From AMS Glossary
Methods used for the solution of integral equations or for the solution of boundary-value problems that can be transformed to integral-equation form.
As an example of the Galerkin method, in the integral equation
the dependent variable y(x) is approximated by a linear combination of the polynomial functions, 1, x, x2, . . ., xn as
The approximating polynomial is substituted in the integral equation and the assumption made that exact equality holds when the resulting relation is multiplied by xi, i = 0, . . ., n, and integrated from a to b. There results a system of n + 1 linear equations in the Aj, the solution for which yields the coefficients in the polynomial approximation of y(x).