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Finite Element Method for Two-Sided Fractional Differential Equations with Variable Coefficients: Galerkin Approach

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Abstract

This paper develops a Galerkin approach for two-sided fractional differential equations with variable coefficients. By the product rule, we transform the problem into an equivalent formulation which additionally introduces the fractional low-order term. We prove the existence and uniqueness of the solutions of the Dirichlet problems of the equations with certain diffusion coefficients. We adopt the Galerkin formulation, and prove its error estimates. Finally, several numerical examples are provided to illustrate the fidelity and accuracy of the proposed theoretical results.

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Acknowledgements

Hao would like to acknowledge the support by National Natural Science Foundation of China (No. 11671083), China Scholarship Council (No. 201506090065). Lin gratefully acknowledges the support from National Science Foundation (DMS-1555072, DMS-1736364, and DMS-1821233). Cai would like to acknowledge the support by the NSF Grant DMS-1522707.

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Authors and Affiliations

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

    Zhaopeng Hao

  2. Department of Mathematics and Statistics, Oakland University, Rochester, MI, 48309, USA

    Moongyu Park

  3. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Guang Lin & Zhiqiang Cai

  4. School of Mechanical Engineering, Purdue University, West Lafayette, IN, 47907, USA

    Guang Lin

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  1. Zhaopeng Hao
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  2. Moongyu Park
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  3. Guang Lin
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  4. Zhiqiang Cai
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Correspondence to Moongyu Park.

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Hao, Z., Park, M., Lin, G. et al. Finite Element Method for Two-Sided Fractional Differential Equations with Variable Coefficients: Galerkin Approach. J Sci Comput 79, 700–717 (2019). https://doi.org/10.1007/s10915-018-0869-5

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  • DOI: https://doi.org/10.1007/s10915-018-0869-5

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