Operator splitting method for the stochastic production-inventory model equation

Stochastic optimal control of an inventory model with a deterministic rate of deteriorating items is first provided by Alshamerni (2013). The main difficulty of solving this partial differential equation is the non-linear term (ux)2 which has no exactly meaning from mathematical view. The normal method to obtain the analytic solution is to give the conjecture that its solution takes the given form (quadratic). Then we solve the ordinary differential equation with initial or terminal conditions. There are two drawbacks for such method: (1) We do not know the solution curve tendency with respect to the time t; (2) Solve the ODE system directly is complicated computing. Instead we apply the operator splitting method after Cole-Hopf transformation for the initial equation. Split the partial differential equation into two parts, each part can be solved with an analytical solution. Numerical application of the method will be presented to verify the result.