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Legendre spectral Galerkin and multi-Galerkin methods for nonlinear Volterra integral equations of Hammerstein type

  • Moumita MandalEmail author
  • Gnaneshwar Nelakanti
Original Research Paper
  • 3 Downloads

Abstract

In this paper, we discuss the superconvergence of the Galerkin solutions for second kind nonlinear integral equations of Volterra–Hammerstein type with a smooth kernel. Using Legendre polynomial bases, we obtain order of convergence \({\mathcal{O}}(n^{-r})\) for the Legendre Galerkin method in both \(L^2\)-norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the kernel. The iterated Legendre Galerkin solutions converge with the order \({\mathcal{O}}(n^{-2r}),\) whose convergence order is the same as that of the multi-Galerkin solutions. We also prove that iterated Legendre multi-Galerkin method has order of convergence \({\mathcal{O}}(n^{-3r})\) in both \(L^2\)-norm and infinity norm. Numerical examples are given to demonstrate the efficacy of Galerkin and multi-Galerkin methods.

Keywords

Volterra–Hammerstein integral equations Smooth kernels Legendre polynomial Galerkin method Multi-Galerkin method Superconvergence rates 

Mathematics Subject Classification

45B05 45G10 65R20 

Notes

Compliance with ethical standards

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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Copyright information

© Forum D'Analystes, Chennai 2019

Authors and Affiliations

  1. 1.Mathematics Department, SASVIT UniversityVelloreIndia
  2. 2.Mathematics DepartmentIIT KharagpurKharagpurIndia

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