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Discrete Galerkin and Iterated Discrete Galerkin Methods for Derivative-Dependent Fredholm–Hammerstein Integral Equations with Green’s Kernel

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Abstract

In this article, we look at a class of two-point nonlinear boundary value problems and transform this into derivative-dependent Fredholm–Hammerstein integral equations, i.e., the integral equation, where the kernel is of Green’s type, and the nonlinear function inside the integral is dependent on the derivative. We obtain the error analysis by replacing all the integrals in the Galerkin method with numerical quadrature. We propose the discrete Galerkin and iterated discrete Galerkin methods by piecewise polynomials to obtain the convergence analysis of these derivative-dependent Fredholm–Hammerstein integral equations. By choosing the numerical quadrature rule appropriately, the convergence rates in Galerkin and iterated Galerkin methods are preserved. We show that the iterated discrete Galerkin method improves over the discrete Galerkin method in terms of the order of convergence. To demonstrate the theoretical results, several numerical examples are provided.

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Acknowledgements

The author would like to thank the referee for the helpful suggestions.

Funding

The research work of Kapil Kant was supported by the ABV-IIITM Gwalior, India, research project: 011/2023.

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Authors and Affiliations

  1. Department of Applied Sciences, ABV-Indian Institute of Information Technology and Management, Gwalior, 474015, India

    Kapil Kant

  2. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Rakesh Kumar

  3. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721 302, India

    Samiran Chakraborty & Gnaneshwar Nelakanti

Authors
  1. Kapil Kant
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  2. Rakesh Kumar
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  3. Samiran Chakraborty
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  4. Gnaneshwar Nelakanti
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Contributions

KK: The author carried out the convergence analysis and proposed method to solve the problem. RK: The author also contributes in convergence analysis. SCh: The author has proposed the problem and provide the technical details. GN: Prof G. Nelakanti helped us to revise the manuscript.

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Correspondence to Kapil Kant.

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Kant, K., Kumar, R., Chakraborty, S. et al. Discrete Galerkin and Iterated Discrete Galerkin Methods for Derivative-Dependent Fredholm–Hammerstein Integral Equations with Green’s Kernel. Mediterr. J. Math. 20, 249 (2023). https://doi.org/10.1007/s00009-023-02444-9

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  • DOI: https://doi.org/10.1007/s00009-023-02444-9

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