Simultaneous inversion of two initial values for a time‐fractional diffusion‐wave equation

Abstract

This study is devoted to recovering two initial values for a time-fractional diffusion-wave equation from boundary Cauchy data. We provide the uniqueness result for recovering two initial values simultaneously by the method of Laplace transformation and analytic continuation. And then we use a nonstationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm to find good approximations to the initial values. Numerical examples in oneand two-dimensional cases are provided to show the effectiveness of the proposed method.

Well-posedness of the stochastic time-fractional diffusion and wave equations and inverse random source problems

“…For example, Zhang and Xu [64] and Li et al [30] discussed the uniqueness of the inverse source problem for the 1D time-fractional diffusion equation; Jiang et al [19] established the results parallel to [64] for the multi-dimensional case by constructing a weak type unique continuation, and proposed the iteration algorithm for the numerical treatment. For some other works about the fractional inverse problems, we refer to [6,24,31,32,38,53,55,57,60,61,63,65] and the references therein.…”

Simultaneous Inversion of the Source Term and Initial Value of the Time Fractional Diffusion Equation

“…In [4], Li et al used a modified optimal perturbation algorithm to deal with simultaneously recovering for the fractional order and diffusion coefficient for the time-fractional diffusion equation. Zhang [27] et al studied the simultaneous identification of two initial values of the fractional wave equation. The topic of simultaneous identification of source item and initial value has been preliminarily studied.…”

Some closed form series solutions for the time-fractional diffusion-wave equation in polar coordinates with a generalized Caputo fractional derivative

“…Fractional differential equations (FDEs) have been applied to an increasing number of fields such as physics, engineering, and other sciences [1][2][3][4][5][6][7]. The time--fractional diffusion models have been used in various fields like biology, physics, chemistry, and finance.…”

“…The time--fractional diffusion models have been used in various fields like biology, physics, chemistry, and finance. The time fractional diffusion equations preferably possess advantages for describing anomalous diffusion phenomena due to the memory property of fractional order derivatives [2][3][4][5][6][7]. Whereas the TFDWEs can be used to model the propagation of diffused waves in viscoelastic media [8].…”