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Extension of the First-Order Recursive Filters Method to Non-Linear Second-Kind Volterra Integral Equations

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  • Rodolphe Heyd

    (Laboratoire Angevin de Mécanique, Procédés et innovAtion (LAMPA), Arts et Métiers ParisTech, 49035 Angers, France)

Abstract

A new numerical method for solving Volterra non-linear convolution integral equations (NLCVIEs) of the second kind is presented in this work. This new approach, named IIRFM-A, is based on the combined use of the Laplace transformation, a first-order decomposition, a bilinear transformation, and the Adomian decomposition. Unlike most numerical methods based on the Laplace transformation, the IIRFM-A method has the dual advantage of requiring neither the calculation of the Laplace transform of the source function nor that of intermediate inverse Laplace transforms. The application of this new method to the case of non-convolutive multiplicative kernels is also introduced in this work. Several numerical examples are presented to illustrate the great flexibility and efficiency of this new approach. A concrete thermal problem, described by a non-linear convolutive Volterra integral equation, is also solved numerically using the new IIRFM-A method. In addition, this new approach extends for the first time the field of use of first-order recursive filters, usually restricted to the case of linear ordinary differential equations (ODEs) with constant coefficients, to the case of non-linear ODEs with variable coefficients. This extension represents a major step forward in the field of recursive filters.

Suggested Citation

  • Rodolphe Heyd, 2024. "Extension of the First-Order Recursive Filters Method to Non-Linear Second-Kind Volterra Integral Equations," Mathematics, MDPI, vol. 12(22), pages 1-22, November.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:22:p:3612-:d:1524505
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    References listed on IDEAS

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    1. Rodolphe Heyd, 2024. "Numerical Solution of Linear Second-Kind Convolution Volterra Integral Equations Using the First-Order Recursive Filters Method," Mathematics, MDPI, vol. 12(15), pages 1-35, August.
    2. Mohamed S. Mohamed & Khaled A. Gepreel & Faisal A. Al-Malki & Maha Al-Humyani, 2015. "Approximate Solutions of the Generalized Abel’s Integral Equations Using the Extension Khan’s Homotopy Analysis Transformation Method," Journal of Applied Mathematics, Hindawi, vol. 2015, pages 1-9, March.
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    1. Rodolphe Heyd, 2024. "Numerical Solution of Linear Second-Kind Convolution Volterra Integral Equations Using the First-Order Recursive Filters Method," Mathematics, MDPI, vol. 12(15), pages 1-35, August.

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