Module

for

Galerkin's Method

Background

To start we need some background regarding an the inner product.

Definition (Inner Product).  Consider the vector space of real functions whose domain is the closed interval  .  We define the inner product of two functions   as follows

.

Remark.  The inner product is a continuous infinite dimensional analog to the ordinary dot product that is studied in linear algebra.  If the inner product is zero then are said to be orthogonal to each other on .  All the functions we use are assumed to be square-integrable, i. e.  .

Mathematica Function (Inner Product). To compute the inner product of two real functions over .

Example 1 (a).  Find the inner product of     and    over .
1 (b).  Find the inner product of     and    over  .
1 (c).  Find the inner product of     and    over  .
Solution 1 (a).
Solution 1 (b).
Solution 1 (c).

Lemma.  If    for any function  ,  then  .

Basis for a Vector Space

A complete basis for a vector space V of functions is a set of linear independent functions   which has the property that any can be written uniquely as a linear combination

.

For example, if V the set of all polynomials and power series, then a complete basis is  .

Property.  If    and    all    then  .

We mention these concepts without proof so as to provide a little background.

Weighted Residual Methods

A weighted residual method uses a finite number of functions .  Consider the differential equation

(1)             over the interval  .

The term denotes a linear differential operator.

Multiplying (1) by any arbitrary weight function and integrating over the interval one obtains

(2)            for any arbitrary  .

Equations (1) and (2) are equivalent, because is any arbitrary function.

We introduce a trial solution to (1) of the form

(3)        ,

and replace with on the left side of  (1).

The residual is defined as follows

(4)

The goal is to construct so that the integral of the residual will be zero for some choices of weight functions.  That is, will partially satisfy (2) in the sense that

(5)            for some choices of  .

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 .

Proof  Galerkin Method

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  .

Proof  Galerkin Method

Computer Programs  Galerkin Method

Example 2.  Solve   ,   with the initial condition   .
Solution 2.

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  .

Proof  Galerkin Method

Computer Programs  Galerkin Method

Example 3.  Solve   .
3 (a).   Use the boundary values    and  .
3 (b).   Use the boundary values    and  .
Solution 3 (a).
Solution 3 (b).

Example 4.  Solve   .
4 (a).   Use the boundary values    and  .
4 (b).   Use the boundary values    and  .
Solution 4 (a).
Solution 4 (b).

Galerkin Method  Internet hyperlinks to web sites and a bibliography of articles.

(c) John H. Mathews 2005