Many people, otherwise numerically knowledgable, imagine that the numerical solution of integral equations must be an extremely arcane topic, since, until recently, it was almost never treated in numerical analysis textbooks. Actually there is a large and growing literature | Chapter 18. Integral Equations and Inverse Theory Introduction Many people otherwise numerically knowledgable imagine that the numerical solution of integral equations must be an extremely arcane topic since until recently it was almost never treated in numerical analysis textbooks. Actually there is a large and growing literature on the numerical solution of integral equations several monographs have by now appeared 1-3 . One reason for the sheer volume of this activity is that there are many different kinds of equations each with many different possible pitfalls often many different algorithms have been proposed to deal with a single case. There is a close correspondence between linear integral equations which specify linear integral relations among functions in an infinite-dimensional function space and plain old linear equations which specify analogous relations among vectors in a finite-dimensional vector space. Because this correspondence lies at the heart of most computational algorithms it is worth making it explicit as we recall how integral equations are classified. Fredholm equations involve definite integrals with fixed upper and lower limits. An inhomogeneous Fredholm equation of the first kind has the form g t i J a K t s f s ds Here f t is the unknown function to be solved for while g t is a known right-hand side. In integral equations for some odd reason the familiar right-hand side is conventionally written on the left The function of two variables K t s is called the kernel. Equation is analogous to the matrix equation K f g whose solution is f K 1 g where K 1 is the matrix inverse. Like equation equation has a unique solution whenever g is nonzero the homogeneous 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 Sample page from NUMERICAL RECIPES IN C THE ART .
