Solving Inverse Problem of Distributed-Order Time-Fractional Diffusion Equations Using Boundary Observations and L 2 Regularization

Lele Yuan, Kewei Liang () and Huidi Wang
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Lele Yuan: School of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China
Kewei Liang: School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, China
Huidi Wang: College of Sciences, China Jiliang University, Hangzhou 310018, China

Mathematics, 2023, vol. 11, issue 14, 1-20

Abstract: This article investigates the inverse problem of estimating the weight function using boundary observations in a distributed-order time-fractional diffusion equation. We propose a method based on L 2 regularization to convert the inverse problem into a regularized minimization problem, and we solve it using the conjugate gradient algorithm. The minimization functional only needs the weight to have L 2 regularity. We prove the weak closedness of the inverse operator, which ensures the existence, stability, and convergence of the regularized solution for the weight in L 2 ( 0 , 1 ) . We propose a weak source condition for the weight in C [ 0 , 1 ] and, based on this, we prove the convergence rate for the regularized solution. In the conjugate gradient algorithm, we derive the gradient of the objective functional through the adjoint technique. The effectiveness of the proposed method and the convergence rate are demonstrated by two numerical examples in two dimensions.

Keywords: weight function recovery; L 2 regularization; distributed-order time-fractional diffusion equation; convergence rate (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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