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We study wetting front (traveling wave) solutions to the Richards equation that describe the vertical infiltration of water through one-dimensional periodically layered unsaturated soils. We prove the existence, uniqueness, and large time asymptotic stability of the traveling wave solutions under prescribed flux boundary conditions and certain constitutive conditions. The traveling waves are connections between two steady state solutions that form near the ground surface and towards the underground water table. We found a closed form expression of the wave speed. The speed of a traveling wave is equal to the ratio of the flux difference and the difference of the spatial averages of the two steady states. We give both analytical and numerical examples showing that the wave speeds in the periodic soils can be larger or smaller than those in the homogeneous soils which have the same mean diffusivity and conductivity

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Information Technology has enabled new levels of convenience to our daily life. While providing people with astonishingly good experiences, computer science has still a long way to meet the needs from the users for various scenarios. As a good example, path selection problem is an important and common problem which finds applications in location based services, internet routing and navigation systems. Choosing the most cost effective and time saving routines plays an essential role in both theory and engineering. Although a plethora of classical solutions have been made to the static shortest path problem, dynamic shortest path problem has not been thoroughly studied. To address this critical issue, this paper investigates the dynamic shortest path problem by optimizing existing work and conducting extensive experiments. The studies fall into three main variants of the problem: unweighted shortest paths, weighted shortest paths, and shortest paths on the map. We present a comprehensive comparison of our optimizations and existing solutions which demonstrates the effectiveness and efficiency of our algorithms.

This paper studies the inventory management problem of dual channels operated by one vendor. Demands of dual channels are inventory-level-dependent.We propose a multi-period stochastic dynamic programming model which shows that under mild conditions, the myopic inventory policy is optimal for the infinite horizon problem. To investigate the importance of capturing demand dependency on inventory levels, we consider a heuristic where the vendor ignores demand dependency on inventory levels, and compare the optimal inventory levels with those recommended by the heuristic. Through numerical examples, we show that the vendor may order less for dual channels than those recommended by the heuristic, and the difference between the inventory levels in the two cases can be so large that the demand dependency on inventory levels cannot be neglected. In the end, we numerically examine the impact of different ways to treat unmet demand and obtain some managerial insights.