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Superconvergent spectral methods for system of modified Volterra Integral Equations

Author

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  • Kumar, Rakesh
  • Kant, Kapil
  • Rathish Kumar, B.V.

Abstract

This study introduces novel approaches, namely the Jacobi Spectral Galerkin (JSG) and Iterated Jacobi Spectral Galerkin (IJSG) techniques, designed specifically to solve a system of linear Volterra integral equations (SLVIEs). These equations involve mixed-type kernels, incorporating both weakly singular (WS) and smooth kernels simultaneously. Jacobi polynomial-based Galerkin and Iterated Galerkin (IG) techniques have been used to tackle these integral equations. Initially, the existence and uniqueness of solutions for both the Galerkin and IG methods are established. Later, the convergence analysis is carried out for smooth and non smooth solution cases. The major attraction of this paper is its exploration of non-smooth solutions, alongside achieving superconvergent results, distinguishing it from other articles. Improved convergence rates are achieved by the IJSG method over the JSG method. The theoretical results are numerically validated.

Suggested Citation

  • Kumar, Rakesh & Kant, Kapil & Rathish Kumar, B.V., 2026. "Superconvergent spectral methods for system of modified Volterra Integral Equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 239(C), pages 115-134.
    • Handle: RePEc:eee:matcom:v:239:y:2026:i:c:p:115-134
    DOI: 10.1016/j.matcom.2025.05.011
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    References listed on IDEAS

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    1. Nigam, Ritu & Nahid, Nilofar & Nelakanti, Gnaneshwar, 2024. "Non-linear integral equations on unbounded domain with global polynomials," Applied Mathematics and Computation, Elsevier, vol. 471(C).
    2. Kumar, Rakesh & Rathish Kumar, B.V., 2024. "Superconvergent scheme for a system of Green nonlinear Fredholm integral equations," Applied Mathematics and Computation, Elsevier, vol. 479(C).
    3. Chakraborty, Samiran & Kant, Kapil & Nelakanti, Gnaneshwar, 2021. "Approximation methods for system of linear Fredholm integral equations of second kind," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    4. Samad Noeiaghdam & Denis Sidorov & Abdul-Majid Wazwaz & Nikolai Sidorov & Valery Sizikov, 2021. "The Numerical Validation of the Adomian Decomposition Method for Solving Volterra Integral Equation with Discontinuous Kernels Using the CESTAC Method," Mathematics, MDPI, vol. 9(3), pages 1-15, January.
    5. Stephen M. Zemyan, 2012. "Linear Systems of Integral Equations," Springer Books, in: The Classical Theory of Integral Equations, edition 127, chapter 0, pages 287-310, Springer.
    Full references (including those not matched with items on IDEAS)

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