A technique for the solution of certain classes of ordinary and partial differential equations that involves the use of a variational principle. That is, the solution of the differential equation is expressed as the solution of a minimization problem that involves an integral expression. The equation is then solved by carrying out an approximate minimization. Variational principles arise naturally in many branches of physics and engineering. As an example, the solution of y″ + q(x)y= f(x), 0≤x≤1 y(0)= y(1) = 0is also the solution of the problem where V is a class of sufficiently differentiable functions that are zero at x= 0 and x= 1. An approximate minimization can be carried out by minimizing over the subspace of functions When the trial functions φj(x) are splines, the resulting method is an example of the finite-element method.
y″ + q(x)y= f(x), 0≤x≤1
y(0)= y(1) = 0