| 释义 |
Integral EquationIf the limits are fixed, an integral equation is called a Fredholm integral equation. If one limit is variable, it iscalled a Volterra integral equation. If the unknown function is only under the integral sign, the equation is said to be ofthe ``first kind.'' If the function is both inside and outside, the equation is called of the ``second kind.'' A Fredholmequation of the first kind is of the form
 | (1) |
 | (2) |
 | (3) |
 | (4) |
 are known as Kernels. Integral equations may be solved directlyif they are Separable. Otherwise, a Neumann Series must be used.
A Kernel is separable if
 | (5) |
where
 | (7) |
 and integrate over  .
 | (8) |
 . Now define
so (8) becomes
 | (11) |
 | (12) |
 | (13) |
 | (14) |
References
 Integral EquationsCorduneanu, C. Integral Equations and Applications. Cambridge, England: Cambridge University Press, 1991. Davis, H. T. Introduction to Nonlinear Differential and Integral Equations. New York: Dover, 1962. Kondo, J. Integral Equations. Oxford, England: Clarendon Press, 1992. Lovitt, W. V. Linear Integral Equations. New York: Dover, 1950. Mikhlin, S. G. Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, 2nd rev. ed. New York: Macmillan, 1964. Mikhlin, S. G. Linear Integral Equations. New York: Gordon & Breach, 1961. Pipkin, A. C. A Course on Integral Equations. New York: Springer-Verlag, 1991. Porter, D. and Stirling, D. S. G. Integral Equations: A Practical Treatment, from Spectral Theory to Applications. Cambridge, England: Cambridge University Press, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Integral Equations and Inverse Theory.'' Ch. 18 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 779-817, 1992. Tricomi, F. G. Integral Equations. New York: Dover, 1957.
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