Ode Boundary Value Problems & Finite Differences


Theorem (Boundary Value Problem).  Assume that  f(t,x,y)   is continuous on the region  matlab121 and  that  matlab122  and  matlab123 are continuous on  R.  If there exists a constant  M > 0  for which  fx and fy  satisfy
matlab124
and
matlab125

then the boundary value problem
matlab126 with    matlab127

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 matlab128   and that  f  and its partial derivatives  matlab129  and matlab1210  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
matlab1211 ,

then the linear boundary value problem
matlab1212
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)          matlab1213

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)        matlab1217

and

(3)        matlab1218

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

matlab1219

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:
matlab1221

for  j = 1,2, …., n-1, where  x0=β and  xn = β. This system has the familiar tridiagonal form.

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