Weighted Residual Methods

where φ(x) is the dependent variable and is unknown and f (x) is a known function. L denotes the differential operator involving spatial derivative of φ , which specifies the actual form of the differential equation. Weighted residual method involves two major steps. In the first step, an approximate solution based on the general behavior of the dependent variable is assumed. The assumed solution is often selected so as to satisfy the boundary conditions for φ . This assumed solution is then substituted in the differential equation. Since the assumed solution is only approximate, it does not in general satisfy the differential equation and hence results in an error or what we call a residual. The residual is then made to vanish in some average sense over the entire solution domain to produce a system of algebraic equations. The second step is to solve the system of equations resulting from the first step subject to the prescribed boundary condition to yield the approximate solution sought. Let ψ(x) ≈ φ(x), is an approximate solution to the differential equation (15). When ψ(x) is substituted in the differential equation (15), it is unlikely that the equation is satisfied. That is, we have L (ψ)+ f 6= 0.