘—โˆถโŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘ˆ๐‘—=Lโˆ’1๎ƒฏโˆ’1๐‘ ๎ƒฉL๎ƒฏ๐‘—โˆ’1โˆ‘๐‘˜=0๎‚€๐‘ˆ๐‘˜๎€ท๐‘ˆ๐‘—โˆ’๐‘˜โˆ’1๎€ธ๐‘ฅ+๐‘‰๐‘˜๎€ท๐‘ˆ๐‘—โˆ’๐‘˜โˆ’1๎€ธ๐‘ฆ๎‚โˆ’1๐‘…๎‚€๎€ท๐‘ˆ๐‘—โˆ’1๎€ธ๐‘ฅ๐‘ฅ+๎€ท๐‘ˆ๐‘—โˆ’1๎€ธ๐‘ฆ๐‘ฆ๎‚,๐‘‰๎ƒฐ๎ƒช๎ƒฐ๐‘—=Lโˆ’1๎ƒฏโˆ’1๐‘ ๎ƒฉL๎ƒฏ๐‘—โˆ’1โˆ‘๐‘˜=0๎‚€๐‘ˆ๐‘˜๎€ท๐‘‰๐‘—โˆ’๐‘˜โˆ’1๎€ธ๐‘ฅ+๐‘‰๐‘˜๎€ท๐‘‰๐‘—โˆ’๐‘˜โˆ’1๎€ธ๐‘ฆ๎‚โˆ’1๐‘…๎‚€๎€ท๐‘‰๐‘—โˆ’1๎€ธ๐‘ฅ๐‘ฅ+๎€ท๐‘‰๐‘—โˆ’1๎€ธ๐‘ฆ๐‘ฆ๎‚,โ‹ฎ๎ƒฐ๎ƒช๎ƒฐ(9)Suppose that the initial approximation has the form ๐‘ˆ(๐‘ฅ,๐‘ฆ,0)=๐‘ข0(๐‘ฅ,๐‘ฆ,๐‘ก)=๐‘“(๐‘ฅ,๐‘ฆ) and ๐‘‰(๐‘ฅ,๐‘ฆ,0)=๐‘ฃ0(๐‘ฅ,๐‘ฆ,๐‘ก)=๐‘”(๐‘ฅ,๐‘ฆ); therefore the exact solution may be obtained as follows ๐‘ข(๐‘ฅ,๐‘ฆ,๐‘ก)=lim๐‘โ†’1๐‘ˆ(๐‘ฅ,๐‘ฆ,๐‘ก)=๐‘ˆ0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ1(๐‘ฅ,๐‘ฆ,๐‘ก)+โ‹ฏ,๐‘ฃ(๐‘ฅ,๐‘ฆ,๐‘ก)=lim๐‘โ†’1๐‘‰(๐‘ฅ,๐‘ฆ,๐‘ก)=๐‘‰0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘‰1(๐‘ฅ,๐‘ฆ,๐‘ก)+โ‹ฏ.(10)

3. Examples

Example 1. Consider the following homogeneous form of a coupled Burgers equation [13]: ๐‘ข๐‘ก+๐‘ข๐‘ข๐‘ฅ+๐‘ฃ๐‘ข๐‘ฆ=1๐‘…๎€ท๐‘ข๐‘ฅ๐‘ฅ+๐‘ข๐‘ฆ๐‘ฆ๎€ธ,๐‘ฃ๐‘ก+๐‘ข๐‘ฃ๐‘ฅ+๐‘ฃ๐‘ฃ๐‘ฆ=1๐‘…๎€ท๐‘ฃ๐‘ฅ๐‘ฅ+๐‘ฃ๐‘ฆ๐‘ฆ๎€ธ,(11) subject to the initial condition ๐‘ฃ๐‘ข(๐‘ฅ,๐‘ฆ,0)=๐‘ฅ+๐‘ฆ,(๐‘ฅ,๐‘ฆ,0)=๐‘ฅโˆ’๐‘ฆ.(12) The exact solution of this equation is ๐‘ข(๐‘ฅ,๐‘ฆ,๐‘ก)=(๐‘ฅ+๐‘ฆโˆ’2๐‘ฅ๐‘ก)/(1โˆ’2๐‘ก2) and ๐‘ฃ(๐‘ฅ,๐‘ฆ,๐‘ก)=(๐‘ฅโˆ’๐‘ฆโˆ’2๐‘ฆ๐‘ก)/(1โˆ’2๐‘ก)2.
Starting with ๐‘ˆ(๐‘ฅ,๐‘ฆ,0)=๐‘ข0=๐‘ฅ+๐‘ฆ, ๐‘‰(๐‘ฅ,๐‘ฆ,0)=๐‘ฃ0=๐‘ฅโˆ’๐‘ฆ and using (9), we obtain

๐‘ˆ0=Lโˆ’1๎‚†1๐‘ ๎‚‡๐‘‰(๐‘ฅ+๐‘ฆ+L{๐‘ฅ+๐‘ฆ})=(๐‘ฅ+๐‘ฆ)(1+๐‘ก),0=Lโˆ’1๎‚†1๐‘ ๎‚‡๐‘ˆ(๐‘ฅโˆ’๐‘ฆ+L{๐‘ฅโˆ’๐‘ฆ})=(๐‘ฅโˆ’๐‘ฆ)(1+๐‘ก),1=Lโˆ’1๎‚†โˆ’1๐‘ ๎‚€L๎‚†๐‘ฅ+๐‘ฆ+๐‘ˆ0๎€ท๐‘ˆ0๎€ธ๐‘ฅ+๐‘‰0๎€ท๐‘ˆ0๎€ธ๐‘ฆโˆ’1๐‘…๎‚€๎€ท๐‘ˆ0๎€ธ๐‘ฅ๐‘ฅ+๎€ท๐‘ˆ0๎€ธ๐‘ฆ๐‘ฆ๎‚๎‚‡๎‚๎‚‡=โˆ’(3๐‘ฅ+๐‘ฆ)๐‘กโˆ’2๐‘ฅ๐‘ก2โˆ’23๐‘ฅ๐‘ก3,๐‘‰1=Lโˆ’1๎‚†โˆ’1๐‘ ๎‚€L๎‚†๐‘ฅโˆ’๐‘ฆ+๐‘ˆ0๎€ท๐‘‰0๎€ธ๐‘ฅ+๐‘‰0๎€ท๐‘‰0๎€ธ๐‘ฆโˆ’1๐‘…๎‚€(๐‘‰0)๐‘ฅ๐‘ฅ+๎€ท๐‘‰0๎€ธ๐‘ฆ๐‘ฆ๎‚๎‚‡๎‚๎‚‡=โˆ’(๐‘ฅ+๐‘ฆ)๐‘กโˆ’2๐‘ฆ๐‘ก2โˆ’23๐‘ฆ๐‘ก3,๐‘ˆ2=Lโˆ’1๎‚†โˆ’1๐‘ ๎‚€L๎‚†๐‘ˆ0๎€ท๐‘ˆ1๎€ธ๐‘ฅ+๐‘ˆ1๎€ท๐‘ˆ0๎€ธ๐‘ฅ+๐‘‰0๎€ท๐‘ˆ1๎€ธ๐‘ฆ+๐‘‰1๎€ท๐‘ˆ0๎€ธ๐‘ฆโˆ’1๐‘…๎‚€๎€ท๐‘ˆ1๎€ธ๐‘ฅ๐‘ฅ+๎€ท๐‘ˆ1๎€ธ๐‘ฆ๐‘ฆ๎‚๎‚‡๎‚๎‚‡=(4๐‘ฅ+2๐‘ฆ)๐‘ก2+๎‚€84๐‘ฅ+3๐‘ฆ๎‚๐‘ก3+๎‚€434๐‘ฅ+3๐‘ฆ๎‚๐‘ก4+๎‚€4415๐‘ฅ+๐‘ฆ๎‚๐‘ก155,๐‘‰2=Lโˆ’1๎‚†โˆ’1๐‘ ๎‚€L๎‚†๐‘ˆ0๎€ท๐‘‰1๎€ธ๐‘ฅ+๐‘ˆ1๎€ท๐‘‰0๎€ธ๐‘ฅ+๐‘‰0๎€ท๐‘‰1๎€ธ๐‘ฆ+๐‘‰1๎€ท๐‘‰0๎€ธ๐‘ฆโˆ’1๐‘…๎‚€๎€ท๐‘‰1๎€ธ๐‘ฅ๐‘ฅ+๎€ท๐‘‰1๎€ธ๐‘ฆ๐‘ฆ๎‚๎‚‡๎‚๎‚‡=2๐‘ฅ๐‘ก2+๎‚€834๐‘ฅโˆ’3๐‘ฆ๎‚๐‘ก3+๎‚€434๐‘ฅโˆ’3๐‘ฆ๎‚๐‘ก4+๎‚€4415๐‘ฅโˆ’๐‘ฆ๎‚๐‘ก155,๐‘ˆ3=Lโˆ’1๎‚†โˆ’1๐‘ ๎‚€L๎‚†๐‘ˆ0๎€ท๐‘ˆ2๎€ธ๐‘ฅ+๐‘ˆ1๎€ท๐‘ˆ1๎€ธ๐‘ฅ+๐‘ˆ2๎€ท๐‘ˆ0๎€ธ๐‘ฅ+๐‘‰0๎€ท๐‘ˆ2๎€ธ๐‘ฆ+๐‘‰1๎€ท๐‘ˆ1๎€ธ๐‘ฆ+๐‘‰2๎€ท๐‘ˆ0๎€ธ๐‘ฆโˆ’1๐‘…๎‚€๎€ท๐‘ˆ2๎€ธ๐‘ฅ๐‘ฅ+๎€ท๐‘ˆ2๎€ธ๐‘ฆ๐‘ฆ๎‚€๎‚๎‚‡๎‚๎‚‡=โˆ’2238๐‘ฅ+3๐‘ฆ๎‚๐‘ก3โˆ’๎‚€2838๐‘ฅ+3๐‘ฆ๎‚๐‘ก4โˆ’๎‚€1634๐‘ฅ+5๐‘ฆ๎‚๐‘ก5โˆ’6845๐‘ฅ๐‘ก6โˆ’68315๐‘ฅ๐‘ก7,๐‘‰3=Lโˆ’1๎‚†โˆ’1๐‘ ๎‚€L๎‚†๐‘ˆ0๎€ท๐‘‰2๎€ธ๐‘ฅ+๐‘ˆ1๎€ท๐‘‰1๎€ธ๐‘ฅ+๐‘ˆ2๎€ท๐‘‰0๎€ธ๐‘ฅ+๐‘‰0๎€ท๐‘‰2๎€ธ๐‘ฆ+๐‘‰1๎€ท๐‘‰1๎€ธ๐‘ฆ+๐‘‰2๎€ท๐‘‰0๎€ธ๐‘ฆโˆ’1๐‘…๎‚€๎€ท๐‘‰2๎€ธ๐‘ฅ๐‘ฅ+๎€ท๐‘‰2๎€ธ๐‘ฆ๐‘ฆ๎‚€8๎‚๎‚‡๎‚๎‚‡=โˆ’3๎‚๐‘ก๐‘ฅ+2๐‘ฆ3โˆ’๎‚€83๎‚๐‘ก๐‘ฅ+4๐‘ฆ4โˆ’๎‚€45๐‘ฅ+56๐‘ฆ๎‚๐‘ก155โˆ’6845๐‘ฆ๐‘ก6โˆ’68315๐‘ฆ๐‘ก7โ‹ฎ(13)

Therefore we gain the solution of (11) as ๐‘ข(๐‘ฅ,๐‘ฆ,๐‘ก)=๐‘ˆ0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ1(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘ˆ3(๐‘ฅ,๐‘ฆ,๐‘ก)+โ‹ฏ=๐‘ฅ+๐‘ฆโˆ’2๐‘ฅ๐‘ก+2๐‘ฅ๐‘ก2+2๐‘ฆ๐‘ก2โˆ’4๐‘ฅ๐‘ก3+4๐‘ฅ๐‘ก4+4๐‘ฆ๐‘ก4โˆ’8๐‘ฅ๐‘ก5๎€ท+โ‹ฏ=๐‘ฅ1+2๐‘ก2+4๐‘ก4๎€ธ๎€ท+โ‹ฏ+๐‘ฆ1+2๐‘ก2+4๐‘ก4๎€ธ๎€ท+โ‹ฏโˆ’2๐‘ฅ๐‘ก1+2๐‘ก2+4๐‘ก4๎€ธ=+โ‹ฏ๐‘ฅ+๐‘ฆโˆ’2๐‘ฅ๐‘ก1โˆ’2๐‘ก2,๐‘ฃ(๐‘ฅ,๐‘ฆ,๐‘ก)=๐‘‰0(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘‰1(๐‘ฅ,๐‘ฆ,๐‘ก)+๐‘‰3(๐‘ฅ,๐‘ฆ,๐‘ก)+โ‹ฏ=๐‘ฅโˆ’๐‘ฆโˆ’2๐‘ฆ๐‘ก+2๐‘ฅ๐‘ก2โˆ’2๐‘ฆ๐‘ก2โˆ’4๐‘ฆ๐‘ก3+4๐‘ฅ๐‘ก4โˆ’4๐‘ฆ๐‘ก4โˆ’8๐‘ฆ๐‘ก5๎€ท+โ‹ฏ=๐‘ฅ1+2๐‘ก2+4๐‘ก4๎€ธ๎€ท+โ‹ฏโˆ’๐‘ฆ1+2๐‘ก2+4๐‘ก4๎€ธ๎€ท+โ‹ฏโˆ’2๐‘ฆ๐‘ก1+2๐‘ก2+4๐‘ก4๎€ธ=+โ‹ฏ๐‘ฅโˆ’๐‘ฆโˆ’2๐‘ฆ๐‘ก1โˆ’2๐‘ก2(14) which is exact solution.

Example 2. Let us consider system of Burgersโ€™ equations (1), with the following initial conditions [14]: 3๐‘ข(๐‘ฅ,๐‘ฆ,0)=4โˆ’14๎€บ๎€ป,๐‘ฃ31+exp(๐‘ฆโˆ’๐‘ฅ)๐‘…/8(๐‘ฅ,๐‘ฆ,0)=4+14๎€บ๎€ป,1+exp(๐‘ฆโˆ’๐‘ฅ)๐‘…/8(15) for which exact solutions are 3๐‘ข(๐‘ฅ,๐‘ฆ,๐‘ก)=4โˆ’14๎€บ๎€ป,๐‘ฃ31+exp(4๐‘ฆโˆ’4๐‘ฅโˆ’๐‘ก)๐‘…/32(๐‘ฅ,๐‘ฆ,๐‘ก)=4+14๎€บ๎€ป.1+exp(4๐‘ฆโˆ’4๐‘ฅโˆ’๐‘ก)๐‘…/32(16) To solve system (1) by LTNHPM, following the same procedure discussed in Section 2 and Example 1, we obtain the iterative relations (9); in this example we take initial approximations (15). The accuracy of LTNHPM for the system of two-dimensional Burgersโ€™ equation agrees good with the exact solution, and absolute errors are very small for the present choice of ๐‘ฅ,๐‘ฆ, and ๐‘ก. These results are listed in Tables 1, 2, 3, and 4 for ๐‘…=0.5 and ๐‘…=1.


( ๐‘ฅ , ๐‘ฆ ) ๐‘ก = 0 . 0 1 ๐‘ก = 0 . 5 ๐‘ก = 2
๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข | ๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข | ๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข |

( 0 . 1 , 0 . 1 ) 0 . 6 2 4 9 9 0 2 3 4 4 9 ๐ธ โˆ’ 6 0 . 6 2 4 5 1 1 7 2 1 3 4 ๐ธ โˆ’ 4 0 . 6 2 3 0 4 7 4 4 5 5 1 ๐ธ โˆ’ 3
( 0 . 5 , 0 . 1 ) 0 . 6 2 3 4 2 7 8 1 7 3 9 ๐ธ โˆ’ 6 0 . 6 2 2 9 4 9 4 0 2 8 2 ๐ธ โˆ’ 3 0 . 6 2 1 4 8 5 7 0 5 7 3 ๐ธ โˆ’ 3
( 0 . 9 , 0 . 1 ) 0 . 6 2 1 8 6 5 8 9 3 7 3 ๐ธ โˆ’ 3 0 . 6 2 1 3 8 7 7 2 7 1 3 ๐ธ โˆ’ 3 0 . 6 1 9 9 2 5 0 6 4 1 5 ๐ธ โˆ’ 3
( 0 . 3 , 0 . 3 ) 0 . 6 2 4 9 9 0 2 3 4 4 9 ๐ธ โˆ’ 6 0 . 6 2 4 5 1 1 7 2 1 3 4 ๐ธ โˆ’ 4 0 . 6 2 3 0 4 7 4 4 5 5 1 ๐ธ โˆ’ 3
( 0 . 7 , 0 . 3 ) 0 . 6 2 3 4 2 7 8 1 7 3 1 ๐ธ โˆ’ 3 0 . 6 2 2 9 4 9 4 0 2 8 2 ๐ธ โˆ’ 3 0 . 6 2 1 4 8 5 7 0 5 7 3 ๐ธ โˆ’ 3
( 0 . 1 , 0 . 5 ) 0 . 6 2 6 5 5 2 6 5 2 8 1 ๐ธ โˆ’ 3 0 . 6 2 6 0 7 4 1 9 0 6 1 ๐ธ โˆ’ 3 0 . 6 2 4 6 0 9 7 9 2 5 3 ๐ธ โˆ’ 4
( 0 . 5 , 0 . 5 ) 0 . 6 2 4 9 9 0 2 3 4 4 9 ๐ธ โˆ’ 6 0 . 6 2 4 5 1 1 7 2 1 3 4 ๐ธ โˆ’ 4 0 . 6 2 3 0 4 7 4 4 5 5 3 ๐ธ โˆ’ 3
( 0 . 9 , 0 . 5 ) 0 . 6 2 3 4 2 7 8 1 7 3 1 ๐ธ โˆ’ 3 0 . 6 2 2 9 4 9 4 0 2 8 2 ๐ธ โˆ’ 3 0 . 6 2 1 4 8 5 7 0 5 7 3 ๐ธ โˆ’ 3
( 0 . 3 , 0 . 7 ) 0 . 6 2 6 5 5 2 6 5 2 8 1 ๐ธ โˆ’ 3 0 . 6 2 6 0 7 4 1 9 0 6 1 ๐ธ โˆ’ 3 0 . 6 2 4 6 0 9 8 7 9 2 5 3 ๐ธ โˆ’ 4
( 0 . 7 , 0 . 7 ) 0 . 6 2 4 9 9 0 2 3 4 4 9 ๐ธ โˆ’ 6 0 . 6 2 4 5 1 1 7 2 1 3 4 ๐ธ โˆ’ 4 0 . 6 2 3 0 4 7 4 4 5 5 1 ๐ธ โˆ’ 3
( 0 . 1 , 0 . 9 ) 0 . 6 2 8 1 1 4 5 9 1 0 3 ๐ธ โˆ’ 3 0 . 6 2 7 6 3 6 3 2 9 2 2 ๐ธ โˆ’ 3 0 . 6 2 6 1 7 2 2 6 5 0 1 ๐ธ โˆ’ 3
( 0 . 5 , 0 . 9 ) 0 . 6 2 6 5 5 2 6 5 2 8 1 ๐ธ โˆ’ 3 0 . 6 2 6 0 7 4 1 9 0 6 1 ๐ธ โˆ’ 3 0 . 6 2 4 6 0 9 7 9 2 5 3 ๐ธ โˆ’ 3
( 0 . 9 , 0 . 9 ) 0 . 6 2 4 9 9 0 2 3 4 4 9 ๐ธ โˆ’ 6 0 . 6 2 4 5 1 1 7 2 1 3 4 ๐ธ โˆ’ 4 0 . 6 2 3 0 4 7 4 4 5 5 1 ๐ธ โˆ’ 3


( ๐‘ฅ , ๐‘ฆ ) ๐‘ก = 0 . 0 1 ๐‘ก = 0 . 5 ๐‘ก = 2
๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ | ๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ | ๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ |

( 0 . 1 , 0 . 1 ) 0 . 8 7 5 0 0 9 7 6 5 6 9 ๐ธ โˆ’ 6 0 . 8 7 5 4 8 8 2 7 8 7 4 ๐ธ โˆ’ 4 0 . 8 7 6 9 5 2 5 5 4 5 1 ๐ธ โˆ’ 3
( 0 . 5 , 0 . 1 ) 0 . 8 7 6 5 7 2 1 8 2 8 1 ๐ธ โˆ’ 3 0 . 8 7 7 0 5 0 5 9 7 3 2 ๐ธ โˆ’ 3 0 . 8 7 8 5 1 4 2 9 4 4 3 ๐ธ โˆ’ 3
( 0 . 9 , 0 . 1 ) 0 . 8 7 8 1 3 4 1 1 1 3 3 ๐ธ โˆ’ 3 0 . 8 7 8 6 1 2 2 7 7 9 3 ๐ธ โˆ’ 3 0 . 8 8 0 0 7 4 9 4 0 9 5 ๐ธ โˆ’ 3
( 0 . 3 , 0 . 3 ) 0 . 8 7 5 0 0 9 7 6 5 6 9 ๐ธ โˆ’ 6 0 . 8 7 5 4 8 8 2 7 8 7 4 ๐ธ โˆ’ 4 0 . 8 7 6 9 5 2 5 5 4 5 1 ๐ธ โˆ’ 3
( 0 . 7 , 0 . 3 ) 0 . 8 7 6 5 7 2 1 8 2 8 1 ๐ธ โˆ’ 3 0 . 8 7 7 0 5 0 5 9 7 3 2 ๐ธ โˆ’ 3 0 . 8 7 8 5 1 4 2 9 4 4 3 ๐ธ โˆ’ 3
( 0 . 1 , 0 . 5 ) 0 . 8 7 3 4 4 7 3 4 2 9 1 ๐ธ โˆ’ 3 0 . 8 7 3 9 2 5 8 0 5 1 1 ๐ธ โˆ’ 3 0 . 8 7 5 3 9 0 2 0 3 2 3 ๐ธ โˆ’ 4
( 0 . 5 , 0 . 5 ) 0 . 8 7 5 0 0 9 7 6 5 6 9 ๐ธ โˆ’ 6 0 . 8 7 5 4 8 8 2 7 8 7 4 ๐ธ โˆ’ 4 0 . 8 7 6 9 5 2 5 5 4 5 1 ๐ธ โˆ’ 3
( 0 . 9 , 0 . 5 ) 0 . 8 7 6 5 7 2 1 8 2 8 1 ๐ธ โˆ’ 3 0 . 8 7 7 0 5 0 5 9 7 3 2 ๐ธ โˆ’ 3 0 . 8 7 8 5 1 4 2 9 4 4 3 ๐ธ โˆ’ 3
( 0 . 3 , 0 . 7 ) 0 . 8 7 3 4 4 7 3 4 2 9 1 ๐ธ โˆ’ 3 0 . 8 7 3 9 2 5 8 0 5 1 1 ๐ธ โˆ’ 3 0 . 8 7 5 3 9 0 2 0 3 2 3 ๐ธ โˆ’ 4
( 0 . 7 , 0 . 7 ) 0 . 8 7 5 0 0 9 7 6 5 6 9 ๐ธ โˆ’ 6 0 . 8 7 5 4 8 8 2 7 8 7 4 ๐ธ โˆ’ 4 0 . 8 7 6 9 5 2 5 5 4 5 1 ๐ธ โˆ’ 3
( 0 . 1 , 0 . 9 ) 0 . 8 7 5 3 9 0 2 0 3 2 3 ๐ธ โˆ’ 3 0 . 8 7 2 3 6 3 6 7 4 1 2 ๐ธ โˆ’ 3 0 . 8 7 3 8 2 7 7 3 8 3 1 ๐ธ โˆ’ 3
( 0 . 5 , 0 . 9 ) 0 . 8 7 3 4 4 7 3 4 2 9 3 ๐ธ โˆ’ 1 0 0 . 8 7 3 9 2 5 8 0 5 1 1 ๐ธ โˆ’ 3 0 . 8 7 5 3 9 0 2 0 3 2 3 ๐ธ โˆ’ 4
( 0 . 9 , 0 . 9 ) 0 . 8 7 5 0 0 9 7 6 5 6 1 ๐ธ โˆ’ 3 0 . 8 7 5 4 8 8 2 7 8 7 4 ๐ธ โˆ’ 4 0 . 8 7 6 9 5 2 5 5 4 5 1 ๐ธ โˆ’ 3


( ๐‘ฅ , ๐‘ฆ ) ๐‘ก = 0 . 0 1 ๐‘ก = 0 . 5 ๐‘ก = 2
๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข | ๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข | ๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข |

( 0 . 1 , 0 . 1 ) 0 . 6 2 4 9 8 0 4 6 8 8 1 ๐ธ โˆ’ 1 0 0 . 6 2 4 0 2 3 4 5 7 6 2 ๐ธ โˆ’ 1 0 0 . 6 2 1 0 9 9 3 7 4 9 4 ๐ธ โˆ’ 6
( 0 . 5 , 0 . 1 ) 0 . 6 2 1 8 5 6 1 3 4 1 2 ๐ธ โˆ’ 9 0 . 6 2 0 8 9 9 9 1 1 3 2 ๐ธ โˆ’ 9 0 . 6 1 7 9 8 0 5 3 2 4 4 ๐ธ โˆ’ 6
( 0 . 9 , 0 . 1 ) 0 . 6 1 8 7 3 5 7 2 0 7 1 ๐ธ โˆ’ 1 0 0 . 6 1 7 7 8 1 4 7 8 0 2 ๐ธ โˆ’ 1 0 0 . 6 1 4 8 7 0 3 8 3 0 4 ๐ธ โˆ’ 6
( 0 . 3 , 0 . 3 ) 0 . 6 2 4 9 8 0 4 6 8 8 1 ๐ธ โˆ’ 1 0 0 . 6 2 4 0 2 3 4 5 7 6 2 ๐ธ โˆ’ 1 0 0 . 6 2 1 0 9 9 3 7 4 9 4 ๐ธ โˆ’ 6
( 0 . 7 , 0 . 3 ) 0 . 6 2 1 8 5 6 1 3 4 1 2 ๐ธ โˆ’ 9 0 . 6 2 0 8 9 9 9 1 1 3 2 ๐ธ โˆ’ 9 0 . 6 1 7 9 8 0 5 3 2 4 4 ๐ธ โˆ’ 6
( 0 . 1 , 0 . 5 ) 0 . 6 2 8 1 0 4 8 3 1 4 1 ๐ธ โˆ’ 9 0 . 6 2 7 1 4 8 2 2 7 6 1 ๐ธ โˆ’ 9 0 . 6 2 4 2 2 3 0 4 8 8 4 ๐ธ โˆ’ 6
( 0 . 5 , 0 . 5 ) 0 . 6 2 4 9 8 0 4 6 8 8 1 ๐ธ โˆ’ 1 0 0 . 6 2 4 0 2 3 4 5 7 6 2 ๐ธ โˆ’ 1 0 0 . 6 2 1 0 9 9 3 7 4 9 4 ๐ธ โˆ’ 6
( 0 . 9 , 0 . 5 ) 0 . 6 2 1 8 5 6 1 3 4 1 2 ๐ธ โˆ’ 9 0 . 6 2 0 8 9 9 9 1 1 3 2 ๐ธ โˆ’ 9 0 . 6 1 7 9 8 0 5 3 2 4 4 ๐ธ โˆ’ 6
( 0 . 3 , 0 . 7 ) 0 . 6 2 8 1 0 4 8 3 1 4 1 ๐ธ โˆ’ 9 0 . 6 2 7 1 4 8 2 2 7 6 1 ๐ธ โˆ’ 9 0 . 6 2 4 2 2 3 0 4 8 8 4 ๐ธ โˆ’ 6
( 0 . 7 , 0 . 7 ) 0 . 6 2 4 9 8 0 4 6 8 8 1 ๐ธ โˆ’ 1 0 0 . 6 2 4 0 2 3 4 5 7 6 2 ๐ธ โˆ’ 1 0 0 . 6 2 1 0 9 9 3 7 4 9 4 ๐ธ โˆ’ 6
( 0 . 1 , 0 . 9 ) 0 . 6 3 1 2 2 5 3 1 4 2 0 ๐ธ โˆ’ 0 0 . 6 3 0 2 7 0 3 1 1 4 2 ๐ธ โˆ’ 1 0 0 . 6 2 7 3 4 7 6 5 4 2 4 ๐ธ โˆ’ 6
( 0 . 5 , 0 . 9 ) 0 . 6 2 8 1 0 4 8 3 1 4 1 ๐ธ โˆ’ 9 0 . 6 2 7 1 4 8 2 2 7 6 1 ๐ธ โˆ’ 9 0 . 6 2 4 2 2 3 0 4 8 8 4 ๐ธ โˆ’ 6
( 0 . 9 , 0 . 9 ) 0 . 6 2 4 9 8 0 4 6 8 8 1 ๐ธ โˆ’ 1 0 0 . 6 2 4 0 2 3 4 5 7 6 2 ๐ธ โˆ’ 1 0 0 . 6 2 1 0 9 9 3 7 4 9 4 ๐ธ โˆ’ 6


( ๐‘ฅ , ๐‘ฆ ) ๐‘ก = 0 . 0 1 ๐‘ก = 0 . 5 ๐‘ก = 2
๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ | ๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ | ๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ |

( 0 . 1 , 0 . 1 ) 0 . 8 7 5 0 1 9 5 3 1 2 1 ๐ธ โˆ’ 1 0 0 . 8 7 5 9 7 6 5 4 2 4 2 ๐ธ โˆ’ 1 0 0 . 8 7 8 9 0 0 6 2 5 1 4 ๐ธ โˆ’ 6
( 0 . 5 , 0 . 1 ) 0 . 8 7 8 1 4 3 8 7 1 3 3 ๐ธ โˆ’ 9 0 . 8 7 9 1 0 0 0 9 4 1 3 ๐ธ โˆ’ 9 0 . 8 8 2 0 1 9 4 7 3 0 4 ๐ธ โˆ’ 6
( 0 . 9 , 0 . 1 ) 0 . 8 8 1 2 6 4 2 7 9 3 1 ๐ธ โˆ’ 1 0 0 . 8 8 2 2 1 8 5 2 2 0 2 ๐ธ โˆ’ 1 0 0 . 8 8 5 1 2 9 6 1 7 0 4 ๐ธ โˆ’ 6
( 0 . 3 , 0 . 3 ) 0 . 8 7 5 0 1 9 5 3 1 2 1 ๐ธ โˆ’ 1 0 0 . 8 7 5 9 7 6 5 4 2 4 2 ๐ธ โˆ’ 1 0 0 . 8 7 8 9 0 0 6 2 5 1 4 ๐ธ โˆ’ 6
( 0 . 7 , 0 . 3 ) 0 . 8 7 8 1 4 3 8 7 1 3 3 ๐ธ โˆ’ 9 0 . 8 7 9 1 0 0 0 9 4 1 3 ๐ธ โˆ’ 9 0 . 8 8 2 0 1 9 4 7 3 0 4 ๐ธ โˆ’ 6
( 0 . 1 , 0 . 5 ) 0 . 8 7 1 8 9 5 1 7 2 2 2 ๐ธ โˆ’ 9 0 . 8 7 2 8 5 1 7 7 6 0 2 ๐ธ โˆ’ 9 0 . 8 7 5 7 7 6 9 5 4 8 4 ๐ธ โˆ’ 6
( 0 . 5 , 0 . 5 ) 0 . 8 7 5 0 1 9 5 3 1 2 1 ๐ธ โˆ’ 1 0 0 . 8 7 5 9 7 6 5 4 2 4 2 ๐ธ โˆ’ 1 0 0 . 8 7 8 9 0 0 6 2 5 1 4 ๐ธ โˆ’ 6
( 0 . 9 , 0 . 5 ) 0 . 8 7 8 1 4 3 8 7 1 3 3 ๐ธ โˆ’ 9 0 . 8 7 9 1 0 0 0 9 4 1 3 ๐ธ โˆ’ 9 0 . 8 8 2 0 1 9 4 7 3 0 4 ๐ธ โˆ’ 6
( 0 . 3 , 0 . 7 ) 0 . 8 7 1 8 9 5 1 7 2 2 2 ๐ธ โˆ’ 1 0 0 . 8 7 2 8 5 1 7 7 6 0 2 ๐ธ โˆ’ 9 0 . 8 7 5 7 7 6 9 5 4 8 4 ๐ธ โˆ’ 6
( 0 . 7 , 0 . 7 ) 0 . 8 7 5 0 1 9 5 3 1 2 1 ๐ธ โˆ’ 1 0 0 . 8 7 5 9 7 6 5 4 2 4 2 ๐ธ โˆ’ 1 0 0 . 8 7 8 9 0 0 6 2 5 1 4 ๐ธ โˆ’ 6
( 0 . 1 , 0 . 9 ) 0 . 8 6 8 7 7 4 6 8 6 4 6 ๐ธ โˆ’ 1 0 0 . 8 6 9 7 2 9 6 8 9 2 4 ๐ธ โˆ’ 1 0 0 . 8 7 2 6 5 2 3 4 6 4 4 ๐ธ โˆ’ 6
( 0 . 5 , 0 . 9 ) 0 . 8 7 1 8 9 5 1 7 2 2 1 ๐ธ โˆ’ 9 0 . 8 7 2 8 5 1 7 7 6 0 2 ๐ธ โˆ’ 9 0 . 8 7 5 7 7 6 9 5 4 8 4 ๐ธ โˆ’ 6
( 0 . 9 , 0 . 9 ) 0 . 8 7 5 0 1 9 5 3 1 2 1 ๐ธ โˆ’ 1 0 0 . 8 7 5 9 7 6 5 4 2 4 2 ๐ธ โˆ’ 1 0 0 . 8 7 8 9 0 0 6 2 5 1 4 ๐ธ โˆ’ 6

Example 3. Let us consider system of Burgersโ€™ equations (8), with the following initial conditions [14]: ๐‘ข(๐‘ฅ,๐‘ฆ,0)=โˆ’4๐œ‹cos(2๐œ‹๐‘ฅ)sin(๐œ‹๐‘ฆ),๐‘…(2+sin(2๐œ‹๐‘ฅ)sin(๐œ‹๐‘ฆ))๐‘ฃ(๐‘ฅ,๐‘ฆ,0)=โˆ’2๐œ‹sin(2๐œ‹๐‘ฅ)cos(๐œ‹๐‘ฆ),๐‘…(2+sin(2๐œ‹๐‘ฅ)sin(๐œ‹๐‘ฆ))(17) for which exact solutions are ๐‘ข๎€ท(๐‘ฅ,๐‘ฆ,๐‘ก)=โˆ’4๐œ‹expโˆ’5๐œ‹2๎€ธ๐‘ก/๐‘…cos(2๐œ‹๐‘ฅ)sin(๐œ‹๐‘ฆ)๐‘…๎€ท๎€ท2+expโˆ’5๐œ‹2๎€ธ๎€ธ,๎€ท๐‘ก/๐‘…sin(2๐œ‹๐‘ฅ)sin(๐œ‹๐‘ฆ)๐‘ฃ(๐‘ฅ,๐‘ฆ,๐‘ก)=โˆ’2๐œ‹expโˆ’5๐œ‹2๎€ธ๐‘ก/๐‘…sin(2๐œ‹๐‘ฅ)cos(๐œ‹๐‘ฆ)๐‘…๎€ท๎€ท2+expโˆ’5๐œ‹2๎€ธ๎€ธ.๐‘ก/๐‘…sin(2๐œ‹๐‘ฅ)sin(๐œ‹๐‘ฆ)(18) To solve system (1) by LTNHPM, following the same procedure discussed in Section 2 and Example 1, we obtain the iterative relations (9); in this example we take initial approximations (17). The accuracy of LTNHPM for the system of two-dimensional Burgersโ€™ equation agrees good with the exact solution, and absolute errors are very small for the present choice of ๐‘ฅ,๐‘ฆ, and ๐‘ก. These results are listed in Tables 5, 6, 7, and 8 for ๐‘…=100 and ๐‘…=500.


( ๐‘ฅ , ๐‘ฆ ) ๐‘ก = 0 . 0 1 ๐‘ก = 0 . 5 ๐‘ก = 2
๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข | ๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข | ๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข |

( 0 . 1 , 0 . 1 ) โˆ’ 0 . 0 1 4 3 3 5 1 5 9 8 0 4 ๐ธ โˆ’ 1 0 โˆ’ 0 . 0 1 1 4 6 3 9 9 7 2 2 3 ๐ธ โˆ’ 6 โˆ’ 0 . 0 4 9 8 5 4 1 4 0 8 1 4 ๐ธ โˆ’ 2
( 0 . 5 , 0 . 1 ) 0 . 0 1 9 3 2 0 5 3 1 7 5 0 ๐ธ โˆ’ 0 0 . 0 1 5 1 7 4 1 7 0 8 3 3 ๐ธ โˆ’ 6 โˆ’ 0 . 0 4 6 2 6 7 4 1 8 5 9 5 ๐ธ โˆ’ 2
( 0 . 9 , 0 . 1 ) โˆ’ 0 . 0 1 7 1 8 3 5 2 2 6 3 4 ๐ธ โˆ’ 1 9 โˆ’ 0 . 0 1 3 1 9 8 4 1 6 7 3 1 ๐ธ โˆ’ 5 0 . 0 6 1 2 5 2 3 1 3 9 9 6 ๐ธ โˆ’ 2
( 0 . 3 , 0 . 3 ) 0 . 0 1 1 3 0 3 4 7 8 6 9 1 ๐ธ โˆ’ 1 1 0 . 0 0 9 4 3 5 0 8 5 6 7 1 1 ๐ธ โˆ’ 6 โˆ’ 0 . 0 1 3 8 9 0 9 5 4 2 2 1 ๐ธ โˆ’ 2
( 0 . 7 , 0 . 3 ) 0 . 0 2 5 3 2 5 7 7 1 5 8 9 ๐ธ โˆ’ 9 0 . 0 1 7 7 7 1 0 9 5 7 2 2 ๐ธ โˆ’ 4 0 . 1 2 1 0 5 6 8 0 9 3 1 ๐ธ โˆ’ 1
( 0 . 1 , 0 . 5 ) โˆ’ 0 . 0 3 9 1 3 6 5 0 0 0 4 8 ๐ธ โˆ’ 1 1 โˆ’ 0 . 0 3 2 2 7 9 5 2 0 6 8 2 ๐ธ โˆ’ 5 0 . 2 7 5 1 6 2 4 4 2 6 2 ๐ธ โˆ’ 1
( 0 . 5 , 0 . 5 ) 0 . 0 6 2 5 2 2 5 5 4 1 1 1 ๐ธ โˆ’ 1 1 0 . 0 4 9 0 1 3 8 7 3 8 8 5 ๐ธ โˆ’ 5 โˆ’ 0 . 8 0 9 2 2 2 0 3 1 3 1 ๐ธ โˆ’ 1
( 0 . 3 , 0 . 7 ) 0 . 0 1 1 3 0 3 4 7 8 7 1 3 ๐ธ โˆ’ 1 1 0 . 0 0 9 4 3 5 0 8 5 6 7 5 1 ๐ธ โˆ’ 6 โˆ’ 0 . 0 1 3 8 9 0 9 5 4 1 7 1 ๐ธ โˆ’ 2
( 0 . 7 , 0 . 7 ) 0 . 0 2 5 3 2 5 8 6 1 1 1 9 ๐ธ โˆ’ 8 0 . 0 1 7 7 7 1 3 8 1 6 1 2 ๐ธ โˆ’ 4 0 . 1 2 1 0 5 6 7 3 6 9 1 ๐ธ โˆ’ 1
( 0 . 1 , 0 . 9 ) โˆ’ 0 . 0 1 4 3 3 5 1 5 8 5 1 8 ๐ธ โˆ’ 1 0 โˆ’ 0 . 0 1 1 4 6 3 9 9 7 0 7 3 ๐ธ โˆ’ 6 โˆ’ 0 . 0 4 9 8 5 4 1 3 7 2 8 4 ๐ธ โˆ’ 2
( 0 . 5 , 0 . 9 ) 0 . 0 1 9 3 2 0 5 3 1 7 0 0 ๐ธ โˆ’ 0 0 . 0 1 5 1 7 4 1 7 0 7 9 3 ๐ธ โˆ’ 6 โˆ’ 0 . 0 4 6 2 6 7 4 1 8 4 9 5 ๐ธ โˆ’ 2
( 0 . 9 , 0 . 9 ) โˆ’ 0 . 0 1 7 1 8 3 5 2 6 9 4 2 ๐ธ โˆ’ 1 0 โˆ’ 0 . 0 1 3 1 9 8 4 2 0 6 3 1 ๐ธ โˆ’ 5 0 . 0 6 1 2 5 2 2 8 2 6 7 6 ๐ธ โˆ’ 2


( ๐‘ฅ , ๐‘ฆ ) ๐‘ก = 0 . 0 1 ๐‘ก = 0 . 5 ๐‘ก = 2
๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ | ๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ | ๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ |

( 0 . 1 , 0 . 1 ) โˆ’ 0 . 0 1 6 0 2 7 1 9 6 8 2 1 ๐ธ โˆ’ 9 โˆ’ 0 . 0 1 2 8 1 3 3 8 0 7 6 5 ๐ธ โˆ’ 7 โˆ’ 0 . 0 1 1 5 8 5 2 8 4 3 0 5 ๐ธ โˆ’ 3
( 0 . 5 , 0 . 1 ) โˆ’ 5 . 5 2 1 2 8 1 3 6 2 ๐ธ โˆ’ 1 6 5 ๐ธ โˆ’ 1 6 โˆ’ 7 . 1 5 9 3 0 9 4 3 1 ๐ธ โˆ’ 6 7 ๐ธ โˆ’ 6 โˆ’ 0 . 0 1 8 1 7 1 0 3 9 3 1 1 ๐ธ โˆ’ 2
( 0 . 9 , 0 . 1 ) 0 . 0 1 9 2 1 1 7 6 6 2 1 7 ๐ธ โˆ’ 1 0 0 . 0 1 4 7 6 9 7 8 8 1 1 2 ๐ธ โˆ’ 7 โˆ’ 0 . 0 6 1 8 5 0 1 5 1 6 6 6 ๐ธ โˆ’ 2
( 0 . 3 , 0 . 3 ) โˆ’ 0 . 0 1 2 6 3 7 6 7 3 3 8 1 ๐ธ โˆ’ 1 1 โˆ’ 0 . 0 1 0 5 4 9 8 0 4 6 1 7 ๐ธ โˆ’ 7 2 . 6 1 5 0 6 6 3 8 ๐ธ โˆ’ 3 8 ๐ธ โˆ’ 3
( 0 . 7 , 0 . 3 ) 0 . 2 8 3 1 5 0 6 7 6 8 5 ๐ธ โˆ’ 9 0 . 0 1 9 6 9 1 6 9 1 9 3 7 ๐ธ โˆ’ 5 0 . 6 1 2 9 3 9 9 5 6 6 6 ๐ธ โˆ’ 1
( 0 . 1 , 0 . 5 ) โˆ’ 0 . 0 0 0 0 0 0 0 0 0 0 0 ๐ธ โˆ’ 0 โˆ’ 0 . 0 0 0 0 0 0 0 0 0 0 0 ๐ธ โˆ’ 0 0 . 0 0 0 0 0 0 0 0 0 0 0 ๐ธ โˆ’ 0
( 0 . 5 , 0 . 5 ) 0 . 0 0 0 0 0 0 0 0 0 0 0 ๐ธ โˆ’ 0 0 . 0 0 0 0 0 0 0 0 0 0 0 ๐ธ โˆ’ 0 0 . 0 0 0 0 0 0 0 0 0 0 0 ๐ธ โˆ’ 0
( 0 . 9 , 0 . 5 ) โˆ’ 0 . 0 0 0 0 0 0 0 0 0 0 0 ๐ธ โˆ’ 0 โˆ’ 0 . 0 0 0 0 0 0 0 0 0 0 0 ๐ธ โˆ’ 0 0 . 0 0 0 0 0 0 0 0 0 0 0 ๐ธ โˆ’ 0
( 0 . 3 , 0 . 7 ) 0 . 0 1 2 6 3 7 6 7 3 3 8 0 ๐ธ โˆ’ 0 0 . 0 1 0 5 4 9 8 0 4 6 1 7 ๐ธ โˆ’ 7 โˆ’ 2 . 6 1 5 0 6 6 3 7 ๐ธ โˆ’ 3 8 ๐ธ โˆ’ 3
( 0 . 7 , 0 . 7 ) โˆ’ 0 . 0 2 8 3 1 5 1 3 9 4 6 7 ๐ธ โˆ’ 8 โˆ’ 0 . 0 1 9 9 6 9 2 0 5 6 7 9 7 ๐ธ โˆ’ 5 โˆ’ 0 . 6 1 2 9 3 7 1 9 7 3 6 ๐ธ โˆ’ 1
( 0 . 1 , 0 . 9 ) 0 . 0 1 6 0 2 7 1 9 5 5 6 1 ๐ธ โˆ’ 1 0 0 . 0 1 2 8 1 3 3 8 0 2 4 5 ๐ธ โˆ’ 7 0 . 0 1 1 5 8 5 2 8 4 7 4 5 ๐ธ โˆ’ 3
( 0 . 5 , 0 . 9 ) โˆ’ 5 . 5 2 1 2 8 1 3 5 1 ๐ธ โˆ’ 1 6 5 ๐ธ โˆ’ 1 6 7 . 1 5 9 3 0 9 4 1 2 ๐ธ โˆ’ 6 7 ๐ธ โˆ’ 6 0 . 0 1 8 1 7 1 0 3 9 2 0 1 ๐ธ โˆ’ 2
( 0 . 9 , 0 . 9 ) โˆ’ 0 . 0 1 1 4 6 3 9 9 7 2 2 3 ๐ธ โˆ’ 6 โˆ’ 0 . 0 1 4 7 6 9 7 8 7 5 5 2 ๐ธ โˆ’ 7 0 . 0 6 1 8 5 0 1 4 6 9 0 6 ๐ธ โˆ’ 2


( ๐‘ฅ , ๐‘ฆ ) ๐‘ก = 0 . 0 1 ๐‘ก = 0 . 5 ๐‘ก = 2
๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข | ๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข | ๐‘ข โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ข โˆ— โˆ’ ๐‘ข |

( 0 . 1 , 0 . 1 ) โˆ’ 0 . 0 0 2 8 7 7 4 2 9 8 2 2 3 ๐ธ โˆ’ 1 0 โˆ’ 0 . 0 0 2 7 5 2 3 9 6 4 6 6 2 ๐ธ โˆ’ 9 โˆ’ 0 . 0 0 2 4 1 2 3 7 3 4 1 4 1 ๐ธ โˆ’ 5
( 0 . 5 , 0 . 1 ) 0 . 0 0 3 8 7 9 3 9 1 3 8 2 0 ๐ธ โˆ’ 0 0 . 0 0 3 6 9 6 2 4 0 1 9 9 3 ๐ธ โˆ’ 9 0 . 0 0 3 1 5 8 2 9 0 0 1 1 2 ๐ธ โˆ’ 5
( 0 . 9 , 0 . 1 ) โˆ’ 0 . 0 0 3 4 5 1 6 5 5 8 1 6 3 ๐ธ โˆ’ 1 0 โˆ’ 0 . 0 0 3 2 7 3 2 7 9 7 7 1 3 ๐ธ โˆ’ 9 โˆ’ 0 . 0 0 2 7 7 4 7 0 8 7 4 9 1 ๐ธ โˆ’ 5
( 0 . 3 , 0 . 3 ) 0 . 0 0 2 2 6 7 1 5 5 5 4 5 2 ๐ธ โˆ’ 1 2 0 . 0 0 2 1 8 8 8 0 9 3 2 0 1 ๐ธ โˆ’ 9 0 . 0 0 1 9 4 7 4 4 8 6 0 3 1 ๐ธ โˆ’ 5
( 0 . 7 , 0 . 3 ) 0 . 0 0 5 0 9 7 6 9 0 5 5 2 5 ๐ธ โˆ’ 9 0 . 0 0 4 7 1 7 9 8 6 7 8 4 1 ๐ธ โˆ’ 9 0 . 0 0 3 6 2 4 1 6 0 3 5 2 1 ๐ธ โˆ’ 4
( 0 . 1 , 0 . 5 ) โˆ’ 0 . 0 0 7 8 5 1 2 3 4 7 1 0 2 ๐ธ โˆ’ 1 2 โˆ’ 0 . 0 0 7 5 6 1 5 8 2 2 3 8 1 ๐ธ โˆ’ 8 โˆ’ 0 . 0 0 6 5 9 9 0 3 8 0 3 1 1 ๐ธ โˆ’ 4
( 0 . 5 , 0 . 5 ) 0 . 0 1 2 5 5 3 9 7 4 2 3 1 ๐ธ โˆ’ 1 1 0 . 0 1 1 9 6 1 2 6 8 5 6 2 ๐ธ โˆ’ 8 0 . 0 1 0 2 0 3 4 5 9 1 4 1 ๐ธ โˆ’ 4
( 0 . 9 , 0 . 5 ) โˆ’ 0 . 0 1 4 3 7 7 7 2 6 1 3 1 ๐ธ โˆ’ 8 โˆ’ 0 . 0 1 3 4 3 5 4 5 4 6 7 1 ๐ธ โˆ’ 7 โˆ’ 0 . 0 1 2 0 1 4 9 3 9 2 4 1 ๐ธ โˆ’ 3
( 0 . 3 , 0 . 7 ) 0 . 0 0 2 2 6 7 1 5 5 5 4 5 3 ๐ธ โˆ’ 1 2 0 . 0 0 2 1 8 8 8 0 9 3 2 4 1 ๐ธ โˆ’ 9 0 . 0 0 1 9 4 7 4 4 8 6 0 6 1 ๐ธ โˆ’ 5
( 0 . 7 , 0 . 7 ) 0 . 0 0 5 0 9 7 7 4 5 6 4 4 4 ๐ธ โˆ’ 8 0 . 0 0 4 7 1 8 0 7 4 7 1 0 8 ๐ธ โˆ’ 8 0 . 0 0 3 6 2 4 1 6 1 9 5 9 1 ๐ธ โˆ’ 4
( 0 . 1 , 0 . 9 ) โˆ’ 0 . 0 0 2 8 7 7 4 2 9 5 7 1 9 ๐ธ โˆ’ 1 1 โˆ’ 0 . 0 0 2 7 5 2 3 9 6 3 7 5 2 ๐ธ โˆ’ 9 โˆ’ 0 . 0 0 2 4 1 2 3 7 3 2 2 6 1 ๐ธ โˆ’ 5
( 0 . 5 , 0 . 9 ) 0 . 0 0 3 8 7 9 3 9 1 3 7 2 0 ๐ธ โˆ’ 0 0 . 0 0 3 6 9 6 2 4 4 0 1 8 9 4 ๐ธ โˆ’ 9 0 . 0 0 3 1 5 8 2 9 0 0 0 3 2 ๐ธ โˆ’ 5
( 0 . 9 , 0 . 9 ) โˆ’ 0 . 0 0 3 4 5 1 6 5 5 8 4 4 3 ๐ธ โˆ’ 9 โˆ’ 0 . 0 0 3 2 7 3 2 7 9 8 0 4 3 ๐ธ โˆ’ 9 โˆ’ 0 . 0 0 2 7 7 4 7 0 8 6 8 5 1 ๐ธ โˆ’ 5


( ๐‘ฅ , ๐‘ฆ ) ๐‘ก = 0 . 0 1 ๐‘ก = 0 . 5 ๐‘ก = 2
๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ | ๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ | ๐‘ฃ โˆ— ( ๐‘ฅ , ๐‘ฆ , ๐‘ก ) | ๐‘ฃ โˆ— โˆ’ ๐‘ฃ |

( 0 . 1 , 0 . 1 ) โˆ’ 0 . 0 0 3 2 1 7 0 6 3 9 9 1 2 ๐ธ โˆ’ 1 1