Publisher Summary This chapter discusses the numerical solution of boundary value problems for ordinary differential equations. It also presents a few recent results on differencemethods. A thorough study of truncated Chebyshev series approximations to the solution of subject to linear multi-points boundary conditions is given by Urabe. For isolated solutions, he has shown existence of a unique approximating polynomial and obtains error estimates. The point of smooth approximation is frequently made by advocates of projection, collocation, or series methods that shooting and difference methods yield only approximate solutions at discrete points as opposed to some piecewise smooth approximating solution defined over the entire interval [a, b]. It is not difficult to show that various shooting and finite difference schemesfor linear two point problems are, in principal, identical, that is, if various linear systems occurring in either procedure are solved exactly, the same numerical approximation results. In such cases, the basic problem is that of determining the most efficient stable procedure for solving the linear system. As regards to the operations in solving the linear systems, the collocation method gets worse, relative to the Box-scheme, as the order of the system increases and as the accuracy increases. Different estimates seem to show that shooting is more efficient than finite differences with the Box-scheme.

Numerical solution of boundary value problems for ordinary differential equations: survey and some recent results on difference methods

Publisher Summary This chapter focuses on the conversion of boundary-value problems into stable initial-value problems via several invariant imbedding algorithms. When the solution of the boundary-value problem is smooth and only a few output points are desired, all the algorithms operate with approximately the same degree of efficiency and return approximately the same accuracy. As a result of the stability of the equations, all of the codes normally produce greater accuracy than requested. The addition formulas, in particular, are especially susceptible to the number and placement of output points and to the strategy of choosing the break points to avoid the singularities. The Scott algorithm must be implemented with the restart process to avoid the loss of significance in subtraction. Since the imbedding algorithms are very stable and employ very sophisticated initial-value codes, which vary the step size, the integration is extremely efficient. The multiple shooting codes can be implemented to use the same initial-value codes. However, if the problem requires many subdivisions of the interval, then the number of equations becomes large and slows down the process. The imbedding will almost always be faster if there is much structure to the solution.

On the conversion of boundary-value problems into stable initial-value problems via several invariant imbedding algorithms

An invariant imbedding algorithm for the solution of inhomogeneous linear two-point boundary value problems

A comparison of several invariant imbedding algorithms for the solution of two-point boundary-value problems

A front-tracking method for the solution of multiphase free boundary problems is presented. Based on the method of lines with invariant imbedding, this method is sufficiently general to allow the solution of problems with three or more phases and problems in which phases appear and disappear. A discussion of the method for the three-phase Stefan problem and for the moving contact problem, as well as a proof of convergence for the latter, is included. In addition, the method is demonstrated for a free boundary problem with a combination of Stefan and ablation boundary conditions and for the two-dimensional contact problem. Numerical results indicate that this front-tracking method is fast enough to solve a wide range of problems on a personal computer.

G.M Wing

Papers

A numerical algorithm suggested by problems of transport in periodic media

A numerical algorithm suggested by problems of transport in periodic media: The matrix case☆☆☆★★★

Generalized trigonometric identities and invariant imbedding

On a generalization of a method of Bellman and Latter for obtaining eigenvalue bounds for integral operators

Some convexity theorems for eigenvalues of Fredholm integral equations