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Multigrid Method for Solving Inverse Problems for Heat Equation

Author

Listed:
  • Hassan K. Ibrahim Al-Mahdawi

    (Department of System Programming, South Ural State University, 454080 Chelyabinsk, Russia)

  • Mostafa Abotaleb

    (Department of System Programming, South Ural State University, 454080 Chelyabinsk, Russia)

  • Hussein Alkattan

    (Department of System Programming, South Ural State University, 454080 Chelyabinsk, Russia)

  • Al-Mahdawi Zena Tareq

    (Ministry of Electricity, The State Company Electricity Production, Baghdad 10053, Iraq)

  • Amr Badr

    (Faculty of Science, School of Science and Technology, University of New England, Armidale, NSW 2350, Australia)

  • Ammar Kadi

    (Department of Food and Biotechnology, South Ural State University, 454080 Chelyabinsk, Russia)

Abstract

In this paper, the inverse problems for the boundary value and initial value in a heat equation are posed and solved. It is well known that those problems are ill posed. The problems are reformulated as integral equations of the first kind by using the separation-of-variables method. The discretization of the integral equation allowed us to reduce the integral equation to a system of linear algebraic equations or a linear operator equation of the first kind on Hilbert spaces. The Landweber-type iterative method was used in order to find an approximation solution. The V-cycle multigrid method is used to obtain more frequent and fast convergence for iteration. The numerical computation examples are presented to verify the accuracy and fast computing of the approximation solution.

Suggested Citation

  • Hassan K. Ibrahim Al-Mahdawi & Mostafa Abotaleb & Hussein Alkattan & Al-Mahdawi Zena Tareq & Amr Badr & Ammar Kadi, 2022. "Multigrid Method for Solving Inverse Problems for Heat Equation," Mathematics, MDPI, vol. 10(15), pages 1-15, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2802-:d:882435
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    Citations

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    Cited by:

    1. Tao Liu & Di Ouyang & Lianjun Guo & Ruofeng Qiu & Yunfei Qi & Wu Xie & Qiang Ma & Chao Liu, 2023. "Combination of Multigrid with Constraint Data for Inverse Problem of Nonlinear Diffusion Equation," Mathematics, MDPI, vol. 11(13), pages 1-15, June.
    2. Sergey I. Martynenko & Aleksey Yu. Varaksin, 2023. "Black-Box Solver for Numerical Simulations and Mathematical Modelling in Engineering Physics," Mathematics, MDPI, vol. 11(16), pages 1-21, August.

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