Numerical Solution of Integral Equations

The principal reformulations of Laplace's equation as boundary integral equations (BIEs) are described, together with results on their solvability and the regularity of their solutions. The numerical methods for solving BIEs are categorized, based on whether the method uses local or global approximating functions, whether the method is of collocation or Galerkin type, and whether the equation being solved is defined on a region whose boundary is smooth or only piecewise smooth. Some of the major ideas in the mathematical analysis of these numerical methods are outlined. Certain problems are associated with all numerical methods for boundary integral equations. Principal among these are numerical integration and the iterative solution of linear systems of equations. The research literature for these topics as they arise in solving BIEs is reviewed, and some of the major ideas are discussed.
This book consists of eight interesting survey papers on current topics in the numerical treatment of integral equations. Combined, these papers cover a vast area. All of them are well organized, clearly written and easy to follow. If a theorem is given without a proof, a suitable reference is supplied for the interested reader. Each paper contains a long reference list which is very useful for further study. This book also has an index, something which cannot always be taken for granted in collections of this type.
1. K. E. Atkinson: A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions.
2. I. H. Sloan: Superconvergence.
3. M. A. Golberg: Perturbed projection methods for various classes of operator and integral equations.
4. G. Miel: Numerical solution on parallel processors of two-point boundary value problems of astrodynamics..
5. M. A. Golberg: Introduction to the numerical solution of Cauchy singular integral equations.
6. D. Elliott: Convergence theorems for singular integral equations.
7. E. 0.Tuck: Planing surfaces.
8. R. S. Anderssen and F. R. de Hoog: Abel integral equations.
In Chapter 1 we find an integral equation reformulation of Laplace's equation. Both direct and indirect boundary element methods are presented and the cases of smooth and piecewise smooth boundaries are dealt with. Numerical solutions are derived by using both global (i.e., when the solution is expressed as a linear combination of some selected basis functions that interpplate globally) and local methods. In most cases a large system of equations is obtained, and the author discusses different approaches for solving this system.
Chapter 2 is devoted to superconvergence, i.e., the phenomenon that the order of convergence for certain functionals of the solution to an integral or a differential equation is higher than for the solution itself. As an example, the accuracy of the solution at the knots may be higher than at other points. The author discusses collocation, Galerkin's and Kantorovich's methods, and presents examples.
Chapter 3 deals with perturbed projection methods. Here, too, we encounter the Kantorovich method as well as the collocation and Galerkin methods. The author also discusses the influence of discretization errors.
In Chapter 4 we encounter a treatment of two-point boundary value problems for ordinary differential equations using a reformulation to an equivalent problem of solving a Hammerstein integral equation. The author gives a lucid presentation of the solution of corresponding operator equations in Banach spaces. Some interesting applications to the movement of space vehicles are described, and an implementation on parallel computers is treated.
Chapter 5 gives a thorough discussion of Cauchy singular integral equations, in particular, the important generalized airfoil equation. We encounter the Galerkin method as well as collocation and quadrature methods.
In Chapter 6 the same class of equations is treated as in the preceding chapter. The author gives an interesting theory for the convergence of important classes of numerical methods.
Chapter 7 deals with the planing equation which is important in ship hydrodynamics. The author reports on problems and progress in this area.
Chapter 8 gives a fairly complete treatment of the theory and the applications of Abel's equation. Although this equation is special, it is important for the applications in many fields. Solutions may be written in analytic form, but there are numerical pitfalls associated with these solutions. The authors discuss these difficulties and give methods for avoiding them. They point out that one is not always interested in the solution itself but rather a functional having the solution as an argument. The latter problem is often better conditioned than the task of determining the solution itself, a fact which may be utilized in the numerical treatment. This important observation is often valid for other ill-conditioned problems, besides the task of solving Abel equations.
The book is strongly recommended for those working in the theory or the applications of integral equations. Thus it should be suitable for graduate students in science or engineering as well as for their teachers.