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A numerical approach to approximation for an ultraparabolic equation

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.