Solving fractional partial differential equations by using the second Chebyshev wavelet operational matrix method

Nonlinear Dynamics - Fractional partial differential equations have many applications in science and engineering. However, not only the analytical solution existed for a limited number of cases,...     

A modified Gaussian moment closure method for nonlinear stochastic differential equations

Parametric excitation of a nonlinear physical pendulum by modulation of its moment of inertia is analyzed in terms of physics as an example of the suggested approach. The modulation is provided by a redistribution of auxiliary masses. The system is investigated both analytically and with the help of computer simulations. The threshold and other characteristics of parametric resonance are found and discussed in detail. The role of nonlinear properties of the physical system in restricting the resonant swinging is emphasized. Phase locking between the drive and oscillations of the pendulum and the phenomenon of parametric autoresonance are investigated. The boundaries of parametric instability are determined as functions of the modulation depth and the quality factor. The feedback providing active optimal control of amplification and attenuation of oscillations is analyzed. An effective method of suppressing undesirable rotary oscillations of suspended constructions is suggested.     

Numerical solutions of differential equations with fractional L-derivative

Fractional differential equations are solved with L-fractional derivatives, using numerical procedures. Two characteristic fractional differential equations are numerically solved. The first equation describes the motion of a thin rigid plate immersed in a Newtonian fluid connected by a massless spring to a fixed point, and the other one the diffusion of gas in a fluid.     

A new approach to the group analysis of one-dimensional stochastic differential equations

Stochastic evolution equations are investigated using a new approach to the group analysis of stochastic differential equations. It is shown that the proposed approach reduces the problem of group analysis for this type of equations to the same problem of group analysis for evolution equations of special form without stochastic integrals.     

Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives

On a series of examples from the field of viscoelasticity we demonstrate that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann–Liouville fractional derivatives, and that it is possible to obtain initial values for such initial conditions by appropriate measurements or observations.Dedicated to professor Hari M. Srivastava on the occasion of his 65th birthday     

A limit theorem for the solutions of slow-fast systems with fractional Brownian motion

A limit theorem which can simplify slow—fast dynamical systems driven by fractional Brownian motion with the Hurst parameter H inside the (1/2, 1) interval has been proved. The slow variables of the original system can be approximated by the solution of the simplified equations in the sense of mean square. An example is presented to illustrate the applications of the limit theorem.     

Suppression of nonlinear vibration system described by nonlinear differential equations using passive controller

In this paper, the linear absorber is proposed to reduce the vibration of a nonlinear dynamical system at simultaneous primary resonance and the presence of 1:1 internal resonance. This leads to a two-degree-of-freedom system subjected to external excitation force. The method of multiple scales perturbation technique is applied throughout to determine the analytical solution up to first-order approximations. The stability of the system near the one of the worst resonance case is studied using the frequency response equations. The effects of the different system and absorber parameters on the behavior of the main system are studied numerically. For validity, the numerical solution is compared with the analytical solution and gets a good agreement. Effectiveness of the absorber ( \(E_{a})\) is about 800 for the nonlinear vibrating system. The simulation results are achieved using MATLAB programs. At the end of the work, the comparison with the available published work is reported.     

Abstract Cauchy problem for fractional differential equations

In this paper, a generalized Darbo’s fixed-point theorem associated with Hausdorff measure of noncompactness is established. Then we apply this new variant fixed-point theorem to study some fractional differential equations in Banach spaces via the technique of measure of noncompactness. Many novel existence and uniqueness results for solutions are obtained under the more general conditions.     

Systems of nonlinear differential equations with bounded solutions

We establish sufficient conditions for the existence of solutions bounded on ℝ for the equation {fx168-01}, in a finite-dimensional Banach space {ie168-01}. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 160–167, April–June, 2008.     

A non-Fickian,particle-tracking diffusion model based on fractional Brownian motion

The work is motivated by the recent discovery that ocean surface drifter trajectories contain fractal properties. This suggests that the dispersion of pollutants in coastal waters may also be described using fractal statistics. The paper describes the development of a fractional Brownian motion model for simulating pollutant dispersion using particle tracking. Numerical test cases are used to compare this new model with the results obtained from a traditional Gaussian particle-tracking model. The results seems to be significantly different, which may have implications for pollution modelling in the coastal zone. © 1997 John Wiley & Sons, Ltd.     

A uniqueness criterion for fractional differential equations with Caputo derivative

We investigate the uniqueness of solutions to an initial value problem associated with a nonlinear fractional differential equation of order α∈(0,1). The differential operator is of Caputo type whereas the nonlinearity cannot be expressed as a Lipschitz function. Instead, the Riemann–Liouville derivative of this nonlinearity verifies a special inequality.     

Modelling the growth rate of a tracer gradient using stochastic differential equations

We develop a model in two dimensions to characterise the growth rate of a tracer gradient mixed by a statistically homogeneous flow that varies on arbitrary timescales. The model is based on the orientation dynamics of the passive-tracer gradient with respect to the straining (compressive) direction of the flow, and involves reducing the dynamics to a set of stochastic differential equations. The statistical properties of the system emerge from solving the associated Fokker–Planck equation. Within the model framework, the tracer gradient aligns with the compressive direction when the mean effective rotation in the flow is zero. At finite values of rotation, the tracer gradient aligns with a different direction, but the mean growth rate of the gradient is positive in all cases. In a certain limiting case, namely temporally decorrelated (rapidly varying) flows, exact, analytical expressions exist for the mean growth rate. Using numerical simulations, we assess the extent to which our model applies to real mixing protocols, and map the stochastic parameters on to flow parameters.     

Periodic solutions of nonlinear delay differential equations using spectral element method

We extend the temporal spectral element method further to study the periodic orbits of general autonomous nonlinear delay differential equations (DDEs) with one constant delay. Although we describe the approach for one delay to keep the presentation clear, the extension to multiple delays is straightforward. We also show the underlying similarities between this method and the method of collocation. The spectral element method that we present here can be used to find both the periodic orbit and its stability. This is demonstrated with a variety of different examples, namely, the delayed versions of Mackey–Glass equation, Van der Pol equation, and Duffing equation. For each example, we show the method’s convergence behavior using both p and h refinement and we provide comparisons between equal size meshes that have different distributions.     

ASYMPTOTIC STABILITIES OF STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

Asymptotic characteristic of solution of the stochastic functional differential equation was discussed and sufficient condition was established by multiple Lyapunov functions for locating the limit set of the solution. Moreover, from them many effective criteria on stochastic asymptotic stability, which enable us to construct the Lyapunov functions much more easily in application, were obtained. The results show that the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. In the end, application in stochastic Hopfield neural networks is given to verify our results.     

Solution of coupled system of nonlinear differential equations using homotopy analysis method

In this article, the homotopy analysis method has been applied to solve a coupled nonlinear diffusion–reaction equations. The validity of this method has been successful by applying it for these nonlinear equations. The results obtained by this method have a good agreement with one obtained by other methods. This work illustrates the validity of the homotopy analysis method for the nonlinear differential equations. The basic ideas of this approach can be widely employed to solve other strongly nonlinear problems.