Localized method of approximate particular solutions with Cole–Hopf transformation for multi-dimensional Burgers equations C.Y. Lin a,b , M.H. Gu c , D.L. Young a,b,n , C.S. Chen d,e a Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan b Hydrotech Research Institute, National Taiwan University, Taipei 10617, Taiwan c National Center for Research on Earthquake Engineering, Taipei 10668, Taiwan d Department of Mathematics, The University of Southern Mississippi, Hattiesburg, MS, USA e Department of Engineering Mechanics, Hohai University, Nanjing, Jiangsu 210098, China article info Article history: Received 23 August 2013 Accepted 28 November 2013 Available online 25 December 2013 Keywords: Burgers equations Cole–Hopf transformation Diffusion equation Meshless methods Irregular domain Localized method of approximate particular solutions abstract The Burgers equations depict propagating wave with quadratic nonlinearity, it can be used to describe nonlinear wave propagation and shock wave, where the nonlinear characteristics cause difficulties for numerical analysis. Although the solution approximation can be executed through iterative methods, direct methods with finite sequence of operation in time can solve the nonlinearity more efficiently. The resolution for nonlinearity of Burgers equations can be resolved by the Cole–Hopf transformation. This article applies the Cole–Hopf transformation to transform the system of Burgers equations into a partial differential equation satisfying the diffusion equation, and uses a combination of finite difference and the localized method of approximate particular solution (FD-LMAPS) for temporal and spatial discretization, respectively. The Burgers equations with behaviors of propagating wave, diffusive N-wave or within multi-dimensional irregular domain have been verified in this paper. Effectiveness of the FD-LMAPS has also been further examined in some experiments, and all the numerical solutions prove that the FD-LMAPS is a promising numerical tool for solving the multi-dimensional Burgers equations. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction The Burgers equations, are quasi-linear partial differential equa- tions (PDEs) consisting an unsteady term, a nonlinear convection term and a diffusion term. The Burgers equations depict propagat- ing wave with quadratic nonlinearity, it can be used to describe nonlinear wave propagation and shock wave [1]. The Burgers equations have been widely applied for mathematic analysis and problems of fluid mechanics. In this paper we approximate the solutions of the Burgers equations by the localized method of approximate particular solutions (LMAPS) [2] which is a novel developed numerical scheme using radial basis functions (RBFs). In order to overcome the nonlinearity of the Burgers equations, the implementation of the LMAPS to solve the resulting diffusion equation of applying Cole–Hopf transformation is considered in this paper. Basic solu- tion approximation for nonlinear equations usually requires itera- tive approaches such as the Newton–Ralphson or Picard method [3,4]. However, the convergence of iterative approaches is not always guaranteed. In contrast to iterative approaches, direct approaches for approximating the solution of the Burgers equa- tions can overcome the nonlinearity through finite time sequence which avoids the issue of convergence. Such direct approaches can be easily found in the literature for solving Burgers equations, for example the well-known Eulerian–Lagrangian method [5], or the direct approach that will be applied in this paper — the Cole–Hopf transformation [6,7]. The Cole–Hopf transformation was introduced by Hopf and Cole [6,7] to deal with the nonlinearity of the viscous Burgers equations. After applying the Cole–Hopf transformation, the sys- tem of Burgers equations can be transformed into a single variable PDE satisfying the diffusion equation. The analytical solution of the diffusion equation can be simply found by solving the Cauchy problem in regular domain. Therefore, it provides a method to obtain analytical solutions for the Burgers equations. Although the Cole–Hopf transformation was originally developed for deriving analytical solutions, in this research the Cole–Hopf transformation is only applied to deal with the nonlinearity of Burgers equations for further numerical processes. For the numerical implementation, it is necessary for the essential conditions, i.e. initial and boundary conditions, to be transformed according to the definition of the Cole-Hopf transfor- mation. This implies both the governing equation and the essential Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements 0955-7997/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enganabound.2013.11.019 n Corresponding author at: Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan. Tel./fax: þ886 2 2362 6114. E-mail address: dlyoung@ntu.edu.tw (D.L. Young). Engineering Analysis with Boundary Elements 40 (2014) 78–92