18.1 Fredholm Equations of the Second Kind 791 special quadrature rules,but they are also sometimes blessings in disguise,since they can spoil a kernel's smoothing and make problems well-conditioned. In 8818.4-18.7 we face up to the issues of inverse problems.$18.4 is an introduction to this large subject. We should note here that wavelet transforms.already discussed in $13.10.are applicable not only to data compression and signal processing,but can also be used to transform some classes of integral equations into sparse linear problems that allow fast solution.You may wish to review $13.10 as part of reading this chapter. Some subjects,such as integro-differential equations,we must simply declare to 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 on 鱼君 a few of the most basic methods involved in this complicated subject. ICAL CITED REFERENCES AND FURTHER READING: Delves,L.M.,and Mohamed,J.L.1985,Computational Methods for Integral Equations (Cam- RECIPES bridge,U.K.:Cambridge University Press).[1] Linz,P.1985,Analytical and Numerical Methods for Volterra Equations(Philadelphia:S.I.A.M.). 9 [2] Atkinson,K.E.1976,A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind (Philadelphia:S.I.A.M.).[3] Brunner,H.1988,in Numerica/Analysis 1987,Pitman Research Notes in Mathematics vol.170, D.F.Griffiths and G.A.Watson,eds.(Harlow,Essex,U.K.:Longman Scientific and Tech- nical),pp.18-38.[4 8S是鸟分 Smithies,F.1958,Integra/Equations(Cambridge,U.K.:Cambridge University Press). Kanwal,R.P.1971,Linear Integra/Equations(New York:Academic Press). IENTIFIC Green,C.D.1969,Integral Equation Methods (New York:Barnes Noble). 18.1 Fredholm Equations of the Second Kind (ISBN Numerica 10.621 We desire a numerical solution for f(t)in the equation 43108 (outside Recipes f(t)=入 K(t,s)f(s)ds+g(t) (18.1.1) The method we describe,a very basic one,is called the Nystrom method.It requires the choice of some approximate quadrature rule: N (s)ds-∑w5(s) (18.1.2) j=1 Here the set [wi}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 integral equations with low-order quadrature rules like the repeated trapezoidal or Simpson's18.1 Fredholm Equations of the Second Kind 791 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). special quadrature rules, but they are also sometimes blessings in disguise, since they can 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 an introduction to this large subject. We should note here that wavelet transforms, already discussed in §13.10, are applicable not only to data compression and signal processing, but can also be used to transform some classes of integral equations into sparse linear problems that allow fast solution. You may wish to review §13.10 as part of reading this chapter. Some subjects, such as integro-differential equations, we must simply declare to 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 on a few of the most basic methods involved in this complicated subject. CITED REFERENCES AND FURTHER READING: Delves, L.M., and Mohamed, J.L. 1985, Computational Methods for Integral Equations (Cam￾bridge, U.K.: Cambridge University Press). [1] Linz, P. 1985, Analytical and Numerical Methods for Volterra Equations (Philadelphia: S.I.A.M.). [2] Atkinson, K.E. 1976, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind (Philadelphia: S.I.A.M.). [3] Brunner, H. 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 Tech￾nical), 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 Kind We desire a numerical solution for f(t) in the equation f(t) = λ
b a K(t, s)f(s) ds + g(t) (18.1.1) The method we describe, a very basic one, is called the Nystrom method. It requires the choice of some approximate quadrature rule:
b a y(s) ds = N j=1 wjy(sj ) (18.1.2) Here the set {wj} are the weights of the quadrature rule, while the N points {s j} are the abscissas. What quadrature rule should we use? It is certainly possible to solve integral equations with low-order quadrature rules like the repeated trapezoidal or Simpson’s