# Galerkin methods

From AMS Glossary

## Galerkin methods

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 The approximating polynomial is substituted in the integral equation and the assumption made that exact equality holds when the resulting relation is multiplied by

*y*(*x*) is approximated by a linear combination of the polynomial functions, 1,*x*,*x*^{2}, . . .,*x*^{n}as*x*^{i},*i*= 0, . . .,*n*, and integrated from*a*to*b*. There results a system of*n*+ 1 linear equations in the*A*_{j}, the solution for which yields the coefficients in the polynomial approximation of*y*(*x*).