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Numerical solution based on two-dimensional orthonormal Bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order

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  • Mirzaee, Farshid
  • Samadyar, Nasrin

Abstract

In this paper, we develop a numerical scheme based on two-dimensional orthonormal Bernstein polynomials (2D-OBPs) to solve two-dimensional nonlinear integral equations of fractional order. The fractional integral considered here is in the Riemann–Liouville sense. By using definition of Riemann–Liouville fractional integral, two-dimensional nonlinear fractional integral equations is transformed into two-dimensional nonlinear ordinary integral equations. Operational matrices method based on 2D-OBPs are applied to obtain an approximate solution with high accuracy for these equations. In addition, error analysis of the proposed method is discussed and an upper error bound is provided under weak assumptions. Some linear and nonlinear examples are given to demonstrate the accuracy, efficiency and speed of the suggested method.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:344-345:y:2019:i::p:191-203
    DOI: 10.1016/j.amc.2018.10.020
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    References listed on IDEAS

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    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. Mirzaee, Farshid & Hadadiyan, Elham, 2015. "Applying the modified block-pulse functions to solve the three-dimensional Volterra–Fredholm integral equations," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 759-767.
    3. Mingxu Yi & Kangwen Sun & Jun Huang & Lifeng Wang, 2013. "Numerical Solutions of Fractional Integrodifferential Equations of Bratu Type by Using CAS Wavelets," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-7, December.
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    Cited by:

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    2. Zhao, Hengzhi & Zhang, Jiwei & Lu, Jing, 2023. "Numerical approximate controllability for unidimensional parabolic integro-differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 575-596.
    3. 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.
    4. Karamollahi, Nasibeh & Heydari, Mohammad & Loghmani, Ghasem Barid, 2021. "Approximate solution of nonlinear Fredholm integral equations of the second kind using a class of Hermite interpolation polynomials," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 414-432.
    5. Wen, Xiaoxia & Huang, Jin, 2021. "A combination method for numerical solution of the nonlinear stochastic Itô-Volterra integral equation," Applied Mathematics and Computation, Elsevier, vol. 407(C).
    6. Sunil Kumar & Ali Ahmadian & Ranbir Kumar & Devendra Kumar & Jagdev Singh & Dumitru Baleanu & Mehdi Salimi, 2020. "An Efficient Numerical Method for Fractional SIR Epidemic Model of Infectious Disease by Using Bernstein Wavelets," Mathematics, MDPI, vol. 8(4), pages 1-22, April.

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