Algorithm 876: Solving Fredholm Integral Equations of the Second Kind in Matlab

ABSTRACT

We present here the algorithms and user interface of a Matlab program, Fie, that solves numerically Fredholm integral equations of the second kind on an interval [a,b] to a specified, modest accuracy. The kernel function K(s,t) is moderately smooth on [a,b] ×[a,b] except possibly across the diagonal s = t. If the interval is finite, provides for kernel functions that behave in a variety of ways across the diagonal, that is, K(s,t) may be smooth, have a discontinuity in a low-order derivative, have a logarithmic singularity, or have an algebraic singularity. Fie also solves a large class of integral equations with moderately smooth kernel function on [0, ∞ ).

REFERENCES

• 1: Atkinson, K. 1967. The numerical solution of Fredholm integral equations of the second kind. SIAM J. Num. Anal. 4, 337--348.
• 2: Atkinson, K. 1976a. A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind. SIAM, Philadelphia, PA.
• 3: Kendall Atkinson, An Automatic Program for Linear Fredholm Integral Equations of the Second Kind, ACM Transactions on Mathematical Software (TOMS), v.2 n.2, p.154-171, June 1976 [doi>10.1145/355681.355686]
• 4: Atkinson, K. 1997. The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge, UK.
• 5: Atkinson, K. and Han, W. 2005. Theoretical Numerical Analysis: A Functional Analysis Framework 2^nd Ed. Springer-Verlag, Berlin, Germany.
• 6: Cubillos, P. 1984. Integral operators with Green's function type kernel. J. Comp. Appl. Math. 10, 25--31.
• 7: El-Gendi, S. E. 1969. Chebyshev solution of differential, integral and integro-differential equations. Comput. J. 12, 282--287.
• 8: Fox, L. 1962. Numerical Solution of Ordinary and Partial Differential Equations. Pergamon Press, London, UK.
• 9: Kirkwood, J. and Riseman, J. 1948. The intrinsic viscosities and diffusion constants of flexible macromolecules in solution. J. Chem. Phys. 16, 565--573.
• 10: Love, E. 1949. The electrostatic field of two equal circular conducting disks. Quart. J. Mech. Appl. Math. 2, 428--451.
• 11: Matlab. The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760.
• 12: Mikhlin, S. G. and Smolitskiy, K. L. 1967. Approximate Methods for Solution of Differential and Integral Equations. Elsevier, London, UK.
• 13: Sloan, I. 1981. Quadrature methods for integral equations of the second kind over infinite intervals. Math. Comp. 36, 511--523.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.9 Integral Equations

Additional Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE

General Terms:
Algorithms

Keywords:
Matlab, Numerical solution