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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Lele Yuan & Kewei Liang & Huidi Wang, 2023. "Solving Inverse Problem of Distributed-Order Time-Fractional Diffusion Equations Using Boundary Observations and L 2 Regularization," Mathematics, MDPI, vol. 11(14), pages 1-20, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:14:p:3101-:d:1193538
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    References listed on IDEAS

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    1. Li, Zhiyuan & Liu, Yikan & Yamamoto, Masahiro, 2015. "Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 381-397.
    2. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2006. "Stochastic model for ultraslow diffusion," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1215-1235, September.
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