Abstract
We use spline quasi-interpolating projectors on a bounded interval for the numerical solution of linear Fredholm integral equations of the second kind by Galerkin, Kantorovich, Sloan and Kulkarni schemes. We get theoretical results related to the convergence order of the methods, in case of quadratic and cubic spline projectors, and we describe computational aspects for the construction of the approximate solutions. Finally, we give several numerical examples, that confirm the theoretical results and show that higher orders of convergence can be obtained by Kulkarni’s scheme.
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Allouch, Ch., Sablonnière, P., Sbibih, D.: Solving Fredholm integral equations by approximating kernels by spline quasi-interpolants. Numer. Algorithms 56, 437–453 (2011)
Allouch, Ch., Sablonnière, P., Sbibih, D.: A modified Kulkarni’s method based on a discrete spline quasi-interpolant. Math. Comput. Simul. 81, 1991–2000 (2011)
Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)
Barrera, D., Sablonnière, P.: Numerical computation of univariate integrals with B-spline weights. I. Gauss type quadrature formulas, II. Product type quadrature formulas. University of Rennes (2013, in preparation)
de Boor, C.: A Practical Guide to Splines, Revised edn. Springer, Berlin (2001)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)
Kulkarni, R.: A superconvergence result for solutions of compact operator equations. Bull. Austral. Math. Soc. 68, 517–528 (2003)
Kulkarni, R.: On improvement of the iterated Galerkin solution of the second kind integral equations. J. Numer. Math. 13, 205–218 (2005)
Kulkarni, R., Grammont, L.: Extrapolation using a modified projection method. Numer. Funct. Anal. Optim. 30, 1339–1359 (2009)
Sablonnière, P.: Univariate spline quasi-interpolants and applications to numerical analysis. Rend. Sem. Mat. Univ. Pol. Torino 63, 211–222 (2005)
Sablonnière, P.: A quadrature formula associated with a univariate quadratic spline quasi-interpolant. BIT Numer. Math. 47, 825–837 (2007)
Sbibih, D., Allouch, Ch., Sablonnière, P., Tahrichi, M.: Product integration methods based on discrete spline quasi-interpolants and applications to weakly singular integral equations. J. Comput. Appl. Math. 233, 2855–2866 (2010)
Sablonnière, P., Sbibih, D., Tahrichi, M.: High-order quadrature rules based on spline quasi-interpolants and application to integral equations. Appl. Numer. Math. 62, 507–520 (2012)
Sloan, I.: Improvement by iteration for compact operator equations. Math. Comp. 30, 758–764 (1976)
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Communicated by Tom Lyche.
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Dagnino, C., Remogna, S. & Sablonnière, P. On the solution of Fredholm integral equations based on spline quasi-interpolating projectors. Bit Numer Math 54, 979–1008 (2014). https://doi.org/10.1007/s10543-014-0486-0
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DOI: https://doi.org/10.1007/s10543-014-0486-0