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ISRN Computational Mathematics
Volume 2012 (2012), Article ID 603280, 8 pages
http://dx.doi.org/10.5402/2012/603280
Research Article

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

Department of Applied Mathematics, School of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht 41938, Iran

Received 14 May 2012; Accepted 18 June 2012

Academic Editors: Y. Peng and V. Rai

Copyright © 2012 Hossein Aminikhah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a new hybrid of the Laplace transform method and new homotopy perturbation method (LTNHPM) that efficiently solves nonlinear two-dimensional Burgers’ 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 𝜕𝑢𝜕𝑡+𝑢𝜕𝑣𝜕𝑥+𝑣𝜕𝑢=1𝜕𝑦𝑅𝜕2𝑢𝜕𝑥2+𝜕2𝑢𝜕𝑦2,𝜕𝑢𝜕𝑡+𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑣=1𝜕𝑦𝑅𝜕2𝑢𝜕𝑥2+𝜕2𝑢𝜕𝑦2,(𝑥,𝑦)Ω,𝑡>0,(1) subject to the initial conditions: 𝑣𝑢(𝑥,𝑦,0)=𝑓(𝑥,𝑦),(𝑥,𝑦)Ω,(𝑥,𝑦,0)=𝑔(𝑥,𝑦),(𝑥,𝑦)Ω,(2) is called the system of two-dimensional Burgers’ equation, where Ω={(𝑥,𝑦)|𝑎𝑥𝑏,𝑎𝑦𝑏}, 𝑢(𝑥,𝑦,𝑡) and 𝑣(𝑥,𝑦,𝑡) are the velocity components to be determined, 𝑓 and 𝑔 are known functions, and 𝑅 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 Burgers’ 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 [25]. 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 Burgers’ equation. Finally, three examples are given to illustrate the proposed approach.

2. Analysis of the Method

For solving system of two-dimensional Burgers’ equation by LTNHPM, we construct the following homotopy: 𝑈𝑡𝑢0𝑢(𝑥,𝑦,𝑡)+𝑝0(𝑥,𝑦,𝑡)+𝑈𝑈𝑥+𝑉𝑈𝑦1𝑅𝑈𝑥𝑥+𝑈𝑦𝑦𝑉=0,𝑡𝑣0𝑣(𝑥,𝑦,𝑡)+𝑝0(𝑥,𝑦,𝑡)+𝑈𝑉𝑥+𝑉𝑉𝑦1𝑅𝑉𝑥𝑥+𝑉𝑦𝑦=0,(3) where 𝑝[0,1] is an embedding parameter and 𝑢0(𝑥,𝑦,𝑡) and 𝑣0(𝑥,𝑦,𝑡) are initial approximation of solution of (1).

By applying Laplace transform on both sides of (3), we have L𝑈𝑡𝑢0𝑢(𝑥,𝑦,𝑡)+𝑝0(𝑥,𝑦,𝑡)+𝑈𝑈𝑥+𝑉𝑈𝑦1𝑅𝑈𝑥𝑥+𝑈𝑦𝑦L𝑉=0,𝑡𝑣0𝑣(𝑥,𝑦,𝑡)+𝑝0(𝑥,𝑦,𝑡)+𝑈𝑉𝑥+𝑉𝑉𝑦1𝑅𝑉𝑥𝑥+𝑉𝑦𝑦=0.(4)

Using the differential property of Laplace transform we have𝑢𝑠L{𝑈}𝑈(𝑥,𝑦,0)=L0𝑢(𝑥,𝑦,𝑡)𝑝0(𝑥,𝑦,𝑡)+𝑈𝑈𝑥+𝑉𝑈𝑦1𝑅𝑈𝑥𝑥+𝑈𝑦𝑦,𝑣𝑠L{𝑉}𝑉(𝑥,𝑦,0)=L0𝑣(𝑥,𝑦,𝑡)𝑝0(𝑥,𝑦,𝑡)+𝑈𝑉𝑥+𝑉𝑉𝑦1𝑅𝑉𝑥𝑥+𝑉𝑦𝑦,(5)

or1L{𝑈}=𝑠𝑢𝑈(𝑥,𝑦,0)+L0𝑢(𝑥,𝑦,𝑡)𝑝0(𝑥,𝑦,𝑡)+𝑈𝑈𝑥+𝑉𝑈𝑦1𝑅𝑈𝑥𝑥+𝑈𝑦𝑦,1L{𝑉}=𝑠𝑣𝑉(𝑥,𝑦,0)+L0𝑣(𝑥,𝑦,𝑡)𝑝0(𝑥,𝑦,𝑡)+𝑈𝑉𝑥+𝑉𝑉𝑦1𝑅𝑉𝑥𝑥+𝑉𝑦𝑦.(6)

By applying inverse Laplace transform on both sides of (6), we have𝑈(𝑥,𝑦,𝑡)=L11𝑠𝑢𝑈(𝑥,𝑦,0)+L0𝑢(𝑥,𝑦,𝑡)𝑝0(𝑥,𝑦,𝑡)+𝑈𝑈𝑥+𝑉𝑈𝑦1𝑅𝑈𝑥𝑥+𝑈𝑦𝑦,𝑉(𝑥,𝑦,𝑡)=L11𝑠𝑣𝑉(𝑥,𝑦,0)+L0𝑣(𝑥,𝑦,𝑡)𝑝0(𝑥,𝑦,𝑡)+𝑈𝑉𝑥+𝑉𝑉𝑦1𝑅𝑉𝑥𝑥+𝑉𝑦𝑦.(7)

According to the HPM, we use the embedding parameter 𝑝 as a small parameter and assume that the solutions of (7) can be represented as a power series in 𝑝 as 𝑈=𝑈0+𝑝𝑈1+𝑝2𝑈2+,𝑉=𝑉0+𝑝𝑉1+𝑝2𝑉2+.(8) Substituting (8) into (7) and equating the terms with the identical powers of 𝑝 lead to𝑝0𝑈0=L11𝑠𝑢𝑈(𝑥,𝑦,0)+L0,𝑉(𝑥,𝑦,𝑡)0=L11𝑠𝑣𝑉(𝑥,𝑦,0)+L0,𝑝(𝑥,𝑦,𝑡)1𝑈1=L11𝑠L𝑢0(𝑥,𝑦,𝑡)+𝑈0𝑈0𝑥+𝑉0𝑈0𝑦1𝑅𝑈0𝑥𝑥+𝑈0𝑦𝑦,𝑉1=L11𝑠L𝑣0(𝑥,𝑦,𝑡)+𝑈0𝑉0𝑥+𝑉0𝑉0𝑦1𝑅𝑉0𝑥𝑥+𝑉0𝑦𝑦,𝑝2𝑈2=L11𝑠L𝑈0𝑈1𝑥+𝑈1𝑈0𝑥+𝑉0𝑈1𝑦+𝑉1𝑈0𝑦1𝑅𝑈1𝑥𝑥+𝑈1𝑦𝑦,𝑉2=L11𝑠L𝑈0𝑉1𝑥+𝑈1𝑉0𝑥+𝑉0𝑉1𝑦+𝑉1𝑉0𝑦1𝑅𝑉1𝑥𝑥+𝑉1𝑦𝑦,𝑝𝑗𝑈𝑗=L11𝑠L𝑗1𝑘=0𝑈𝑘𝑈𝑗𝑘1𝑥+𝑉𝑘𝑈𝑗𝑘1𝑦1𝑅𝑈𝑗1𝑥𝑥+𝑈𝑗1𝑦𝑦,𝑉𝑗=L11𝑠L𝑗1𝑘=0𝑈𝑘𝑉𝑗𝑘1𝑥+𝑉𝑘𝑉𝑗𝑘1𝑦1𝑅𝑉𝑗1𝑥𝑥+𝑉𝑗1𝑦𝑦,(9)Suppose that the initial approximation has the form 𝑈(𝑥,𝑦,0)=𝑢0(𝑥,𝑦,𝑡)=𝑓(𝑥,𝑦) and 𝑉(𝑥,𝑦,0)=𝑣0(𝑥,𝑦,𝑡)=𝑔(𝑥,𝑦); therefore the exact solution may be obtained as follows 𝑢(𝑥,𝑦,𝑡)=lim𝑝1𝑈(𝑥,𝑦,𝑡)=𝑈0(𝑥,𝑦,𝑡)+𝑈1(𝑥,𝑦,𝑡)+,𝑣(𝑥,𝑦,𝑡)=lim𝑝1𝑉(𝑥,𝑦,𝑡)=𝑉0(𝑥,𝑦,𝑡)+𝑉1(𝑥,𝑦,𝑡)+.(10)

3. Examples

Example 1. Consider the following homogeneous form of a coupled Burgers equation [13]: 𝑢𝑡+𝑢𝑢𝑥+𝑣𝑢𝑦=1𝑅𝑢𝑥𝑥+𝑢𝑦𝑦,𝑣𝑡+𝑢𝑣𝑥+𝑣𝑣𝑦=1𝑅𝑣𝑥𝑥+𝑣𝑦𝑦,(11) subject to the initial condition 𝑣𝑢(𝑥,𝑦,0)=𝑥+𝑦,(𝑥,𝑦,0)=𝑥𝑦.(12) The exact solution of this equation is 𝑢(𝑥,𝑦,𝑡)=(𝑥+𝑦2𝑥𝑡)/(12𝑡2) and 𝑣(𝑥,𝑦,𝑡)=(𝑥𝑦2𝑦𝑡)/(12𝑡)2.
Starting with 𝑈(𝑥,𝑦,0)=𝑢0=𝑥+𝑦, 𝑉(𝑥,𝑦,0)=𝑣0=𝑥𝑦 and using (9), we obtain

𝑈0=L11𝑠𝑉(𝑥+𝑦+L{𝑥+𝑦})=(𝑥+𝑦)(1+𝑡),0=L11𝑠𝑈(𝑥𝑦+L{𝑥𝑦})=(𝑥𝑦)(1+𝑡),1=L11𝑠L𝑥+𝑦+𝑈0𝑈0𝑥+𝑉0𝑈0𝑦1𝑅𝑈0𝑥𝑥+𝑈0𝑦𝑦=(3𝑥+𝑦)𝑡2𝑥𝑡223𝑥𝑡3,𝑉1=L11𝑠L𝑥𝑦+𝑈0𝑉0𝑥+𝑉0𝑉0𝑦1𝑅(𝑉0)𝑥𝑥+𝑉0𝑦𝑦=(𝑥+𝑦)𝑡2𝑦𝑡223𝑦𝑡3,𝑈2=L11𝑠L𝑈0𝑈1𝑥+𝑈1𝑈0𝑥+𝑉0𝑈1𝑦+𝑉1𝑈0𝑦1𝑅𝑈1𝑥𝑥+𝑈1𝑦𝑦=(4𝑥+2𝑦)𝑡2+84𝑥+3𝑦𝑡3+434𝑥+3𝑦𝑡4+4415𝑥+𝑦𝑡155,𝑉2=L11𝑠L𝑈0𝑉1𝑥+𝑈1𝑉0𝑥+𝑉0𝑉1𝑦+𝑉1𝑉0𝑦1𝑅𝑉1𝑥𝑥+𝑉1𝑦𝑦=2𝑥𝑡2+834𝑥3𝑦𝑡3+434𝑥3𝑦𝑡4+4415𝑥𝑦𝑡155,𝑈3=L11𝑠L𝑈0𝑈2𝑥+𝑈1𝑈1𝑥+𝑈2𝑈0𝑥+𝑉0𝑈2𝑦+𝑉1𝑈1𝑦+𝑉2𝑈0𝑦1𝑅𝑈2𝑥𝑥+𝑈2𝑦𝑦=2238𝑥+3𝑦𝑡32838𝑥+3𝑦𝑡41634𝑥+5𝑦𝑡56845𝑥𝑡668315𝑥𝑡7,𝑉3=L11𝑠L𝑈0𝑉2𝑥+𝑈1𝑉1𝑥+𝑈2𝑉0𝑥+𝑉0𝑉2𝑦+𝑉1𝑉1𝑦+𝑉2𝑉0𝑦1𝑅𝑉2𝑥𝑥+𝑉2𝑦𝑦8=3𝑡𝑥+2𝑦383𝑡𝑥+4𝑦445𝑥+56𝑦𝑡1556845𝑦𝑡668315𝑦𝑡7(13)

Therefore we gain the solution of (11) as 𝑢(𝑥,𝑦,𝑡)=𝑈0(𝑥,𝑦,𝑡)+𝑈1(𝑥,𝑦,𝑡)+𝑈3(𝑥,𝑦,𝑡)+=𝑥+𝑦2𝑥𝑡+2𝑥𝑡2+2𝑦𝑡24𝑥𝑡3+4𝑥𝑡4+4𝑦𝑡48𝑥𝑡5+=𝑥1+2𝑡2+4𝑡4++𝑦1+2𝑡2+4𝑡4+2𝑥𝑡1+2𝑡2+4𝑡4=+𝑥+𝑦2𝑥𝑡12𝑡2,𝑣(𝑥,𝑦,𝑡)=𝑉0(𝑥,𝑦,𝑡)+𝑉1(𝑥,𝑦,𝑡)+𝑉3(𝑥,𝑦,𝑡)+=𝑥𝑦2𝑦𝑡+2𝑥𝑡22𝑦𝑡24𝑦𝑡3+4𝑥𝑡44𝑦𝑡48𝑦𝑡5+=𝑥1+2𝑡2+4𝑡4+𝑦1+2𝑡2+4𝑡4+2𝑦𝑡1+2𝑡2+4𝑡4=+𝑥𝑦2𝑦𝑡12𝑡2(14) which is exact solution.

Example 2. Let us consider system of Burgers’ equations (1), with the following initial conditions [14]: 3𝑢(𝑥,𝑦,0)=414,𝑣31+exp(𝑦𝑥)𝑅/8(𝑥,𝑦,0)=4+14,1+exp(𝑦𝑥)𝑅/8(15) for which exact solutions are 3𝑢(𝑥,𝑦,𝑡)=414,𝑣31+exp(4𝑦4𝑥𝑡)𝑅/32(𝑥,𝑦,𝑡)=4+14.1+exp(4𝑦4𝑥𝑡)𝑅/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 Burgers’ equation agrees good with the exact solution, and absolute errors are very small for the present choice of 𝑥,𝑦, and 𝑡. These results are listed in Tables 1, 2, 3, and 4 for 𝑅=0.5 and 𝑅=1.

tab1
Table 1: The LTNHPM results for 𝑢(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=0.5.
tab2
Table 2: The LTNHPM results for 𝑣(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=0.5.
tab3
Table 3: The LTNHPM results for 𝑢(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=1.
tab4
Table 4: The LTNHPM results for 𝑣(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=1.

Example 3. Let us consider system of Burgers’ equations (8), with the following initial conditions [14]: 𝑢(𝑥,𝑦,0)=4𝜋cos(2𝜋𝑥)sin(𝜋𝑦),𝑅(2+sin(2𝜋𝑥)sin(𝜋𝑦))𝑣(𝑥,𝑦,0)=2𝜋sin(2𝜋𝑥)cos(𝜋𝑦),𝑅(2+sin(2𝜋𝑥)sin(𝜋𝑦))(17) for which exact solutions are 𝑢(𝑥,𝑦,𝑡)=4𝜋exp5𝜋2𝑡/𝑅cos(2𝜋𝑥)sin(𝜋𝑦)𝑅2+exp5𝜋2,𝑡/𝑅sin(2𝜋𝑥)sin(𝜋𝑦)𝑣(𝑥,𝑦,𝑡)=2𝜋exp5𝜋2𝑡/𝑅sin(2𝜋𝑥)cos(𝜋𝑦)𝑅2+exp5𝜋2.𝑡/𝑅sin(2𝜋𝑥)sin(𝜋𝑦)(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 Burgers’ equation agrees good with the exact solution, and absolute errors are very small for the present choice of 𝑥,𝑦, and 𝑡. These results are listed in Tables 5, 6, 7, and 8 for 𝑅=100 and 𝑅=500.

tab5
Table 5: The LTNHPM results for 𝑢(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=100.
tab6
Table 6: The LTNHPM results for 𝑣(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=100.
tab7
Table 7: The LTNHPM results for 𝑢(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=500.
tab8
Table 8: The LTNHPM results for 𝑣(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=500.

4. Conclusions

In this work, we considered a new hybrid of Laplace transform method and homotopy perturbation method (LTNHPM) for solving system of two-dimensional Burgers’ equation. Using this method we obtained new efficient relations to solve these systems. New method is a powerful straightforward method. The LTNHPM is apt to be utilized as an alternative approach to current techniques being employed to a wide variety of mathematical problems.

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