Recovering the potential term in a fractional diffusion equation

2017 ◽  
Vol 82 (3) ◽  
pp. 579-600 ◽  
Author(s):  
Zhidong Zhang ◽  
Zhi Zhou
Keyword(s):  
Diffusion Equation ◽  
Noisy Data ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Convergence Theory ◽  
Unique Determination ◽  
Time Data ◽  
Final Time ◽  
Potential Term ◽  
Reconstruction Operator

Abstract In this article, we consider an inverse problem of recovering the potential term in a 1D time-fractional diffusion equation from the overdetermined final time data. We introduce a reconstruction operator and show its contractivity and monotonicity, which give the unique determination and an efficient algorithm. Further, for noisy data, we propose a regularized iterative algorithm based on mollification and derive error estimates for the approximation. Extensive numerical experiments for both smooth and nonsmooth potential data are provided to illustrate the efficiency and stability of the algorithm, and to verify the convergence theory.

An Inverse Source Problem with Sparsity Constraint for the Time-Fractional Diffusion Equation

2015 ◽  
Vol 8 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Zhousheng Ruan ◽  
Zhijian Yang ◽  
Xiliang Lu
Keyword(s):  
Diffusion Equation ◽  
Optimization Problem ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Inverse Source Problem ◽  
Final Time ◽  
Elastic Net Regularization ◽  
Semismooth Newton ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, an inverse source problem for the time-fractional diffusion equation is investigated. The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure. Here the sparsity is understood with respect to the pixel basis, i.e., the source has a small support. By an elastic-net regularization method, this inverse source problem is formulated into an optimization problem and a semismooth Newton (SSN) algorithm is developed to solve it. A discretization strategy is applied in the numerical realization. Several one and two dimensional numerical examples illustrate the efficiency of the proposed method.

Recovering a time-dependent potential function in a multi-term time fractional diffusion equation by using a nonlinear condition

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Su Zhen Jiang ◽  
Yu Jiang Wu
Keyword(s):  
Diffusion Equation ◽  
Potential Function ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Time Dependent ◽  
Potential Term ◽  
Nonlinear Condition ◽  
Dependent Potential ◽  
Time Fractional Diffusion Equation ◽  
The Fixed Point Theorem

AbstractIn the present paper, we devote our effort to a nonlinear inverse problem for recovering a time-dependent potential term in a multi-term time fractional diffusion equation from an additional measurement in the form of an integral over the space domain. First we study the existence, uniqueness, regularity and stability of the solution for the direct problem by using the fixed point theorem. And we obtain the uniqueness of the inverse time-dependent potential term problem. Numerically, we use the Levenberg–Marquardt method to find the approximate potential function. Four different examples are presented to show the feasibility and efficiency of the proposed method.

scholarly journals Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abderrazak Nabti ◽  
Ahmed Alsaedi ◽  
Mokhtar Kirane ◽  
Bashir Ahmad
Keyword(s):  
Diffusion Equation ◽  
Global Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Laplacian Operator ◽  
Nonlocal Source ◽  
Time Space ◽  
Nonexistence Of Solutions ◽  
Beta 2 ◽  
Fractional Laplacian Operator

Abstract We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source $$\begin{aligned} u_{t} + (-\Delta )^{\frac{\beta }{2}} u =\bigl(1+ \vert x \vert \bigr)^{ \gamma } \int _{0}^{t} (t-s)^{\alpha -1} \vert u \vert ^{p} \bigl\Vert \nu ^{ \frac{1}{q}}(x) u \bigr\Vert _{q}^{r} \,ds \end{aligned}$$ u t + ( − Δ ) β 2 u = ( 1 + | x | ) γ ∫ 0 t ( t − s ) α − 1 | u | p ∥ ν 1 q ( x ) u ∥ q r d s for $(x,t) \in \mathbb{R}^{N}\times (0,\infty )$ ( x , t ) ∈ R N × ( 0 , ∞ ) with initial data $u(x,0)=u_{0}(x) \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})$ u ( x , 0 ) = u 0 ( x ) ∈ L loc 1 ( R N ) , where $p,q,r>1$ p , q , r > 1 , $q(p+r)>q+r$ q ( p + r ) > q + r , $0<\gamma \leq 2 $ 0 < γ ≤ 2 , $0<\alpha <1$ 0 < α < 1 , $0<\beta \leq 2$ 0 < β ≤ 2 , $(-\Delta )^{\frac{\beta }{2}}$ ( − Δ ) β 2 stands for the fractional Laplacian operator of order β, the weight function $\nu (x)$ ν ( x ) is positive and singular at the origin, and $\Vert \cdot \Vert _{q}$ ∥ ⋅ ∥ q is the norm of $L^{q}$ L q space.

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