Two homoclinic solutions for a nonperiodic fourth‐order differential equation without coercive condition

In this paper, we investigate the existence of homoclinic solutions for a class of fourth‐order nonautonomous differential equations where w is a constant, and . By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that a(x) and f(x,u) are nonperiodic in x,a does not fulfil the coercive condition, and f does not satisfy the well‐known (AR)‐condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth‐order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.     

Stochastic partial functional differential equations with locally monotone coefficients,locally Lipschitz non-linearity and delay

In this paper, we study the existence and uniqueness of strong solutions for stochastic partial functional differential equations with locally monotone coefficients, locally Lipschitz non-linearity, and time delay. Our results extend previous results obtained by Liu–Röckner, Caraballo et al. and Taniguchi et al. Examples are given to illustrate the wide applicability of our results.     

Low-Storage Runge-Kutta Method for Simulating Time-Dependent Quantum Dynamics

A wide range of quantum systems are time-invariant and the corresponding dynamics is dictated by linear differential equations with constant coefficients.Although simple in mathematical concept,the integration of these equations is usually complicated in practice for complex systems,where both the computational time and the memory storage become limiting factors.For this reason,low-storage Runge-Kutta methods become increasingly popular for the time integration.This work suggests a series of s-stage sth-order explicit RungeKutta methods specific for autonomous linear equations,which only requires two times of the memory storage for the state vector.We also introduce a 13-stage eighth-order scheme for autonomous linear equations,which has optimized stability region and is reduced to a fifth-order method for general equations.These methods exhibit significant performance improvements over the previous general-purpose low-stage schemes.As an example,we apply the integrator to simulate the non-Markovian exciton dynamics in a 15-site linear chain consisting of perylene-bisimide derivatives.     

The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion

In this paper, we consider a class of impulsive stochastic differential equations driven by G-Brownian motion (IGSDEs in short). By means of the G-Lyapunov function method, some criteria on p-th moment stability and p-th moment asymptotical stability for the trivial solutions of IGSDEs are established. An example is presented to illustrate the efficiency of the obtained results.     

On the construction of non-affine jump-diffusion models

We describe a method for construction of jump analogues of certain one-dimensional diffusion processes satisfying solvable stochastic differential equations. The method is based on the reduction of the original stochastic differential equations to the ones with linear diffusion coefficients, which are reducible to the associated ordinary differential equations, by using the appropriate integrating factor processes. The analogues are constructed by means of adding the jump components linearly into the reduced stochastic differential equations. We illustrate the method by constructing jump analogues of several diffusion processes and expand the notion of market price of risk to the resulting non-affine jump-diffusion models.     

High‐order compact schemes for fractional differential equations with mixed derivatives

In this article, we consider two‐dimensional fractional subdiffusion equations with mixed derivatives. A high‐order compact scheme is proposed to solve the problem. We establish a sufficient condition and show that the scheme converges with fourth order in space and second order in time under this condition.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2141–2158, 2017     

Some inverse problems for the nonlocal heat equation with Caputo fractional derivative

A class of inverse problems for restoring the right‐hand side of a fractional heat equation with involution is considered. The results on existence and uniqueness of solutions of these problems are presented.     

Generalized Bessel functions: Theory and their applications

This paper presents 2 new classes of the Bessel functions on a compact domain [0,T] as generalized‐tempered Bessel functions of the first‐ and second‐kind which are denoted by GTBFs‐1 and GTBFs‐2. Two special cases corresponding to the GTBFs‐1 and GTBFs‐2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self‐adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.     

Time Discrete Approximation of Weak Solutions to Stochastic Equations of Geophysical Fluid Dynamics and Applications

As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans,the time discretization of these equations by an implicit Euler scheme is studied.From the deterministic point of view,the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions.From the probabilistic viewpoint,this paper deals with a wide class of nonlinear,state dependent,white noise forcings which may be interpreted in either the It6 or the Stratonovich sense.The proof of convergence of the Euler scheme,which is carried out within an abstract framework,covers the equations for the oceans,the atmosphere,the coupled oceanic-atmospheric system as well as other related geophysical equations.The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense,a result which is new by itself to the best of our knowledge.     

一类1 < α < 2非局部条件下的脉冲分数阶微分方程mild解的存在性

本文研究了一类1 < α < 2非局部条件下的脉冲分数阶偏微分方程mild解的存在性问题.利用解算子的相关性质及Krasnoselskii不动点理论的方法,获得了这类方程的mild解并予以证明,且得到了解的存在性结果.