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
.  

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
.  

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  
.