Boundary-Value Problems for ODE
This chapter first deals with various finite-difference methods for scalar boundary-value problems involving ordinary differential equations and for systems of such problems. The concepts of consistency, stability and convergence are introduced, as well as methods to increase the local solution precision by extrapolation. Shooting methods are offered as a clear alternative to difference methods in the case of non-linear equations and their systems. A separate section is devoted to various types of discretizations that mirror the asymptotic physics regimes of the underlying differential equation. Collocation and weighted-residual methods are presented. Several approaches to boundary-value problems with eigenvalues are attempted: finite-difference methods, shooting methods involving the Prüfer transformation, and the Pruess method. The treatment of problems in which the eigenvalues appear in the boundary conditions is also illustrated. Examples and Problems include the non-linear Gelfand–Bratu equation, diffusion and reaction in a catalytic pellet, deflection of an inhomogeneous beam, the one-dimensional Schrödinger equation, and a boundary-layer problem.
KeywordsDifference Scheme Collocation Method Collocation Point Uniform Mesh Shooting Method
- 10.D. Knoll, J. Morel, L. Margolin, M. Shashkov, Physically motivated discretization methods. Los Alamos Sci. 29, 188 (2005) Google Scholar
- 16.J.E. Flaherty, Finite element analysis. CSCI, MATH 6860 Lecture Notes, Rensselaer Polytechnic Institute, Troy, 2000 Google Scholar
- 35.A.N. Tikhonov, A.S. Leonov, A.G. Yagola, Nonlinear Ill-Posed Problems, Vols. I and II (Chapman and Hall, London, 1998) Google Scholar
- 38.M.E. Davis, Numerical Methods and Modeling for Chemical Engineers (Wiley, New York, 1984) Google Scholar