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Boundary Value Problems

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Scientific Computing

Part of the book series: Texts in Computational Science and Engineering ((TCSE,volume 20))

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Abstract

This chapter is devoted to boundary value problems for ordinary differential equations. It begins with analysis of the existence and uniqueness of solutions to these problems, and the effect of perturbations to the problem. The first numerical approach is the shooting method. This is followed by finite differences and collocation. Finite elements allow for the development of very high order methods for many boundary value problems, but their analysis typically requires sophisticated ideas from real analysis. The chapter ends with the application of deferred correction to both collocation and finite elements.

…years ago many considered BVPs as some sort of a subclass of IVPs, wherein one fiddles with the initial conditions in order to get things right at the other end. It gradually became clear that IVPs are actually a special and in some sense relatively simple subclass of BVPs. The fundamental difference is that for IVPs one has complete information about the solution at one point (the initial point), so one may consider using a marching algorithm which is always local in nature. For BVPs, on the other hand, no complete information is available at any point, so the end points have to be connected by the solution algorithm in a global way. Only after stepping through the entire domain can the solution at any point be determined.

Uri M. Ascher, Robert M. M Mattheij and Robert D. Russell[7, p. xvii].

Science is a differential equation. Religion is a boundary condition.

Alan Turing in epigram to Robin Gandy [104, p. 513]

Additional Material: The details of the computer programs referred in the text are available in the Springer website (http://extras.springer.com/2018/978-3-319-69110-7) for authorized users.

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Authors and Affiliations

  1. Professor Emeritus of Mathematics, Department of Mathematics, Duke University, Durham, North Carolina, USA

    John A. Trangenstein

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  1. John A. Trangenstein
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Trangenstein, J.A. (2017). Boundary Value Problems. In: Scientific Computing . Texts in Computational Science and Engineering, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-69110-7_4

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