scholarly journals Fractional Relaxation and Fractional Oscillation Models Involving Erdélyi-Kober Integrals

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Moreno Concezzi ◽  
Roberto Garra ◽  
Renato Spigler
Keyword(s):  
Differential Equations ◽  
Numerical Approach ◽  
Numerical Results ◽  
Cauchy Type ◽  
Time Varying ◽  
Varying Coefficients ◽  
Fractional Relaxation ◽  
Time Varying Coefficients ◽  
Damped Oscillator

AbstractWe consider fractional relaxation and fractional oscillation equations involving Erdélyi-Kober integrals. In terms of the Riemann-Liouville integrals, the equations we analyze can be understood as equations with time-varying coefficients. Replacing the Riemann-Liouville integrals with Erdélyi-Kober-type integrals in certain fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchy-type problems can be solved numerically, and, in some cases analytically, in terms of the Saigo-Kilbas Mittag-Leffler functions. The numerical results are obtained by a treatment similar to that developed by K. Diethelm and N.J. Ford to solve the Bagley-Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.

Computing Numerical Solutions of Delayed Fractional Differential Equations With Time Varying Coefficients

2014 ◽  
Vol 10 (1) ◽  
Author(s):  
Venkatesh Suresh Deshmukh
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Metal Cutting ◽  
Numerical Solutions ◽  
Viscoelastic Damping ◽  
Time Varying ◽  
Varying Coefficients ◽  
Fractional Differential ◽  
Time Varying Coefficients ◽  
Cutting Processes

Fractional differential equations with time varying coefficients and delay are encountered in the analysis of models of metal cutting processes such as milling and drilling with viscoelastic damping elements. Viscoelastic damping is modeled as a fractional derivative. In the present paper, delayed fractional differential equations with bounded time varying coefficients in four different forms are analyzed using series solution and Chebyshev spectral collocation. A fractional differential equation with a known exact solution is then solved by the methodology presented in the paper. The agreement between the two is found to be excellent in terms of point-wise error in the trajectories. Solutions to the described fractional differential equations are computed next in state space and second order forms.

Solutions of Delayed Partial Differential Equations With Space-Time Varying Coefficients

2011 ◽  
Vol 7 (2) ◽  
Author(s):  
Venkatesh Deshmukh
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Initial Boundary ◽  
Time Varying ◽  
Initial Boundary Value Problems ◽  
Weakly Nonlinear ◽  
Varying Coefficients ◽  
Time Varying Coefficients ◽  
Initial Boundary Value

A constructive algorithm using Chebyshev spectral collocation is proposed for computing trustworthy approximate solutions of linear and weakly nonlinear delayed partial differential equations or initial boundary value problems, with continuous and bounded coefficients. The boundary conditions are assumed to be Dirichlet. The solution of linear problems is obtained at Chebyshev grid points in space and a given interval of time. The algorithm is then extended to systems with weak nonlinearities using perturbation series, which yields nonhomogeneous initial boundary value problems without delay. The proposed methodology is illustrated using examples of linear and weakly nonlinear heat and wave equations with bounded continuous space-time varying coefficients.

Dynamics of Nonautonomous Ordinary Differential Equations with Quasi-Periodic Coefficients

2017 ◽  
Vol 27 (06) ◽  
pp. 1750092 ◽  
Author(s):  
Xu Zhang
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
Nonlinear Term ◽  
Time Varying ◽  
Periodic Functions ◽  
Periodic Coefficients ◽  
Varying Coefficients ◽  
Nonautonomous Systems ◽  
Time Varying Coefficients

We investigate the dynamics of two types of nonautonomous ordinary differential equations with quasi-periodic time-varying coefficients and nonlinear terms. The vector fields for the nonautonomous systems are written as [Formula: see text], [Formula: see text], where [Formula: see text] is the spacial part and [Formula: see text] is the time-varying part, and [Formula: see text] and [Formula: see text] are real parameters. The first type has a polynomial as the nonlinear term, another type has a continuous periodic function as the nonlinear term. The polynomials and periodic functions have simple zeros. Several examples with numerical experiments are given. It is found by numerical calculation that there might exist only one attractor for the systems with polynomials as nonlinear terms and [Formula: see text], and there might exist infinitely many attractors for systems with periodic functions as nonlinear terms and [Formula: see text]. For [Formula: see text] sufficiently small, the parameter regions for [Formula: see text] are roughly divided into three parts: the spacial region ([Formula: see text]), the balance region ([Formula: see text]), and the time-varying region ([Formula: see text]); (i) for [Formula: see text], the orbits approach some planes depending on the zeros of the polynomials or the periodic functions; (ii) for [Formula: see text], there exist attractors with the number no less than the number of zeros of the polynomials or the periodic functions, implying the existence of infinitely many attractors for systems with periodic functions as nonlinear terms; (iii) for [Formula: see text], the orbits wind around some region depending on the choice of the initial position. The shape of the attractors might be strange or regular for different parameters, and we obtain the existence of ball-like (regular) attractors, two-wings (strange) attractors, and other attractors with different shapes. The Lyapunov exponents are negative. These results reveal an intrinsic relationship between the existence of attractors (or strange dynamics) and the parameters [Formula: see text] and [Formula: see text] for nonautonomous systems with quasi-periodic coefficients. These results will be very useful in the understanding of the dynamics of general nonautonomous systems, nonautonomous control theory and other related fields.

Linear Time-Varying Dynamic Systems Optimization via Higher-Order Method Using Shifted Chebyshev’s Polynomials

1999 ◽  
Vol 121 (2) ◽  
pp. 258-261 ◽  
Author(s):  
Xiaochun Xu ◽  
Sunil K. Agrawal
Keyword(s):  
Differential Equations ◽  
Dynamic Systems ◽  
Linear Time ◽  
Optimal Solution ◽  
Higher Order ◽  
Time Varying ◽  
Varying Coefficients ◽  
Higher Order Method ◽  
Time Varying Coefficients ◽  
Higher Order Differential Equations

For optimization of classes of linear time-varying dynamic systems with n states and m control inputs, a new higher-order procedure was presented by the authors that does not use Lagrange multipliers. In this new procedure, the optimal solution was shown to satisfy m 2p-order differential equations with time-varying coefficients. These differential equations were solved using weighted residual methods. Even though solution of the optimization problem using this procedure was demonstrated to be computation efficient, shifted Chebyshev’s polynomials are used in the paper to solve the higher-order differential equations. This further reduces the computations and makes this algorithm more appropriate for real-time implementation.

scholarly journals The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

Eng ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 99-125
Author(s):  
Edward W. Kamen
Keyword(s):  
Discrete Time ◽  
Initial Conditions ◽  
Linear Time ◽  
Order Term ◽  
Initial Time ◽  
Time Varying ◽  
Varying Coefficients ◽  
Time Signals ◽  
Time Varying Systems ◽  
Time Varying Coefficients

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

scholarly journals Estimating latent group structure in time-varying coefficient panel data models

2019 ◽  
Author(s):  
Jia Chen
Keyword(s):  
Panel Data ◽  
Data Models ◽  
Panel Data Models ◽  
Time Varying ◽  
Clustering Method ◽  
Varying Coefficient ◽  
Varying Coefficients ◽  
Time Varying Coefficients ◽  
Latent Group ◽  
Group Structures

Summary This paper studies the estimation of latent group structures in heterogeneous time-varying coefficient panel data models. While allowing the coefficient functions to vary over cross-sections provides a good way to model cross-sectional heterogeneity, it reduces the degree of freedom and leads to poor estimation accuracy when the time-series length is short. On the other hand, in a lot of empirical studies, it is not uncommon to find that heterogeneous coefficients exhibit group structures where coefficients belonging to the same group are similar or identical. This paper aims to provide an easy and straightforward approach for estimating the underlying latent groups. This approach is based on the hierarchical agglomerative clustering (HAC) of kernel estimates of the heterogeneous time-varying coefficients when the number of groups is known. We establish the consistency of this clustering method and also propose a generalised information criterion for estimating the number of groups when it is unknown. Simulation studies are carried out to examine the finite-sample properties of the proposed clustering method as well as the post-clustering estimation of the group-specific time-varying coefficients. The simulation results show that our methods give comparable performance to the penalised-sieve-estimation-based classifier-LASSO approach by Su et al. (2018), but are computationally easier. An application to a panel study of economic growth is also provided.

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