A Spectral Cut‐Off Regularization Method for the Nonlinear Space‐Time Fractional Diffusion Equation With Noisy Data

ABSTRACT

In this paper, we investigate the initial value problem for a nonlinear space‐time‐fractional diffusion equation with a Caputo time‐fractional derivative of order , a fractional Laplacian of order , and a nonlinear source term . Since the problem is ill‐posed, we propose a spectral cut‐off regularization method to solve the problem in a stable way. The regularized problem is proved to be well‐posed. Then, we obtain some error estimates between the exact solution and the regularized solution. We also propose a numerical algorithm to solve the regularization problem numerically. The numerical algorithm is proved to converge to the regularized solution with an error of , where is the time step, is the number of Fourier modes, and . Numerical examples in one and two dimensional spaces confirm our theoretical results of convergent rates, show the robustness of the algorithm under significant noise levels.