Zero‐coupon bond prices in the Vasicek and CIR models: Their computation as group‐invariant solutions

We compute prices of zero‐coupon bonds in the Vasicek and Cox–Ingersoll–Ross interest rate models as group‐invariant solutions. Firstly, we determine the symmetries of the valuation partial differential equation that are compatible with the terminal condition and then seek the desired solution among the invariant solutions arising from these symmetries. We also point to other possible studies on these models using the symmetries admitted by the valuation partial differential equations. Copyright © 2007 John Wiley & Sons, Ltd.     

A Navier–Stokes–Voight model with memory

In this article, we consider a three‐dimensional Navier–Stokes–Voight model with memory where relaxation effects are described through a distributed delay. We prove the existence of uniform global attractors , where ? ∈ (0,1) is the scaling parameter in the memory kernel. Furthermore, we prove that the model converges to the classical three‐dimensional Navier–Stokes–Voight system in an appropriate sense as ? → 0. In particular, we construct a family of exponential attractors Ξ_? that is robust as ? → 0. Copyright © 2013 John Wiley & Sons, Ltd.     

Ion acoustic solitary wave solutions of two‐dimensional nonlinear Kadomtsev–Petviashvili–Burgers equation in quantum plasma

Propagation of two‐dimensional nonlinear ion‐acoustic solitary waves and shocks in a dissipative quantum plasma is analyzed. By applying the reductive perturbation theory, the two‐dimensional ion acoustic solitary waves in a dissipative quantum plasma lead to a nonlinear Kadomtsev–Petviashvili–Burgers (KPB) equation. By implementing extended direct algebraic mapping, extended sech‐tanh, and extended direct algebraic sech methods, the ion solitary traveling wave solutions of the two‐dimensional nonlinear KPB equation are investigated. An analytical as well as numerical solution of the two‐dimensional nonlinear KPB equation is obtained and analyzed with the effects of external electric field and ion pressure. Copyright © 2016 John Wiley & Sons, Ltd.     

Global weak solutions to the Cauchy problem of compressible Navier–Stokes–Vlasov–Fokker–Planck equations

A fluid–particles system of the compressible Navier‐Stokes equations and Vlasov‐Fokker‐Planck equation (including the case of Vlasov equation) in three‐dimensional space is considered in this paper. The coupling arises from a drag force exerted by the fluid onto the particles. We study a Cauchy problem with large data, and establish the existence of global weak solutions through an approximation scheme, energy estimates, and weak convergence. Copyright © 2016 John Wiley & Sons, Ltd.     

Blow‐up of smooth solutions to the Navier–Stokes–Poisson equations

The main purpose of this paper is concerned with blow‐up smooth solutions to Navier–Stokes–Poisson (N‐S‐P) equations. First, we present a sufficient condition on the blow up of smooth solutions to the N‐S‐P system. Then we construct a family of analytical solutions that blow up in finite time. Copyright © 2010 John Wiley & Sons, Ltd.     

Discrete Jacobi sub‐equation method for nonlinear differential–difference equations

We will propose a unified algebraic method to construct Jacobi elliptic function solutions to differential–difference equations (DDEs). The solutions to DDEs in terms of Jacobi elliptic functions sn, cn and dn have a unified form and can be presented through solving the associated algebraic equations. To illustrate the effectiveness of this method, we apply the algorithm to some physically significant DDEs, including the discrete hybrid equation, semi‐discrete coupled modified Korteweg–de Vries and the discrete Klein–Gordon equation, thereby generating some new exact travelling periodic solutions to the discrete Klein–Gordon equation. A procedure is also given to determine the polynomial expansion order of Jacobi elliptic function solutions to DDEs. Copyright © 2010 John Wiley & Sons, Ltd.     

Asymptotic dynamics of the one‐dimensional attraction–repulsion Keller–Segel model

The asymptotic behavior of the attraction–repulsion Keller–Segel model in one dimension is studied in this paper. The global existence of classical solutions and nonconstant stationary solutions of the attraction–repulsion Keller–Segel model in one dimension were previously established by Liu and Wang (2012), which, however, only provided a time‐dependent bound for solutions. In this paper, we improve the results of Liu and Wang (2012) by deriving a uniform‐in‐time bound for solutions and furthermore prove that the model possesses a global attractor. For a special case where the attractive and repulsive chemical signals have the same degradation rate, we show that the solution converges to a stationary solution algebraically as time tends to infinity if the attraction dominates. Copyright © 2014 John Wiley & Sons, Ltd.     

Travelling waves of a predator–prey model with a spatiotemporal delay

This paper is concerned with the existence of traveling waves to a predator–prey model with a spatiotemporal delay. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states are established, and the existence of Hopf bifurcation at the positive steady state is also discussed. By constructing a pair of upper–lower solutions and by using the cross‐iteration method as well as the Schauder's fixed‐point theorem, the existence of a traveling wave solution connecting the semi‐trivial steady state and the positive steady state is proved. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Well‐posedness and asymptotic analysis for a Penrose–Fife type phase field system

In this paper, an asymptotic analysis of the (non‐conserved) Penrose–Fife phase field system for two vanishing time relaxation parameters ε and δ is developed, in analogy with the similar analyses for the phase field model proposed by G. Caginalp (Arch. Rational Mech. Anal. 1986; 92 :205–245), which were carried out by Rossi and Stoth (Adv. Math. Sci. Appl. 2003; 13 :249–271; Quart. Appl. Math. 1995; 53 :695–700). Although formally the singular limits for ε ↓ 0 and for ε and δ ↓ 0 are, respectively, the viscous Cahn–Hilliard equation and the Cahn–Hilliard equation, it turns out that the Penrose–Fife system is indeed a bad approximation for these equations. Therefore, we consider an alternative approximating phase field system, which could be viewed as a generalization of the classical Penrose–Fife phase field system, featuring a double non‐linearity given by two maximal monotone graphs. A well‐posedness result is proved for such a system, and it is shown that the solutions converge to the unique solution of the viscous Cahn–Hilliard equation as ε ↓ 0, and of the Cahn–Hilliard equation as ε ↓ 0 and δ ↓ 0. Copyright © 2004 John Wiley & Sons, Ltd.     

Periodic wave solutions and asymptotic analysis of the Hirota–Satsuma shallow water wave equation

In this paper, we devise a simple way to explicitly construct the Riemann theta function periodic wave solution of the nonlinear partial differential equation. The resulting theory is applied to the Hirota–Satsuma shallow water wave equation. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function. We obtain the one‐periodic and two‐periodic wave solutions of the equation. The relations between the periodic wave solutions and soliton solutions are rigorously established. Copyright © 2014 John Wiley & Sons, Ltd.     

Zero‐viscosity‐capillarity limit to rarefaction waves for the 1D compressible Navier–Stokes–Korteweg equations

In this paper, we study the zero viscosity and capillarity limit problem for the one‐dimensional compressible isentropic Navier–Stokes–Korteweg equations when the corresponding Euler equations have rarefaction wave solutions. In the case that either the effects of initial layer are ignored or the rarefaction waves are smooth, we prove that the solutions of the Navier–Stokes–Korteweg equation with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity and capillarity number vanish, and we also obtain a rate of convergence, which is valid uniformly for all time. These results are showed by a scaling argument and elementary energy analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Wave features of a hyperbolic prey–predator model

A hyperbolic predator–prey model is proposed within the context of extended thermodynamics. The nature of the steady state solutions for the uniform and non‐uniform perturbations are analyzed. The existence of smooth traveling wave‐like solutions, related to the invasion of the predator population into a prey‐only state is discussed. Validation of the model in point is also accomplished by searching for numerical solutions of the system, which also points out limit cycles in the populations. Copyright © 2010 John Wiley & Sons, Ltd.     

Solutions of two-factor models with variable interest rates

The focus of this work is on numerical solutions to two-factor option pricing partial differential equations with variable interest rates. Two interest rate models, the Vasicek model and the Cox–Ingersoll–Ross model (CIR), are considered. Emphasis is placed on the definition and implementation of boundary conditions for different portfolio models, and on appropriate truncation of the computational domain. An exact solution to the Vasicek model and an exact solution for the price of bonds convertible to stock at expiration under a stochastic interest rate are derived. The exact solutions are used to evaluate the accuracy of the numerical simulation schemes. For the numerical simulations the pricing solution is analyzed as the market completeness decreases from the ideal complete level to one with higher volatility of the interest rate and a slower mean-reverting environment. Simulations indicate that the CIR model yields more reasonable results than the Vasicek model in a less complete market.     

Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions

We consider in this article the Cahn–Hilliard equation endowed with dynamic boundary conditions. By interpreting these boundary conditions as a parabolic equation on the boundary and by considering a regularized problem, we obtain, by the Leray–Schauder principle, the existence and uniqueness of solutions. We then construct a robust family of exponential attractors. Copyright © 2005 John Wiley & Sons, Ltd.     

The Cauchy–Neumann problem for a multi‐dimensional isentropic hydrodynamic model for semiconductors

We investigate a multi‐dimensional isentropic hydrodynamic (Euler–Poisson) model for semiconductors, where the energy equation is replaced by the pressure–density relation p(n) . We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and homogeneous Neumann boundary conditions. We show that, as t→+∞, the solutions converge to the non‐constant stationary solutions of the corresponding drift–diffusion equations. Moreover, we also investigate the existence and uniqueness of the stationary solutions for the corresponding drift–diffusion equations. Copyright © 2005 John Wiley & Sons, Ltd.     

An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model

We propose an original scheme for the time discretization of a triphasic Cahn–Hilliard/Navier–Stokes model. This scheme allows an uncoupled resolution of the discrete Cahn–Hilliard and Navier‐Stokes system, which is unconditionally stable and preserves, at the discrete level, the main properties of the continuous model. The existence of discrete solutions is proved, and a convergence study is performed in the case where the densities of the three phases are the same. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq. 2013     

Existence of global weak solutions for a phase–field model of a vesicle moving into a viscous incompressible fluid

We consider in this article a model of vesicle moving into a viscous incompressible fluid. The vesicle is described through a phase–field equation and through a transport equation modeling the local incompressibility of its membrane. The equations for the fluid are the classical Navier–Stokes equations with a force resulting from the presence of the vesicle. Our main result states the existence of weak solutions for the corresponding system. The proof is based on compactness/monotonicity arguments. Copyright © 2013 John Wiley & Sons, Ltd.     

A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2‐D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub‐domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time‐stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.     

Conservation laws for Lotka–Volterra models

We derive necessary and sufficient conditions on a Lotka–Volterra model to admit a conservation law of Volterra's type. The result and the proof for the corresponding linear algebra problem are given in graph‐theoretical terms; they refer to the directed graph which is defined by the coefficients of the differential equation system. Copyright © 2003 John Wiley & Sons, Ltd.     

Long-time behaviour for solutions of the Vlasov–Poisson–Fokker–Planck equation

We study the long-time behaviour of solutions of the Vlasov–Poisson–Fokker–Planck equation for initial data small enough and satisfying some suitable integrability conditions. Our analysis relies on the study of the linearized problems with bounded potentials decaying fast enough for large times. We obtain global bounds in time for the fundamental solutions of such problems and their derivatives. This allows to get sharp bounds for the decay of the difference between the solutions of the Vlasov–Poisson–Fokker–Planck equation and the solution of the free equation with the same initial data. Thanks to these bounds, we get an explicit form for the second term in the asymptotic expansion of the solutions for large times. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.