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Identifying the Unknown Source in Linear Parabolic Equation by a Convoluting Equation Method

Author

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  • Zhenping Li

    (Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, China
    Department of Mathematics, Northwest Normal University, Lanzhou 730070, China)

  • Xiangtuan Xiong

    (Department of Mathematics, Northwest Normal University, Lanzhou 730070, China)

  • Qiang Cheng

    (Department of Mathematics, Northwest Normal University, Lanzhou 730070, China)

Abstract

This article is devoted to identifying a space-dependent source term in linear parabolic equations. Such a problem is ill posed, i.e., a small perturbation in the input data may cause a dramatically large error in the solution (if it exists). The conditional stability of the solution is analyzed. Based on a convoluting equation method, we can deal with the problem under the a priori parameter choice rule. Meanwhile, a modified version of Morozov’s discrepancy principle is provided to decide on an a posteriori regularization parameter choice strategy and a log-type error estimate is obtained. Two numerical results show that our proposed method works well.

Suggested Citation

  • Zhenping Li & Xiangtuan Xiong & Qiang Cheng, 2022. "Identifying the Unknown Source in Linear Parabolic Equation by a Convoluting Equation Method," Mathematics, MDPI, vol. 10(13), pages 1-17, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:13:p:2191-:d:845978
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    References listed on IDEAS

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    1. Ma, Yong-Ki & Prakash, P. & Deiveegan, A., 2019. "Optimization method for determining the source term in fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 155(C), pages 168-176.
    2. S. Li, Y. & Wei, T., 2018. "An inverse time-dependent source problem for a time–space fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 257-271.
    3. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
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