Multipoint solution of two-point boundary-value problems
- 81 Downloads
The purpose of this paper is to report on the application of multipoint methods to the solution of two-point boundary-value problems with special reference to the continuation technique of Roberts and Shipman. The power of the multipoint approach to solve sensitive two-point boundary-value problems with linear and nonlinear ordinary differential equations is exhibited. Practical numerical experience with the method is given.
Since employment of the multipoint method requires some judgment on the part of the user, several important questions are raised and resolved. These include the questions of how many multipoints to select, where to specify the multipoints in the interval, and how to assign initial values to the multipoints.
Three sensitive numerical examples, which cannot be solved by conventional shooting methods, are solved by the multipoint method and continuation. The examples include (1) a system of two linear, ordinary differential equations with a boundary condition at infinity, (2) a system of five nonlinear ordinary differential equations, and (3) a system of four linear ordinary equations, which isstiff.
The principal results are that multipoint methods applied to two-point boundary-value problems (a) permit continuation to be used over a larger interval than the two-point boundary-value technique, (b) permit continuation to be made with larger interval extensions, (c) converge in fewer iterations than the two-point boundary-value methods, and (d) solve problems that two-point boundary-value methods cannot solve.
KeywordsBoundary Condition Differential Equation Numerical Experience Ordinary Differential Equation Special Reference
Unable to display preview. Download preview PDF.
- 1.Keller, H. B.,Numerical Methods for Two-Point Boundary-Value Problems, Blaisdell Publishing Company, Waltham, Massachusetts, 1968.Google Scholar
- 2.Morrison, D. D., Riley, J. D., andZancanaro, J. F.,Multiple Shooting Method for Two-Point Boundary-Value Problems, Communications of the Association for Computing Machinery, Vol. 5, No. 12, 1962.Google Scholar
- 3.Osborne, M. R.,On Shooting Methods for Boundary-Value Problems, Journal of Mathematical Analysis and Applications, Vol. 27, No. 2, 1969.Google Scholar
- 4.Roberts, S. M., andShipman, J. S.,Continuation in Shooting Methods for Two-Point Boundary-Value Problems, Journal of Mathematical Analysis and Applications, Vol. 18, No. 1, 1967.Google Scholar
- 5.Roberts, S. M., andShipman, J. S.,Justification for the Continuation Method in Two-Point Boundary-Value Problems, Journal of Mathematical Analysis and Applications, Vol. 21, No. 1, 1968.Google Scholar
- 6.Roberts, S. M., andShipman, J. S.,Two-Point Boundary-Value Problems: Shooting Methods, American Elsevier, New York (to appear).Google Scholar
- 7.Goodman, T. R., andLance, G. N.,The Numerical Solution of Two-Point Boundary-Value Problems, Mathematical Tables and Other Aids to Computation, Vol. 10, No. 54, 1956.Google Scholar
- 8.Holt, J. F.,Numerical Solution of Nonlinear Two-Point Boundary-Value Problems by Finite-Difference Methods, Communications of the Association for Computing Machinery, Vol. 7, No. 6, 1964.Google Scholar
- 9.Conte, S. D.,The Numerical Solution of Linear Boundary-Value Problems, SIAM Review, Vol. 8, No. 3, 1966.Google Scholar