Sine‐cosine wavelet method for fractional oscillator equations

Purpose In this article, a novel computational method is introduced for solving the fractional nonlinear oscillator differential equations on the semi‐infinite domain. The purpose of the proposed method is to get better and more accurate results. Design/methodology/approach The proposed method is the combination of the sine‐cosine wavelets and Picard technique. The operational matrices of fractional‐order integration for sine‐cosine wavelets are derived and constructed. Picard technique is used to convert the fractional nonlinear oscillator equations into a sequence of discrete fractional linear differential equations. Operational matrices of sine‐cosine wavelets are utilized to transformed the obtained sequence of discrete equations into the systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear oscillator equations. Findings The convergence and supporting analysis of the method are investigated. The operational matrices contains many zero entries, which lead to the high efficiency of the method, and reasonable accuracy is achieved even with less number of collocation points. Our results are in good agreement with exact solutions and more accurate as compared with homotopy perturbation method, variational iteration method, and Adomian decomposition method. Originality/value Many engineers can utilize the presented method for solving their nonlinear fractional models.     

Multiscale behavior and fractional kinetics from the data of solar wind–magnetosphere coupling

Multiscale phenomena are ubiquitous in nature as well as in laboratories. A broad range of interacting space and time scales determines the dynamics of many systems which are inherently multiscale. In many systems multiscale phenomena are not only prominent, but also they often play the dominant role. In the solar wind–magnetosphere interaction, multiscale features coexist along with the global or coherent features. Underlying these phenomena are the mathematical and theoretical approaches such as phase transitions, turbulence, self-organization, fractional kinetics, percolation, etc. The fractional kinetic equations provide a suitable mathematical framework for multiscale behavior. In the fractional kinetic equations the multiscale nature is described through fractional derivatives and the solutions of these equations yield infinite moments, showing strong multiscale behavior. Using a Lévy flights approach, we analyze the correlated data of the solar wind–magnetosphere coupling. Based on this analysis a model of the multiscale features is proposed and compared with the solutions of diffusion-type equations. The equation with fractional spatial derivative shows strong multiscale behavior with infinite moments. On the other hand, the equation with space dependent diffusion coefficients yield finite moments, indicating Gaussian type solutions and absence of long tails typically associated with multiscale behavior.     

Stochastic Evolution Equations Driven by a Fractional White Noise

Abstract

In this paper we study stochastic evolution equations driven by a fractional white noise with arbitrary Hurst parameter in infinite dimension. We establish the existence and uniqueness of a mild solution for a nonlinear equation with multiplicative noise under Lipschitz condition by using a fixed point argument in an appropriate inductive limit space. In the linear case with additive noise, a strong solution is obtained. Those results are applied to stochastic parabolic partial differential equations perturbed by a fractional white noise.     

Optimal Solutions of Fractional Nonlinear Impulsive Neutral Stochastic Functional Integro-Differential Equations

Abstract

In this article, we consider a new class of fractional impulsive neutral stochastic functional integro-differential equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, fractional calculus, analytic α-resolvent operator and suitable fixed point theorems, we prove the existence of mild solutions and optimal mild solutions for these equations. Second, the existence of optimal pairs of system governed by fractional impulsive partial stochastic integro-differential equations is also presented. The results are obtained under weaker conditions in the sense of the fractional power arguments. Finally, an example is given for demonstration.     

Approximation by Chebyshevian Bernstein Operators versus Convergence of Dimension Elevation

On a closed bounded interval, consider a nested sequence of Extended Chebyshev spaces possessing Bernstein bases. This situation automatically generates an infinite dimension elevation algorithm transforming control polygons of any given level into control polygons of the next level. The convergence of these infinite sequences of polygons towards the corresponding curves is a classical issue in computer-aided geometric design. Moreover, according to recent work proving the existence of Bernstein-type operators in such Extended Chebyshev spaces, this nested sequence is automatically associated with an infinite sequence of Bernstein operators which all reproduce the same two-dimensional space. Whether or not this sequence of operators converges towards the identity on the space of all continuous functions is a natural issue in approximation theory. In the present article, we prove that the two issues are actually equivalent. Not only is this result interesting on the theoretical side, but it also has practical implications. For instance, it provides us with a Korovkin-type theorem of convergence of any infinite dimension elevation algorithm. It also enables us to tackle the question of convergence of the dimension elevation algorithm for any nested sequence obtained by repeated integration of the kernel of a given linear differential operator with constant coefficients.     

Random Dynamical Systems and Stationary Solutions of Differential Equations Driven by the Fractional Brownian Motion

Abstract

Linear and semilinear stochastic evolution equations with additive noise, where the forcing term is an infinite dimensional fractional Brownian motion are studied. Under usual dissipativity conditions the equations are shown to define random dynamical systems which have unique, exponentially attracting fixed points. The results are applied to stochastic parabolic PDE's. They are also applicable to standard finite-dimensional dissipative stochastic equation driven by fractional Brownian motion.     

Dimension Reduction methods for Infinite Dimensional Systems

From experiments and also from computer simulation of dynamical systems it is well known that for many dynamical phenomena in physics or engineering, which are modelled by infinite dimensional dynamical systems, the asymptotic behavior can be accurately described by replacing the original infinite dimensional system by a low dimensional system represented by so‐called essential variables. Such a dimension reduction of a dynamical system turns out to be central, both for a qualitative and quantitative understanding of its behaviour. In [1] Approximate Inertial Manifolds are presented, which perform extremely well for nonlinear evolution equations, but don't work as expected for the dynamics of a fluid conveying tube. By comparing the results for different internal damping values it can be seen that the larger gaps and the location of the cluster point in the spectrum for the weaker damping improve the approximation quality considerably.     

Existence of solutions for nonlinear fractional stochastic differential equations

The fractional stochastic differential equations have wide applications in various fields of science and engineering. This paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in Hilbert spaces. Using fractional calculations, fixed point technique, stochastic analysis theory and methods adopted directly from deterministic fractional equations, new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay. Further, we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions. Examples are provided to illustrate the obtained theory.     

Equivalent embeddings of the dynamics on an invariant manifold

A dynamical system admitting an invariant manifold can be interpreted as a single element of an infinite class of dynamical systems that all exhibit the same behaviour on the invariant manifold. This observation is used in the context of autonomous ordinary differential equations to generalize a global stability result of Li and Muldowney. The new result is demonstrated on an epidemiological model.     

On the geometry of a four-parameter rational planar system of difference equations

In this paper, we consider geometric aspects of a rational, planar system of difference equations defined on the open first quadrant and whose behaviour is governed by four independent, non-negative parameters. This system, indexed as (23, 23) in the notation of Ladas (Open problems and conjectures, J. Differential Equ. Appl. 15(3) 2009, pp. 303–323), is one of the 200 systems from Ladas about which little is known. Using geometric techniques, we answer several questions concerning the behaviour of this system.     

Geometric configurations of singularities for quadratic differential systems with three distinct real simple finite singularities

In this work we classify, with respect to the geometric equivalence relation, the global configurations of singularities, finite and infinite, of quadratic differential systems possessing exactly three distinct finite simple singularities. This relation is finer than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci (or saddles) of different orders. Such distinctions are, however, important in the production of limit cycles close to the foci (or loops) in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows us to incorporate all these important geometric features which can be expressed in purely algebraic terms. The geometric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in a work published in 2013 when the classification was done for systems with total multiplicity m _f of finite singularities less than or equal to one. That work was continued in an article which is due to appear in 2014 where the geometric classification of configurations of singularities was done for the case m _f = 2. In this article we go one step further and obtain the geometric classification of singularities, finite and infinite, for the subclass mentioned above. We obtain 147 geometrically distinct configurations of singularities for this family. We give here the global bifurcation diagram of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for this class of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants, a fact which gives us an algorithm for determining the geometric configuration of singularities for any quadratic system in this particular class.     

Solution of a new model of fractional telegraph point reactor kinetics using differential transformation method

A new model of fractional telegraph point reactor kinetics FTPRK is introduced to approximate the time dependent Boltzmann transport equation considering new terms that contain time derivative of the reactivity and fractional integral of the neutron density. Caputo fractional derivatives and fractional Leibniz rule are used for such derivation. Cattaneoequation is applied to overcome the flaw of infinite neutron velocity and to describe the anomalous transport. Effect of the new term on the neutron behaviour is discussed. The new model is applied to both TRIGA reactor and to commercial pressured water reactor of a Three Mile Island type reactor, TMI-type PWR. Results for step, ramp and sinusoidal excess reactivities with thermal hydraulic feedback are presented and discussed for different values of anomalous sub-diffusion exponent, the fractional order, 0 < µ ≤ 1. To maintain the reactor safe at start-up after insertion of step reactivity and based on the concept of prompt jump approximation, the FTPRK model is simplified and solved analytically by Mittag–Liffler function. Physical interpretations of the fractional order µ and relaxation time τ and their effects on the behaviour of the neutron population are discussed. Also, the effect of a small perturbation in the geometric buckling on the neutron behaviour is discussed for finite reactor core. The new model is solved numerically using the fractional order multi-step differential transform method MDTM. The MDTM constitutes an easy algorithm based on Taylor's formula and Caputo fractional derivative. Two theorems with their proofs are introduced to solve the fractional system. Two major disadvantages of the method about the choice of the fractional order values and the step size length are addressed. We present a procedure which enables us to solve the system with appropriate values of fraction orders.     

Solution of one class of systems of dual summation equations for associated Legendre functions

We have obtained analytical solutions of one class of systems of dual summation equations for associated Legendre functions with fractional indices. Such equations appear in studying the interaction of vector electromagnetic fields with the circular edge of a conductive open cone in the low-frequency region. We have derived formulas for the reexpansion of Legendre functions, which are used for passage from summation equations to infinite systems of linear algebraic equations, containing convolution-type matrix operators. The operators inverse to them are applied for finding a solution in the required class of sequences. We give an example of determining the effect of interaction of TM- and TE-waves with the edge of a finite cone.     

Realizing nonholonomic dynamics as limit of friction forces

The classical question whether nonholonomic dynamics is realized as limit of friction forces was first posed by Carathéodory. It is known that, indeed, when friction forces are scaled to infinity, then nonholonomic dynamics is obtained as a singular limit.Our results are twofold. First, we formulate the problem in a differential geometric context. Using modern geometric singular perturbation theory in our proof, we then obtain a sharp statement on the convergence of solutions on infinite time intervals. Secondly, we set up an explicit scheme to approximate systems with large friction by a perturbation of the nonholonomic dynamics. The theory is illustrated in detail by studying analytically and numerically the Chaplygin sleigh as an example. This approximation scheme offers a reduction in dimension and has potential use in applications.     

Fractional integral and its physical interpretation

A relationship is established between Cantor's fractal set (Cantor's bars) and a fractional integral. The fractal dimension of the Cantor set is equal to the fractional exponent of the integral. It follows from analysis of the results that equations in fractional derivatives describe the evolution of physical systems with loss, the fractional exponent of the derivative being a measure of the fraction of the states of the system that are preserved during evolution timet. Such systems can be classified as systems with residual memory, and they occupy an intermediate position between systems with complete memory, on the one hand, and Markov systems, on the other. The use of such equations to describe transport and relaxation processes is discussed. Some generalizations that extent the domain of applicability of the fractional derivative concept are obtained.Kazan State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 354–368, March, 1992.     

Infinite Systems of Quasilinear Differential Difference Inequalities and Applications

The article deals with the initial boundary value problem for an infinite system of first order quasilinear functional differential equations. A comparison result concerning infinite systems of differential difference inequalities is proved. A function satisfying such inequalities is estimated by a solution of a suitable Cauchy problem for an ordinary functional differential system. The comparison result is used in an existence theorem and in the investigation of the stability of the numerical method of lines for the original problem. A theorem on the error estimate of the method is given. The infinite system of first order functional differential equations contains, as particular cases, equations with a deviated argument and integral differential equations of the Volterra type.     

Local dynamics of spatially distributed Hutchinson equation

The local dynamics of spatially distributed Hutchinson equation is studied. In critical cases, which all have infinite dimension, systems of parabolic equations are built, which play the role of normal forms.     

Nesting inertial manifolds for reaction and diffusion equations with large diffusivity

We study the asymptotic behaviour in large diffusivity of inertial manifolds governing the long time dynamics of a semilinear evolution system of reaction and diffusion equations. A priori, we review both local and global dynamics of the system in scales of Banach spaces of Hilbert type and we prove the existence of a universal compact attractor for the equations. Extensions yield the existence of a family of nesting inertial manifolds dependent on the diffusion of the system of equations. It is introduced an upper semicontinuity notion in large diffusivity for inertial manifolds. The limit inertial manifold whose dimension is strictly less than those of the infinite dimensional system of semilinear evolution equations is obtained.     

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.