A collocation method for Fredholm integral equations of the second kind

Authors:
E. N. Houstis and T. S. Papatheodorou

Journal:
Math. Comp. **32** (1978), 159-173

MSC:
Primary 65R05; Secondary 45B05

DOI:
https://doi.org/10.1090/S0025-5718-1978-0458967-8

MathSciNet review:
0458967

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Abstract | References | Similar Articles | Additional Information

Abstract: An interpolation scheme based on piecewise cubic polynomials with Gaussian points as interpolation points is analyzed and applied to the solution of Fredholm equations of the second kind.

**[1]**Kendall E. Atkinson,*A survey of numerical methods for the solution of Fredholm integral equations of the second kind*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. MR**0483585****[2]**Kendall Atkinson,*An automatic program for linear Fredholm integral equations of the second kind*, ACM Trans. Math. Software**2**(1976), no. 2, 154–171. MR**0418489**, https://doi.org/10.1145/355681.355686**[3]**G. Birkhoff, M. H. Schultz, and R. S. Varga,*Piecewise Hermite interpolation in one and two variables with applications to partial differential equations*, Numer. Math.**11**(1968), 232–256. MR**0226817**, https://doi.org/10.1007/BF02161845**[4]**Carl de Boor and Blâir Swartz,*Collocation at Gaussian points*, SIAM J. Numer. Anal.**10**(1973), 582–606. MR**0373328**, https://doi.org/10.1137/0710052**[5]**L. M. Delves (ed.),*Numerical solution of integral equations*, Clarendon Press, Oxford, 1974. A collection of papers based on the material presented at a joint Summer School in July 1973, organized by the Department of Mathematics, University of Manchester, and the Department of Computational and Statistical Science, University of Liverpool. MR**0464624****[6]**Jim Douglas Jr. and Todd Dupont,*A finite element collocation method for quasilinear parabolic equations*, Math. Comp.**27**(1973), 17–28. MR**0339508**, https://doi.org/10.1090/S0025-5718-1973-0339508-8**[7]**Jim Douglas Jr. and Todd Dupont,*A finite element collocation method for the heat equation*, Symposia Mathematica, Vol. X (Convegno di Analisi Numerica, INDAM, Rome, 1972) Academic Press, London, 1972, pp. 403–410. MR**0373329****[8]**Jim Douglas Jr. and Todd Dupont,*Collocation methods for parabolic equations in a single space variable*, Lecture Notes in Mathematics, Vol. 385, Springer-Verlag, Berlin-New York, 1974. Based on 𝐶¹-piecewise-polynomial spaces. MR**0483559****[9]**Elias Houstis,*A collocation method for systems of nonlinear ordinary differential equations*, J. Math. Anal. Appl.**62**(1978), no. 1, 24–37. MR**0488785**, https://doi.org/10.1016/0022-247X(78)90215-9**[10]**E. N. Houstis,*Application of method of collocation on lines for solving nonlinear hyperbolic problems*, Math. Comp.**31**(1977), no. 138, 443–456. MR**0443379**, https://doi.org/10.1090/S0025-5718-1977-0443379-2**[11]**E. N. Houstis,*Collocation methods for linear elliptic problems*, BIT**18**(1978), no. 3, 301–310. MR**508331**, https://doi.org/10.1007/BF01930899**[12]**E. N. HOUSTIS & T. S. PAPATHEODOROU,*Piecewise Cubic Hermite Interpolation at the Gaussian Points*, Purdue University, CSD-TR 199, July 1976.**[13]**Martin H. Schultz,*Spline analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR**0362832**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0458967-8

Keywords:
Collocation method,
integral equations

Article copyright:
© Copyright 1978
American Mathematical Society