Numerical Algorithms For Two Kinds Of Inverse Problems Of Diffusion Equations

This thesis studies numerical algorithms for two kinds of inverse problems of diffusion equations.The first problem is the inverse source problem for the spacefractional diffusion equation;and the second problem is the Cauchy problem for the integer-order diffusion equation.In this thesis,meshless numerical method for solving these two kinds of problems are proposed.The numerical solution is approximated by using the fundamental solution of the space-fractional diffusion equation as a basis function.Fundamental solutions of the integer-order diffusion equation and radial basis functions(RBFs)are used to obtain a numerical solution.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.Finally some typical numerical examples are given to illustrate the efficiency and accuracy of the proposed method.