On Some Numerical Methods for Solving Fredholm Integral Equations with Continuous Kernel
Keywords:
Fredholm integral equation, Collocation method, Galerkin method, Boundary integral equationsAbstract
Fredholm integral equations with continues kernels arise for instance in the boundary integral equations either directly or as a result of treating different kinds of singular kernels using some regularization techniques. Here, we show a comparison of the convergence of two well-known numerical methods for solving integral equations. These methods are the collocation method and the Galerkin method. An illustrated examples for second kind Fredholm integral equations of continuous kernel show that the collocation method seems to converge faster than the Galerkin method.
References
H. Hochstadt. Integral Equations. Wiley, New York, 1975.
N. I. Muskhelishvili. Singular Integral Equations. Mir Publisher, Moscow, 1953.
G. Ya. Popov. Contact Problems for a Linearly Deformable Base. Kiev. Odessa, 1982.
M. A. Abdou. Integral Equation of Mixed Type and Integrals of Orthogonal Polynomials. J. comp. Appl. Math., vol. 138, pp.273-285, 2002.
R. P. Kanwal. Linear Integral Equations Theory and Technique. Boston, 1996.
V. M. Alexandrov, E. V. Kovalenko. Problems with Mixed Boundary Conditions in Continum Mechanics. Nauka, Moscow.1986.
C. D. Green. Integral Equation Methods. Nelson N. Y. 1960.
M. A. Abdou, N. Y. Ezz-Eldin. Kreins Method with Certain Singular Kernel for Solving the Integral Equation of the First Kind. Peri. Math. Hung., vol. 28 no.2, pp.143-149, 1994.
M. A. Golberg. Solution Methods for Integral Equations. Camberdge, N. Y. 1979.
P. Schiavone, C. Costanda, A. Mioduchowski. Integral Methods in Science and Engineering. Birkhauser, Boston, 2002.
L. Hacia. Approximate Solution of Hammerstein Equations. Appl. Anal., vol. 50, pp.277-284, 1993.
L. Hacia. Solving Nonlinear Integral Equations by Projection-iteration methods, Nonlin. Vib. Probp., vol. 25, pp.135-141, 1993.
S. Kummar. A Discrete Collection-Type Method for Hammerstein Equations. SIAMJ. Number. Anal., vol. 25, pp. 328-341, 1988.
S. Kummar, I. H. Sloan. A New Collection Type Method for Hammerstein Integral Equations, Math. Comput., vol. 48, pp.585-593, 1987.
H. Brunner. A. Makroglou. R. K. Miller. On Mixed Collection Method for Volterra Integral Equation with Periodic Solution. Appl. Number. Math., vol. 24, pp. 115-130, 1997.
A. Yu. Lucka. Projection – Iteration Methods For Solving Differential And Integral Equations, Dakl. Acad. Nauka SSSR, vol. 45, pp. 113-118, 1980.
H. Kaneko, A Projection Method for Solving Fredholm Integral Equations of The Second Kind, Appl. Num. Math., vol. 5, pp.333-344, 1989.
J. I. Frankel. A Galerkin Solution to a Regularized Cauchy Singular Integral-Differential Equation. Quart. Appl. Math. L. vol. 111, no.2, pp. 245-258, 1995.
M. A. Abdou. Integral Equations with Cauchy Kernel in The Contact Problem. Korean J. Comut. Appl. Math., vol. 7, no.3, pp.663-672, 2000.
K. E. Atkinson . A Survey of Numerical Method for the Solution of Fredholm Integral Equation of the Second Kind. Philadelphia, 1976.
K. E. Atkinson. The Numerical Solution of Integral Equation of the Second Kind, Cambridge, 1997.
L. M. Delves, J. L. Mohamed. Computational Methods for Integral equations. Cambridge, London 1985.
C. T. H. Baker. Treatment of Integral Equations by Numerical Methods. Academic Press, 1982.
M. A. Golberg. Numerical Solution of Integral Equations. SIAM. London, 1990.
C.A. Brebbia, J.C.F. Tells, L.C. Wrobel. Boundary Element Techniques. Springer- Verlag, Berlin and New York, 1984.
H. Ben Hamdin. On Some Techniques of Treating Weakly Singular Integrals. Sirte University Scientific Journal (Applied Sciences), vol. 5 , no. 2, pp.23-36, 2015.
J. A. Dzhuraev. Methods of Singular Integral Equations. London, New York, 1992.
M. Tanaka, V. Sladek, J. Sladek. Regularization Techniques Applied to Boundary Element Methods. Applied Mechanics Reviews, vol.47, no.10, pp. 457-499, 1994.
