Uniformly stable explicitly solvable finite difference method for fractional diffusion equations
A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grunwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discrete l 2 norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.
In this paper, we study the asymptotical optimal solutions for the general airline revenue management problem, considering both customer cancelations and no-shows. We study a simple case where there is only one fare-class; the management controls the opening and closing of the fare-class, with the objective of maximizing the total revenue. We first develop a fluid control problem, where the customer arrival and cancelation processes are treated as deterministic, whereas the no. of no-shows is random. The optimal fluid solution is shown to have a simple structure of accepting up to a single time point t* and rejecting afterwards. We show that this policy acts as an upper bound for the MDP problem. Applying the optimal fluid solution to the stochastic environment, we show that this policy is asymptotically optimal on the fluid scaling.