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Double Approximation Methods for the Solution of Fredholm Integral Equations

  • P. M. Anselone
  • J. W. Lee
Part of the International Series of Numerical Mathematics book series (ISNM, volume 30)

Abstract

A double approximation method with error bounds is presented for the numerical solution of Fredholm integral equations. The method embraces standard finite rank and projections methods which, when applied to the Fredholm equation (I − K)x = y, yield approximations (I − Km)xm = y equivalent to matrix problems. However, the matrix elements are integrals which almost always must be done numerically. This is equivalent to a double approximation scheme (I − K)x n m = y, where x n m is the solution computed numerically. Under typical conditions, K n m → K pointwise as m, n → = ∞, K is compact, {K n m } is collectively compact, and an available operator approximation theory yields convergence theorems and error bounds for the approximate solutions. Optimal choices for m relative to n are also considered.

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References

  1. 1.
    Anselone, P. M.: Collectively Compact Operator Approximation Theory and Applications to Integral Equations. Englewood Cliffs, New Jersey, Prentice Hall, Inc. 1971.zbMATHGoogle Scholar
  2. 2.
    Atkinson, K. E.: The Numerical Solution of Fredholm Integral Equations of the Second Kind. SIAM J. Num. Anal. 4 (1967), 337-348.zbMATHCrossRefGoogle Scholar
  3. 3.
    Ikebe, Y.: The Galerkin Method for the Numerical Solution of Fredholm Integral Equations of the Second Kind. SIAM Review 14 (1972), 465-491.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Isaacson, E. and Keller, H. B.: Analysis of Numerical Methods. New York, John Wiley and Sons, Inc. 1966.zbMATHGoogle Scholar
  5. 5.
    Phillips, J. L.: Error Analysis for Direct Linear Integral Equations Methods. Math. of Comp. 27 (1973), 849-859.zbMATHCrossRefGoogle Scholar
  6. 6.
    Prenter, P. M.: A Collocation Method for the Numerical Solution of Integral Equations. SIAM J. Numer. Anal. 10 (1973), 570-581.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1976

Authors and Affiliations

  • P. M. Anselone
    • 1
  • J. W. Lee
    • 1
  1. 1.Department of MathematicsOregon State UniversityCorvallisUSA

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