Numerical method for solving time inverse problem of reaction-diffusion equation using radial basis functions

Time inverse problems of reaction-diffusion equation, both linear and nonlinear, are considered in this paper. A meshless method that combines time discretization and Radial Basis Functions (RBFs) approximation is constructed, and a new kind of RBF, Power-Generalized Multiquadric (Power-GMQ), is used in the calculation. The highly ill-posed system is solved by the standard Tikhonov regularization technique, the regularized parameter is chosen by L-curve method. Numerical examples are given to verify the accuracy and effectiveness, and to demonstrate that the new RBF can also be used to solve the inverse problem. Even though the final condition is disturbed by some noise, the method can still recover the initial solution well.