scholarly journals A problem with an integral boundary condition for a time fractional diffusion equation and an inverse problem

2016 ◽  
pp. 133-145 ◽  
Author(s):  
Halyna Lopushanska
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Integral Boundary Condition ◽  
Integral Boundary ◽  
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.

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