## ABSTRACT

For numerical approximation of solutions of ordinary differential equations (ODEs) our methods have included finite differences in Section 8.8 and spline functions in Section 8.9. In Chapter 12, we will see analogous methods for approximating solutions of partial differential equations (PDEs). For a heat equation

∂T ∂t

(x, t) = α ∂ 2T

∂x2 (x, t) + g(x, t), 0 < x < L, t > 0, (12.1)

we can use finite differences for both the space variable, x, and the time variable, t. Partition the interval [ 0,L ] using

0 = x0 < x1 < · · · < xN = L.