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Galerkin finite element method for nonlinear fractional differential equations

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Abstract

In this paper, we study the existence, regularity, and approximation of the solution for a class of nonlinear fractional differential equations. In order to do this, suitable variational formulations are defined for nonlinear boundary value problems with Riemann-Liouville and Caputo fractional derivatives together with the homogeneous Dirichlet condition. We investigate the well-posedness and also the regularity of the corresponding weak solutions. Then, we develop a Galerkin finite element approach for the numerical approximation of the weak formulations and drive a priori error estimates and prove the stability of the schemes. Finally, some numerical experiments are provided to demonstrate the accuracy of the proposed method.

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Acknowledgments

We are greatly indebted to the editor, Claude Brezinski and anonymous referees for providing helpful comments which improved the manuscript.

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Authors and Affiliations

  1. Institute for Advanced Studies in Basic Sciences, Zanjan, Iran

    Khadijeh Nedaiasl

  2. School of Mathematics, Iran University of Science and Technology, Tehran, Iran

    Raziyeh Dehbozorgi

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  1. Khadijeh Nedaiasl
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  2. Raziyeh Dehbozorgi
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Correspondence to Khadijeh Nedaiasl.

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Nedaiasl, K., Dehbozorgi, R. Galerkin finite element method for nonlinear fractional differential equations. Numer Algor 88, 113–141 (2021). https://doi.org/10.1007/s11075-020-01032-2

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  • DOI: https://doi.org/10.1007/s11075-020-01032-2

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