APPROXIMATION OF WEAKLY SINGULAR NON-LINEAR VOLTERRA-URYSOHN INTEGRAL EQUATIONS BY PIECEWISE POLYNOMIAL PROJECTION METHODS BASED ON GRADED MESH

Ritu Nigam; Kapil Kant; BV Rathish Kumar; Gnaneshwar Nelakanti

doi:10.11948/20220147

2023 Volume 13 Issue 3

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Ritu Nigam, Kapil Kant, BV Rathish Kumar, Gnaneshwar Nelakanti. APPROXIMATION OF WEAKLY SINGULAR NON-LINEAR VOLTERRA-URYSOHN INTEGRAL EQUATIONS BY PIECEWISE POLYNOMIAL PROJECTION METHODS BASED ON GRADED MESH[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1359-1387. doi: 10.11948/20220147

Citation:

Ritu Nigam, Kapil Kant, BV Rathish Kumar, Gnaneshwar Nelakanti. APPROXIMATION OF WEAKLY SINGULAR NON-LINEAR VOLTERRA-URYSOHN INTEGRAL EQUATIONS BY PIECEWISE POLYNOMIAL PROJECTION METHODS BASED ON GRADED MESH[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1359-1387. doi: 10.11948/20220147

APPROXIMATION OF WEAKLY SINGULAR NON-LINEAR VOLTERRA-URYSOHN INTEGRAL EQUATIONS BY PIECEWISE POLYNOMIAL PROJECTION METHODS BASED ON GRADED MESH

1.
Department of Mathematics, Indian Institute of Technology Kharagpur, India-721302
2.
Department of Applied Sciences, ABV-Indian Institute of Information Technology and Management, Gwalior, 474015, India

Author Bio: Email: ritunigamiitkgp@gmail.com(R. Nigam); Email: bvrk@iitk.ac.in(B. Kumar); Email: gnanesh@maths.iitkgp.ac.in(G. Nelakanti)
Corresponding author: Email: sahukapil8@gmail.com(K. Kant)

Abstract

Abstract

In this article, we address the approximation solution of Volterra-Urysohn integral equations which involves weakly singular kernels. In order to get better convergence rates, projection methods namely Galerkin and multi Galerkin methods, along with their iterated versions are used in the space of piecewise polynomials subspaces based on the graded mesh. In addition, we compute the superconvergence results for the proposed integral equation and show that iterated Galerkin method outperforms Galerkin method in terms of order of convergence. Further, we demonstrate numerical examples to verify the proposed theoretical framework.
- Superconvergence results /
- Volterra integral equations with weakly singular kernel /
- Galerkin method /
- Multi-Galerkin method /
- piecewise polynomials
MSC: 45B05, 45G10, 65R20

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References

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