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A new regularization method for a Cauchy problem of the time fractional diffusion equation

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Abstract

In this paper, we consider a Cauchy problem of the time fractional diffusion equation (TFDE). Such problem is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α (0 < α ≤ 1). We show that the Cauchy problem of TFDE is severely ill-posed and further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates in the interior and on the boundary of solution domain are obtained respectively under different a-priori bound assumptions for the exact solution and suitable choices of regularization parameters. Finally, numerical examples are given to show that the proposed numerical method is effective.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, People’s Republic of China

    G. H. Zheng & T. Wei

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  1. G. H. Zheng
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  2. T. Wei
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Correspondence to T. Wei.

Additional information

Communicated by Yuesheng Xu and Hongqi Yang.

The work described in this paper was supported by the NSF of China (10971089), the Fundamental Research Funds for the Central Universities (lzujbky-2010-k10) and the Funds for the Ph.D. academic newcomer award of Lanzhou University.

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Zheng, G.H., Wei, T. A new regularization method for a Cauchy problem of the time fractional diffusion equation. Adv Comput Math 36, 377–398 (2012). https://doi.org/10.1007/s10444-011-9206-3

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  • DOI: https://doi.org/10.1007/s10444-011-9206-3

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