INITIAL-VALUE METHOD FOR TWO-POINT BOUNDARY-VALUE PROBLEMS

Abstract

In this thesis, we consider two problems: (i) linear, two-point boundary-value problems with differential constraints and general boundary conditions; and (ii) nonlinear, two-point boundary-value problems with differential constraints, nondifferential constraints, and general boundary conditions. Since linear, two-point boundary-value problems play an essential role in the study of nonlinear two-point boundary-value problems, we develop three computationally efficient algorithms for the numerical solution of linear problems. These algorithms are developed in conjunction with the following techniques: (a) method of complementary functions; (b) method of particular solutions; and (c) method of adjoint variables. Then, we turn our attention to nonlinear, two-point boundary-value problems. We develop an initial-value method such that the differential constraints and the nondifferential constraints are satisfied at each iteration of the process. As a consequence, the two-point boundary-value problem is reduced to solving a system of n nonlinear equations in which the unknowns are the n components of the state vector of the initial point. Next, Miele's modified quasilinearization algorithm is applied to finding the solution of the nonlinear equations. Once the correct initial values are found, the solution of the two-point boundary-value problem is obtained by forward integration of the differential constraints, subject to the nondifferential constraints. Nine numerical examples (two linear and seven nonlinear) are presented. The numerical results demonstrate the quadratic convergence property of the shooting method.