Journal ArticleDOI
Backward problem for time-space fractional diffusion equations in Hilbert scales
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TLDR
A modified version of quasi-boundary value method is applied to construct stable approximation problem for time-space fractional diffusion equations associated with the observed data measured in Hilbert scales to overcome the ill-posedness of the problem.Abstract:
This work is concerned with a mathematical study of backward problem for time-space fractional diffusion equations associated with the observed data measured in Hilbert scales. Transforming the original problem into an operator equation, we investigate the existence, the uniqueness and the instability for the problem. In order to overcome the ill-posedness of the problem, we apply a modified version of quasi-boundary value method to construct stable approximation problem. Using a Holder-type smoothness assumption of the exact solution it is shown that estimates achieve optimal rates of convergence in Hilbert scales both for an a-priori and for an a-posteriori parameter choice strategies.read more
Citations
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Global boundedness and asymptotic behavior of time-space fractional nonlocal reaction-diffusion equation
Hui Zhan,Fei Gao,Liujie Guo +2 more
TL;DR: In this article , the authors investigated the global boundedness and asymptotic behavior for the solution of time-space fractional non-local reaction-diffusion equation.
Journal ArticleDOI
Galerkin Method for a Backward Problem of Time-Space Fractional Symmetric Diffusion Equation
Hongwu Zhang,Yong Lv +1 more
TL;DR: In this paper , a Galerkin regularization method was proposed to overcome the ill-posedness of the considered problem, where the negative Laplace operator −Δ contained in the main equation belongs to the category of uniformly symmetric elliptic operators.
Journal ArticleDOI
Recovering a Space-Dependent Source Term in the Fractional Diffusion Equation with the Riemann–Liouville Derivative
TL;DR: In this article , an unknown source term in the fractional diffusion equation with the Riemann-Liouville derivative was determined and a fractional Tikhonov regularization method was applied to regularize the inverse source problem.
Journal ArticleDOI
An Inverse Source Problem for A One-dimensional Time-Space Fractional Diffusion Equation
TL;DR: In this paper , a source term with time-dependence of the time-space fractional diffusion equation with additional observation data is determined, and a numerical method is proposed based on the optimal perturbation algorithm with optimized Tikhonov regularization.
Journal ArticleDOI
Numerical Solution of Backward Problem of a Multi-term Time-space Fractional Diffusion Equation
Yuxuan Yang,Yushan Li,Xianru Qin +2 more
TL;DR: In this article , a numerical solution based on a Tikhonov regularization technique combined with an optimal perturbation algorithm (OPA) is proposed to solve the backward problem of the fractional diffusion equation.
References
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Book
Semigroups of Linear Operators and Applications to Partial Differential Equations
TL;DR: In this article, the authors considered the generation and representation of a generator of C0-Semigroups of Bounded Linear Operators and derived the following properties: 1.1 Generation and Representation.
Book
Theory and Applications of Fractional Differential Equations
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Book
An Introduction to the Fractional Calculus and Fractional Differential Equations
Kenneth S. Miller,Bertram Ross +1 more
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Book
Regularization of Inverse Problems
TL;DR: Inverse problems have been studied in this article, where Tikhonov regularization of nonlinear problems has been applied to weighted polynomial minimization problems, and the Conjugate Gradient Method has been used for numerical realization.