NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS: SURVEY AND SOME RECENT RESULTS ON DIFFERENCE METHODS

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.

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