Oscillation analysis of numerical solutions in the θ‐methods for differential equation of advanced type

The paper deals with the oscillation analysis of numerical solution in the θ‐methods for differential equations with piecewise constant arguments of advanced type. The conditions of the oscillation for the θ‐method are obtained. It is proved that the oscillation of the analytic solution is preserved by the θ‐ method. Some numerical experiments are given. Copyright © 2015 John Wiley & Sons, Ltd.     

The improved split‐step θ methods for stochastic differential equation

Two improved split‐step θ methods, which, respectively, named split‐step composite θ method and modified split‐step θ‐Milstein method, are proposed for numerically solving stochastic differential equation of Itô type. The stability and convergence of these methods are investigated in the mean‐square sense. Moreover, an approach to improve the numerical stability is illustrated by choices of parameters of these two methods. Some numerical examples show the accordance between the theoretical and numerical results. Further numerical tests exhibit not only the Hamiltonian‐preserving property of the improved split‐step θ methods for a stochastic differential system but also the positivity‐preserving property of the modified split‐step θ‐Milstein method for the Cox–Ingersoll–Ross model. Copyright © 2013 John Wiley & Sons, Ltd.     

Oscillation analysis of θ‐methods for the Nicholson's blowflies model

This paper is concerned with the oscillation of numerical solution for the Nicholson's blowflies model. Using two kinds of θ‐methods, namely, the linear θ‐method and the one‐leg θ‐method, several conditions under which the numerical solution oscillates are derived. Moreover, it is shown that every non‐oscillatory numerical solution tends to equilibrium point of the original continuous‐time model. Finally, numerical experiments are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Global C∞‐solutions to 1D compressible Navier–Stokes equations with density‐dependent viscosity

In this paper, we consider the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries. The initial density ρ₀∈W^1,2n is bounded below away from zero and the initial velocity u₀∈L²ⁿ. The viscosity coefficient µ is proportional to ρ^θ with 0<θ?1, where ρis the density. The existence and uniqueness of global solutions in Hⁱ([0,1])(i = 1,2,4) have been established in (J. Math. Phys. 2009; 50 :023101; Meth. Appl. Anal. 2005; 12 :239–252; J. Differ. Equations 2008; 245:3956–3973; Commun. Pure Appl. Anal. 2008; 7 :373–381). By mathematical induction method, we will establish the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries when the initial data ρ₀ and u₀ are smooth. Copyright © 2011 John Wiley & Sons, Ltd.     

Well‐posedness of time‐space fractional stochastic evolution equations driven by α‐stable noise

This paper is devoted to the well‐posedness for time‐space fractional Ginzburg‐Landau equation and time‐space fractional Navier‐Stokes equations by α‐stable noise. The spatial regularity and the temporal regularity of the nonlocal stochastic convolution are firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag‐Leffler functions, respectively. Numerical simulations for time‐space fractional Ginzburg‐Landau equation are provided to verify the analysis results.     

Stepanov‐like doubly weighted pseudo almost automorphic processes and its application to Sobolev‐type stochastic differential equations driven by G‐Brownian motion

This paper investigates the properties of the p‐mean Stepanov‐like doubly weighted pseudo almost automorphic (S^pDWPAA) processes and its application to Sobolev‐type stochastic differential equations driven by G‐Brownian motion. We firstly prove the equivalent relation between the S^pDWPAA and Stepanov‐like asymptotically almost automorphic stochastic processes based on ergodic zero set. We further establish the completeness of the space and the composition theorem for S^pDWPAA processes. These results obtained improve and extend previous related conclusions. As an application, we show the existence and uniqueness of the S^p DWPAA solution for a class of nonlinear Sobolev‐type stochastic differential equations driven by G‐Brownian motion and present a decomposition of this unique solution. Moreover, an example is given to illustrate the effectiveness of our results.     

Non‐well‐founded extensions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {V}$\end{document}

This paper describes a new and user‐friendly method for constructing models of non‐well‐founded set theory. Given a sufficiently well‐behaved system θ of non‐well‐founded set‐theoretic equations, we describe how to construct a model M_θ for $\mathsf {ZFC}^-$ in which θ has a non‐degenerate solution. We shall prove that this M_θ is the smallest model for $\mathsf {ZFC}^-$ which contains $\mathbf {V}$ and has a non‐degenerate solution of θ.     

A remark on free boundary problem of 1‐D compressible Navier–Stokes equations with density‐dependent viscosity

In this paper, we prove the existence and uniqueness of the weak solution of the one‐dimensional compressible Navier–Stokes equations with density‐dependent viscosity µ(ρ)=ρ^θ with θ∈(0, γ?2], γ>1. The initial data are a perturbation of a corresponding steady solution and continuously contact with vacuum on the free boundary. The obtained results apply for the one‐dimensional Siant–Venant model of shallow water and generalize ones in (Arch. Rational Mech. Anal. 2006; 182: 223–253). Copyright © 2009 John Wiley & Sons, Ltd.     

Two grid finite element discretization method for semi‐linear hyperbolic integro‐differential equations

In this paper, we will investigate a two grid finite element discretization method for the semi‐linear hyperbolic integro‐differential equations by piecewise continuous finite element method. In order to deal with the semi‐linearity of the model, we use the two grid technique and derive that once the coarse and fine mesh sizes H, h satisfy the relation h = H² for the two‐step two grid discretization method, the two grid method achieves the same convergence accuracy as the ordinary finite element method. Both theoretical analysis and numerical experiments are given to verify the results.     

Compressible Navier–Stokes equations with vacuum state in the case of general pressure law

In this paper, we consider the one‐dimensional compressible isentropic Navier–Stokes equations with a general ‘pressure law’ and the density‐dependent viscosity coefficient when the density connects to vacuum continuously. Precisely, the viscosity coefficient µ is proportional to ρ^θ and 0<θ<1, where ρ is the density. And the pressure P = P(ρ) is a general ‘pressure law’. The global existence and the uniqueness of weak solutions is proved, and a decay result for the pressure as t→ + ∞ is given. It is also proved that no vacuum states and no concentration states develop, and the free boundary do not expand to infinite. Copyright © 2006 John Wiley & Sons, Ltd.     

Exponential stabilization and L2‐gain for uncertain switched nonlinear systems with interval time‐varying delay

This paper is concerned with the exponential stabilization and L₂‐gain for a class of uncertain switched nonlinear systems with interval time‐varying delay. Based on Lyapunov–Krasovskii functional method, novel delay‐dependent sufficient conditions of exponential stabilization for a class of uncertain switched nonlinear delay systems are developed under an average dwell time scheme. Then, novel criteria to ensure the exponential stabilization with weighted L₂‐gain performance for a class of uncertain switched nonlinear delay systems are established. Furthermore, an effective method is proposed for the designing of a stabilizing feedback controller with L₂‐gain performance. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

On measure‐valued solutions to a two‐dimensional gravity‐driven avalanche flow model

This paper concerns measure‐valued solutions for the two‐dimensional granular avalanche flow model introduced by Savage and Hutter. The system is similar to the isentropic compressible Euler equations, except for a Coulomb–Mohr friction law in the source term. We will partially follow the study of measure‐valued solutions given by DiPerna and Majda. However, due to the multi‐valued nature of the friction law, new more sensitive measures must be introduced. The main idea is to consider the class of x‐dependent maximal monotone graphs of non‐single‐valued operators and their relation with 1‐Lipschitz, Carathéodory functions. This relation allows to introduce generalized Young measures for x‐dependent maximal monotone graph. Copyright © 2005 John Wiley & Sons, Ltd.     

A defect correction method for the time‐dependent Navier‐Stokes equations

A method for solving the time dependent Navier‐Stokes equations, aiming at higher Reynolds' number, is presented. The direct numerical simulation of flows with high Reynolds' number is computationally expensive. The method presented is unconditionally stable, computationally cheap, and gives an accurate approximation to the quantities sought. In the defect step, the artificial viscosity parameter is added to the inverse Reynolds number as a stability factor, and the system is antidiffused in the correction step. Stability of the method is proven, and the error estimations for velocity and pressure are derived for the one‐ and two‐step defect‐correction methods. The spacial error is O(h) for the one‐step defect‐correction method, and O(h²) for the two‐step method, where h is the diameter of the mesh. The method is compared to an alternative approach, and both methods are applied to a singularly perturbed convection–diffusion problem. The numerical results are given, which demonstrate the advantage (stability, no oscillations) of the method presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Self‐Intersection Local Time for 𝒮′(ℝd)‐Ornstein‐Uhlenbeck Processes Arising from Immigration Systems

Fluctuation limits of an immigration branching particle system and an immigration branching measure‐valued process yield different types of 𝒮′(ℝ^d)‐valued Ornstein‐Uhlenbeck processes whose covariances are given in terms of an excessive measure for the underlying motion in Rd, which is taken to be a symmetric α‐stable process. In this paper we prove existence and path continuity results for the self‐intersection local time of these Ornstein‐Uhlenbeck processes. The results depend on relationships between the dimension d and the parameter α.     

Stability of a combined finite element ‐ finite volume discretization of convection‐diffusion equations

We consider a time‐dependent and a stationary convection‐diffusion equation. These equations are approximated by a combined finite element – finite volume method: the diffusion term is discretized by Crouzeix‐Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the nonstationary case, we use an implicit Euler approach for time discretization. This scheme is shown to be L²‐stable uniformly with respect to the diffusion coefficient. In addition, it turns out that stability is unconditional in the time‐dependent case. These results hold if the underlying grid satisfies a condition that is fulfilled, for example, by some structured meshes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 402–424, 2012     

S‐asymptotically ω‐periodic solutions for semilinear Volterra equations

We study S‐asymptotically ω‐periodic mild solutions of the semilinear Volterra equation u′(t)=(a* Au)(t)+f(t, u(t)), considered in a Banach space X, where A is the generator of an (exponentially) stable resolvent family. In particular, we extend the recent results for semilinear fractional integro‐differential equations considered in (Appl. Math. Lett. 2009; 22:865–870) and for semilinear Cauchy problems of first order given in (J. Math. Anal. Appl. 2008; 343(2): 1119–1130). Applications to integral equations arising in viscoelasticity theory are shown. Copyright © 2010 John Wiley & Sons, Ltd.     

‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

We introduce, characterise and provide a combinatorial interpretation for the so‐called q‐Jacobi–Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order q‐differential operator having the q‐classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the q‐version of those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a q‐version of the Jacobi–Stirling numbers given by Gelineau and the second author.     

Global well‐posedness of 2D nonlinear Boussinesq equations with mixed partial viscosity and thermal diffusivity

In this paper, we discuss with the global well‐posedness of 2D anisotropic nonlinear Boussinesq equations with any two positive viscosities and one positive thermal diffusivity. More precisely, for three kinds of viscous combinations, we obtain the global well‐posedness without any assumption on the solution. For other three difficult cases, under the minimal regularity assumption, we also derive the unique global solution. To the authors' knowledge, our result is new even for the simplified model, that is, F(θ) = θe₂. Copyright © 2017 John Wiley & Sons, Ltd.     

On the approximation of the solutions of stochastic equations with θ-integrals

The paper investigates the problem of approximation of stochastic θ-integrals and the solutions of stochastic differential equations. The complete classification of the methods of approximation of stochastic θ-integrals in the convolution algebra is proposed. It is proved that the solutions of stochastic integral equations with θ-integral can be approximated by the solutions of finite-difference equations with averaging.     

High‐order split‐step theta methods for non‐autonomous stochastic differential equations with non‐globally Lipschitz continuous coefficients

In this paper, we first propose the so‐called improved split‐step theta methods for non‐autonomous stochastic differential equations driven by non‐commutative noise. Then, we prove that the improved split‐step theta method is convergent with strong order of one for stochastic differential equations with the drift coefficient satisfying a superlinearly growing condition and a one‐sided Lipschitz continuous condition. Finally, the obtained results are verified by numerical experiments. Copyright © 2016 John Wiley & Sons, Ltd.