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18.0 IntroductionMany people, otherwise numerically knowledgable, imagine that the numericalsolution of integral equations must be an extremely arcane topic, since, until recently,it was almost never treated in numerical analysis textbooks. Actually there is alarge and growing literature on the numerical solution of integral equations; severalmonographs have by now appeared [1-3]. One reason for the sheer volume of thisactivity is that there are many different kinds of equations, each with many differentpossible pitfalls; often many different algorithms have been proposed to deal witha single case.There is a close correspondence between linear integral equations, which specifylinear, integral relations among functions in an infinite-dimensional function space,and plain old linear equations, which specify analogous relations among vectorsin a finite-dimensional vector space.
Because this correspondence lies at the heartof most computational algorithms, it is worth making it explicit as we recall howintegral equations are classified.Fredholm equations involve definite integrals with fixed upper and lower limits.An inhomogeneous Fredholm equation of the first kind has the formZ bg(t) =K(t, s)f(s) ds(18.0.1)aHere f(t) is the unknown function to be solved for, while g(t) is a known “right-handside.” (In integral equations, for some odd reason, the familiar “right-hand side” isconventionally written on the left!) The function of two variables, K(t, s) is calledthe kernel. Equation (18.0.1) is analogous to the matrix equationK·f=g(18.0.2)whose solution is f = K−1 · g, where K−1 is the matrix inverse. Like equation(18.0.2), equation (18.0.1) has a unique solution whenever g is nonzero (thehomogeneous case with g = 0 is almost never useful) and K is invertible.
However,as we shall see, this latter condition is as often the exception as the rule.The analog of the finite-dimensional eigenvalue problem(K − σ1) · f = g788(18.0.3)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).Chapter 18.
Integral Equationsand Inverse Theory18.0 Introduction789is called a Fredholm equation of the second kind, usually writtenZbK(t, s)f(s) ds + g(t)f(t) = λ(18.0.4)aZb[K(t, s) − σδ(t − s)]f(s) ds = −σg(t)(18.0.5)awhere δ(t − s) is a Dirac delta function (and where we have changed from λ to itsreciprocal σ for clarity). If σ is large enough in magnitude, then equation (18.0.5)is, in effect, diagonally dominant and thus well-conditioned. Only if σ is small dowe go back to the ill-conditioned case.Homogeneous Fredholm equations of the second kind are likewise not particularly ill-posed.
If K is a smoothing operator, then it will map many f’s to zero,or near-zero; there will thus be a large number of degenerate or nearly degenerateeigenvalues around σ = 0 (λ → ∞), but this will cause no particular computationaldifficulties. In fact, we can now see that the magnitude of σ needed to rescue theinhomogeneous equation (18.0.5) from an ill-conditioned fate is generally much lessthan that required for diagonal dominance. Since the σ term shifts all eigenvalues,it is enough that it be large enough to shift a smoothing operator’s forest of nearzero eigenvalues away from zero, so that the resulting operator becomes invertible(except, of course, at the discrete eigenvalues).Volterra equations are a special case of Fredholm equations with K(t, s) = 0for s > t. Chopping off the unnecessary part of the integration, Volterra equations arewritten in a form where the upper limit of integration is the independent variable t.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).Again, the notational conventions do not exactly correspond: λ in equation (18.0.4)is 1/σ in (18.0.3), while g is −g/λ. If g (or g) is zero, then the equation is saidto be homogeneous.
If the kernel K(t, s) is bounded, then, like equation (18.0.3),equation (18.0.4) has the property that its homogeneous form has solutions forat most a denumerably infinite set λ = λn , n = 1, 2, . . . , the eigenvalues. Thecorresponding solutions fn (t) are the eigenfunctions. The eigenvalues are real ifthe kernel is symmetric.In the inhomogeneous case of nonzero g (or g), equations (18.0.3) and (18.0.4)are soluble except when λ (or σ) is an eigenvalue — because the integral operator(or matrix) is singular then.
In integral equations this dichotomy is called theFredholm alternative.Fredholm equations of the first kind are often extremely ill-conditioned. Applying the kernel to a function is generally a smoothing operation, so the solution,which requires inverting the operator, will be extremely sensitive to small changesor errors in the input. Smoothing often actually loses information, and there is noway to get it back in an inverse operation.
Specialized methods have been developedfor such equations, which are often called inverse problems. In general, a methodmust augment the information given with some prior knowledge of the nature of thesolution. This prior knowledge is then used, in one way or another, to restore lostinformation. We will introduce such techniques in §18.4.Inhomogeneous Fredholm equations of the second kind are much less oftenill-conditioned.
Equation (18.0.4) can be rewritten as790Chapter 18.Integral Equations and Inverse TheoryThe Volterra equation of the first kindZtK(t, s)f(s) dsg(t) =(18.0.6)ahas as its analog the matrix equation (now written out in components)Kkj fj = gk(18.0.7)j=1Comparing with equation (18.0.2), we see that the Volterra equation corresponds toa matrix K that is lower (i.e., left) triangular, with zero entries above the diagonal.As we know from Chapter 2, such matrix equations are trivially soluble by forwardsubstitution.
Techniques for solving Volterra equations are similarly straightforward.When experimental measurement noise does not dominate, Volterra equations of thefirst kind tend not to be ill-conditioned; the upper limit to the integral introduces asharp step that conveniently spoils any smoothing properties of the kernel.The Volterra equation of the second kind is writtenZtf(t) =K(t, s)f(s) ds + g(t)(18.0.8)awhose matrix analog is the equation(K − 1) · f = g(18.0.9)with K lower triangular.
The reason there is no λ in these equations is that (i) inthe inhomogeneous case (nonzero g) it can be absorbed into K, while (ii) in thehomogeneous case (g = 0), it is a theorem that Volterra equations of the second kindwith bounded kernels have no eigenvalues with square-integrable eigenfunctions.We have specialized our definitions to the case of linear integral equations.The integrand in a nonlinear version of equation (18.0.1) or (18.0.6) would beK(t, s, f(s)) instead of K(t, s)f(s); a nonlinear version of equation (18.0.4) or(18.0.8) would have an integrand K(t, s, f(t), f(s)). Nonlinear Fredholm equationsare considerably more complicated than their linear counterparts. Fortunately, theydo not occur as frequently in practice and we shall by and large ignore them in thischapter. By contrast, solving nonlinear Volterra equations usually involves only aslight modification of the algorithm for linear equations, as we shall see.Almost all methods for solving integral equations numerically make use ofquadrature rules, frequently Gaussian quadratures.
This would be a good timefor you to go back and review §4.5, especially the advanced material towards theend of that section.In the sections that follow, we first discuss Fredholm equations of the secondkind with smooth kernels (§18.1). Nontrivial quadrature rules come into thediscussion, but we will be dealing with well-conditioned systems of equations. Wethen return to Volterra equations (§18.2), and find that simple and straightforwardmethods are generally satisfactory for these equations.In §18.3 we discuss how to proceed in the case of singular kernels, focusinglargely on Fredholm equations (both first and second kinds).
Singularities requireSample 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).kX18.1 Fredholm Equations of the Second Kind791CITED REFERENCES AND FURTHER READING:Delves, L.M., and Mohamed, J.L.
1985, Computational Methods for Integral Equations (Cambridge, 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 IntegralEquations of the Second Kind (Philadelphia: S.I.A.M.). [3]Brunner, H.
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