The Method Of Particular Solutions For Solving A Class Of Reaction-Diffusion Partial Differential Equations

Reaction-diffusion partial differential equations is a kind of important parabolic e-quations, comprised a reaction term and a diffusion term, came from the reaction phe-nomenon widespread in the nature. Reaction-diffusion partial differential equations have a wide range of applications in most fields of mathematical physics, chemistry, biology and so on, such as in space to describe the spread of populations, in chemical physics to describe concentration and temperature distributions. In the latter case, the reaction term describes the heat and material transformations, and the diffusion term describes heat and the rate of material to produce. Nonlinear term often describes the the rules of material behavior.The aim of this artical is to apply the Method of Particular Solutions (MPS), a newly proposed messless method, selecting Multiquadrics (MQ), Inverse Multiquadrics (IMQ) and the Thin Plate Splines (TPS) Radial Basis Functions (RBFs), by using collocation points and then approximating the solutions, for solving a class of reaction-diffusion par-tial differential equations. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.The structure of the article is as follows. In chapter1, we first general compare two kinds of methods for solving the partial differential equations, mesh methods and mesh-less methods. Then we introduce the development history of meshless methods in detail. Following this the background of reaction-diffusion partial differential equations are p- resented. Finally, summarizing the work had done. In chapter2, we provide correlation concepts of radial basis functions, followed by the theory of approximating of radial basis functions. In chapter3, we introduce three kinds of meshless methods, based on radial basis functions, such as Kansa method, the Method of Fundamental Solutions(MFS) and the Method of Particular Solutions(MPS). In chapter4, the basic principle of a class of linear reaction-diffusion partial differential equations using the Method of Particular So-lutions is described in detail, followed by two and three-dimensional numerical examples of a class of linear reaction-diffusion partial differential equations to illustrate the stability and accuracy of the proposed method. In comparison, the testing result is consistent with that of the numerical analysis. In chapter5, we start with the basic principle of a class of linear reaction-diffusion partial differential equations using the Method of Particular Solu-tions. Following this two dimensional numerical examples are presented. Verifies that the simulation result satisfactorily fits with data. In chapter6, the main research conclusions are summarized and the following work is forecasted.