A New Efficient Method for Solving Two-Dimensional Burgers' Equation
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.
The system of partial differential equation of the following form subject to the initial conditions: is called the system of two-dimensional Burgers’ equation, where , and are the velocity components to be determined, and are known functions, and 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 . 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 [2–5]. 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 , who found its steady solutions, descriptive of certain viscous flows. It was later proposed by Burgers  as one of a class of equations describing mathematical models of turbulence. In the context of gas dynamics, it was discussed by Hopf  and Cole . They also illustrated independently that the Burgers equation can be solved exactly for an arbitrary initial condition. Benton and Platzman  have surveyed the analytical solutions of the one-dimensional Burgers equation. It can be considered as a simplified form of the Navier-Stokes equation  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  and in the Burgers model of turbulence . It can be solved analytically for arbitrary initial conditions . 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  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: where is an embedding parameter and and are initial approximation of solution of (1).
By applying Laplace transform on both sides of (3), we have
Using the differential property of Laplace transform we have
By applying inverse Laplace transform on both sides of (6), we have
According to the HPM, we use the embedding parameter as a small parameter and assume that the solutions of (7) can be represented as a power series in as Substituting (8) into (7) and equating the terms with the identical powers of lead toSuppose that the initial approximation has the form and ; therefore the exact solution may be obtained as follows
Example 1. Consider the following homogeneous form of a coupled Burgers equation :
subject to the initial condition
The exact solution of this equation is and
Starting with , and using (9), we obtain
Therefore we gain the solution of (11) as which is exact solution.
Example 2. Let us consider system of Burgers’ equations (1), with the following initial conditions : for which exact solutions are 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 , and . These results are listed in Tables 1, 2, 3, and 4 for and .
Example 3. Let us consider system of Burgers’ equations (8), with the following initial conditions : for which exact solutions are 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 , and . These results are listed in Tables 5, 6, 7, and 8 for and .
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.
J. M. Burgers, “Application of a model system to illustrate some points of the statistical theory of free turbulence,” The Royal Netherlands Academy of Arts and Sciences, vol. 43, pp. 2–12, 1940.View at: Google Scholar
E. N. Aksan, “Quadratic B-spline finite element method for numerical solution of the Burgers' equation,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 884–896, 2006.View at: Publisher Site | Google Scholar
S. Kutluay and A. Esen, “A lumped galerkin method for solving the burgers equation,” International Journal of Computer Mathematics, vol. 81, no. 11, pp. 1433–1444, 2004.View at: Publisher Site | Google Scholar
S. Abbasbandy and M. T. Darvishi, “A numerical solution of Burgers' equation by modified Adomian method,” Applied Mathematics and Computation, vol. 163, no. 3, pp. 1265–1272, 2005.View at: Publisher Site | Google Scholar
Sirendaoreji, “Exact solutions of the two-dimensional Burgers equation,” Journal of Physics A, vol. 32, no. 39, pp. 6897–6900, 1999.View at: Publisher Site | Google Scholar
H. Bateman, “Some recent researches on the motion of fluids,” Monthly Weather Review, vol. 43, pp. 163–170, 1915.View at: Google Scholar
E. Hopf, “The partial differential equation ut C uux D uxx,” Communications on Pure and Applied Mathematics, vol. 3, pp. 201–230, 1950.View at: Google Scholar
J. D. Cole, “On a quasilinear parabolic equation occurring in aerodynamics,” Quarterly of Applied Mathematics, vol. 9, pp. 225–236, 1951.View at: Google Scholar
E. R. Benton and G. W. Platzman, “A table of solutions of the one-dimensional Burgers' equation,” Quarterly of Applied Mathematics, vol. 30, pp. 195–212, 1972.View at: Google Scholar
V. I. Karpman, Nonlinear Waves in Dispersive Media, Pergamon Press, Oxford, UK, 1975.
J. Burgers, Advances in Applied Mechanics, Academic Press, New York, NY, USA, 1948.
H. Aminikhah and M. Hemmatnezhad, “An efficient method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 835–839, 2010.View at: Publisher Site | Google Scholar
J. Biazar and H. Ghazvini, “Exact solutions for nonlinear burgers' equation by homotopy perturbation method,” Numerical Methods for Partial Differential Equations, vol. 25, no. 4, pp. 833–842, 2009.View at: Publisher Site | Google Scholar
M. Tamsir and V. K. Srivastava, “A semi-implicit finite-difference approach for two-dimensional coupled Burgers’ equations,” International Journal of Scientific & Engineering Research, vol. 2, pp. 1–6, 2011.View at: Google Scholar