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1988, in Numerical Analysis 1987, Pitman Research Notes in Mathematics vol. 170,D.F. Griffiths and G.A. Watson, eds. (Harlow, Essex, U.K.: Longman Scientific and Technical), pp. 18–38. [4]Smithies, F. 1958, Integral Equations (Cambridge, U.K.: Cambridge University Press).Kanwal, R.P.
1971, Linear Integral Equations (New York: Academic Press).Green, C.D. 1969, Integral Equation Methods (New York: Barnes & Noble).18.1 Fredholm Equations of the Second KindWe desire a numerical solution for f(t) in the equationZbK(t, s)f(s) ds + g(t)f(t) = λ(18.1.1)aThe method we describe, a very basic one, is called the Nystrom method. It requiresthe choice of some approximate quadrature rule:Zby(s) ds =aNXwj y(sj )(18.1.2)j=1Here the set {wj } are the weights of the quadrature rule, while the N points {sj }are the abscissas.What quadrature rule should we use? It is certainly possible to solve integralequations with low-order quadrature rules like the repeated trapezoidal or Simpson’sSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).special quadrature rules, but they are also sometimes blessings in disguise, since theycan spoil a kernel’s smoothing and make problems well-conditioned.In §§18.4–18.7 we face up to the issues of inverse problems. §18.4 is anintroduction to this large subject.We should note here that wavelet transforms, already discussed in §13.10, areapplicable not only to data compression and signal processing, but can also be usedto transform some classes of integral equations into sparse linear problems that allowfast solution.
You may wish to review §13.10 as part of reading this chapter.Some subjects, such as integro-differential equations, we must simply declareto be beyond our scope. For a review of methods for integro-differential equations,see Brunner [4].It should go without saying that this one short chapter can only barely touch ona few of the most basic methods involved in this complicated subject..
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