Spectral approximation methods for Fredholm integral equations with non-smooth kernels
Abstract
In this paper, polynomially based projection and modified projection methods for approximating the solution of Fredholm integral equations with a kernel of Green’s function type are studied. The projection is either an orthogonal projection or an interpolatory projection using Legendre polynomial basis. The orders of convergence of these methods and those of superconvergence of the iterated versions are analysed. A numerical example is given to illustrate the theoretical results.
Keyword : Fredholm integral equation, orthogonal projection, interpolatory projection, Legendre polynomial, superconvergence
How to Cite
Allouch, C., Sbibih, D., & Tahrichi, M. (2022). Spectral approximation methods for Fredholm integral equations with non-smooth kernels. Mathematical Modelling and Analysis, 27(4), 652–667. https://doi.org/10.3846/mma.2022.16029
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
M. Ahues, A. Largillier and B. Limaye. Spectral Computations for Bounded Operators. Chapman and Hall/CRC, New York, 2001. https://doi.org/10.1201/9781420035827
C. Allouch, D. Sbibih and M. Tahrichi. Legendre superconvergent Galerkin-collocation type methods for Hammerstein equations. Journal of Computational and Applied Mathematics, 353:253–264, 2019. https://doi.org/10.1016/j.cam.2018.12.040
K. Atkinson. The numerical solution of integral equations of the second kind. Cambridge University Press, 1997. https://doi.org/10.1017/CBO9780511626340
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. On the discrete Galerkin method for Fredholm integral equations of the second kind. IMA Journal of Numerical Analysis, 9(3):385–403, 1989.
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang. Spectral Methods, Fundamentals in Single Domains. Springer-Verlag, Berlin, 2006. https://doi.org/10.1007/978-3-540-30726-6
F. Chatelin and R. Lebbar. The iterated projection solution for the Fredholm integral equation of second kind. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 22(4):439–451, 1981. https://doi.org/10.1017/S0334270000002782
F. Chatelin and R. Lebbar. Superconvergence results for the iterated projection method applied to a Fredholm integral equation of the second kind and the corresponding eigenvalue problem. Journal of Integral Equations, 6(1):71–91, 1984.
M. Golberg. Improved convergence rates for some discrete Galerkin methods. Journal of Integral Equations and Applications, 8(3):307–335, 1996. https://doi.org/10.1216/jiea/1181075955
M.A. Golberg. Discrete polynomial-based Galerkin methods for Fredholm integral equations. Journal of Integral Equations and Applications, 6(2):197–211, 1994. https://doi.org/10.1216/jiea/1181075804
M.A. Golberg and C.S. Chen. Discrete projection methods for integral equations. Computational Mechanics Publications, Boston, 1997.
R.P. Kulkarni. A superconvergence result for solutions of compact operator equations. Bulletin of the Australian Mathematical Society, 68(3):517–528, 2003. https://doi.org/10.1017/S0004972700037916
R.P. Kulkarni. On improvement of the iterated Galerkin solution of the second kind integral equations. Journal of Numerical Mathematics, 13(3):205–218, 2005. https://doi.org/10.1515/156939505774286139
R.P. Kulkarni and N. Gnaneshwar. Iterated discrete polynomially based Galerkin methods. Applied Mathematics and Computation, 146(1):153–165, 2003. https://doi.org/10.1016/S0096-3003(02)00533-7
G. Long, M.M. Sahani and G. Nelakanti. Polynomially based multiprojection methods for Fredholm integral equations of the second kind. Applied Mathematics and Computation, 215(1):147–155, 2009. https://doi.org/10.1016/j.amc.2009.04.053
B.L. Panigrahi and G. Nelakanti. Legendre Galerkin method for weakly singular Fredholm integral equations and the corresponding eigenvalue problem. Journal of Applied Mathematics and Computing, 43(1):175–197, 2013. https://doi.org/10.1007/s12190-013-0658-0
L.L. Schumaker. Theory of Ordinary Differential Equations. John Wiley & Sons, New York, 1981.
I. Sloan. Improvement by iteration for compact operator equations. Mathematics of Computation, 30(136):758–764, 1976.
T. Tang, X. Xu and J. Cheng. On spectral methods for Volterra type integral equations and the convergence analysis. Journal of Computational Mathematics, 26(6):825–837, 2008.
C. Allouch, D. Sbibih and M. Tahrichi. Legendre superconvergent Galerkin-collocation type methods for Hammerstein equations. Journal of Computational and Applied Mathematics, 353:253–264, 2019. https://doi.org/10.1016/j.cam.2018.12.040
K. Atkinson. The numerical solution of integral equations of the second kind. Cambridge University Press, 1997. https://doi.org/10.1017/CBO9780511626340
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. On the discrete Galerkin method for Fredholm integral equations of the second kind. IMA Journal of Numerical Analysis, 9(3):385–403, 1989.
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang. Spectral Methods, Fundamentals in Single Domains. Springer-Verlag, Berlin, 2006. https://doi.org/10.1007/978-3-540-30726-6
F. Chatelin and R. Lebbar. The iterated projection solution for the Fredholm integral equation of second kind. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 22(4):439–451, 1981. https://doi.org/10.1017/S0334270000002782
F. Chatelin and R. Lebbar. Superconvergence results for the iterated projection method applied to a Fredholm integral equation of the second kind and the corresponding eigenvalue problem. Journal of Integral Equations, 6(1):71–91, 1984.
M. Golberg. Improved convergence rates for some discrete Galerkin methods. Journal of Integral Equations and Applications, 8(3):307–335, 1996. https://doi.org/10.1216/jiea/1181075955
M.A. Golberg. Discrete polynomial-based Galerkin methods for Fredholm integral equations. Journal of Integral Equations and Applications, 6(2):197–211, 1994. https://doi.org/10.1216/jiea/1181075804
M.A. Golberg and C.S. Chen. Discrete projection methods for integral equations. Computational Mechanics Publications, Boston, 1997.
R.P. Kulkarni. A superconvergence result for solutions of compact operator equations. Bulletin of the Australian Mathematical Society, 68(3):517–528, 2003. https://doi.org/10.1017/S0004972700037916
R.P. Kulkarni. On improvement of the iterated Galerkin solution of the second kind integral equations. Journal of Numerical Mathematics, 13(3):205–218, 2005. https://doi.org/10.1515/156939505774286139
R.P. Kulkarni and N. Gnaneshwar. Iterated discrete polynomially based Galerkin methods. Applied Mathematics and Computation, 146(1):153–165, 2003. https://doi.org/10.1016/S0096-3003(02)00533-7
G. Long, M.M. Sahani and G. Nelakanti. Polynomially based multiprojection methods for Fredholm integral equations of the second kind. Applied Mathematics and Computation, 215(1):147–155, 2009. https://doi.org/10.1016/j.amc.2009.04.053
B.L. Panigrahi and G. Nelakanti. Legendre Galerkin method for weakly singular Fredholm integral equations and the corresponding eigenvalue problem. Journal of Applied Mathematics and Computing, 43(1):175–197, 2013. https://doi.org/10.1007/s12190-013-0658-0
L.L. Schumaker. Theory of Ordinary Differential Equations. John Wiley & Sons, New York, 1981.
I. Sloan. Improvement by iteration for compact operator equations. Mathematics of Computation, 30(136):758–764, 1976.
T. Tang, X. Xu and J. Cheng. On spectral methods for Volterra type integral equations and the convergence analysis. Journal of Computational Mathematics, 26(6):825–837, 2008.