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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