Convergence and continuous dependence for the Brinkman–Forchheimer equations

This paper investigates the flow of fluid in a porous medium which is described in the Brinkman–Forchheimer equations and obtains the structural stability results for the coefficients.     

Continuous dependence for the convective Brinkman–Forchheimer equations

In this article, we have considered the convective Brinkman–Forchheimer equations with Dirichlet's boundary conditions. The continuous dependence of solutions on the Forchheimer coefficient in H ¹ norm is proved.     

Approximation of the incompressible convective Brinkman–Forchheimer equations

This paper studies the approximation of solutions for the incompressible convective Brinkman–Forchheimer (CBF) equations via the artificial compressibility method. We first introduce a family of perturbed compressible CBF equations that approximate the incompressible CBF equations. Then, we prove the existence and convergence of solutions for the compressible CBF equations to the solutions of the incompressible CBF equations.     

Reflected forward–backward stochastic differential equations with continuous monotone coefficients

In this work, we prove that there exists at least one solution for the reflected forward–backward stochastic differential equations satisfying the obstacle constraint with continuous monotone coefficients. The distinct character of our result is that the coefficient of the forward SDEs contains the solution variable of the reflected BSDEs.     

On the asymptotic behavior of solutions of Emden–Fowler equations on time scales

Consider the Emden-Fowler dynamic equation $$ x^{\Delta\Delta}(t)+p(t)x^\alpha(t)=0,\:\:\alpha >0 , \qquad \qquad \qquad \qquad (0.1) $$ where ${p\in C_{rd}([t_0,\infty)_{\mathbb{T}},\mathbb{R}), \alpha}$ is the quotient of odd positive integers, and ${\mathbb{T}}$ denotes a time scale which is unbounded above and satisfies an additional condition (C) given below. We prove that if ${\int^\infty_{t_0}t^\alpha |p(t)|\Delta t<\infty}$ (and when ???=?1 we also assume lim _t???? tp(t)??(t)?=?0), then (0.1) has a solution x(t) with the property that $$ \lim_{t\rightarrow\infty} \frac{x(t)}{t}=A\neq 0.$$      

On the almost-periodic solution of Hasegawa–Wakatani equations

In order to describe the resistive drift wave turbulence appearing in nuclear fusion plasma, the Hasegawa–Wakatani equations were proposed in 1983. In this paper, we consider the zero-resistivity limit for the Hasegawa–Wakatani equations in a cylindrical domain when the initial data are Stepanov-almost-periodic in the axial direction. We prove two results: one is the existence and uniqueness of a strong global in time Stepanov-almost-periodic solution to the initial boundary value problem for the Hasegawa–Mima-like equation; another is the convergence of the solution of the Hasegawa–Wakatani equations to that of the Hasegawa–Mima-like equation established at the first stage as the resistivity tends to zero. In order to obtain a priori estimates of the Stepanov-almost-periodic solutions to our problems, we have to overcome some difficulties. In the proof, we prepare some lemmas for the Stepanov-almost-periodic functions and then obtain a priori estimates.     

On the solutions of time-fractional reaction–diffusion equations

In this paper, a new application of generalized differential transform method (GDTM) has been used for solving time-fractional reaction–diffusion equations. To illustrate the reliability of the method, some examples are provided.     

On the relation between the continuous and discrete Painlevé equations

A method for deriving difference equations (the discrete Painlevé equations in particular) from the Bäcklund transformations of the continuous Painlevé equations is discussed. This technique can be used to derive several of the known discrete painlevé equations (in particular, the first and second discrete Painlevé equations and some of their alternative versions). The Painlevé equations possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions for special values of the parameters. Hence, the aforementioned relations can be used to generate hierarchies of exact solutions for the associated discrete Painlevé equations. Exact solutions of the Painlevé equations simultaneously satisfy both a differential equation and a difference equation, analogously to the special functions.     

On the coefficients of the asymptotic expansion of the kernel of Berezin–Toeplitz quantization

We give new methods for computing the coefficients of the asymptotic expansions of the kernel of Berezin?CToeplitz quantization obtained recently by Ma?CMarinescu, and of the composition of two Berezin?CToeplitz quantizations. Our main tool is the stationary phase formula of Melin?CSj?strand.     

On Fourier–Haar coefficients of the function from the Marcinkiewicz space

The main objective of the paper is to describe asymptotic behaviour of Fourier–Haar coefficients of functions from Marcinkiewicz spaces. We also discuss the Cesàro summability and the almost convergence of sequences related to Fourier–Haar coefficients and generalise a result from [17]. Some analogues of Mercer's theorem for Fourier coefficients are proved.     

On some properties of the Navier–Stokes system of equations

We discuss the initial and boundary value problems for the system of dimensionless Navier–Stokes equations describing the dynamics of a viscous incompressible fluid using the method of characteristics and the geometric method developed by the authors. Some properties of the formulation of these problems are considered. We study the effect of the Reynolds number on the flow of a viscous fluid near the surface of a body.     

Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada–Watanabe theorem (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain $D$ of the phase space $\mathbb{R}$ and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called free energy function on the density function space, we prove that the density functions, which are solutions of the Fokker–Planck equation, converge to the stationary density function exponentially under the Kullback–Leibler divergence, thus also in the total variation norm. The results show that there is a relation between the Bakry–Émery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher–Lamperti transformation. Several applications are discussed, including the Cox–Ingersoll–Ross model and the Ait-Sahalia model in finance and the Wright–Fisher model in population genetics.     

On the Hamilton–Poisson realizations of the integrable deformations of the Maxwell–Bloch equations

In this note, we construct integrable deformations of the three-dimensional real valued Maxwell–Bloch equations by modifying their constants of motions. We obtain two Hamilton–Poisson realizations of the new system. Moreover, we prove that the obtained system has infinitely many Hamilton–Poisson realizations. Particularly, we present a Hamilton–Poisson approach of the system obtained considering two concrete deformation functions.     

On the stability of the generalized Cauchy–Jensen set-valued functional equations

In this paper, we prove the Hyers–Ulam–Rassias stability of the generalized Cauchy–Jensen set-valued functional equation defined by

$$\begin{aligned} \alpha f\left( \frac{x+y}{\alpha } + z\right) = f(x) \oplus f(y)\oplus \alpha f(z) \end{aligned}$$

for all \(x,y,z \in X\) and \(\alpha \ge 2\) on a Banach space by using the fixed point alternative theorem.