Numerical computation for backward time-fractional diffusion equation

Based on kernel-based approximation technique, we devise in this paper an efficient and accurate numerical scheme for solving a backward problem of time-fractional diffusion equation (BTFDE). The kernels used in the approximation are the fundamental solutions of the time-fractional diffusion equation which can be expressed in terms of the M-Wright functions. To stably and accurately solve the resultant highly ill-conditioned system of equations, we successfully combine the standard Tikhonov regularization technique and the L-curve method to obtain an optimal choice of the regularization parameter and the location of source points. Several 1D and 2D numerical examples are constructed to demonstrate the superior accuracy and efficiency of the proposed method for solving both the classical backward heat conduction problem (BHCP) and the BTFDE.