Determination of a distribution in a source term of a time fractional diffusion-wave equation

Abstract

We study the inverse Cauchy problem to a time fractional diffusion-wave equation with distributions in right-hand sides. This problem is to find a generalized solution of direct problem and an unknown time-dependent part of a source from the space of distributions. The unique solvability of the problem is established.

Identifying a fractional order and a space source term in a time-fractional diffusion-wave equation simultaneously

“…From examples 1-3, we can see that the the Morozov discrepancy principle plays an important role, but the role of regularization parameters cannot be omitted. For example, if γ 0 = 0 or µ 0 = 0, the computing process is often broken in examples 1 and 2, because β k is out of the range (1,2) at some steps. In example 3, when γ 0 = 0 , µ 0 10 −10 , the approximate solutions become worse, see table 5.…”

“…In [16], Liu et al considered the reconstruction of the time-dependent source problem by observation at a single point, and used a fixed-point iteration for the numerical reconstruction and proved its convergence. For the time-fractional diffusion-wave equations, Lopushansky et al in [2] studied an inverse source problem on an unbounded domain. And Siskova et al in [26] studied an inverse time-dependent source problem for a semilinear time-fractional wave equation by the Rothe method on a bounded domain.…”

The backward problem for radially symmetric time‐fractional diffusion‐wave equation under Robin boundary condition

“…The direct problems for time-fractional diffusion-wave equation have been investigated extensively. 2,3 And the inverse problems connected with time-fractional diffusion-wave equation have some papers such as inverse source problems, 4,5 inverse initial value problems, 6,7 and inverse coefficient problems. 8 In this paper, we are interested in studying the backward problem of the radially symmetric time-fractional diffusion-wave equation.…”

The quasi-reversibility regularization method for backward problems of the time-fractional diffusion-wave equation

“…Several inverse problems of time-fractional diffusion-wave equation have attracted some authors' interest in recent literature, such as inverse source problems [14][15][16][17] and initial inverse problems [18,19]. Besides, we can refer to the inverse source Cauchy problem proposed by Lopushansky and Lopushanska [20], and the recognition of a time-dependent source by using Rothe method in [21]. In the past years, the inverse problems of fractional diffusion/wave equations are hot topic all the time, several scholars studied the inverse problems by different methods.…”