Meshless numerical solutions for Burgers' equations by Multiquadrics method and Cole-Hopf transformation

Abstract

In this paper, the numerical solutions of the Burgers' equations are acquired by the proposed meshless scheme, which is a combination of the Multiquadrics method and the Cole-Hopf transformation. The quasi-linear Burgers' equations are converted to the linear diffusion equation by the Cole-Hopf transformation and then the Multiquadrics method, which is one of the promising meshless methods, is adopted to solve the resultant diffusion equation. Due to the features of the Multiquadrics method, the Robin boundary conditions in the resultant diffusion problem can be handled effectively. Besides, the number of unknowns in the two-dimensional problem can be reduced to one, so the computer memory and computational cost can be lessened to the minimum. One- and two-dimensional Burgers' equations are examined by the proposed meshless method and the numerical results are in good agreement with the analytical solutions. Therefore, the proposed numerical method can be considered as a simple and efficient numerical tool in dealing with the problems of Burgers' equations.