Computer aided symbolic solution of multi-point boundary value problems occurring in physics and engineering

Abstract

The author has elsewhere published papers demonstrating applicability of computer algebraic and symbol manipulation in obtaining solutions to ordinary and partial differential equations by Piccard's method, steepest descent and various forms of the Newton-Kantorovich theorems and applying them to non-trivial problems in engineering, physics and, especially, celestial mechanics. In this paper, the Taylor series will be developed permitting expansion about any point and for any boundary conditions for any order derivative at arbitrary points, i.e., the general multi-point boundary valve problem (MPBVP) will be solved. The symbolic algorithm developed is written in PL/1-Formac and produces the Taylor series solution for any nonlinear differential equation in which the highest order derivative may be algebraically isolated. This program permits the continuation of this solution on intervals of the independent variable, in the manner of polynomial splines. This program permits symbolic solutions, e.g. in terms of a symbolic initial condition. However, such a solution requires enormous main storage. PL/1-Formac was used due to its general availability and its compatibility with relatively small main storage, even as small as 200 K bytes. Two parameters are available for attaining a given numerical accuracy; the order of the Taylor expansion and the number of continuation intervals into which the solution range is divided. Experiments show that high accuracy can be obtained by judiciously selecting these two parameters in order to counterbalance truncation error against numerical round-off error. Extensive additional documentation of this procedure has been performed on scores of problems occurring in the applied mathematics literature.