scholarly journals Existence and uniqueness of a weak solution to a non-autonomous time-fractional diffusion equation (of distributed order)

2020 ◽  
Vol 109 ◽  
pp. 106540 ◽  
Author(s):  
Karel Van Bockstal
Keyword(s):  
Weak Solution ◽  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Time Fractional Diffusion Equation ◽  
Distributed Order

Inverse source problem for a distributed-order time fractional diffusion equation

2020 ◽  
Vol 28 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Xiaoliang Cheng ◽  
Lele Yuan ◽  
Kewei Liang
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Series Representation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Time Fractional Diffusion Equation ◽  
Distributed Order ◽  
Regularized Solution

AbstractThis paper studies an inverse source problem for a time fractional diffusion equation with the distributed order Caputo derivative. The space-dependent source term is recovered from a noisy final data. The uniqueness, ill-posedness and a conditional stability for this inverse source problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization method. Further, based on the series representation of the regularized solution, we give convergence rates of the regularized solution under an a-priori and an a-posteriori regularization parameter choice rule. With an adjoint technique for computing the gradient of the regularization functional, the conjugate gradient method is applied to reconstruct the space-dependent source term. Two numerical examples illustrate the effectiveness of the proposed method.

Identification of the diffusion coefficient in a time fractional diffusion equation

2020 ◽  
Vol 28 (2) ◽  
pp. 299-306
Author(s):  
Amir Hossein Salehi Shayegan ◽  
Ali Zakeri ◽  
Soheila Bodaghi ◽  
M. Heshmati
Keyword(s):  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Stability Result ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Cost Functional ◽  
Time Fractional Diffusion Equation ◽  
The Cost ◽  
The Uniqueness Theorem ◽  
Admissible Functions

AbstractIn this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to {{}^{C}D_{t}^{\alpha}u-\nabla\cdot(k(x)\nabla u)=f}, where the function {k=k(x)} is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function k. At the first step of the methodology, we give a stability result corresponding to connectivity of k and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given.

Variational method for a backward problem for a time-fractional diffusion equation

2019 ◽  
Vol 53 (4) ◽  
pp. 1223-1244 ◽  
Author(s):  
Ting Wei ◽  
Jun Xian
Keyword(s):  
Weak Solution ◽  
Variational Method ◽  
Diffusion Equation ◽  
Variational Problem ◽  
Fractional Diffusion ◽  
Adjoint Problem ◽  
Fractional Diffusion Equation ◽  
Tikhonov Regularization Method ◽  
Backward Problem ◽  
Time Fractional Diffusion Equation

This paper is devoted to solve a backward problem for a time-fractional diffusion equation by a variational method. The regularity of a weak solution for the direct problem as well as the existence and uniqueness of a weak solution for the adjoint problem are proved. We formulate the backward problem into a variational problem by using the Tikhonov regularization method, and obtain an approximation to the minimizer of the variational problem by using a conjugate gradient method. Four numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed algorithm.

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