18.3 Integral Equations with Singular Kernels 797 This procedure can be repeated as with Romberg integration. The general consensus is that the best of the higher order methods is the block-by-block method (see [1)).Another important topic is the use of variable stepsize methods,which are much more efficient if there are sharp features in K or f.Variable stepsize methods are quite a bit more complicated than their counterparts for 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 find them.then look to 818.3 for additional ideas. CITED REFERENCES AND FURTHER READING: 8 Linz,P.1985,Analytical and Numerical Methods for Volterra Equations (Philadelphia:S.I.A.M.). [1] Delves,L.M.,and Mohamed,J.L.1985,Computationa/Methods for Integral Equations (Cam- bridge,U.K.:Cambridge University Press).[2] Cam ICAL RECIPES 18.3 Integral Equations with Singular Kernels Many integral equations have singularities in either the kernel or the solution or both. 巴互2。 A simple quadrature method will show poor convergence with N if such singularities are ignored.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,the singular behavior K(t,s)s1/2 or s-1/2 near s=0 can be removed by the transformation =s1/2.Note that we are assuming that the singular behavior is confined to K,whereas SCIENTIFIC the quadrature actually involves the product K(t,s)f(s),and it is this product that must be 6 "fixed."Ideally,you must deduce the singular nature of the product before you try a numerical solution,and take the appropriate action.Commonly,however,a singular kernel does not produce a singular solution f(t).(The highly singular kernel K(t,s)=6(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)is smooth,then a Gaussian quadrature based on w(s)as a weight function will work well.Even if the factorization is only approximate,the convergence is often improved dramatically.All you have to do is replace gauleg in the routine fred2 by another quadrature routine.Section Numerica 10621 4.5 explained how to construct such quadratures;or you can find tabulated abscissas and 431 weights 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 out Recipes a singular term p(t,s)depending on both t and s from K and constructs suitable weights for (outside its Gaussian quadrature.The calculations in the general case are quite cumbersome,because the weights depend on the chosent as well as the form of p(t,s). Software. 首 We prefer to implement the product Nystrom method on a uniform grid,with a quadrature scheme that generalizes the extended Simpson's 3/8 rule (equation 4.1.5)to arbitrary weight functions.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 much smaller than the scale on which the solution f(t)varies.In that case,a quadrature formula can be based on locally approximating f(s)by a polynomial or spline,while calculating the first few moments of the kernel K(t,s)at the tabulation points ti.In such a scheme the narrow width of the kernel becomes an asset,rather than a liability:The quadrature becomes exact 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 a large finite value should be used only as a last resort.If the kernel goes rapidly to zero,then18.3 Integral Equations with Singular Kernels 797 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). This procedure can be repeated as with Romberg integration. The general consensus is that the best of the higher order methods is the block-by-block method (see [1]). Another important topic is the use of variable stepsize methods, which are much more efficient if there are sharp features in K or f. Variable stepsize methods are quite a bit more complicated than their counterparts for 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 find them, then look to §18.3 for additional ideas. CITED REFERENCES AND FURTHER READING: Linz, P. 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 (Cam￾bridge, U.K.: Cambridge University Press). [2] 18.3 Integral Equations with Singular Kernels Many 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 are ignored. 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, the singular behavior K(t, s) ∼ s1/2 or s−1/2 near s = 0 can be removed by the transformation z = s1/2. Note that we are assuming that the singular behavior is confined to K, whereas the quadrature actually involves the product K(t, s)f(s), and it is this product that must be “fixed.” Ideally, you must deduce the singular nature of the product before you try a numerical solution, and take the appropriate action. Commonly, however, a singular kernel does 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) is smooth, then a Gaussian quadrature based on w(s) as a weight function will work well. Even if the factorization is only approximate, the convergence is often improved dramatically. All you have to do is replace gauleg in the routine fred2 by another quadrature routine. Section 4.5 explained how to construct such quadratures; or you can find tabulated abscissas and weights 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 out a singular term p(t, s) depending on both t and s from K and constructs suitable weights for its Gaussian quadrature. The calculations in the general case are quite cumbersome, because the 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 quadrature scheme that generalizes the extended Simpson’s 3/8 rule (equation 4.1.5) to arbitrary weight functions. 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 much smaller than the scale on which the solution f(t) varies. In that case, a quadrature formula can be based on locally approximating f(s) by a polynomial or spline, while calculating the first few moments of the kernel K(t, s) at the tabulation points ti. In such a scheme the narrow width of the kernel becomes an asset, rather than a liability: The quadrature becomes exact 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 a large finite value should be used only as a last resort. If the kernel goes rapidly to zero, then