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Optimal error estimate of an accurate second-order scheme for Volterra integrodifferential equations with tempered multi-term kernels

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Abstract

In this paper, we investigate and analyze numerical solutions for the Volterra integrodifferential equations with tempered multi-term kernels. Firstly, we derive some regularity estimates of the exact solution. Then a temporal-discrete scheme is established by employing Crank-Nicolson technique and product integration (PI) rule for discretizations of the time derivative and tempered-type fractional integral terms, respectively, from which, nonuniform meshes are applied to overcome the singular behavior of the exact solution at \(t=0\). Based on deduced regularity conditions, we prove that the proposed scheme is unconditionally stable and possesses accurately temporal second-order convergence in \(L_2\)-norm. Numerical examples confirm the effectiveness of the proposed method.

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Acknowledgements

The first author would like to thank the reviewers for their helpful suggestions and comments to improve the quality of this paper. In addition, the first author is also very grateful to his girlfriend Dr. Kexin Li for her support in scientific research and care in life.

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  1. MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, 410081, People’s Republic of China

    Wenlin Qiu

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  1. Wenlin Qiu
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Communicated by: Bangti Jin

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Qiu, W. Optimal error estimate of an accurate second-order scheme for Volterra integrodifferential equations with tempered multi-term kernels. Adv Comput Math 49, 43 (2023). https://doi.org/10.1007/s10444-023-10050-2

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