## Tuesday, February 2, 2010

### Galerkin's Method

One of the most important weighted residual methods was invented by the Russian mathematician Boris Grigoryevich Galerkin (February 20, 1871 - July 12, 1945). Galerkin's method selects the weight function functions in a special way: they are chosen from the basis functions, i.e. . It is required that the following equations hold true

(6) for .

To apply the method, all we need to do is solve these equations for the coefficients .

Galerkin's Method for solving an I. V. P.

Suppose we wish to solve the initial value problem

(i) ,
with
over the interval .

We use the trial function

(ii) .

There are equations to solve for , i.e.

(iii) for .

Remark: For the solution of an I. V. P. we choose .

Galerkin's Method for solving an a B. V. P.

Suppose we wish to solve a boundary value problem over the interval ,

(I) ,
with

We define and use the trial function

(II) .

There are equations to solve for , i.e.

(III) for .

Remark: The functions must all be chosen with the boundary properties

and for .