Recovering the historical distribution for nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space

Abstract In this paper, we investigate the problem of recovering the historical distribution for a nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space. This problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a fractional Laplacian of order α ∈ ( 1 ∕ 2 , 1 ] , which is usually used to model the anomalous diffusion. The problem is severely ill-posed. To regularize the problem, we propose a modified version of the Tikhonov regularization method. A stability estimate of Holder type is established. Finally, several numerical examples based on the finite difference approximation and the discrete Fourier transform are presented to illustrate the theoretical results.

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