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An Analytical‐Numerical Method for the Solution of Nonlinear Fractional Fredholm Integro‐Differential Equations With Logarithmic Weakly Singular Kernel

Author

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  • Ali Edham Awadh
  • Esmaeil Najafi

Abstract

In this work, we investigate a numerical method for solving nonlinear fractional Fredholm integro‐differential equations with logarithmic weakly singular kernels. Since the direct solution of these equations using classical methods results in low accuracy and high computational cost due to the singular behavior of the exact solution at both endpoints of the interval, we consider two approaches for the numerical solution. The first is the use of an analytical‐iterative method to transform the nonlinear equation into a sequence of linear equations, which avoids the occurrence of nonlinear systems. The second is the use of a regularization technique, which regularizes the exact solution of the equation, making it possible to achieve high accuracy using common numerical methods. To test the accuracy and performance of the proposed method and to compare theoretical and numerical results, several test problems are solved using the presented method, and the results obtained from them are analyzed.

Suggested Citation

  • Ali Edham Awadh & Esmaeil Najafi, 2025. "An Analytical‐Numerical Method for the Solution of Nonlinear Fractional Fredholm Integro‐Differential Equations With Logarithmic Weakly Singular Kernel," Journal of Mathematics, John Wiley & Sons, vol. 2025(1).
    • Handle: RePEc:wly:jjmath:v:2025:y:2025:i:1:n:8183229
    DOI: 10.1155/jom/8183229
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    References listed on IDEAS

    as
    1. Qays Atshan Almusawi & Esmaeil Najafi & Birendra Nath Mandal, 2024. "Quasilinearization-Collocation Method for the Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations With Logarithmic Weakly Singular Kernel," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2024, pages 1-19, November.
    2. Wang, Yanxin & Zhu, Li, 2016. "SCW method for solving the fractional integro-differential equations with a weakly singular kernel," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 72-80.
    3. Ivano Colombaro & Andrea Giusti & Silvia Vitali, 2018. "Storage and Dissipation of Energy in Prabhakar Viscoelasticity," Mathematics, MDPI, vol. 6(2), pages 1-9, January.
    4. Marcello Artioli & Giuseppe Dattoli & Silvia Licciardi & Simonetta Pagnutti, 2017. "Fractional Derivatives, Memory Kernels and Solution of a Free Electron Laser Volterra Type Equation," Mathematics, MDPI, vol. 5(4), pages 1-9, December.
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