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Comparative analysis of nonlinear Urysohn functional integral equations via Nyström method

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  • Bhat, Imtiyaz Ahmad
  • Mishra, Lakshmi Narayan
  • Mishra, Vishnu Narayan

Abstract

In this sequel, the investigation of a diverse range of Urysohn type nonlinear functional integral equations falling under the Fredholm type is examined. This equation encompasses various integral and functional equations as specific instances. By imposing smoothness conditions on the involved functions, both the solution's existence and its uniqueness using the fixed point method are established. Subsequently, we employ the Nyström method for solution's approximation, leading to a set of algebraic equations of nonlinear form. The Picard iterative method is then applied to the solution's approximation for this system of algebraic equations. Additionally, the trapezoidal method is applied to approximate the solution and a novel Grönwall inequality is used to establish the convergence of method, providing a reliable theoretical foundation. Numerical examples and a comparative analysis are presented to demonstrate the convergence, effectiveness, and superiority of the Nyström method compared to the trapezoidal method, highlighting its improved practicality and versatility.

Suggested Citation

  • Bhat, Imtiyaz Ahmad & Mishra, Lakshmi Narayan & Mishra, Vishnu Narayan, 2025. "Comparative analysis of nonlinear Urysohn functional integral equations via Nyström method," Applied Mathematics and Computation, Elsevier, vol. 494(C).
    • Handle: RePEc:eee:apmaco:v:494:y:2025:i:c:s0096300325000141
    DOI: 10.1016/j.amc.2025.129287
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    References listed on IDEAS

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    1. Abdel Hamid, Haydar & Al Sayed, Waad, 2016. "Integrable solutions of a generalized mixed-type functional integral equation," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 356-366.
    2. Abbasbandy, S., 2007. "Application of He’s homotopy perturbation method to functional integral equations," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1243-1247.
    3. Bhat, Imtiyaz Ahmad & Mishra, Lakshmi Narayan, 2024. "A comparative study of discretization techniques for augmented Urysohn type nonlinear functional Volterra integral equations and their convergence analysis," Applied Mathematics and Computation, Elsevier, vol. 470(C).
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