Exact Traveling Wave Solutions of the Generalized Fractional Differential mBBM Equation

By using the fractional complex transform and the bifurcation theory to the generalized fractional differential mBBM equation, we first transform this fractional equation into a plane dynamic system, and then find its equilibrium points and first integral. Based on this, the phase portraits of the corresponding plane dynamic system are given. According to the phase diagram characteristics of the dynamic system, the periodic solution corresponds to the limit cycle or periodic closed orbit. Therefore, according to the phase portraits and the properties of elliptic functions, we obtain exact explicit parametric expressions of smooth periodic wave solutions. This method can also be applied to other fractional equations.

On Algebroid Solutions of Generalized Complex Differential Equations

Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized complex algebraic differential equations and obtain some results.

Generalized Higher-Order Algebraic Differential Equations with Admissible Algebroid Solutions

Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized higher order algebraic differential equations.

New lump solutions and several interaction solutions and their dynamics of a generalized(3+1)-dimensional nonlinear differential equation

In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a(3+1)-dimensional nonlinear differential equation.Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions.Moreover,the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves.In addition,the breath-wave solutions and several interaction solutions of the(3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.

A Set of Almost Automorphic Functions and Applications

For a set S of real numbers, we introduce the concept of S-almost automorphic functions valued in a Banach space. It generalizes in particular the space of Z-almost automorphic functions. Considering the space of S-almost automorphic functions, we give sufficient conditions of the existence and uniqueness of almost automorphic solutions of a differential equation with a piecewise constant argument of generalized type. This is done using the Banach fixed point theorem.

Fully Coupled Forward-Backward Stochastic Functional Differential Equations and Applications to Quadratic Optimal Control

This paper considers the fully coupled forward-backward stochastic functional differential equations(FBSFDEs) with stochastic functional differential equations as the forward equations and the generalized anticipated backward stochastic differential equations as the backward equations. The authors will prove the existence and uniqueness theorem for FBSFDEs. As an application, we deal with a quadratic optimal control problem for functional stochastic systems, and get the explicit form of the optimal control by virtue of FBSFDEs.

MAXIMUM PRINCIPLE FOR STOCHASTIC OPTIMAL CONTROL PROBLEM WITH DISTRIBUTED DELAYS

This paper is concerned with a Pontryagin's maximum principle for the stochastic optimal control problem with distributed delays given by integrals of not necessarily linear functions of state or control variables.By virtue of the duality method and the generalized anticipated backward stochastic differential equations,we establish a necessary maximum principle and a sufficient verification theorem.In particular,we deal with the controlled stochastic system where the distributed delays enter both the state and the control.To explain the theoretical results,we apply them to a dynamic advertising problem.

Asymptotically efficient parameter estimation for ordinary differential equations

Parameter estimation for ordinary differential equations arises in many fields of science and engineering. To be the best of our knowledge, traditional methods are often either computationally intensive or inaccurate for statistical inference. Ramsay et al.(2007) proposed a generalized profiling procedure. It is easily implementable and has been demonstrated to have encouraging numerical performance. However, little is known about statistical properties of this procedure. In this paper, we provide a theoretical justification of the generalized profiling procedure. Under some regularity conditions, the procedure is shown to be consistent for a broad range of tuning parameters. When the tuning parameters are sufficiently large, the procedure can be further shown to be asymptotically normal and efficient.

Maximum Principle for Non-Zero Sum Stochastic Differential Game with Discrete and Distributed Delays

This technical note is concerned with the maximum principle for a non-zero sum stochastic differential game with discrete and distributed delays.Not only the state variable,but also control variables of players involve discrete and distributed delays.By virtue of the duality method and the generalized anticipated backward stochastic differential equations,the author establishes a necessary maximum principle and a sufficient verification theorem.To explain theoretical results,the author applies them to a dynamic advertising game problem.

Stability in terms of two measures for population growth models with impulsive perturbations

In this paper,we establish some criteria for the stability of trivial solution of population growth models with impulsive perturbations.The working tools are based on the theory of generalized ordinary differential equations.Here,the conditions concerning the functions are more general than the classical ones.

Nonlinear Doob-Meyer Decomposition for General Filtration

The comparison theorem for generalized backward stochastic differential equations is discussed. Some topics related to equations of this type are also investigated.