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New numerical methods for solving the partial fractional differential equations with uniform and non-uniform meshes

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Abstract

In this paper, we design and develop some algorithms by using the piecewise linear interpolation polynomial for solving the partial fractional differential equations involving Caputo derivative, with uniform and non-uniform meshes. For designing new methods, we select the mesh points based on the two equal-height and equal-area distribution. Furthermore, the error bounds of proposed methods with uniform and equidistributing meshes are obtained. We also show that our numerical method is stable and convergent with the accuracy of \(O(\kappa ^2 + h)\). Also, some numerical examples are constructed to demonstrate the efficacy and usefulness of the numerical methods. Finally, a comparative study for different values of parameters is also presented.

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Acknowledgements

The authors would like to express special thanks to the referees for carefully reading, constructive comments, and valuable remarks which significantly improved the quality from this paper. This research is supported by a research grant of the University of Tabriz (Number 940).

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  1. Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

    Mohammad Javidi & Mahdi Saedshoar Heris

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  1. Mohammad Javidi
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Javidi, M., Saedshoar Heris, M. New numerical methods for solving the partial fractional differential equations with uniform and non-uniform meshes. J Supercomput 79, 14457–14488 (2023). https://doi.org/10.1007/s11227-023-05198-z

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