  We study the following ultraparabolic equation
  \[ \frac{\partial}{\partial t}u\left(t,s\right)+\frac{\partial}{\partial
s}u\left(t,s\right)+\mathcal{L}u\left(t,s\right)=f\left(u\left(t,s\right),t,s\right),\quad\left(t,s\right)\in\left(0,T\right)\times\left(0,T\right),
\]
  where $\mathcal{L}$ is a positive-definite, self-adjoint operator with
compact inverse and $f$ is a nonlinear function. Mathematically, the
bibliography on initial-boundary value problems for ultraparabolic equations is
not extensive although the problems have many applications related to option
pricing, multi parameter Brownian motion, population dynamics and so forth. In
this paper, we present the approximate solution by virtue of finite difference
scheme and Fourier series. For the linear case, we give the approximate
solution and obtain a stability result. For the nonlinear case, we use an
iterative scheme by linear approximation to get the approximate solution and
obtain error estimates. Some numerical examples are given to demonstrate the
efficiency of the method.
