A general global approximation method for the solution of boundary value problems

A general global approximation scheme is developed and its generality is demonstrated by the derivation of classical Lagrange and Hermite interpolation and finite difference and finite element approximations as its special cases. It is also shown that previously reported general approximation techniques which use the idea of moving least square are also special cases of the present method. The combination of the developed general global approximation technique with the weighted residual methods provides a very powerful scheme for the solution of the boundary value problems formulated in terms of differential equations. Although this application is the main purpose of the this project, nevertheless, the power and flexibility of the developed approximation allows it to be used in many other areas. In this study the following applications of the described approximation are developed: 1- data fitting (including curve and surface fitting) 2- plane mapping (both in cases where a conformal mapping exists and for non-conformal mapping) 3- solution of eigenvalue problems with particular application to spectral expansions used in the modal representation of shallow water equations 4- solution of ordinary differential equations (including Sturm-Liouville equations, non-homogeneous equations with non-smooth right hand sides and 4th order equations) 5- elliptic partial differential equations (including Poisson equations with non-smooth right hand sides) A computer program which can handle the above applications is developed. This program utilises symbolic, numerical and graphical and the programming language provided by the Mathematica package.