10.1007/s40096-022-00499-5

An inverse fractional diffusion problem of source identification type

  • A. Janmohammadi1,
  • J. Damirchi*1,
  • S. M. Mahmoudi2,
  1. Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, IR
  2. Department of Statistics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, IR

Published in Issue 2022-11-20

How to Cite

Janmohammadi, A., Damirchi, J., & Mahmoudi, S. M. (2022). An inverse fractional diffusion problem of source identification type. Mathematical Sciences, 18(2 (June 2024). https://doi.org/10.1007/s40096-022-00499-5
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Abstract

Abstract One of the major objectives in the field of inverse problems is to construct a space-dependent term of an unknown source in a stable manner. Many different fields of science have used this source term, especially when the extra condition is accompanied by noise. We focus on a one-dimensional situation in a fractional diffusion problem to recover the source term that is unknown. In order to accomplish this, the major problem was transformed into an equation of operator form in a way that allowed the unique solvability of this equation to be established. The Ritz-Galerkin method is then used to implement a numerical solution to the inverse problem. In conjunction with the Galerkin method, shifted Bernoulli wavelets (BWs) are used as basis functions to reduce the main problem to an algebraic equation. It is essential to include some kind of regularization method within the numerical algorithm to obtain a stable solution to the resulting linear system. We concluded by giving numerical examples that demonstrate the proposed algorithm’s validity and efficiency in the presence of noise.

Keywords

  • Caputo derivative,
  • Ill-posed problem,
  • Regularization approach,
  • Ritz-Galerkin procedure,
  • Time-fractional inverse problem

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