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1985, Analytical and Numerical Methods for Volterra Equations (Philadelphia: S.I.A.M.).[1]Delves, L.M., and Mohamed, J.L. 1985, Computational Methods for Integral Equations (Cambridge, U.K.: Cambridge University Press). [2]18.3 Integral Equations with Singular KernelsMany integral equations have singularities in either the kernel or the solution or both.A simple quadrature method will show poor convergence with N if such singularities areignored. There is sometimes art in how singularities are best handled.We start with a few straightforward suggestions:1.

Integrable singularities can often be removed by a change of variable. For example, thesingular behavior K(t, s) ∼ s1/2 or s−1/2 near s = 0 can be removed by the transformationz = s1/2 . Note that we are assuming that the singular behavior is confined to K, whereasthe quadrature actually involves the product K(t, s)f (s), and it is this product that mustbe “fixed.” Ideally, you must deduce the singular nature of the product before you try anumerical solution, and take the appropriate action. Commonly, however, a singular kerneldoes not produce a singular solution f (t). (The highly singular kernel K(t, s) = δ(t − s)is simply the identity operator, for example.)2. If K(t, s) can be factored as w(s)K(t, s), where w(s) is singular and K(t, s) issmooth, then a Gaussian quadrature based on w(s) as a weight function will work well.

Evenif the factorization is only approximate, the convergence is often improved dramatically. Allyou have to do is replace gauleg in the routine fred2 by another quadrature routine. Section4.5 explained how to construct such quadratures; or you can find tabulated abscissas andweights in the standard references [1,2] . You must of course supply K instead of K.This method is a special case of the product Nystrom method [3,4], where one factors outa singular term p(t, s) depending on both t and s from K and constructs suitable weights forits Gaussian quadrature. The calculations in the general case are quite cumbersome, becausethe weights depend on the chosen {ti } as well as the form of p(t, s).We prefer to implement the product Nystrom method on a uniform grid, with a quadraturescheme that generalizes the extended Simpson’s 3/8 rule (equation 4.1.5) to arbitrary weightfunctions.

We discuss this in the subsections below.3. Special quadrature formulas are also useful when the kernel is not strictly singular,but is “almost” so. One example is when the kernel is concentrated near t = s on a scale muchsmaller than the scale on which the solution f (t) varies. In that case, a quadrature formulacan be based on locally approximating f (s) by a polynomial or spline, while calculating thefirst few moments of the kernel K(t, s) at the tabulation points ti . In such a scheme thenarrow width of the kernel becomes an asset, rather than a liability: The quadrature becomesexact as the width of the kernel goes to zero.4.

An infinite range of integration is also a form of singularity. Truncating the range at alarge finite value should be used only as a last resort. If the kernel goes rapidly to zero, thenSample 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).This procedure can be repeated as with Romberg integration.The general consensus is that the best of the higher order methods is theblock-by-block method (see [1]). Another important topic is the use of variablestepsize methods, which are much more efficient if there are sharp features in K orf.

Variable stepsize methods are quite a bit more complicated than their counterpartsfor differential equations; we refer you to the literature [1,2] for a discussion.You should also be on the lookout for singularities in the integrand. If you findthem, then look to §18.3 for additional ideas..

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