Identifying the space source term problem for time-space-fractional diffusion equation

Abstract

In this paper, we consider an inverse source problem for the time-space-fractional diffusion equation. Here, in the sense of Hadamard, we prove that the problem is severely ill-posed. By applying the quasi-reversibility regularization method, we propose by this method to solve the problem (1.1). After that, we give an error estimate between the sought solution and regularized solution under a prior parameter choice rule and a posterior parameter choice rule, respectively. Finally, we present a numerical example to find that the proposed method works well.

On the unique solvability of a Cauchy problem with a fractional derivative

“…Finding analytical solutions to equations with fractional derivatives is accompanied by certain diculties, so some researchers deal with the existence and uniqueness of the solution to the problem [2,15,29], while others use numerical methods to nd approximate solutions [20].…”

Local Fractional Aboodh Transform and its Applications to Solve Linear Local Fractional Differential Equations

“…The search for analytical solutions to fractional dierential equations is often dicult, so in many cases researchers focus on studying the existence, uniqueness and properties of solutions [5,12,18,21,27,28], while others tend to employ numerical methods to search for approximate solutions. Transformations dened by integrals play an important role in the resolution of ordinary dierential equations, partial dierential equations and in the resolution of integral dierential equations with integer order or fractional order.…”

Hyers–Ulam stability of impulsive Volterra delay integro-differential equations

“…In addition, some estimations were for the convergence of the Picard iteration sequence provided an estimate for Green's function was related to the problem and employed in the proof of the existence and uniqueness theorem for the solution of the given problem. Karapınar et al in [19] considered an inverse-source-time-space-fraction problem diffusion equation. Actually, in the Hadamard sense, they proved that the problem is very bad posture.…”