Integral Equations

  • B. Davies
Part of the Applied Mathematical Sciences book series (AMS, volume 25)


Integral equations in which the unknown function appears in a convolution occur in some important situations. The equation
$$g(x) = f(x) + \lambda \int_a^b {k(x - y)\;g(y)\;dy} $$
, where f(x) and k(x) are given functions and λ a given constant, is an example of a Fredholm integral equation of the second kind. (An equation of the first kind is one in which the unknown function g does not appear outside the integral.) If the upper limit of integration b is replaced by the variable x, then (1) is said to be of Volterra, rather than Fredholm, type. By the change of variables x′ = x − a, y′ = y − a, (1) may then he written
$$G(p) = F(p) + \lambda \;K(p)\;G(p)$$


Integral Equation LAPLACE Transform Fredholm Integral Equation Pair Distribution Function Convolution Equation 
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  1. 1.
    See J. H. Giese, SIAM Review (1963), 5, 1.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Some numerical values for the case a = -2 computed by Padé approximation may be found in L. Fox and E. J. Goodwin, Phil. Trans. Roy. Soc. Lond. (1953), A245 501.Google Scholar
  3. 3.
    N. Mullineux and J. R. Reed, Q. Appl. Math. (1967), 25, 327.zbMATHGoogle Scholar
  4. 4.
    Equations of this type may be solved by the Weiner-Hopf technique (see Section 18). However, we are interested here in a class of problems which can be solved by more elementary methods.Google Scholar
  5. 5.
    We must first take Re(p) a, and then use analytic continuation on the final result to extend it to Re(p) a.Google Scholar
  6. 6.
    As with (29), a process of analytic continuation may be involved.Google Scholar
  7. 7.
    This is the probability of finding two particles at the stated positions. For an infinite uniform system it is a function only of the relative positions of the two.Google Scholar
  8. 8.
    This identification is only valid in the Percus-Yevick approximation.Google Scholar
  9. 9.
    M. S. Wertheim, J. Math. Phys. (1964), 5, 643. The more general case where V(x) ≠ 0 for a ≤ |x| ≤ ℓ R is also analyzed using Laplace transforms.Google Scholar
  10. 10.
    The ensuing procedure is a simple example of the type of argument which is used in the Wiener-Hopf technique (Section 18).Google Scholar
  11. 11.
    Problems 8–13 and some related material may be found in D. O. Reudink, SIAM Review (1967), 9, 4.Google Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • B. Davies
    • 1
  1. 1.The Australian National UniversityCanberraAustralia

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