Weighted Residual Method

In this method, we use the integral form corresponding to the weighted residual formulation (equation 2.5b) while obtaining an approximate solution. We rearrange equation (2.5b) as follows :

|

(3.68)

As in the Ritz method, an approximate solution is assumed as a series of N terms involving the unknown coefficients and a set of linearly independent functions defined over the interval [0,L] :

(3.69)

The application of the Dirichlet boundary condition on (equation 2.1b) to this expression leads to

at

(3.70)

If we assume that all except satisfy the condition

at

(3.71)

then, from equation (3.70), we get

(3.72)

Then, the expression (3.69) starts from i = 1 rather then from i = 0. Thus,

(3.73)

Substitution of this expression in equation (3.68) leads to :

(3.74)

To convert the above equation into a set of N algebraic equations in the unknown coefficients , we choose N weight functions as follows :

for

(3.75)

Here, the functions represent another set of linearly independent functions defined over the interval [0, L ]. Note that the functions must satisfy the constraints arising out of the admissibility conditions of the weight function . These conditions are stated in section 2.2.

When the functions are different from , the corresponding weighted residual method is called as the Petrov-Galerkin Method. On the other hand, when the functions are the same as , then the method is called as the Galerkin Method. The next section describes the details of the Galerkin method.