New method for the numerical solution of the Fredholm linear integral equation on a large interval

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Abstract

The traditional numerical process to tackle a linear Fredholm integral equation on a large interval is divided into two parts, the first is discretization, and the second is the use of the iterative scheme to approach the solutions of the huge algebraic system. In this paper we propose a new method based on constructing a generalization of the iterative scheme, which is adapted to the system of linear bounded operators. Then we don’t discretize the whole system, but only the diagonal part of the system. This system is built by transforming our integral equation. As discretization we consider the product integration method and the Gauss-Seidel iterative method as iterative scheme. We also prove the convergence of this new method. The numerical tests developed show its effectiveness.

About the authors

Samir Lemita

Higher Normal School of Ouargla

Email: lem.samir@gmail.com
PhD, Assistant Professor B.P. 398, Ennacer St., Ouargla 30000, Algeria

Hamza Guebbai

University May 8, 1945 - Guelma

Email: guebaihamza@yahoo.fr
Full Professor B.P. 401, Guelma 24000, Algeria

Ilyes Sedka

University May 8, 1945 - Guelma

Email: di_sedka@esi.dz
Post-Graduate Student B.P. 401, Guelma 24000, Algeria

Mohamed Zine Aissaoui

University May 8, 1945 - Guelma

Email: aissaouizine@gmail.com
Full Professor B.P. 401, Guelma 24000, Algeria

References

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  7. S. Lemita, H. Guebbai, “New process to approach linear Fredholm integral equations defined on large interval”, Asian Eur. J. Math., 12:01 (2019), 1950009
  8. S. Lemita, H. Guebbai, M. Z. Aissaoui, “Generalized Jacobi method for linear bounded operators system”, Comput. Appl. Math., 37:3 (2018), 3967-3980
  9. M. Ahues, A. Largillier, O. Titaud, “The roles of a weak singularity and the grid uniformity in relative error bounds”, Numerical Functional Analysis and Optimization, 22 (2001), 789-814
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