Regularizing an inverse problem for the nonlinear ultraparabolic equation

We investigate a backward problem for the ultraparabolic equation with nonlinear source function, which aims to determine the initial status of some physical field such as option pricing, multi-parameter Brownian motion, population dynamics, and so forth. This problem is well-known to be ill posed because the solution exhibits unstable dependence on the given data functions. To obtain the approximate solution, we apply the central difference method and establish a convergence rate of H"{o}lder type under some a priori assumptions on the exact solution. Eventually, a numerical example is given to justify the theoretical analysis.