Finite Difference Method for ODE's

Background

Theorem  (Boundary Value Problem).  Assume that  
  is continuous on the region  
  and  that  
  and  
  are continuous on  
.  If there exists a constant  
  for which  
  satisfy

    
  
and
    
,  

then the boundary value problem

    
  with  
  

has a unique solution   
.  

    The notation  
  has been used to distinguish the third variable of the function    
.   Finally, the special case of linear differential equations is worthy of mention.

Corollary (Linear Boundary Value Problem).  Assume that  
  in the theorem has the form  
  and that  f  and its partial derivatives  
  and  
are continuous on  
.  If there exists a constant  
  for which  p(t)  and   q(t)  satisfy

    
  
and
    
,  

then the linear boundary value problem

    
  with  
  

has a unique solution   
.  

Finite-Difference Method

    Methods involving difference quotient approximations for derivatives can be used for solving certain second-order boundary value problems.  Consider the linear equation
    
(1)        
  

over  [a,b]  with  
.  Form a partition of [a, b] using the points  
,  where  
  and
  for  
.  The central-difference formulas discussed in Chapter 6 are used to approximate the derivatives  

(2)        


and

(3)        

Use the notation
for the terms
on the right side of (2) and (3) and drop the two terms
.  Also, use the notations  
,    
,  and  
this produces the difference equation

        


which is used to compute numerical approximations to the differential equation (1).  This is carried out by multiplying each side  by
and then collecting terms involving  
  and arranging them in a system of linear equations:

        
  

for  
, where
and  
. This system has the familiar tridiagonal form.