Nondegeneracy and uniqueness of positive solutions for Robin problem of second order ordinary differential equations and its applications

This article studies positive solutions of Robin problem for semi-linear second order ordinary differential equations. Nondegeneracy and uniqueness results are proven for homogeneous differential equations. Necessary and sufficient conditions for the existence of one or two positive solutions for inhomogeneous differential equations or differential equations with concave-convex nonlinearities are obtained by making use of the nondegeneracy and uniqueness results for positive solutions of homogeneous differential equations.     

Properties of set-valued stochastic differential equations

The paper is devoted to properties of set-valued stochastic differential equations. The main result of the paper deals with existence and uniqueness of solutions. Furthermore, a connection between solutions of stochastic differential inclusions and solutions of set-valued stochastic differential equations are given. The result of the paper extends a lot of particular results dealing with such type equations.     

Global bifurcation of waves

In this note it will be shown how a theorem of Alexander [1] and Ize [9] together with computational results of Alexander and Yorke [4] and Alexander and Fitzpatrick [2] may be used to generalize the existence theorem for, and to prove some global results about, certain wave-like solutions of nonlinear systems of partial differential equations.The equations to be studied are weakly coupled parabolic systems of equations defined on a bounded axisymmetric domain. Such equations are often called reaction-diffusion equations (or interaction-diffusion equations) and arise in many parts of biology and chemistry. The question as to how wave-like solutions of these equations may bifurcate from a family of trivial solutions was studied by Auchmuty [5] and the results will be considerably extended here.     

Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping

In this paper, a class of systems of matrix nonlinear differential equations containing as particular cases the systems of coupled Riccati differential equations arising in connection with control of some linear stochastic systems is considered.The system of differential equations considered in this paper are converted in a suitable nonlinear differential equation on a finite-dimensional Hilbert space adequately choosen.This allows us to use the positivity properties of the linear evolution operator defined by the linear differential equations of Lyapunov type.Our aim is to investigate properties of stabilizing and bounded solutions of the considered differential equations and to obtain some conditions ensuring the existence of such solutions.Conditions providing the existence of a maximal solution (minimal solution respectively) with respect to some classes of global solutions are presented. It is shown that if the coefficients of the equations are periodic functions all these special solutions (stabilizing, maximal, minimal) are periodic functions, too.Whenever possible the probabilistic arguments were avoided and so the results proved in the paper appear as results in the field of differential equations with interest in themselves.     

On Some Model Equations of Euler and Navier-Stokes Equations

The author proposes a two-dimensional generalization of Constantin-Lax-Majda model. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line(vorticity formulation), the author presents some further model equations. He possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. Some discussions on the possible connection between tur...     

Second Order Scalar Functional Differential Equations with Piecewise Constant Arguments

The study of functional differential equations with piecewise constant arguments usually results in a study of certain related difference equations. In this paper we consider certain neutral functional differential equations of this type and the associated difference equations. We give conditions under which such equations with almost periodic time dependence will have unique almost periodic solutions, and for certain autonomous cases, we obtain certain stability results and also conditions for chaotic behavior of solutions. We are particularly concerned with such equations which are partially discretized versions of non-forced Duffing equations.     

On meromorphic solutions of linear partial differential equations of second order

This paper is concerned with entire and meromorphic solutions of linear partial differential equations of second order with polynomial coefficients. We will characterize entire solutions for a class of partial differential equations associated with the Jacobi differential equations, and give a uniqueness theorem for their meromorphic solutions in the sense of the value distribution theory, which also applies to general linear partial differential equations of second order. The results are complemented by various examples for completeness.     

一类具有高阶奇性解的Hilbert核奇异积分方程

本文利用指数变换研究了一类解具高阶奇性的周期黎曼边值问题,通过转化法研究了一类解具有高阶奇性的Hilber核奇异积分方程,获得了相应的解和可解条件表达式.推广了Hilber核奇异积分方程的结果.     

Regularity for solutions of nonlinear second order evolution equations

This paper deals with the existence of solutions for the class of nonlinear second order evolution equations. The regularity and a variation of solutions of the given equations are also given. As particular cases of our general formulation, some results for Volterra integrodifferential equations of the hyperbolic type are given.     

Homogenization for semilinear hyperbolic systems with oscillatory data

The behavior of multi-dimensional discrete Boltzmann systems with highly oscillatory data is studied. Homogenized equations for the mean solutions are obtained. Uniform convergence of the oscillatory solutions of the discrete Boltzmann equations to the solutions of the corresponding homogenized equations is established. Moreover, we find that the weak limits of the oscillatory solutions for a model of Broadwell type are not continuous functions of the discrete velocities. Generalization of the above results to problems with multiple-scale initial data is also established.     

On solutions of Emden-Fowler-type equations

The paper deals with solutions to Emden-Fowler-type equations of any arbitrary order. The asymptotic properties of solutions to these equations are studied, and a systematic survey of numerous uncoordinated results of analysis of continuable and noncontinuable solutions is given.     

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

Approximate analytical solutions for Kolmogorov’s equations

This paper reports the explicit analytical solutions for Kolmogorov’s equations. Kolmogorov’s equations are commonly used to describe the structure of local isotropic turbulence, but their exact analytical solutions have not yet been found. In this paper, the closed-form solutions for two kinds of Kolmogorov’s equations are obtained. The derivations of the approximate solutions are based on the homotopy analysis method, which is a new tool for obtaining the approximate analytical solutions of both strong and weak nonlinear differential equations. To examine the validity of the approximate solutions, numerical comparisons between results from the homotopy analysis method and the fourth-order Runge-Kutta method are carried out. It is shown that the results are in good agreement.     

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

Hyperbolic Type Stochastic Evolution Equations with Lévy Noise

The existence and uniqueness of the solutions for a class of hyperbolic type stochastic evolution equations driven by some non-Gaussian Lévy processes are obtained. Moreover, an energy equality for the solutions of the equations is established. As examples, theses results are applied to a couple of stochastic wave type equations with jumps.     

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.     

Positive fixed points for a class of nonlinear operators and applications

In this paper, we study the existence and uniqueness of positive solutions for a class of nonlinear operator equations on ordered Banach spaces. Various applications are also considered to illustrate our obtained results (existence of solutions to quadratic integral equations with a linear modification of the argument, positive solution of second-order Neumann boundary value problem, and positive definite solutions of a class of nonlinear matrix equations).     

Existence of solutions of laminar boundary layer equations with decelerating external flows

New results on the existence and nonexistence of solutions of laminar boundary layer equations with decelerating external flows are obtained. Some previous results treated these equations with accelerating or constant external flows. Our approach is to establish a system of two integral equations with singularities which will be proved to be equivalent to the laminar boundary layer equations and prove that the system has solutions by using the Leray–Schauder fixed point theorem and the Helly selection principle.     

Solving systems of fractional differential equations using differential transform method

This paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of systems of fractional differential equations. The solutions of our model equations are calculated in the form of convergent series with easily computable components. Some examples are solved as illustrations, using symbolic computation. The numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations.     

Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method

In this paper, we introduce a spectral collocation method based on Lagrange polynomials for spatial derivatives to obtain numerical solutions for some coupled nonlinear evolution equations. The problem is reduced to a system of ordinary differential equations that are solved by the fourth order Runge–Kutta method. Numerical results of coupled Korteweg–de Vries (KdV) equations, coupled modified KdV equations, coupled KdV system and Boussinesq system are obtained. The present results are in good agreement with the exact solutions. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations.