Ergodicity for the Dissipative Boussinesq Equations with Random Forcing

We study the stationary measure for the two-dimensional Boussinesq equation with random forcing. We prove the ergodicity for the two-dimensional stochastically forced Boussinesq equation. We also study the Galerkin truncations of the three-dimensional Boussinesq equations under degenerate stochastic forcing. We follow closely the previous results on the stochastically forced Navier–Stokes equations.     

Ergodicity of the Finite Dimensional Approximation of the 3D Navier–Stokes Equations Forced by a Degenerate Noise

We prove ergodicity of the finite dimensional approximations of the three dimensional Navier–Stokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential.     

Dynamical analysis of low-temperature monte carlo cluster algorithms

We present results on the Swendsen-Wang dynamics for the Ising ferromagnet in the low-temperature case without external field in the thermodynamic limit. We discuss in particular the rate of convergence to the equilibrium Gibbs state in finite and infinite volume, the absence of ergodicity in the infinite volume, and the long-time behavior of the probability distribution of the dynamics for various starting configurations. Our results are purely dynamical in nature in the sense that we never use the reversibility of the process with respect to the Gibbs state, and they apply to a stochastic particle system withnon- Gibbsian invariant measure.     

Stieltjes Integrals of Hölder Continuous Functions with Applications to Fractional Brownian Motion

We give a new estimate on Stieltjes integrals of Hölder continuous functions and use it to prove an existence-uniqueness theorem for solutions of ordinary differential equations with Hölder continuous forcing. We construct stochastic integrals with respect to fractional Brownian motion, and establish sufficient conditions for its existence. We prove that stochastic differential equations with fractional Brownian motion have a unique solution with probability 1 in certain classes of Hölder-continuous functions. We give tail estimates of the maximum of stochastic integrals from tail estimates of the Hölder coefficient of fractional Brownian motion. In addition we apply the techniques used for ordinary Brownian motion to construct stochastic integrals of deterministic functions with respect to fractional Brownian motion and give tail estimates of its maximum.     

A test model for fluctuation-dissipation theorems with time-periodic statistics

The recently developed time-periodic fluctuation-dissipation theorem (FDT) provides a very convenient way of addressing the climate change of atmospheric systems with seasonal cycle by utilizing statistics of the present climate. A triad nonlinear stochastic model with exactly solvable first and second order statistics is introduced here as an unambiguous test model for FDT in a time-periodic setting. This model mimics the nonlinear interaction of two Rossby waves forced by baroclinic processes with a zonal jet forced by a polar temperature gradient. Periodic forcing naturally introduces the seasonal cycle into the model. The exactly solvable first and second order statistics are utilized to compute both the ideal mean and variance response to the perturbations in forcing or dissipation and the quasi-Gaussian approximation of FDT (qG-FDT) that uses the mean and the covariance in the equilibrium state. The time-averaged mean and variance qG-FDT response to perturbations of forcing or dissipation is compared with the corresponding ideal response utilizing the triad test model in a number of regimes with various dynamical and statistical properties such as weak or strong non-Gaussianity and resonant or non-resonant forcing. It is shown that even in a strongly non-Gaussian regime, qG-FDT has surprisingly high skill for the mean response to the changes in forcing. On the other hand, the performance of qG-FDT for the variance response to the perturbations of dissipation is good in the near-Gaussian regime and deteriorates in the strongly non-Gaussian regime. The results here on the test model should provide useful guidelines for applying the time-periodic FDT to more complex realistic systems such as atmospheric general circulation models.     

Thermal and non-thermal fluctuations in active polar gels

We discuss general features of noise and fluctuations in active polar gels close to and away from equilibrium. We use the single-component hydrodynamic theory of active polar gels built by Kruse and coworkers to describe the cytoskeleton in cells. Close to equilibrium, we calculate the response function of the gel to external fields and introduce Langevin forces in the constitutive equations with correlation functions respecting the fluctuation-dissipation theorem. We then discuss the breakage of the fluctuation-dissipation theorem due to an external field such as the activity of the motors. Active gels away from equilibrium are considered at the scaling level. As an example of application of the theory, we calculate the density correlation function (the dynamic structure factor) of a compressible active polar gel and discuss possible instabilities.     

Fluctuation-dissipation relations in the activated regime of simple strong-glass models

We study the out-of-equilibrium fluctuation-dissipation (FD) relations in the low-temperature, finite-time, physical-aging regime of two simple models with strong glass behavior, the Fredrickson-Andersen model and the square-plaquette interaction model. We explicitly show the existence of unique, waiting-time independent dynamical FD relations. While in the Fredrickson-Andersen model the FD theorem is obeyed at all times, the plaquette model displays piecewise-linear FD relations, similar to what is found in disordered mean-field models and in simulations of supercooled liquids, and despite the fact that its static properties are trivial. We discuss the wider implications of these results.     

Gibbsian Dynamics and Invariant Measures for Stochastic Dissipative PDEs

We present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative PDEs. It consists of two main steps. The first step is the reduction to a finite dimensional Gibbsian dynamics of the low modes. The second step is to prove the equivalence between measures induced by different past histories using Girsanov theorem. As applications, we prove ergodicity for Ginzburg–Landau, Kuramoto–Sivashinsky and Cahn–Hilliard equations with stochastic forcing.     

Finite difference methods for two-dimensional fractional dispersion equation

Fractional order partial differential equations, as generalizations of classical integer order partial differential equations, are increasingly used to model problems in fluid flow, finance and other areas of application. In this paper we discuss a practical alternating directions implicit method to solve a class of two-dimensional initial-boundary value fractional partial differential equations with variable coefficients on a finite domain. First-order consistency, unconditional stability, and (therefore) first-order convergence of the method are proven using a novel shifted version of the classical Grünwald finite difference approximation for the fractional derivatives. A numerical example with known exact solution is also presented, and the behavior of the error is examined to verify the order of convergence.     

Exact solutions for nonlinear partial fractional differential equations

In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved(G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.     

Existence of solutions for sequential fractional differential equations with four-point nonlocal fractional integral boundary conditions

This paper investigates the existence of solutions for a nonlinear boundary value problem of sequential fractional differential equations with four-point nonlocal Riemann-Liouville type fractional integral boundary conditions. We apply Banach’s contraction principle and Krasnoselskii’s fixed point theorem to establish the existence of results. Some illustrative examples are also presented.     

Glassy properties of the Kawasaki dynamics of two-dimensional ferromagnets

We study numerically the Kawasaki dynamics of the 2D Ising model. At large time we recover the coarsening growth as l(c)(t) proportional t(1/3). At shorter time however, the system enters a metastable glassy regime that displays an extremely slow growth and nontrivial violations of the fluctuation-dissipation theorem similar to those observed in spin glasses: this is one of the simplest systems in which such violations occur. We also consider Potts models, where a similar behavior is observed, and the model of Shore and Sethna where the domain growth is also slow, but where violations of the fluctuation-dissipation theorem are trivial. We finally comment on these violations in the context of activated coarsening, and on similarities and differences with the glass transition phenomenology.     

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.     

A Fourier method for the fractional diffusion equation describing sub-diffusion

In this paper, a fractional partial differential equation (FPDE) describing sub-diffusion is considered. An implicit difference approximation scheme (IDAS) for solving a FPDE is presented. We propose a Fourier method for analyzing the stability and convergence of the IDAS, derive the global accuracy of the IDAS, and discuss the solvability. Finally, numerical examples are given to compare with the exact solution for the order of convergence, and simulate the fractional dynamical systems.     

A Fourier method for the fractional diffusion equation describing sub-diffusion

In this paper, a fractional partial differential equation (FPDE) describing sub-diffusion is considered. An implicit difference approximation scheme (IDAS) for solving a FPDE is presented. We propose a Fourier method for analyzing the stability and convergence of the IDAS, derive the global accuracy of the IDAS, and discuss the solvability. Finally, numerical examples are given to compare with the exact solution for the order of convergence, and simulate the fractional dynamical systems.     

Solutions to Class of Linear and Nonlinear Fractional Differential Equations

In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Kd V equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the(3+1)-spacetime fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.     

Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology

In this study, the modified Kudryashov method is used to construct new exact solutions for some conformable fractional differential equations. By implementing the conformable fractional derivative and compatible fractional complex transforms, the fractional generalized reaction duffing (RD) model equation, the fractional biological population model and the fractional diffusion reaction (DR) equation with quadratic and cubic nonlinearity are discussed. As an outcome, some new exact solutions are formally established. All solutions have been verified back into its corresponding equation with the aid of maple package program. We assure that the employed method is simple and robust for the estimation of the new exact solutions, and practically capable for reducing the size of computational work for solving a various class of fractional differential equations arising in applied mathematics, mathematical physics and biology.     

A systematic literature review of Burgers’ equation with recent advances

Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’ equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations.