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Existence, uniqueness, and numerical solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations in a Banach space

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Abstract

The purpose of this research is to provide sufficient conditions for the local and global existence of solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations, based on the Schauder’s and Tychonoff’s fixed-point theorems. Also, we provide sufficient conditions for the uniqueness of the solutions. Moreover, we use operational matrices of hybrid of two-dimensional block-pulse functions and two-variable shifted Legendre polynomials via collocation method to find approximate solutions of the mentioned equations. In addition, a discussion on error bound and convergence analysis of the proposed method is presented. Finally, the accuracy and efficiency of the presented method are confirmed by solving three illustrative examples and comparing the results of the proposed method with other existing numerical methods in the literature.

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Acknowledgements

The authors express their sincere thanks to the reviewer for his valuable comments and suggestions that improved the content of the paper.

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Authors and Affiliations

  1. School of Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran, 16846 13114, Iran

    Khosrow Maleknejad, Jalil Rashidinia & Tahereh Eftekhari

Authors
  1. Khosrow Maleknejad
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  2. Jalil Rashidinia
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  3. Tahereh Eftekhari
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Correspondence to Khosrow Maleknejad.

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Communicated by José Tenreiro Machado.

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Maleknejad, K., Rashidinia, J. & Eftekhari, T. Existence, uniqueness, and numerical solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations in a Banach space. Comp. Appl. Math. 39, 271 (2020). https://doi.org/10.1007/s40314-020-01322-4

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  • DOI: https://doi.org/10.1007/s40314-020-01322-4

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