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18.3 Integral Equations with Singular Kernels797CITED 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 (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.798Chapter 18.Integral Equations and Inverse Theorya Gauss-Laguerre [w ∼ exp(−αs)] or Gauss-Hermite [w ∼ exp(−s2 )] quadrature shouldwork well. Long-tailed functions often succumb to the transformation2α−α(18.3.1)z+1which maps 0 < s < ∞ to 1 > z > −1 so that Gauss-Legendre integration can be used.Here α > 0 is a constant that you adjust to improve the convergence.5.
A common situation in practice is that K(t, s) is singular along the diagonal linet = s. Here the Nystrom method fails completely because the kernel gets evaluated at (ti , si ).Subtraction of the singularity is one possible cure:Z bZ bZ bK(t, s)f (s) ds =K(t, s)[f (s) − f (t)] ds +K(t, s)f (t) dsaaa(18.3.2)Z bs=aRbwhere r(t) = a K(t, s) ds is computed analytically or numerically. If the first term onthe right-hand side is now regular, we can use the Nystrom method. Instead of equation(18.1.4), we getfi = λNXwj Kij [fj − fi] + λri fi + gi(18.3.3)j=1j6=iSometimes the subtraction process must be repeated before the kernel is completely regularized.See [3] for details.
(And read on for a different, we think better, way to handle diagonalsingularities.)Quadrature on a Uniform Mesh with Arbitrary WeightIt is possible in general to find n-point linear quadrature rules that approximate theintegral of a function f (x), times an arbitrary weight function w(x), over an arbitrary rangeof integration (a, b), as the sum of weights times n evenly spaced values of the function f (x),say at x = kh, (k + 1)h, . . . , (k + n − 1)h. The general scheme for deriving such quadraturerules is to write down the n linear equations that must be satisfied if the quadrature rule isto be exact for the n functions f (x) = const, x, x2 , .
. . , xn−1 , and then solve these for thecoefficients. This can be done analytically, once and for all, if the moments of the weightfunction over the same range of integration,Z b1Wn ≡ nxn w(x)dx(18.3.4)h aare assumed to be known. Here the prefactor h−n is chosen to make Wn scale as h if (asin the usual case) b − a is proportional to h.Carrying out this prescription for the four-point case gives the resultZbaw(x)f (x)dx =1f (kh) (k + 1)(k + 2)(k + 3)W0 − (3k2 + 12k + 11)W1 + 3(k + 2)W2 − W361+ f ([k + 1]h) − k(k + 2)(k + 3)W0 + (3k2 + 10k + 6)W1 − (3k + 5)W2 + W321+ f ([k + 2]h) k(k + 1)(k + 3)W0 − (3k2 + 8k + 3)W1 + (3k + 4)W2 − W321+ f ([k + 3]h) − k(k + 1)(k + 2)W0 + (3k2 + 6k + 2)W1 − 3(k + 1)W2 + W36(18.3.5)Sample 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).K(t, s)[f (s) − f (t)] ds + r(t)f (t)=18.3 Integral Equations with Singular Kernels799While the terms in brackets superficially appear to scale as k2 , there is typically cancellationat both O(k2 ) and O(k).Equation (18.3.5) can be specialized to various choices of (a, b). The obvious choiceis a = kh, b = (k + 3)h, in which case we get a four-point quadrature rule that generalizesSimpson’s 3/8 rule (equation 4.1.5). In fact, we can recover this special case by settingw(x) = 1, in which case (18.3.4) becomesBack to the case of general w(x), some other choices for a and b are also useful.
Forexample, we may want to choose (a, b) to be ([k + 1]h, [k + 3]h) or ([k + 2]h, [k + 3]h),allowing us to finish off an extended rule whose number of intervals is not a multipleof three, without loss of accuracy: The integral will be estimated using the four valuesf (kh), . . . , f ([k + 3]h). Even more useful is to choose (a, b) to be ([k + 1]h, [k + 2]h), thususing four points to integrate a centered single interval.
These weights, when sewed togetherinto an extended formula, give quadrature schemes that have smooth coefficients, i.e., withoutthe Simpson-like 2, 4, 2, 4, 2 alternation. (In fact, this was the technique that we used to deriveequation 4.1.14, which you may now wish to reexamine.)All these rules are of the same order as the extended Simpson’s rule, that is, exactfor f (x) a cubic polynomial. Rules of lower order, if desired, are similarly obtained. Thethree point formula isZ b1w(x)f (x)dx = f (kh) (k + 1)(k + 2)W0 − (2k + 3)W1 + W22a(18.3.8)+ f ([k + 1]h) − k(k + 2)W0 + 2(k + 1)W1 − W21+ f ([k + 2]h) k(k + 1)W0 − (2k + 1)W1 + W22Here the simple special case is to take, w(x) = 1, so thath[(k + 2)n+1 − kn+1 ]n+1Then equation (18.3.8) becomes Simpson’s rule,Z (k+2)hh4hhf (x)dx = f (kh) +f ([k + 1]h) + f ([k + 2]h)333khWn =(18.3.9)(18.3.10)For nonconstant weight functions w(x), however, equation (18.3.8) gives rules of one orderless than Simpson, since they do not benefit from the extra symmetry of the constant case.The two point formula is simplyZ (k+1)hw(x)f (x)dx = f (kh)[(k + 1)W0 − W1 ] + f ([k + 1]h)[−kW0 + W1 ] (18.3.11)khHere is a routine wwghts that uses the above formulas to return an extended N -pointquadrature rule for the interval (a, b) = (0, [N − 1]h).
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