ABSTRACT
This chapter addresses problems that may not be easily solved by analytic techniques, if solvable at all. It discusses techniques that provide approximate solutions for such problems. The approximate extremum is slightly higher than the analytic extremum, but by only a very acceptable error. The difference between Ritz’s method and that of Galerkin’s is in the fact that the latter addresses the differential equation form of a variational problem. Galerkin’s method minimizes the residual of the differential equation integrated over the domain with a weight function; hence, it is also called the method of weighted residuals. Both the Ritz and Galerkin methods are restricted in their choices of basis functions, because their basis functions are required to satisfy the boundary conditions. The boundary integral method is related to Kantorovich’s method in the sense that both make use of the boundary-interior distinction of a variational problem.
