On the Design of a Simple Solver for Nonlinear Two-Point Boundary Value Problems

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The underlying principles and structure of an easy-to-use solver for two-point boundary-value problems described by sets of nonlinear ordinary differential equations is presented. The solution approach is based on the finite difference approximations for the derivatives. The Newton—Raphson iterative scheme with analytical Jacobians is used for solving the resulting large-scale nonlinear set of algebraic equations. Part 1 of the paper presents an in-depth analysis of the general problem and the solution method which results in a split of the overall problem into two distinct parts, a pure mathematical analytical part, which does not depend on the chosen numerical solution method, and a numerical part, which implements all the details of the numerical procedure. The prototype implementation, presented in Part 2, is based on this separation. It makes use of different programs which are specialized for solving particular subproblems identified in the analysis. An algebraic manipulator is used to aid in generating the Jacobians analytically and a matrix-oriented environment is used to implement the numerical matrix operations. The resulting package requires only the essential information from the user, namely the model equations and the solution domain variables as well as initial guesses of the solution. The package is a prototype that can be used to solve second-order problems based on three-point polynomial approximations of the derivatives.

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