S. M. Roberts

Multipoint solution of two-point boundary-value problems

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.

Scheduling of plant maintenance personnel

Given (i) a set of maintenance jobs to be processed over a fixed time horizon, (ii) the breakdown of each job into finite time intervals in which the skills required are known, and (iii) the pool of available manpower for each skill type over the horizon, we formulate and solve the problem of scheduling personnel and jobs to minimize personnel idle time, by integer programming.

Continuation in quasilinearization

A continuation method is described for extending the applicability of quasilinearization to numerically unstable two-point boundary-value problems. Since quasilinearization is a realization of Newtons method, one might expect difficulties in finding satisfactory initial trialpoints, which actually are functions over the specified interval that satisfy the boundary conditions. A practical technique for generating suitable initial profiles for quasilinearization is described. Numerical experience with these techniques is reported for two numerically unstable problems.

Minimum problem-size formulation for the scheduling of plant maintenance personnel

The problem of scheduling plant maintenance personnel has been recast to give the minimum problem-size formulation.

Solution of źYź+YYźźY=0 by a nonasymptotic method

The solution of εy″(x)+y(x)y′(x)−y(x)=0, with the boundary conditionsy(0)=α,y(1)=β, is obtained by a nonasymptotic method. It is shown that the nature of the inner solution for both left-hand and right-hand boundary layers depends on the roots of a transcendental equation. From sketches of this function, the location of the roots can be found. For the left-hand boundary layer, depending on the relative size and signs of α and β, 13 cases exist for the possible solution of the transcendental equation. Of these cases, only five correspond to acceptable solutions. Similar remarks apply to the right-hand boundary layer solutions. Numerical experience with the method is also reported to confirm the theoretical analysis.

Fundamental matrix and two-point boundary-value problems

Theoretically, the solution of all linear ordinary differential equation problems, whether initial-value or two-point boundary-value problems, can be expressed in terms of the fundamental matrix. The examination of well-known two-point boundary-value methods discloses, however, the absence of the fundamental matrix in the development of the techniques and in their applications. This paper reveals that the fundamental matrix is indeed present in these techniques, although its presence is latent and appears in various guises.

The epsilon variation method in two-point boundary-value problems

This paper discusses the solution of two-point boundary-value problems by the combined technique of the introduction of partitioned equations employing the perturbation parameter ∈ and the utilization of the variational equations relative to epsilon. The combined technique is called the epsilon variation method. Two practical realizations of the method are presented with numerical examples to illustrate how in practice each of these realizations perform.

Variational perturbation method and power-series approximation method

The power-series approximation method for solving regular perturbation problems is reexamined to show why the method works. As another way to approach these problems, the variational perturbation technique is described. Although the assumptions on which each method is based and the mechanism of deriving their differential equations are different, the system of differential equations developed by each method is identical.

On the formulation of invariant imbedding problems

The formulation of an invariant imbedding problem from a given linear, two-point boundary-value problem is not unique. In this paper, we illustrate how the formulation of the problem by partitioning the original vectory(z) into [u(z),v(z)], can affect the numerical accuracy. In fact, the partitioning, the choice of theR, O system orS, T system of equations in Scotts method, the location and number of switch points, and the switching procedure, all influence the numerical results and the ease of obtaining solutions. A new method of switching and the appropriate formulas are described, namely, the repeated switching from theR, Q system to theR, Q system of equations or from theS, T system to theS, T system of equations.

An approach to singular perturbation problems insoluble by asymptotic methods

Singular perturbation problems not amenable to solution by asymptotic methods require special treatment, such as the method of Carrier and Pearson. Rather than devising special methods for these problems, this paper suggests that there may be a uniform way to solve singular perturbation problems, which may or may not succumb to asymptotic methods. A potential mechanism for doing this is the authors boundary-value technique, a nonasymptotic method, which previously has only been applied to singular perturbation problems that lend themselves to asymptotic techniques. Two problems, claimed by Carrier and Pearson to be insoluble by asymptotic methods, are solved by the boundary-value method.