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Integral Equations

  • George Adomian
Chapter
Part of the Fundamental Theories of Physics book series (FTPH, volume 60)

Abstract

Integral equations of Volterra type arise quite naturally in physical applications modelled by initial-value problems. Consider the linear Volterra equation of the second kind. (Fredholm equations of the second kind which are associated with boundary value problems for a fmite interval [a,b], are similar except that the upper limit is b.)
with a ≤ x, y ≤ b, and let λ = 1. Using decomposition , we identify φ 0 = f(x) assuming f(x) ≠ 0,
then and write as the solution, or with an m-term approximant . In some cases, exact solutions are determinable. Consider an example:
Then
Thus the two-term approximant is
or φ = sin x as is easily verified either by calculating more terms or by substitution.

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Reference

  1. 1.
    Y. Cherruault, G. Saccomandi, and B. Some, New Results for the Convergence of Adomian’s Method Applied to Integral Equations, Math. Comput. Modelling, 16, (8593) (1992).Google Scholar

Suggested Reading

  1. 1.
    G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, New York (1986).Google Scholar
  2. 2.
    B. Some, Some Recent Numerical Methods for Solving Hammersteins Integral Equations, Math. Comput. Modelling, to appear.Google Scholar
  3. 3.
    B. Some, A New Computational Method for Solving Integral Equations, submitted for publication.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1994

Authors and Affiliations

  • George Adomian
    • 1
  1. 1.General Analytics CorporationAthensUSA

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