# Ode Boundary Value Problems & Finite Differences

Theorem (Boundary Value Problem).  Assume that  f(t,x,y)   is continuous on the region   and  that    and   are continuous on  R.  If there exists a constant  M > 0  for which  fx and fy  satisfy

and

then the boundary value problem
with

has a unique solution   x=x(t) for a<=t<=b .

The notation  y = x `(t)  has been used to distinguish the third variable of the function    f(t, x, x`).   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  R.  If there exists a constant    for which  p(t)  and   q(t)  satisfy

q (t) > 0 for all t € [a, b]
and
,

then the linear boundary value problem

with   x (a) = α and x (b) =β

has a unique solution   x =x(t) over a≤ t ≤b.

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 tj = a + jh  for  j=0,1,2, ….., n.  The central-difference formulas discussed in Chapter 6 are used to approximate the derivatives

(2)

and

(3)

Use the notation  xj for the terms  x (tj) on the right side of (2) and (3) and drop the two terms  0(h2).  Also, use the notations  pj = p (tj),    qj = q (tj),  and   rj = r (tj) 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  j = 1,2, …., n-1, where  x0=β and  xn = β. This system has the familiar tridiagonal form.

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