Conservation laws of (3+α)-dimensional time-fractional diffusion equation

The concept of Lie–Backlund symmetry plays a fundamental role in applied mathematics. It is clear that in order to find conservation laws for a given partial differential equations (PDEs) or fractional differential equations (FDEs) by using Lagrangian function, firstly, we need to obtain the symmetries of the considered equation.Fractional derivation is an efficient tool for interpretation of mathematical methods. Many applications of fractional calculus can be found in various fields of sciences as physics (classic, quantum mechanics and thermodynamics), biology, economics, engineering and etc. So in this paper, we present some effective application of fractional derivatives such as fractional symmetries and fractional conservation laws by fractional calculations. In the sequel, we obtain our results in order to find conservation laws of the time-fractional equation in some special cases.     

Resonant soliton and complexiton solutions for (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

In this letter, the linear superposition principle is used to discuss the $(3+1)$-dimensional Boiti–Leon–Manna–Pempinelli equation with bilinear derivatives. As a result, we obtain new resonant soliton and complexiton solutions by discussing two different cases involved the parameters. These solutions are a class of $N$-wave solutions of linear combinations of exponential traveling waves.     

Invariant analysis and conservation laws of (2+1) dimensional time-fractional ZK–BBM equation in gravity water waves

This paper intends to make an in-depth study on the symmetry properties and conservation laws of the ($2+1$) dimensional time fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZK–BBM) equation with Riemann–Liouville fractional derivative. Symmetry properties have been investigated here via Lie symmetry analysis method. In view of Erdélyi-Kober fractional differential operator, the reduction of ($2+1$) dimensional time fractional ZK–BBM equation has been done into fractional ordinary differential equation. To analyse the conservation laws, new theorem of conservation law has been proposed here for constructing the new conserved vectors for ($2+1$) dimensional time fractional ZK–BBM equation with the help of formal Lagrangian.     

Continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system on RN

In this article, we use the so-called difference estimate method to investigate the continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system with a general nonlinearity. Firstly, under weak assumptions on the noise coefficient, we prove the existence of a pullback attractor in $L^2\left({\mathbb{R}}^N\right)\times L^2\left({\mathbb{R}}^N\right)$ by using the tail estimate method and a certain compact embedding on bounded domains. Secondly, although the difference of the first component of solutions possesses at most $p$-times integrability where $p$ is the growth exponent of the nonlinearity, we overcome the absence of higher-order integrability and establish the continuity of solutions in $(L^p\left({\mathbb{R}}^N\right)\cap H^1\left({\mathbb{R}}^N\right))\times L^2\left({\mathbb{R}}^N\right)$ with respect to the initial values belonging to $L^2\left({\mathbb{R}}^N\right)\times L^2\left({\mathbb{R}}^N\right)$. As an application of the result on the continuity, the existence of a pullback attractor in $(L^p\left({\mathbb{R}}^N\right)\cap H^1\left({\mathbb{R}}^N\right))\times L^2\left({\mathbb{R}}^N\right)$ is proved for arbitrary $N\ge 1$ and $p>2$.     

A sensitivity study of the Navier–Stokes-α model

We present a sensitivity study of the Navier Stokes-$\alpha$ model with respect to perturbations of the differential filter length $\alpha$. The parameter-sensitivity is evaluated using the sensitivity equations method. Once formulated, the sensitivity equations are discretized and computed alongside the NS$\alpha$ model using the same finite elements in space, and Crank–Nicolson in time. We provide a complete stability analysis of the scheme, along with the results of several benchmark problems in both 2D and 3D. We further demonstrate a practical technique to utilize sensitivity calculations to determine the reliability of the NS$\alpha$ model in problem-specific settings. Lastly, we investigate the sensitivity and reliability of important functionals of the velocity and pressure solutions.     

The new exact solitary and multi-soliton solutions for the (2+1)-dimensional Zakharov–Kuznetsov equation

In this paper, by employing two different simplest equation methods, the (2$+$1)-dimensional Zakharov–Kuznetsov (ZK) equation derived for describing weakly nonlinear ion-acoustic waves in the plasma is investigated. With the aid of the Bernoulli equation and the coupled Burgers’ equations, the electric field potential of ZK equation are formally obtained, which are presented as the new solitary and multi-soliton solutions. Meanwhile, the electric field and magnetic field can be accordingly obtained. In addition, the significant features of the variable coefficient and parameter are discovered. The results show that the solitary and multi-soliton solutions are precisely obtained and the efficiency of the methods is demonstrated. These new exact solutions will extend previous results and help to explain the features of nonlinear ion-acoustic waves in the plasma.     

影讯简

杰瑞·尤斯曼与玛姬·泰勒夫妇的作品里，任何一个单位元素，云朵、湖水、树木、蜜蜂、斑马……都是我们再熟悉不过的，这些都是构成真实世界的真实元素。但是两位摄影师打乱了物体原本的规律，在视觉上运用真实的元素把我们带入了另外一个内心的“真实”世界。