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
then the boundary value problem
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]
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.
Methods involving difference quotient approximations for derivatives can be used for solving certain second-order boundary value problems. Consider the linear equation
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
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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