Advanced Numerical and Semi-Analytical Methods for Differential Equations

3Weighted Residual Methods

3.1 Introduction

Weighted residual is treated as another powerful method for computation of solution to differential equations subject to boundary conditions referred to as boundary value problems (BVPs). Weighted residual method (WRM) is an approximation technique in which solution of differential equation is approximated by linear combination of trial or shape functions having unknown coefficients. The approximate solution is then substituted in the governing differential equation resulting in error or residual. Finally, in the WRM the residual is forced to vanish at average points or made as small as possible depending on the weight function in order to find the unknown coefficients. WRMs, viz. collocation and Galerkin methods, have been discussed by Gerald and Wheatley [1]. Further discussion of various WRMs may be found in standard books viz. [2–4]. As regards, least‐square method for solving BVPs has been given by Locker [5]. Weighted residual‐based finite‐element methods are discussed in Refs. 1 –3, 6 , 7 . In Chapter 6, finite‐element discretization approach using Galerkin WRM has been introduced. Moreover, sometimes trial or shape functions taken as boundary characteristic orthogonal polynomials are advantageous. So, Chapter 4 is dedicated in solving BVP using boundary characteristic orthogonal polynomials incorporated into Galerkin and Rayleigh–Ritz methods.

In this regard, this chapter is organized such that various WRMs, viz. collocation, ...