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Analysis of finite difference schemes for Volterra integro-differential equations involving arbitrary order derivatives

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Abstract

In this work, we consider a Volterra integro-differential equation involving Caputo fractional derivative of order \( \alpha \in (0,1). \) To approximate the solution, we propose two finite difference schemes that use L1 and L1-2 discretization to approximate the differential part and a composite trapezoidal rule to approximate an integral part. The error estimates for both schemes are established. It is shown that the approximate solution obtained by using the L1-2 scheme converges to the exact solution more rapidly than the L1 scheme. Finally, some numerical experiments are carried out to show the validity and accuracy of the proposed schemes.

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The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The first author would like to thank the Council of Scientific & Industrial Research (CSIR), Government of India (File No.: 09/983(0046)/2020-EMR-I), for financial support to carry out his research work at NIT Rourkela.

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  1. Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha, 769008, India

    Bappa Ghosh & Jugal Mohapatra

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  1. Bappa Ghosh
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Ghosh, B., Mohapatra, J. Analysis of finite difference schemes for Volterra integro-differential equations involving arbitrary order derivatives. J. Appl. Math. Comput. 69, 1865–1886 (2023). https://doi.org/10.1007/s12190-022-01817-9

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