An inverse problem of identifying the control function in two and three-dimensional parabolic equations through the spectral meshless radial point interpolation

This study proposes a kind of spectral meshless radial point interpolation (SMRPI) for solving two and three-dimensional parabolic inverse problems on regular and irregular domains. The SMRPI is developed for identifying the control parameter which satisfies the semilinear time-dependent two and three-dimensional diffusion equation with both integral overspecialization and overspecialization at a point in the spatial domain. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions are used to construct shape functions which act as basis functions in the frame of SMRPI. It is proved that the scheme is stable with respect to the time variable in H1 and convergent by the order of convergence O(δt). The results of numerical experiments are compared to analytical solutions to confirm the accuracy and efficiency of the presented scheme.