Finite Difference Method for ODE's

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 for . The central-difference formulas discussed in Chapter 6 are used to approximate the derivatives

(2)

and

(3)

Use the notation for the terms on the right side of (2) and (3) and drop the two terms . Also, use the notations , , and 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 , where and . This system has the familiar tridiagonal form.