Conservation laws for a modified lubrication equation

In this work we study the conservation laws of a modified lubrication equation, which describes the dynamics of the interfacial motion in phase transition. We show that the equation is nonlinear self-adjoint and has an exact Lagrangian with an auxiliary function. As a result, by a general theorem on conservation laws proved by Nail Ibragimov recently and Noether’s theorem, some new conservation laws for the equation are obtained. Our results show that the non-locally defined conservation laws generated by Noether’s theorem are equivalent to the local ones given by Ibragimov’s theorem.     

Analysis of the time fractional nonlinear diffusion equation from diffusion process

Under investigation in this paper is a time fractional nonlinear diffusion equation which can be utilized to express various diffusion processes. The symmetry of this considered equation has been obtained via fractional Lie group approach with the sense of Riemann-Liouville (R-L) fractional derivative. Based on the symmetry, this equation can be changed into an ordinary differential equation of fractional order. Moreover, some new invariant solutions of this considered equation are found. Lastly, utilising the Noether theorem and the general form of Noether type theorem, the conservation laws are yielded to the time fractional nonlinear diffusion equation, respectively. Our discovery that there are no conservation laws under the general form of Noether type theorem case. This result tells us the symmetry of this equation is not variational symmetry of the considered functional. These rich results can give us more information to interpret this equation.     

On the nonlinear self-adjointness of the Zakharov–Kuznetsov equation

A new general theorem, which does not require the existence of Lagrangians, allows to compute conservation laws for an arbitrary differential equation. This theorem is based on the concept of self-adjoint equations for nonlinear equations. In this paper we show that the Zakharov–Kuznetsov equation is self-adjoint and nonlinearly self-adjoint. This property is used to compute conservation laws corresponding to the symmetries of the equation. In particular the property of the Zakharov–Kuznetsov equation to be self-adjoint and nonlinearly self-adjoint allows us to get more conservation laws.     

Integral condition for oscillation of half-linear differential equations with damping

The purpose of this paper is to provide an oscillation theorem that can be applied to half-linear differential equations with time-varying coefficients. A parametric curve by the coefficients is focused in order to obtain our theorem. This parametric curve is a generalization of the curve given by the characteristic equation of the second-order linear differential equation with constant coefficients. The obtained theorem is proved by transforming the half-linear differential equation to a standard polar coordinates system and using phase plane analysis carefully.     

Functional differential inequalities and estimation of the Cauchy function of an equation with aftereffect

We consider scalar functional differential inequalities that are used to estimate solutions to differential equations with deviating argument. A theorem on positiveness of the Cauchy function of a differential equation with aftereffect is derived from a theorem on a functional differential inequality with nonlinearmonotone operator, which is a direct generalization of the simplest classical theorem on a differential inequality. The suggested proofs rely on local properties of continuous functions only.     

A completeness theorem for the linear stability problem of nearly parallel flows

A completeness/expansion theorem, analogous to that of DiPrima and Habetler (D-H), is proved for the equation governing the linear stability of nearly parallel flows, to which the D-H theorem does not apply. It is also proved that only a finite number of eigenvalues with negative real parts can occur. Both results are based on a theorem of Gohberg and Kreǐn.     

构造求方程重根高阶迭代公式的基本定理

在[1]中已经介绍了构造高阶多点迭代公式的基本定理:设φ(x)d是p阶的,则φ(x)=φ(x)-f(φ(x))/f′(x)是P 1阶的,[2]中又给出了[1]的一个改进了的基本定理,但这些定理仅适用于方程f(x)是单根的情况.本文针对φ′(x_*)的性质,提出了在重根情况下亦适用的多点迭代构造定理.     

A Generalization of a Volterra Integral Equation

Sufficient conditions for the existence of a solution to a non-linear Volterra integral equation are given for special cases of the general equation. In the generality given here, this equation has, apparently, not been studied before. The major technique used is the classical fixed point theorem of Banach. An apparent innovation of this article is the use of Banach's theorem to prove both the existence and find the location of a solution to the integral equation and prove the existence and find the location of the derivative to this solution, which exists almost everywhere. Furthermore, it is shown that for some particular choices of the constants, multiple solutions exist to this equation.     

Existence of a solution of the cauchy problem for an integrodifferential equation of viscoelasticity

Mechanics of Composite Materials - The Cauchy problem for a nonlinear integrodifferential equation is reduced to a nonlinear integral equation for which a theorem of existence and uniqueness is...     

A new composition theorem for square-mean almost automorphic functions and applications to stochastic differential equations

In this paper, we establish a new composition theorem for square-mean almost automorphic functions under conditions which are different from Lipschitz conditions in the literature. We apply this new composition theorem together with Schauder’s fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions for a stochastic differential equation in a real separable Hilbert space. Finally, an interesting corollary is also given for the sub-linear growth cases.     

New method for constructing Chernoff functions

We state and, for the first time, prove a theorem in the theory of strongly continuous operator semigroups. This theorem, which has essentially been suggested by O.G. Smolyanov, in particular, enables one to reduce solving the Schr¨odinger equation to solving the heat equation.     

Some stability theorems of uncertain differential equation

Canonical process is a type of uncertain process with stationary and independent increments which are normal uncertain variables, and uncertain differential equation is a type of differential equation driven by canonical process. This paper will give a theorem on the Lipschitz continuity of canonical process based on which this paper will also provide a sufficient condition for an uncertain differential equation being stable.     

A new conservation theorem

A general theorem on conservation laws for arbitrary differential equations is proved. The theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables. The new theorem does not require existence of a Lagrangian and is based on a concept of an adjoint equation for non-linear equations suggested recently by the author. It is proved that the adjoint equation inherits all symmetries of the original equation. Accordingly, one can associate a conservation law with any group of Lie, Lie-Bäcklund or non-local symmetries and find conservation laws for differential equations without classical Lagrangians.     

局部凸空间中集值映射的极小不动点定理及其应用

利用局部凸空间中Fan-Kakutani不动点定理,得到局部凸空间中集值映射的极小不动点定理,应用此定理,证明了半线性不适定的算子方程的最小范数极值解的存在性.此结果可以应用到不适定常微方程的两点边值问题,不适定偏微方程的边值问题.     

A shallow-water sedimentation model with friction and Coriolis: An existence theorem

We present an existence theorem of a two-dimensional sedimentation model coupling a shallow water system with a sediment transport equation. The shallow water system includes Coriolis and friction terms. A Galerkin method is used to obtain a finite-dimensional problem which is solved using a Brouwer fixed point theorem. We prove that the limits of the resulting solution sequences satisfy the model equations.     

On the Modification of Rouche's Theorem for the Queueing Theory Problems

In analytic queueing theory, Rouche's theorem is frequently used to prove the existence of a certain number of zeros in the domain of regularity of a given function. If the theorem can be applied it leads in a simple way to results concerning the ergodicity condition and the construction of the solution of the functional equation for the generating function of the stationary distribution. Unfortunately, the verification of the conditions needed to apply Rouche's theorem is frequently quite difficult. We prove the theorem which allows to avoid some difficulties arising in applying classical Rouche's theorem to an analysis of queueing models.     

A sensitivity analysis for linear systems involving M-matrices and its application to the Leontif model

A sensitivity analysis is made for solutions to linear equation systems involving M-matrices. We present a theorem which tells about relative changes of elements of the solution vector when the coefficients of a given M-matrix shift. The Metzler theorem and the Morishima theorem are generalized, and applied to the Leontief model.     

Martingale solutions of stochastic fractional nonlinear Schrödinger equation on a bounded interval*

This paper is concerned with stochastic fractional nonlinear Schrödinger equation, which plays a very important role in fractional nonrelativistic quantum mechanics. Due to disturbing and interacting of the fractional Laplacian operator on a bounded interval with white noise, the stochastic fractional nonlinear Schrödinger equation is too complicated to be understood. This paper would explore and analyze this stochastic fractional system. Using a suitable weighted space with some fractional operator skills, it overcame the difficulties coming from the fractional Laplacian operator on a bounded interval. Applying the tightness instead of the common compactness, and combining Prokhorov theorem with Skorokhod embedding theorem, it solved the convergence problem in the case of white noise. It finally established the existence of martingale solutions for the stochastic fractional nonlinear Schrödinger equation on a bounded interval.     

Existence of Solutions for Impulsive Neutral Integro-Differential Inclusions with Nonlocal Initial Conditions via Fractional Operators

This paper is mainly concerned with existence of mild solutions for first-order impulsive neutral integro-differential inclusions with nonlocal initial conditions in α-norm. We assume that the undelayed part generates an analytic resolvent operator and transforms it into an integral equation. By using a fixed point theorem for condensing multivalued maps, a main existence theorem is established. As an application of this main theorem, a practical consequence is derived for the sublinear growth case. Finally, we present an application to a neutral partial integro-differential equation with Dirichlet and nonlocal initial conditions.