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

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Erdal Karapinar ◽  
Devendra Kumar ◽  
Rathinasamy Sakthivel ◽  
Nguyen Hoang Luc ◽  
N. H. Can
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Choice Rule ◽  
Parameter Choice ◽  
Time Space ◽  
Space Fractional Diffusion Equation ◽  
Ill Posed ◽  
Regularized Solution

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.

scholarly journals Recovering the space source term for the fractional-diffusion equation with Caputo–Fabrizio derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Le Nhat Huynh ◽  
Nguyen Hoang Luc ◽  
Dumitru Baleanu ◽  
Le Dinh Long
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Regularization Parameter ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Choice Rule ◽  
Parameter Choice ◽  
Landweber Method ◽  
Regularized Solution

AbstractThis article is devoted to the study of the source function for the Caputo–Fabrizio time fractional diffusion equation. This new definition of the fractional derivative has no singularity. In other words, the new derivative has a smooth kernel. Here, we investigate the existence of the source term. Through an example, we show that this problem is ill-posed (in the sense of Hadamard), and the fractional Landweber method and the modified quasi-boundary value method are used to deal with this inverse problem and the regularized solution is also obtained. The convergence estimates are addressed for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. In addition, we give a numerical example to illustrate the proposed method.

Quasi solution of a backward space fractional diffusion equation

2019 ◽  
Vol 27 (6) ◽  
pp. 795-814 ◽  
Author(s):  
Amir Hossein Salehi Shayegan ◽  
Ali Zakeri
Keyword(s):  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Linear Equations ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
The Finite Element Method ◽  
System Of Linear Equations ◽  
Solution Approach ◽  
Space Fractional Diffusion Equation ◽  
Ill Posed

Abstract In this paper, based on a quasi solution approach, i.e., a methodology involving minimization of a least squares cost functional, we study a backward space fractional diffusion equation. To this end, we give existence and uniqueness theorems of a quasi solution in an appropriate class of admissible initial data. In addition, in order to approximate the quasi solution, the finite element method is used. Since the obtained system of linear equations is ill-posed, we apply TSVD regularization. Finally, three numerical examples are given. Numerical results reveal the efficiency and applicability of the proposed method.

scholarly journals Finite Element Solutions for the Space Fractional Diffusion Equation with a Nonlinear Source Term

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Y. J. Choi ◽  
S. K. Chung
Keyword(s):  
Finite Element ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Source Term ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Nonlinear Source ◽  
Backward Euler ◽  
Space Fractional Diffusion Equation

We consider finite element Galerkin solutions for the space fractional diffusion equation with a nonlinear source term. Existence, stability, and order of convergence of approximate solutions for the backward Euler fully discrete scheme have been discussed as well as for the semidiscrete scheme. The analytical convergent orders are obtained asO(k+hγ˜), whereγ˜is a constant depending on the order of fractional derivative. Numerical computations are presented, which confirm the theoretical results when the equation has a linear source term. When the equation has a nonlinear source term, numerical results show that the diffusivity depends on the order of fractional derivative as we expect.

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.

scholarly journals Backward problem for a time-space fractional diffusion equation

2018 ◽  
Vol 12 (3) ◽  
pp. 773-799 ◽  
Author(s):  
Junxiong Jia ◽  
◽  
Jigen Peng ◽  
Jinghuai Gao ◽  
Yujiao Li ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Backward Problem ◽  
Time Space ◽  
Space Fractional Diffusion Equation
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