Interaction of fundamental parametric resonances with subharmonic resonances of order one-half

The interaction of fundamental parametric resonances with subharmonic resonances of order one-half in a single-degree-of-freedom system with quadratic and cubic nonlinearities is investigated. The method of multiple scales is used to derive two first-order ordinary differential equations that describe the modulation of the amplitude and the phase of the response with the non-linearity and both resonances. These equations are used to determine the steady state solutions and their stability. Conditions are derived for the quenching or enhancement of a parametric resonance by the addition of a subharmonic resonance of order one-half. The degree of quenching or enhancement depends on the relative amplitudes and phases of the excitations. The analytical results are verified by numerically integrating the original governing differential equation.     

A stochastic multiscale framework for modeling flow through random heterogeneous porous media

Flow through porous media is ubiquitous, occurring from large geological scales down to the microscopic scales. Several critical engineering phenomena like contaminant spread, nuclear waste disposal and oil recovery rely on accurate analysis and prediction of these multiscale phenomena. Such analysis is complicated by inherent uncertainties as well as the limited information available to characterize the system. Any realistic modeling of these transport phenomena has to resolve two key issues: (i) the multi-length scale variations in permeability that these systems exhibit, and (ii) the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochastic processes.A stochastic variational multiscale formulation is developed to incorporate uncertain multiscale features. A stochastic analogue to a mixed multiscale finite element framework is used to formulate the physical stochastic multiscale process. Recent developments in linear and non-linear model reduction techniques are used to convert the limited information available about the permeability variation into a viable stochastic input model. An adaptive sparse grid collocation strategy is used to efficiently solve the resulting stochastic partial differential equations (SPDEs). The framework is applied to analyze flow through random heterogeneous media when only limited statistics about the permeability variation are given.     

Spatially adaptive stochastic numerical methods for intrinsic fluctuations in reaction–diffusion systems

Stochastic partial differential equations are introduced for the continuum concentration fields of reaction–diffusion systems. The stochastic partial differential equations account for fluctuations arising from the finite number of molecules which diffusively migrate and react. Spatially adaptive stochastic numerical methods are developed for approximation of the stochastic partial differential equations. The methods allow for adaptive meshes with multiple levels of resolution, Neumann and Dirichlet boundary conditions, and domains having geometries with curved boundaries. A key issue addressed by the methods is the formulation of consistent discretizations for the stochastic driving fields at coarse-refined interfaces of the mesh and at boundaries. Methods are also introduced for the efficient generation of the required stochastic driving fields on such meshes. As a demonstration of the methods, investigations are made of the role of fluctuations in a biological model for microorganism direction sensing based on concentration gradients. Also investigated, a mechanism for spatial pattern formation induced by fluctuations. The discretization approaches introduced for SPDEs have the potential to be widely applicable in the development of numerical methods for the study of spatially extended stochastic systems.     

Bright and singular soliton solutions of the conformable time-fractional Klein–Gordon equations with different nonlinearities

Finding the exact solutions of nonlinear fractional differential equations has gained considerable attention, during the past two decades. In this paper, the conformable time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities are studied. Several exact soliton solutions, including the bright (non-topological) and singular soliton solutions are formally extracted by making use of the ansatz method. Results demonstrate that the method can efficiently handle the time-fractional Klein–Gordon equations with different nonlinearities.     

Numerical method based on the lattice Boltzmann model for the Fisher equation

In this paper, a lattice Boltzmann model for the Fisher equation is proposed. First, the Chapman-Enskog expansion and the multiscale time expansion are used to describe higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. Second, the modified partial differential equation of the Fisher equation with the higher-order truncation error is obtained. Third, comparison between numerical results of the lattice Boltzmann models and exact solution is given. The numerical results agree well with the classical ones.     

Direct quadrature method of moments solution of the Fokker–Planck equation

The direct quadrature method of moments is presented as an efficient and accurate means of numerically computing solutions of the Fokker–Planck equation corresponding to stochastic nonlinear dynamical systems. The theoretical details of the solution procedure are first presented. The method is then used to solve Fokker–Planck equations for both 1D and 2D (noisy van der Pol oscillator) processes which possess nonlinear stochastic differential equations. Higher-order moments of the stationary solutions are computed and prove to be very accurate when compared to analytic (1D process) and Monte Carlo (2D process) solutions.     

Numerical upscaling for the eddy-current model with stochastic magnetic materials

This paper deals with the upscaling of the time-harmonic Maxwell equations for heterogeneous media. We analyze the eddy-current approximation of Maxwell’s equations to describe the electric field for heterogeneous, isotropic magnetic materials. The magnetic permeability of the materials is assumed to have random heterogeneities described by a Gaussian random field. We apply the so-called Coarse Graining method to develop a numerical upscaling of the eddy-current model. The upscaling uses filtering and averaging procedures in Fourier space which results in a formulation of the eddy-current model on coarser resolution scales where the influence of sub-scale fluctuations is modeled by effective scale- and space-dependent reluctivity tensors. The effective reluctivity tensors can be obtained by solving local partial differential equations which contain a Laplacian as well as a curl–curl operator. We present a computational method how the equation of the combined operators can be discretized and solved numerically using an extended variational formulation compared to standard discretizations. We compare the results of the numerical upscaling of the eddy-current model with theoretical results of Eberhard [J.P. Eberhard, Upscaling for the time-harmonic Maxwell equations with heterogeneous magnetic materials, Physical Review E 72 (3), (2005)] and obtain a very good agreement.     

A new auxiliary equation and exact travelling wave solutions of nonlinear equations

A new auxiliary ordinary differential equation and its solutions are used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the auxiliary equation which has more new exact solutions. More new exact travelling wave solutions are obtained for the quadratic nonlinear Klein–Gordon equation, the combined KdV and mKdV equation, the sine-Gordon equation and the Whitham–Broer–Kaup equations.     

NON-LINEAR BEAM OSCILLATIONS EXCITED BY LATERAL FORCE AT COMBINATION RESONANCE

Non-linear oscillations of a beam subjected to a periodic force at a combination resonance are considered. Using the Galerkin method, a partial differential equation of oscillations is reduced to a system of ordinary differential equations with a small parameter. A system of three autonomous differential equations is derived, the multiple scales method being used. Qualitative properties of trajectories are analyzed. The Naimark-Sacker bifurcations at the combination resonance are analyzed by the center manifold method. Almost-periodic oscillations of a beam arise due to these bifurcations. These oscillations are investigated qualitatively and numerically.     

Probability Density and Joint Probability Density of SDE with General Nonlinear Gaussian Noise

We consider a stochastic differential equation with a general nonlinearity in Gaussian noise; With both D (fluctuation intensity) and γ (correlation time) small quantities with D/γ < 1/10, approximate equations for the probability density p(q, t) and the joint probability density p(q, t, q^t, t^p) are derived. As applications of our general equations, quadratic noise, exponential noise and triangle function noise are studied.     

Averaging Principle for the Higher Order Nonlinear Schrödinger Equation with a Random Fast Oscillation

This work concerns the problem associated with averaging principle for a higher order nonlinear Schrödinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction–diffusion equation evolving with respect to the fast time. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear Schrödinger equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single higher order nonlinear Schrödinger equation with a modified coefficient.     

On the well-posedness of the stochastic Allen–Cahn equation in two dimensions

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical systems in space dimensions d = 1, 2, 3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d ? 2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen–Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that shrinking the mesh size in simulations of the two-dimensional white noise-driven Allen–Cahn equation does not lead to the recovery of a physically meaningful limit.     

Periodic solutions of Wick-type stochastic Korteweg–de Vries equations

Nonlinear stochastic partial differential equations have a wide range of applications in science and engineering. Finding exact solutions of the Wick-type stochastic equation will be helpful in the theories and numerical studies of such equations. In this paper, Kudrayshov method together with Hermite transform is implemented to obtain exact solutions of Wick-type stochastic Korteweg–de Vries equation. Further, graphical illustrations in two- and three-dimensional plots of the obtained solutions depending on time and space are also given with white noise functionals.     

ON THE NONLOCAL SYMMETRIES OF THE μ-CAMASSA–HOLM EQUATION

The μ-Camassa–Holm (μCH) equation is a nonlinear integrable partial differential equation closely related to the Camassa–Holm and the Hunter–Saxton equations. This equation admits quadratic pseudo-potentials which allow us to compute some first-order nonlocal symmetries. The found symmetries preserve the mean of solutions. Finally, we discuss also the associated μCH equation.     

随机微分方程计算方法及其应用

介绍随机微分方程离散化格式的构造、收敛性法则、强收敛格式、弱收敛格式、带跳跃的随机微分方程的计算方法,偏微分方程的概率求解以及它们在物理、工程和金融等领域中的一些应用.     

Matrix approach to discrete fractional calculus II: Partial fractional differential equations

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny’s matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359–386]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.     

Variable-stepsize Runge–Kutta methods for stochastic Schrödinger equations

Stochastic wave equations of Schrödinger type are widely employed in physics and have numerous potential applications in chemistry. While some accurate numerical methods exist for particular classes of stochastic differential equations they cannot generally be used for Schrödinger equations. Efficient and accurate methods for their numerical solution therefore need to be developed. Here we show that existing Runge–Kutta methods for ordinary differential equations (odes) can be modified to solve stochastic wave equations provided that appropriate changes are made to the way stepsizes are selected. The order of the resulting stochastic differential equation (sde) scheme is half the order of the ode scheme. Specifically, we show that an explicit 9th order Runge–Kutta method (with an embedded 8th order method) for odes yields an order 4.5 method for sdes which can be implemented with variable stepsizes. This method is tested by solving systems of equations originating from master equations and from the many-body Schrödinger equation.     

Soliton-like solutions of certain types of nonlinear diffusion-reaction equations with variable coefficient

With a view to exploring new soliton-like solutions of certain types of nonlinear diffusion-reaction (DR) equations with a variable coefficient, we demonstrate the viability of a method which is the combination of both the symbolic computation technique of Gao and Tian [Y.T. Gao, B. Tian, Comput. Phys. Commun. 133 (2001) 158] and auxiliary equation method of Sirendaoreji [Sirendaoreji, Phys. Lett. A 356 (2006) 124] and used recently for the KdV equation. In particular, the DR equations with quadratic and cubic nonlinearities with a time-dependent velocity in the convective flux term are studied and the existence of soliton-like solutions is shown.     

Solitary Excitations in Antiferrornagnets AFeBr3

Introducing the Holstein-Primakoff transformation, the coherent-state ansatz and the timedependent variation principle, we obtain two partial differential equations of motion. Employing the method of multiple scales, we reduce these equations into the envelope function equations and force the amplitude function to satisfy a nonlinear Schrödinger equation. Using the inyerse-scattering transformation, we obtain the single-, two- and N-soliton solutions and discuss the gap soliton in antiferromagnets AFeB₃.