Superconvergence Results for Volterra-Urysohn Integral Equations of Second Kind

  • Moumita MandalEmail author
  • Gnaneshwar Nelakanti
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 655)


In this paper, we consider the Galerkin method to approximate the solution of Volterra-Urysohn integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We show that the exact solution is approximated with the order of convergence \(\mathcal {O}(h^{r})\) for the Galerkin method, whereas the iterated Galerkin solutions converge with the order \(\mathcal {O}(h^{2r})\) in uniform norm, where h is the norm of the partition and r is the smoothness of the kernel. For improving the accuracy of the approximate solution of the integral equation, the multi-Galerkin method is also discussed here and we prove that the exact solution is approximated with the order of convergence \(\mathcal {O}(h^{3r})\) in uniform norm for iterated multi-Galerkin method. Numerical examples are given to illustrate the theoretical results.


Volterra-Urysohn integral equations Smooth kernels Galerkin method Multi-Galerkin method Piecewise polynomials Superconvergence rates 


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© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of TechnologyKharagpurIndia

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