Document Type : Research Paper

Authors

1 University Mohammed I, Team MSC, FPN, LAMAO Laboratory, Nador, 62000, Morocco.

2 University Mohammed I, Team ANAA, EST, LANO Laboratory, Oujda, 60000, Morocco.

Abstract

In this paper, polynomial-based superconvergent degenerate kernel and {Nyström} methods for solving {Fredholm} integral equations of the second kind with  the smooth kernel are studied. By using an interpolatory projection based on Legendre polynomials of degree $\leq n,$ we analyze the convergence of these methods and we establish superconvergence results for their iterated versions. Two numerical examples are given to illustrate the theoretical estimates.

Keywords

[1] M. Ahues, A. Largillier and B. Limaye, Spectral Computations for Bounded Operators, CRC Press, (2001).
[2] C. Allouch, A. Boujraf and M. Tahrichi, Discrete superconvergent degenerate kernel method for Fredholm integral equations, Math. Comput. Simul., 164 (2019), pp. 24-32.
[3] C. Allouch, P. Sablonnière, D. Sbibih and M. Tahrichi, Superconvergent Nyström and degenerate kernel methods for the numerical solution of integral equations of the second kind, J. Int. Eqns. Appl., 24 (2012), pp. 463-485.
[4] C. Allouch and M. Tahrichi, Discrete superconvergent Nyström method for integral equations and eigenvalue problems, Math. Comput. Simul., 118 (2015), pp. 17-29.
[5] K.E. Atkinson, The numerical solution of integral equations of the second kind, Cambridge University Press, (1997).
[6] K.E. Atkinson and W. Han, Theoretical numerical analysis, Springer Verlag, Berlin (2nd edition) (2005).
[7] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods, Springer-Verlag, Berlin, Heidelberg, (2006).
[8] M. Golberg, Discrete Polynomial-Based Galerkin Methods for Fredholm Integral Equations, J. Int. Eqns. Appl., 6 (1994), pp. 197-211.
[9] P.Junghanns, G.Mastroianni and I.Notarangelo, Weighted Polynomial Approximation and Numerical Methods for Integral Equations, Springer International Publishing, (2021).
[10] R.P. Kulkarni, A superconvergence result for solutions of compact operator equations, Bull. Aust. Math. Soc., 68 (2003), pp. 517-528.
[11] R.P. Kulkarni and G. Nelakanti, Iterated discrete polynomially based Galerkin methods J. Appl. Math. Comp., 146 (2003), pp. 153-165.
[12] G. Long, G. Nelakanti and M.M. Sahani, Polynomially based multi-projection methods for Fredholm integral equations of the second kind, Appl. Math. Comput., 215 (2009), pp. 147-155.
[13] G.Mastroianni and G.V. Milovanovic, Interpolation processes: basic theory and applications, Springer Monographs in Mathematics, Springer, Berlin, (2009).
[14] G. Nelakanti and B.L. Panigrahi, Legendre Galerkin method for weakly singular Fredholm integral equations and the corresponding eigenvalue problem., Appl. Math. Comput., 43 (2013), pp. 175-197.
[15] B.L. Panigrahi, G. Long and G. Nelakanti, Legendre multi-projection methods for solving eigenvalue problems for a compact integral operator, Math. Comput. Simul., 239 (2013), pp. 135-151.
[16] I.H. Sloan, Improvement by iteration for compact operator equations, Math. Comput., 30 (136), (1976), pp. 758-764.