# 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 **x _{0}**

**=β and x**. This system has the familiar tridiagonal form.

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