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Numerical Algorithms with High Spatial Accuracy for the Fourth-Order Fractional Sub-Diffusion Equations with the First Dirichlet Boundary Conditions

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Abstract

In this paper, a compact algorithm for the fourth-order fractional sub-diffusion equations with first Dirichlet boundary conditions, which depict wave propagation in intense laser beams, is investigated. Combining the average operator for the spatial fourth-order derivative, the L1 formula is applied to approximate the temporal Caputo fractional derivative. A novel technique is introduced to deal with the first Dirichlet boundary conditions. Using mathematical induction method, we prove that the presented difference scheme is unconditionally stable and convergent by the energy method. The convergence order is \(O(\tau ^{2-\alpha }+h^4)\) in \(L_2\)-norm. The outline for the two-dimensional problem is also considered. Finally, some numerical examples are provided to confirm the theoretical results.

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Acknowledgments

We would like to express our gratitude to the editor and the reviewers for their many valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Cui-cui Ji, Zhi-zhong Sun & Zhao-peng Hao

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  1. Cui-cui Ji
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  2. Zhi-zhong Sun
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  3. Zhao-peng Hao
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Corresponding author

Correspondence to Zhi-zhong Sun.

Additional information

The research is supported by National Natural Science Foundation of China (No. 11271068).

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Ji, Cc., Sun, Zz. & Hao, Zp. Numerical Algorithms with High Spatial Accuracy for the Fourth-Order Fractional Sub-Diffusion Equations with the First Dirichlet Boundary Conditions. J Sci Comput 66, 1148–1174 (2016). https://doi.org/10.1007/s10915-015-0059-7

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