Meshless Methods For Three Kinds Of Inverse Problems For Diffusion Equations

Numerical methods for three kinds of inverse diffusion equation problems are studied in this thesis,including simultaneous determination of the heat source and initial temperature of inverse heat conduction problem,nonhomogeneous backward heat conduction problem and Cauchy problem for the Poisson equation.In this the-sis,meshless numerical methods for solving three kinds of problems are proposed.As a result of the fundamental solution method and the method of boundary collo-cation are used to solve the homogeneous equations,and the above three problems are homogeneous,therefore,for the first kind of problem,we used the fundamental solution-radial basis function method to simultaneously determinating a heat source and initial temperature.We used the fundamental solution-radial basis function method to solve initial temperature of second problem.The boundary collocation ra-dial basis functions method is used to obtain a numerical solution for third problem.Since the resulting matrix equations are extremely ill-conditioned,the regularized solutions are obtained by adopting the Tikhonov regularization scheme,in which the choice of the regularization parameters are based on generalized cross-validation criterion and L-curve.Finally some typical one-dimensional and two-dimensional numerical examples are given to illustrate the efficiency and accuracy of the proposed method.