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Efficient numerical schemes based on the cubic B-spline collocation method for time-fractional partial integro-differential equations of Volterra type

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Abstract

This paper proposes two efficient numerical schemes to deal with time-fractional integro-differential reaction-convection-diffusion equations of the Volterra type in a rectangular domain. The fractional derivative is taken in the Caputo sense of order \( \alpha \in (0,1).\) In order to construct the computational semi-discrete schemes, we discretize the fractional operator using finite difference techniques, namely the L1 and the L1-2, on a uniform mesh in the time direction. A composite trapezoidal rule approximates the integral part. The semi-discrete problems become a set of two-point boundary value problems (BVPs) in the spatial domain. The cubic B-spline collocation method is employed to solve the corresponding BVPs. The optimal error estimation and convergence analysis are established under suitable regularity assumptions on the initial data and the true solution of the considered model problem. It is shown in the theoretical establishment that the L1 based scheme achieves \( (2-\alpha ) \) order of accuracy, while the rate of convergence of the L1-2 based scheme is quadratic. Finally, a numerical experiment is performed in support of the theoretical findings.

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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 acknowledges the financial support from the Council of Scientific & Industrial Research (CSIR), India (File No.: 09/983(0046)/2020-EMR-I).

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

    Bappa Ghosh & Jugal Mohapatra

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  1. Bappa Ghosh
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Ghosh, B., Mohapatra, J. Efficient numerical schemes based on the cubic B-spline collocation method for time-fractional partial integro-differential equations of Volterra type. J. Appl. Math. Comput. 70, 741–769 (2024). https://doi.org/10.1007/s12190-023-01981-6

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