The Backward Problem for a Nonlinear Riesz-Feller Diffusion Equation

Abstract

In this paper, we reconstruct the solution u(x,t) of the backward space-fractional diffusion problem with a locally Lipschitzian nonlinear source

This problem is severely ill-posed in the Hadamard sense, hence, a regularization is in order. In the paper, we introduce one spectral regularization method and establish stability error estimates with optimal order under an a priori choice of regularization parameter. Finally, numerical implementations are given to show the effectiveness of the proposed regularization methods.

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Acknowledgements

The authors would like to thank the referees for the careful reading, helpful comments and suggestions leading to the improved version of the paper.

Funding

This paper is supported by National Foundation of Scientific and Technology Development (NAFOSTED)-Project 101.02-2016.26.

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Correspondence to Dang Duc Trong.

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Hai, D.N.D., Trong, D.D. The Backward Problem for a Nonlinear Riesz-Feller Diffusion Equation. Acta Math Vietnam 43, 449–470 (2018). https://doi.org/10.1007/s40306-018-0255-2

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Keywords

  • Space-fractional backward diffusion problem
  • Ill-posed problem
  • Regularization
  • Error estimate

Mathematics Subject Classification (2010)

  • 26A33
  • 47A52
  • 47J06
  • 65M32