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General Introduction

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Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems

Part of the book series: Lecture Notes in Engineering ((LNENG,volume 7))

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Abstract

The mathematical formulation of many problems in physics and engineering involving rates of change with respect to two or more independent variables, leads either to a partial differential equation or to a set of such equations. These equations are supplemented by a set of prescribed boundary conditions to constitute a boundary value problem (BVP), the solution of which, in general, lies beyond the reach of analytical approaches. Consequently a variety of numerical schemes have been developed in order to provide approximate solutions to such BVP’s. Certainly the most widely used are the finite difference (FD) and finite element (FE) “space discretisation” techniques in which the governing partial differential equations are approximated by a set of discretised equations whose solution is subsequently obtained numerically at a finite number of prespecified points in the solution domain. An alternative approach, upon which the present work is based, has been to employ integral equation techniques, which are formulated so that the governing equations are solved only on the boundary of the required solution domain. Various such techniques were originally suggested in the latter half of the last century by Helmholtz [1], Kelvin [2] and Kirchoff [3], primarily for the study of wave propagation in unbounded media. In 1903, Fredholm [4] presented the first rigorous investigation of the existence and uniqueness of solutions to integral equations, although neither he nor his immediate successors envisaged the range of problems to which his formulations could be applied.

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Authors and Affiliations

  1. Department of Applied Mathematical Studies, University of Leeds, Leeds, LS 2 9 JT, England

    Derek B. Ingham & Mark A. Kelmanson

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  1. Derek B. Ingham
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© 1984 Springer-Verlag Berlin, Heidelberg

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Ingham, D.B., Kelmanson, M.A. (1984). General Introduction. In: Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems. Lecture Notes in Engineering, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82330-5_1

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  • DOI: https://doi.org/10.1007/978-3-642-82330-5_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13646-0

  • Online ISBN: 978-3-642-82330-5

  • eBook Packages: Springer Book Archive

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