Averaging Theory of Arbitrary Order for Piecewise Smooth Differential Systems and Its Application

The averaging theory for studying periodic orbits of smooth differential systems has a long history. Whereas the averaging theory for piecewise smooth differential systems appeared only in recent years, where the unperturbed systems are smooth. When the unperturbed systems are only piecewise smooth, there is not an existing averaging theory to study existence of periodic orbits of their perturbed systems. Here we establish such a theory for one dimensional perturbed piecewise smooth periodic differential equations. Then we show how to transform planar perturbed piecewise smooth differential systems to one dimensional piecewise smooth periodic differential equations when the unperturbed planar piecewise smooth differential systems have a family of periodic orbits. Finally as application of our theory we study limit cycle bifurcation of planar piecewise differential systems which are perturbation of a \(\Sigma \)-center.     

Refined differential equations of deflections in axial symmetrical bending problems of spherical shell and their singular perturbation solutions

This paper deals with the research of accuracy of differential equations of deflections.The basic idea is as follows.Firstly,considering the boundary effect the meridianmidsurface displacement u=0,thus we derive the deflection differential equations;secondly we accurately prove that by use of the deflection differential equations or theoriginal differential equations the same inner forces solutions are obtained;finally,weaccurately prove that considering the boundary effect the meridian surface displacementu=0 is an exact solution.In this paper we give the singular perturbation solution of thedeflection differential equations.Finally we check the equilibrium condition and prove theinner forces solved by perturbation method and the outer load are fully equilibrated.Itshows that perturbation solution is accurate.On the other hand,it shows again that thedeflection differential equation is an exact equation.The features of the new differential equations are as follows:1.The accuracies of the new differentia     

Piecewise linear differential systems without equilibria produce limit cycles?

In this article, we study the planar piecewise differential systems formed by two linear differential systems separated by a straight line, such that both linear differential have no equilibria, neither real nor virtual. When the piecewise differential system is continuous, we show that the system has no limit cycles. But when the piecewise differential system is discontinuous, we show that it can have at most one limit cycle.     

New Theory of Hybrid-Upwind Technique and Discrete Del Operator for Finite-Element Method

In this paper, we present a new hybrid-streamline-upwind finite-element method, which is based on the finite analytic method developed in the field of the finite difference method. In order to obtain an optimal shape function, we introduce an adjoint differential operator to the differential operator for a steady advection-diffusion equation. The shape functions which satisfy these differential operators are mutually dual. One of them interpolates accurately the functions appearing in convection-dominated flows, the other becomes a hybrid-streamline-upwind weighting function. Furthermore, we define a discrete del operator for the reduction of memory storage in the computer. As a result, we achieve simplicity of formulation and high-speed calculation of the finite-element method.     

The solution of the second part of the 16th Hilbert problem for nine families of discontinuous piecewise differential systems

We provide the maximum number of limit cycles of some classes of discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are linear centers or three families of cubic isochronous centers, giving rise to ten different classes of discontinuous piecewise differential systems. These maximum number of limit cycles vary from 0, 1, 2, 3, 5, 7 and 12 depending on the chosen class. For nine of these classes, we prove that the corresponding maximum number of limit cycles are reached. In particular, we have solved the extension of the second part of the 16th Hilbert problem to these classes of discontinuous piecewise differential systems. The main tool used for proving these results is based on the first integrals of the systems which form the discontinuous piecewise differential systems.

     

Establishment of several differential equations for the case of variable solid thermal conductivity

In this paper we propose that the solid thermal conductivity varies first with the space coordinate according to a linear law and then with an exponential one. In this way we have set up six differential equations successfully; and for variable density, variable specific heat as well as variable conductivity, another six differential equations have also been deduced.     

Correcting the initialization of models with fractional derivatives via history-dependent conditions

Fractional differential equations are more and more used in modeling memory(history-dependent,nonlocal,or hereditary) phenomena.Conventional initial values of fractional differential equations are define at a point,while recent works defin initial conditions over histories.We prove that the conventional initialization of fractional differential equations with a Riemann–Liouville derivative is wrong with a simple counter-example.The initial values were assumed to be arbitrarily given for a typical fractional differential equation,but we fin one of these values can only be zero.We show that fractional differential equations are of infinit dimensions,and the initial conditions,initial histories,are define as functions over intervals.We obtain the equivalent integral equation for Caputo case.With a simple fractional model of materials,we illustrate that the recovery behavior is correct with the initial creep history,but is wrong with initial values at the starting point of the recovery.We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.     

Identification of momentum life cycle stage of stock price

In this article, we study the continuous and discontinuous planar piecewise differential systems formed only by linear centers separated by one or two parallel straight lines. When these piecewise differential systems are continuous, they have no limit cycles. Also if they are discontinuous separated by a unique straight line, they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel straight lines, we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle.     

Numerical approximation of a class of discontinuous systems of fractional order

In this paper we investigate the possibility to formulate an implicit multistep numerical method for fractional differential equations, as a discrete dynamical system to model a class of discontinuous dynamical systems of fractional order. For this purpose, the problem is continuously transformed into a set-valued problem, to which the approximate selection theorem for a class of differential inclusions applies. Next, following the way presented in the book of Stewart and Humphries (Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996) for the case of continuous differential equations, we prove that a variant of Adams?CBashforth?CMoulton method for fractional differential equations can be considered as defining a discrete dynamical system, approximating the underlying discontinuous fractional system. For this purpose, the existence and uniqueness of solutions are investigated. One example is presented.     

Constructions of the general solution for a system of partial differential equations with variable coefficients

Solving partial differential equations has not only theoretical significance,but alsopractical value.In this paper.by the property of conjugate operator.we give a method toconstruct the general solutions of a system of partial differential equations.     

Differential characteristic set algorithm for the complete symmetry classification of partial differential equations

In this paper, we present a differential polynomial characteristic set algorithm for the complete symmetry classification of partial differential equations （PDEs） with some parameters. It can make the solution to the complete symmetry classification problem for PDEs become direct and systematic. As an illustrative example, the complete potential symmetry classifications of nonlinear and linear wave equations with an arbitrary function parameter are presented. This is a new application of the differential form characteristic set algorithm, i.e., Wu＇s method, in differential equations.     

Uniform convergence for the difference scheme in the conservation form of ordinary differential equation with a small parameter

In this paper, we consider a singular perturbation boundary problem for a self-adjoint ordinary differential equaiton. We construct a class of difference schemes with fitted factors, and give the sufficient conditions under which the solution of difference scheme converges uniformly to the solution of differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence.     

Existence of solutions for a singularly perturbed boundary value problem for the third order semi-linear differential equation with a turning point

In this paper,by using the techniques of differential inequalities,we prove the existence of the solutions of a singularly perturbed boundary value problem for the third order semilinear differential equation with a turning point.     

The index of lyapunov stable fixed points in two dimensions

In this paper, we prove that a stable isolated fixed point of an orientation preserving local homeomorphism onR ₂ has fixed point index 1. We also give a number of applications to differential equations. In particular, we deduce that a number of existence methods for producing periodic solutions of differential equations in the plane always produce unstable solutions.     

Nonlinear travelling waves in a hyperelastic rod composed of a compressible Mooney-Rivlin material

In this paper, we study strongly nonlinear axisymmetric waves in a circular cylindrical rod composed of a compressible Mooney-Rivlin material. To consider the travelling wave solutions for the governing partial differential system, we first reduce it to a nonlinear ordinary differential equation. By using the bifurcation theory of planar dynamical systems, we show that the reduced system has seven periodic annuluses with different boundaries which depend on four parameters. We further consider the bifurcation behavior of the phase portraits for the reduced one-parameter vector fields when other three parameters are fixed. Corresponding to seven different periodic annuluses, we obtain seven types of travelling wave solutions, including solitary waves of radial contraction, solitary waves of radial expansion, solitary shock waves of radial contraction, solitary shock waves of radial expansion, periodic waves and two types of periodic shock waves. These are physically acceptable solutions by the governing partial differential system. The rigorous parameter conditions for the existence of these waves are given.     

A Hilbert Space Perspective on Ordinary Differential Equations with Memory Term

We discuss ordinary differential equations with delay and memory terms in Hilbert spaces. By introducing a time derivative as a normal operator in an appropriate Hilbert space, we develop a new approach to a solution theory covering integro-differential equations, neutral differential equations and general delay differential equations within a unified framework. We show that reasonable differential equations lead to causal solution operators.     

Similarity solutions of Prandtl mixing length modelled two dimensional turbulent boundary layer equations

The exact similarity solutions of two dimensional laminar boundary layer were obtained by Blasius in 1908, however, for two dimensional turbulent boundary layers, no Blasius type similarity solutions (special exact solutions) have ever been found. In the light of Blasius’ pioneer works, we extend Blasius similarity transformation to the two dimensional turbulent boundary layers, and for a special case of flow modelled by Prandtl mixing-length, we successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation. The ordinary differential equation is numerically solved and some useful quantities are produced. For numerical calculations, a complete Maple code is provided.     

The uniformly convergent difference schemes for a singular perturbation problem of a self-adjoint ordinary differential equation

In this paper, we construct a class of difference schemes with fitted factors for a singular perturbation problem of a self-adjoint ordinary differential equation. Using a different method from [1], by analyzing the truncation errors of schemes, we give the sufficient conditions under which the solution of the difference scheme converges uniformly to the solution of the differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence, and applying them to example, obtain the numerical results.     

FORCED OSCILLATIONS OF BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS

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Fractional generalized synchronization in a class of nonlinear fractional order systems

Generalized synchronization in nonlinear fractional order systems occurs whether the states of one system by means of a functional mapping are identical to states of another. This mapping can be obtained if there exists a fractional differential primitive element whose elements are fractional derivatives which generate a differential transcendence basis. In this contribution we investigate the fractional generalized synchronization (FGS) problem for a class of strictly different nonlinear fractional order systems and we consider the master-slave synchronization scheme. As well as, of a natural manner we construct a fractional generalized observability canonical form, we introduce a fractional algebraic observability property, and we design a fractional dynamical controller able to achieve synchronization. These particular forms of FGS are illustrated with numerical results.