The weighted residual method was developed as an approximation technique for solving differential equations and served as a prelude to the finite elements method. The basic concept of the WRM can be described as follows. Consider a linear differential operator L acting on a function f(x) to produce a function g(x):

L[f(x)] = g(x) (9.1)

Let the unknown function f(x) be approximated by a finite series,

≈ =

∑ α ϕ 1


where φi are ‘trial’ or ‘basis’ functions selected from a linearly independent set and αi are unknown constants. Substitution of the approximated function (Equation 9.2) into the differential operator (Equation 9.1) results in an error:

R x L f x g x( ) [ ( )] ( )= − ≠ 

0 (9.3)

where R(x) is the error or residual. The main concept of the WRM is to minimize the error by forcing the residual to zero in some average sense over the solution domain D. Thus,

R x w dxi D

( )∫ = 0 (9.4) where wi are the weight functions. This procedure results in a set of n-number of algebraic equations for the n-number of αi coefficients. During the selection of the trial functions, special attention should be given ensuring they satisfy the boundary conditions. Depending on the choice of the weight functions, there are different variations of the WRM, as discussed in the following sections.