ABSTRACT
In 1915 B. G. Galerkin [1] developed a new extremely general and effec tive method for finding an approximate solution to boundary value problems. The idea of the method consists in the following. Suppose we are to solve the equation L(u) = 0 under certain (for the sake of being specific, say ho mogeneous) boundary conditions. We seek an approximate solution in the form
where the (fk are known functions satisfying the same boundary conditions. The requirement that the expression L(u) vanish identically will be replaced by the weaker requirement that L(un) is orthogonal to all the functions (pk(k = 1 ,2 ,. . . , n). Then in order to determine the constant cfc, we obtain the eauation
The Galerkin method is intimately related to the Ritz method but by comparison to the latter it has two important advantages: it is much more general, it may be applied to several cases when the Ritz method is not applicable, and it is much simpler; in cases when the Ritz method may be applied, it leads to the same approximate solution, but the necessary equations may be found in a much shorter way.
