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### Open Access Journals

We consider an Internet congestion control system which is presented as a group of differential equations with time delay, modeling the random early detection (RED) algorithm. Although this model achieves success in many aspects, some basic problems are not clear. We provide the result on the existence of the equilibrium and the positivity and boundedness of the solution. Also, we implement the model by route switch mechanism, based on the minimum delay principle, to model the dynamic routing. For the simple network topology, we show that the Filippov solution exists under some restrictions on parameters. For the case with a single user group and two alternative links, we prove that the discontinuous boundary, or equivalently the sliding region, always exists and is locally attractive. This result implies that for some cases this type of routing may deviate from the purpose of the original design.

We propose a model of foam cell formation accounting for macrophage RCT. This model is presented as a system of non-linear ordinary differential equations. Motivated by experimental observations regarding time scales for oxidation of lipids and MRCT, we impose a quasi-steady state assumption and analyze the resulting systems of equations. We focus on the existence and stability of equilibrium solutions as determined by the governing parameters with the results interpreted in terms of their potential bio-medical implications.

In this paper, we present an optimal feedback control model to deal with the problem of energy efficiency management. Especially, an emission permits trading scheme is considered in our model, in which the decision maker can trade the emission permits flexibly. We make use of the optimal control theory to derive a Hamilton-Jacobi-Bellman (HJB) equation satisfied by the value function, and then propose an upwind finite difference method to solve it. The stability of this method is demonstrated and the accuracy, as well as the usefulness, is shown by the numerical examples. The optimal management strategies, which maximize the discounted stream of the net revenue, together with the value functions, are obtained. The effects of the emission permits price and other parameters in the established model on the results have been also examined. We find that the influences of emission permits price on net revenue for the economic agents with different initial quotas are quite different. All the results demonstrate that the emission permits trading scheme plays an important role in the energy efficiency management.

It becomes increasingly important to manage water and improve the efficiency of irrigation under higher temperatures and irregular precipitation patterns. The choice of investment in water saving technologies and its timing play key roles in improving efficiency of water use. In this paper, we use a real option approach to establish a model to handle future uncertainties about the water price. In addition, to match the practical situation, the expiration of the real option is considered to be finite in our model, such that it is difficult to solve the model. Therefore, we reformulate the problem into a linear parabolic variational inequality (Ⅵ) and develop a power penalty method to solve it numerically. Thus, a nonlinear partial differential equation (PDE) is obtained, which is shown to be uniquely solvable and the solution of the nonlinear PDE converges to that of the Ⅵ at the rate of $O(λ^{-\frac{k}{2}})$ with $λ$ being the penalty number. Furthermore, a so-called fitted finite volume method is proposed to solve the nonlinear PDE. Finally, several numerical experiments are performed. It is shown that the subjective discount rate will affect the investment boundary mostly, and the flexibility to suspend operation will enlarge the investment region.

In this paper, we present a combined homotopy interior point method for solving multiobjective programs with equilibrium constraints. Under suitable conditions, we prove the existence and convergence of a smooth homotopy path from almost any interior point to a solution of the K-K-T system. Numerical results are presented to show the effectiveness of this algorithm.

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