## ABSTRACT

As usual, we approximate the spatial derivatives by finite differences on a uniform lattice of N + 1 points having spacing h = 1/iV, while the time derivative is approximated by the simplest first-order difference formula assuming a time step At. Using the superscript n to label the time step (that is, 0n = 0(tn), tn = 7iA/), we can approximate (7.3) as

j 7 i + l __ in -j ■ * W f l - = ^ { S 2r ) i + S f , (7.4)

At i = 1 and i = N — 1, this equation involves the Dirichlet boundary conditions specifying 0o and 0jy.