1 Background

Linear systems that arise in practice can become quite large in size. The purpose of this computer project is to understand why this often occurs and why there is frequently a special structure to the linear systems that come from practical applications.

Given an interval [a, b] and two numbers α and β, consider the general problem of trying to find a function
y(t) that satisfies the differential equation

u(t)y
′′(t) + v(t)y
′
(t) + w(t)y(t) = f(t), with y(a) = α and y(b) = β

The functions u, v, w, and f are assumed to be known functions on [a, b]. Because the unknown function
y(t) is specified at the boundary points a and b, Equation (1) is referred to as a two-point boundary value
problem. Such problems abound in nature and are frequently very hard to handle because it is often not
possible to express y(t) in terms of elementary functions. Numerical methods are usually employed to
approximate y(t) at discrete points inside [a, b]. Approximations are produced by subdividing the interval
[a, b] into n + 1 equal subintervals, each of length h = (b − a)/(n + 1) as shown in Figure 1.

Derivative approximations at the interior nodes (grid points) ti = a + ih are made by using Taylor series
expansions y(t) = P∞
k=0 y
(k)
(ti)(t − ti)
k/k! to write

y(ti + h) = y(ti) + y
′
(ti)h +
y
′′(ti)
2! h
2 +
y
′′′(ti)
3! h
3 +

and

y(ti − h) = y(ti) − y
′
(ti)h +
y
′′(ti)
2! h
2 −
y
′′′(ti)
3! h
3 + . . .

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