On the Solution Structure of Nonlinear Hill's Equation I,Global Results

The solution structure in W^2,p(?) × ? of nonlinear Hill's equation is discussed in full detail. In a recent article, the existence of unbounded solution components was shown for values of the branching parameter in the gaps of the continuous spectrum of the linearized problem. This result is the starting point of further investigations concerning the existence region of the corresponding solution components. In particular, new phenomena such as asymptotic bifurcation from infinity at a specific parameter value inside of each gap of the spectrum can be shown, if the nonlinearity satisfies a growth condition. The main assumption is the concentration of the nonlinearity to a compact interval which allows the reduction to an equivalent nonlinear Sturm - Liouville problem with parameter dependent boundary conditions, if the parameter does not belong to the continuous spectrum. Extending this problem to all real parameter values makes it possible to get information about the existence region of the solution components with the help of a priori bounds for solutions of the Sturm-Liouville problem.     

A Liouville Theorem for the Planer Navier-Stokes Equations with the No-Slip Boundary Condition and Its Application to a Geometric Regularity Criterion

We establish a Liouville type result for a backward global solution to the Navier-Stokes equations in the half plane with the no-slip boundary condition. No assumptions on spatial decay for the vorticity nor the velocity field are imposed. We study the vorticity equations instead of the original Navier-Stokes equations. As an application, we extend the geometric regularity criterion for the Navier-Stokes equations in the three-dimensional half space under the no-slip boundary condition.     

ON THE QUASI-RANDOM CHOICE METHOD FOR THE LIOUVILLE EQUATION OF GEOMETRICAL OPTICS WITH DISCONTINUOUS WAVE SPEED

We study the quasi-random choice method （QRCM） for the Liouville equation of ge- ometrical optics with discontinuous locM wave speed. This equation arises in the phase space computation of high frequency waves through interfaces, where waves undergo partial transmissions and reflections. The numerical challenges include interface, contact discon- tinuities, and measure-valued solutions. The so-called QRCM is a random choice method based on quasi-random sampling （a deterministic alternative to random sampling）. The method not only is viscosity-free but also provides faster convergence rate. Therefore, it is appealing for the prob!em under study which is indeed a Hamiltonian flow. Our analy- sis and computational results show that the QRCM 1） is almost first-order accurate even with the aforementioned discontinuities; 2） gives sharp resolutions for all discontinuities encountered in the problem; and 3） for measure-valued solutions, does not need the level set decomposition for finite difference/volume methods with numerical viscosities.     

The Finite Difference Methods for Fractional Ordinary Differential Equations

Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis.     

一类非线性椭圆型方程的一个Liouville定理

设 (Mn,g)是一个 n维的完备黎曼流形 ,其 Ricci曲率满足 Ric M(x)≥ - A(1 r2 (x) ln2 (2 r(x) ) ) ,其中 A是非负常数 ,r(x)表示点 x∈ M到某固定点 x0 ∈ M的测地距离 .则 M上方程 Δu Su Kuα=0在下述条件“ (i)在 M上 S≤ 0 ;(ii)在 M上 K<0且有常数 a>0使在一个紧集之外 K≤ - a2 ;(iii)常数 α>1”下的 C2 -非负解只有零解 .     

Analytical and approximate solutions of fractional‐order susceptible‐infected‐recovered epidemic model of childhood disease

In this work, we make use of the conformable fractional differential transform method (CFDTM) in order to compute an approximate solution of the fractional‐order susceptible‐infected‐recovered (SIR) epidemic model of childhood disease. The method provides the solution in the form of a rapidly convergent series. Two explanatory and illustrative examples are given to represent the efficacy of the obtained results.     

Remarks on Liouville type theorems for the 3D steady axially symmetric Navier–Stokes equations

In this note, we investigate the 3D steady axially symmetric Navier–Stokes equations, and obtain Liouville type theorems if the velocity or the vorticity satisfies some a priori decay assumptions, which improve the recent result of Seregin's in [15].     

Geometrical conditions for oscillation of second-order half-linear differential equations

This paper deals with the second-order half-linear differential equation

关于椭圆型方程解的Liouville定理

在ｂ（ｘ）的较弱条件下，对下面的椭圆型方程，证整解的Ｌｉｏｕｖｉｌｌｅ 定理成立。