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An Investigation on Existence, Uniqueness, and Approximate Solutions for Two-Dimensional Nonlinear Fractional Integro-Differential Equations

Author

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  • Tahereh Eftekhari

    (School of Mathematics, Iran University of Science & Technology (IUST), Narmak, Tehran 16846 13114, Iran)

  • Jalil Rashidinia

    (School of Mathematics, Iran University of Science & Technology (IUST), Narmak, Tehran 16846 13114, Iran)

Abstract

In this research, we provide sufficient conditions to prove the existence of local and global solutions for the general two-dimensional nonlinear fractional integro-differential equations. Furthermore, we prove that these solutions are unique. In addition, we use operational matrices of two-variable shifted Jacobi polynomials via the collocation method to reduce the equations into a system of equations. Error bounds of the presented method are obtained. Five test problems are solved. The obtained numerical results show the accuracy, efficiency, and applicability of the proposed approach.

Suggested Citation

  • Tahereh Eftekhari & Jalil Rashidinia, 2023. "An Investigation on Existence, Uniqueness, and Approximate Solutions for Two-Dimensional Nonlinear Fractional Integro-Differential Equations," Mathematics, MDPI, vol. 11(4), pages 1-29, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:4:p:824-:d:1059433
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    References listed on IDEAS

    as
    1. Najafalizadeh, S. & Ezzati, R., 2016. "Numerical methods for solving two-dimensional nonlinear integral equations of fractional order by using two-dimensional block pulse operational matrix," Applied Mathematics and Computation, Elsevier, vol. 280(C), pages 46-56.
    2. Hesameddini, Esmail & Shahbazi, Mehdi, 2018. "Two-dimensional shifted Legendre polynomials operational matrix method for solving the two-dimensional integral equations of fractional order," Applied Mathematics and Computation, Elsevier, vol. 322(C), pages 40-54.
    3. Maleknejad, Khosrow & Rashidinia, Jalil & Eftekhari, Tahereh, 2018. "Numerical solution of three-dimensional Volterra–Fredholm integral equations of the first and second kinds based on Bernstein’s approximation," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 272-285.
    4. Abdon Atangana & Adem Kilicman, 2013. "Analytical Solutions of the Space-Time Fractional Derivative of Advection Dispersion Equation," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-9, April.
    5. Mirzaee, Farshid & Samadyar, Nasrin, 2019. "Numerical solution based on two-dimensional orthonormal Bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order," Applied Mathematics and Computation, Elsevier, vol. 344, pages 191-203.
    6. Abdon Atangana & Adem Kılıçman, 2013. "A Possible Generalization of Acoustic Wave Equation Using the Concept of Perturbed Derivative Order," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-6, April.
    7. Eftekhari, Tahereh & Rashidinia, Jalil, 2022. "A novel and efficient operational matrix for solving nonlinear stochastic differential equations driven by multi-fractional Gaussian noise," Applied Mathematics and Computation, Elsevier, vol. 429(C).
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