Weak or Weighted Residual Formulation

Consider a function , defined over the interval [0, L ], which satisfies both the boundary conditions 18.1(b) and 18.1(c). In general, this function will not satisfy the differential equation 18.1(a). Instead, when is substituted in the left hand side of eq. 18.1(a), it will lead to the following error called as the residue and denoted by R(x) :

(18.2)

To make the function an approximate solution of the problem (18.1(a), 18.1(b), 18.1(c)), we minimize the above residue by setting the integral of the product of R (x) and weight function w (x) to zero :

(18.3)

The weight function w ( x ) must belong to a class of admissible functions. For the present problem, this class consists of the functions which satisfy the following conditions :

1. At the boundary where and are specified, w and must be zero. Thus, in the present problem, w = 0, and at x = 0.
2. At the boundary where and are specified, w and must be unconstrained. Thus, in the present problem, w and are unconstrained at x = L .
3. The function w should be smooth enough for the integral of the weighted residue to be finite.

We rewrite eq. (18.3) as :

(18.4)

To relax the smoothness requirements on the choice of the approximation function , we integrate by parts twice the left side of eq. (18.4). It also makes the order of derivatives of and w to be the same. Thus, the integral form becomes symmetric in and w . The first integration by parts of eq. (18.4) leads to

(18.5)

Since, w is zero at x = 0, the last term is zero. Further, satisfies the boundary condition 18.1(c). Therefore, the second last term can be expressed in terms of . Then, eq. (18.5) becomes :

(18.6)

By carrying out the second integration by parts, eq.(18.6) becomes :

(18.7)

Since, is zero at x = 0, the last term is zero. Further, satisfies the boundary condition 18.1(c). Therefore, the second last term can be expressed in terms of M . Thus, eq. (18.7) becomes :

(18.8)

This is called as the Weighted Residual Integral . This is the integral form used in the Weighted Residual Formulation . Now, the condition 3 of the class of admissible functions can be made explicit. For all the integrals of eq. (18.8) to be finite, must be piecewise continuous on the interval (0, L ) with only finite discontinuities.