In this monograph we present a brief, elementary yet rigorous account of practical numerical methods for solving very general two-point boundary-value problems. Three techniques are studied in detail: initial value or “shooting” methods in Chapter 2, finite difference methods in Chapter 3, integral equation methods in Chapter 4. Each method is applied to nonlinear second-order problems, and the first two methods are applied to first-order systems of nonlinear equations. Sturm-Liouville eigenvalue problems are treated with all three techniques in Chapter 5. The application of shooting to generalized or nonlinear eigenvalue problems is also discussed in Chapter 5.
Initial-value methods are seldom advocated in the literature, but we find them extremely practical and theoretically powerful. A modification, called parallel shooting, is introduced to treat those “unstable” cases (with rapidly growing solutions) for which ordinary shooting may be inadequate. High-order accurate methods are stressed, and the well-developed theory of numerical methods for initial-value problems is employed to obtain corresponding results for boundary-value problems. Continuity techniques (related to imbedding) are discussed and illustrated in the examples.
To help maintain an elementary level, we include in Chapter 1 brief ac-counts of some of the basic prerequisites : existence theorems for initial-value and two-point boundary-value problems, numerical methods for initial-value problems and iterative methods for solving nonlinear systems. In addition, several other areas of numerical analysis enter our study, such as numerical quadrature, eigenvalue-eigenvector calculation, and solution of linear algebraic systems. Since so many diverse numerical procedures play an important role, the subject matter of this monograph is useful in motivating, introducing, and/or reviewing more general studies in numerical analysis. The level of presentation is such that the text could be used by well-prepared undergraduate mathematics students. Advanced calculus, basic numerical analysis, some ordinary differential equations, and a smattering of linear algebra are assumed.
Many of the problems contain extensions of the text material and so should at least be read, if not worked out. More difficult problems are starred.
It is a pleasure to acknowledge the helpful advice received from Professor Eugene Isaacson and Mr. Richard Swenson, both of whom read the entire manuscript. The computations for the examples were accomplished by the superior work of W. H. Mitchell at Caltech and J. Steadman at the Courant Institute, N.Y.U. The typing was done by Miss Connie Engle in her usual flawless manner.
Finally, the support of the U.S. Army Research Office (Durham) and the U.S. Atomic Energy Commission is gratefully acknowledged.
H. B. K.