ABSTRACT
Variational formulation of boundary value problems originates from the fact that weighted variational methods provide approximate solutions of such problems. Variational methods for solving boundary value problems are based on the techniques developed in the calculus of variations. They deal with the problem of minimizing a functional and thus reducing the given problem to the solution of Euler-Lagrange differential equations. If the functional to be minimized has more than one independent variable, the Euler-Lagrange equations are partial differential equations. Conversely, a boundary value problem can be formulated as a minimizing problem. The functional which corresponds to the partial differential equation is generally known as the energy function. In the case when the solution is not available in a simple form, an approximate solution is found such that it minimizes the energy equation. The approximating function is a linear combination of the form
∞∑
ciφi, c0 = 1, of
specially chosen functions φi which are known as the test functions or interpolation functions. The function φ0 satisfies the same boundary conditions as the original unknown function, while the remaining functions φi, i = 0, satisfy the homogeneous boundary conditions. The constants ci, i = 0, are then determined by minimizing the energy function.
