Discrete modified projection methods for Urysohn integral equations with Green’s function type kernels
In the present paper we consider discrete versions of the modified projection methods for solving a Urysohn integral equation with a kernel of the type of Green’s function. For r ≥ 0, a space of piecewise polynomials of degree ≤ r with respect to an uniform partition is chosen to be the approximating space. We define a discrete orthogonal projection onto this space and replace the Urysohn integral operator by a Nyström approximation. The order of convergence which we obtain for the discrete version indicates the choice of numerical quadrature which preserves the orders of convergence in the continuous modified projection methods. Numerical results are given for a specific example.
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K.E. Atkinson, I. Graham and I. Sloan. Piecewise continuous collocation for integral equations. SIAM Journal on Numerical Analysis, 20(1):172–186, 1983. https://doi.org/10.1137/0720012
K.E. Atkinson and F.A. Potra. Projection and iterated projection methods for nonlinear integral equations. SIAM Journal on Numerical Analysis, 24(6):1352– 1373, 1987. https://doi.org/10.1137/0724087
K.E. Atkinson and F.A. Potra. The discrete Galerkin method for nonlinear integral equations. Journal of Integral Equations and Applications, 1(1):17–54, 1988. https://doi.org/10.1216/JIE-1988-1-1-17
L. Grammont. A Galerkin’s perturbation type method to approximate a fixed point of a compact operator. International Journal of Pure and Applied Mathematics, 69(1):1–14, 2011.
L. Grammont and R.P. Kulkarni. A superconvergent projection method for nonlinear compact operator equations. C.R. Acad. Sci. Paris, 342(3):215–218, 2006. https://doi.org/10.1016/j.crma.2005.11.011
L. Grammont, R.P. Kulkarni and T.J. Nidhin. Modified projection method for Urysohn integral equations with non-smooth kernels. Journal of Computational and Applied Mathematics, 294:309–322, 2016. https://doi.org/10.1016/j.cam.2015.08.020
M.A. Krasnosel’skii. Topological methods in the theory of nonlinear integral equations. Macmillan, New York, 1964.
M.A. Krasnosel’skii, G.M. Vainikko, P.P. Zabreiko, Y.B. Ruticki and V.V. Stet’senko. Approximate solution of operator equations. P. Noordhoff, Groningen, 1972.
M.A. Krasnosel’skii and P.P. Zabreiko. Geometrical methods of nonlinear analysis. Springer-Verlag, Berlin, 1984.
R.P. Kulkarni and G. Rakshit. Discrete modified projection method for Urysohn integral equations with smooth kernels. Applied Numerical Mathematics, 126:180–198, 2018. https://doi.org/10.1016/j.apnum.2017.12.008