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An efficient numerical approach for solving variable-order fractional partial integro-differential equations

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Abstract

This paper reports an efficient numerical method based on the Bernoulli polynomials for solving the variable-order fractional partial integro-differential equation(V-O-FPIDEs). The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann–Liouville integral operator are used in our paper. First, we prove the existence and uniqueness of the original equation by Gronwall inequality. Second, we use two-dimensional Bernoulli polynomials to approximate the unknown function of the original equation, and use the Gauss–Jacobi quadrature formula to deal with the variable-order Caputo fractional derivative operator and Riemann–Liouville integral operator. Then, the original equation is transformed into the corresponding system of algebraic equations. Furthermore, we give the convergence analysis and error estimation of the proposed method. Finally, some numerical examples illustrate the effectiveness of the method.

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Acknowledgements

The authors would like to acknowledge the support provided by the National Natural Science Foundation of China (12101089) and the Natural Science Foundation of Sichuan Province (2022NSFSC1844).

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Authors and Affiliations

  1. University of Electronic Science and Technology of China, Chengdu, Sichuan, China

    Yifei Wang, Jin Huang & Ting Deng

  2. School of Mathematics, Chengdu Normal University, Chengdu, China

    Hu Li

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  1. Yifei Wang
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  2. Jin Huang
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  3. Ting Deng
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  4. Hu Li
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Correspondence to Jin Huang.

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Wang, Y., Huang, J., Deng, T. et al. An efficient numerical approach for solving variable-order fractional partial integro-differential equations. Comp. Appl. Math. 41, 411 (2022). https://doi.org/10.1007/s40314-022-02131-7

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