Simultaneous uniqueness for an inverse problem in a time-fractional diffusion equation

2020 ◽  
Vol 109 ◽  
pp. 106558
Author(s):  
Xiaohua Jing ◽  
Jigen Peng
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Time Fractional Diffusion Equation

scholarly journals Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source

2020 ◽  
Vol 28 (2) ◽  
pp. 211-235
Author(s):  
Tran Bao Ngoc ◽  
Nguyen Huy Tuan ◽  
Mokhtar Kirane
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Regularization Method ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Nonlinear Source ◽  
Numerical Example ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e., the solution does not depend continuously on the data. The problem is ill-posed in the sense of Hadamard. Under some weak a priori assumptions on the sought solution, we propose a new regularization method for stabilizing the ill-posed problem. We also provide a numerical example to illustrate our results.

scholarly journals An inverse source problem for a two dimensional time fractional diffusion equation with involution

2021 ◽  
Author(s):  
Batirkhan Turmetov ◽  
B. J. Kadirkulov
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Diffusion Equation ◽  
Fourier Method ◽  
Dirichlet Condition ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Inverse Source Problem ◽  
Two Dimensional ◽  
Time Fractional Diffusion Equation

In this paper, we consider a two-dimensional generalization of the parabolic equation. Using the Fourier method, we study the solvability of the inverse problem with the Dirichlet condition and periodic conditions.

Inverse coefficient problem for the time-fractional diffusion equation

2021 ◽  
Vol 9 (1) ◽  
pp. 44-54
Author(s):  
D. K. Durdiev ◽  
Keyword(s):  
Inverse Problem ◽  
Fundamental Solution ◽  
Diffusion Equation ◽  
Direct Problem ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Unknown Coefficient ◽  
Single Observation ◽  
Uniqueness Results ◽  
Time Fractional Diffusion Equation

We study the inverse problem of determining the time depending reaction diffu- sion coefficient in the Cauchy problem for the time-fractional diffusion equation by a single observation at the point x = 0 of the diffusion process. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation is used and properties of this solution are investigated. The fundamental solution contains the Fox’s H− functions widely used in fractional calculus. In particular, using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient which will be used in study inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven. Also the stability estimate is obtained.

scholarly journals SIMULTANEOUS DETERMINATION OF A SOURCE TERM AND DIFFUSION CONCENTRATION FOR A MULTI-TERM SPACE-TIME FRACTIONAL DIFFUSION EQUATION

2021 ◽  
Vol 26 (3) ◽  
pp. 411-431
Author(s):  
Salman A. Malik ◽  
Asim Ilyas ◽  
Arifa Samreen
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Source Term ◽  
Fractional Diffusion ◽  
Adjoint Problem ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Diffusion Temperature ◽  
Non Local ◽  
Time Fractional Diffusion Equation

An inverse problem of determining a time dependent source term along with diffusion/temperature concentration from a non-local over-specified condition for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation involve Caputo fractional derivative in space and Hilfer fractional derivatives in time of different orders between 0 and 1. Under certain conditions on the given data we proved that the inverse problem is locally well-posed in the sense of Hadamard. Our method of proof based on eigenfunction expansion for which the eigenfunctions (which are Mittag-Leffler functions) of fractional order spectral problem and its adjoint problem are considered. Several properties of multinomial Mittag-Leffler functions are proved.

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