# General Introduction

• Derek B. Ingham
• Mark A. Kelmanson
Part of the Lecture Notes in Engineering book series (LNENG, volume 7)

## 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.

## Keywords

Boundary Value Problem Boundary Integral Equation Solution Domain Integral Equation Method Biharmonic Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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