Control synthesis by full state vector in systems with fractional-order derivatives using Caputo-Fabrizio operator

2020 ◽  
Vol 8 (1) ◽  
pp. 106-115
Author(s):  
A. O. Lozynskyy ◽  
◽  
O. Yu. Lozynskyy ◽  
L. V. Kasha ◽  
◽  
...  
Keyword(s):  
Characteristic Polynomial ◽  
Fundamental Matrix ◽  
State Vector ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Control Synthesis ◽  
Multiple Roots ◽  
System Operation ◽  
Full State ◽  
Control System Synthesis

In the paper, the control system synthesis by means of the full state vector is considered when using fractional derivatives in the description of this system. To conduct research in the synthesized system with fractional derivatives in the Caputo--Fabrizio representation, a fundamental matrix of the system is formed, which also allows us to analyze the influence of initial conditions on the processes within the system. In particular, the finding of the fundamental matrix of the system in the case of multiple roots of a characteristic polynomial, which are obtained by transforming the synthesized system to the binomial form, is demonstrated. The influence of the fractional derivative index and the location of the roots of the characteristic polynomial transformed to the binomial form on the system operation is analyzed.

Initialization of Riemann-Liouville and Caputo Fractional Derivatives

2011 ◽  
Author(s):  
Jean-Claude Trigeassou ◽  
Nezha Maamri ◽  
Alain Oustaloup
Keyword(s):  
Numerical Simulations ◽  
State Vector ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Fractional Integration ◽  
Initial State ◽  
Caputo Fractional Derivatives ◽  
Infinite Dimensional ◽  
Fractional Integrator ◽  
Initial Conditions Problem

Riemann-Liouville and Caputo fractional derivatives are fundamentally related to fractional integration operators. Consequently, the initial conditions of fractional derivatives are the frequency distributed and infinite dimensional state vector of fractional integrators. The paper is dedicated to the estimation of these initial conditions and to the validation of the initialization problem based on this distributed state vector. Numerical simulations applied to Riemann-Liouville and Caputo derivatives demonstrate that the initial conditions problem can be solved thanks to the estimation of the initial state vector of the fractional integrator.

scholarly journals Necessary conditions for local and global existence to a reaction-diffusion system with fractional derivatives

2006 ◽  
Vol 2006 ◽  
pp. 1-8 ◽  
Author(s):  
Kamel Haouam ◽  
Mourad Sfaxi
Keyword(s):  
Global Existence ◽  
Fractional Derivatives ◽  
Necessary Conditions ◽  
Initial Conditions ◽  
Reaction Diffusion ◽  
Single Equation ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Existence Of A Solution

We give some necessary conditions for local and global existence of a solution to reaction-diffusion system of type (FDS) with temporal and spacial fractional derivatives. As in the case of single equation of type (STFE) studied by M. Kirane et al. (2005), we prove that these conditions depend on the behavior of initial conditions for large|x|.

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

A robust technique based solution of time-fractional seventh-order Sawada–Kotera and Lax’s KdV equations

2021 ◽  
pp. 2150265
Author(s):  
Rajarama Mohan Jena ◽  
Snehashish Chakraverty ◽  
Dumitru Baleanu ◽  
Waleed Adel ◽  
Hadi Rezazadeh
Keyword(s):  
Convergence Analysis ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Absolute Error ◽  
Transform Method ◽  
Cpu Time ◽  
Kdv Equations ◽  
Differential Transform ◽  
Seventh Order

In this paper, the fractional reduced differential transform method (FRDTM) is used to obtain the series solution of time-fractional seventh-order Sawada–Kotera (SSK) and Lax’s KdV (LKdV) equations under initial conditions (ICs). Here, the fractional derivatives are considered in the Caputo sense. The results obtained are contrasted with other previous techniques for a specific case, [Formula: see text] revealing that the presented solutions agree with the existing solutions. Further, convergence analysis of the present results with an increasing number of terms of the solution and absolute error has also been studied. The behavior of the FRDTM solution and the effects on different values [Formula: see text] are illustrated graphically. Also, CPU-time taken to obtain the solutions of the title problems using FRDTM has been demonstrated.

scholarly journals Mathematical Modeling of Linear Fractional Oscillators

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1879 ◽  
Author(s):  
Roman Parovik
Keyword(s):  
Cauchy Problem ◽  
Fractional Derivatives ◽  
Model Equation ◽  
Initial Conditions ◽  
Memory Function ◽  
Analytical Formula ◽  
Forced Oscillations ◽  
Caputo Fractional Derivatives ◽  
Specific Test ◽  
The Cauchy Problem

In this work, based on Newton’s second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov–Caputo fractional derivatives is carried out. For the resulting model equation, local initial conditions are set (the Cauchy problem). Numerical methods for solving the Cauchy problem using an explicit non-local finite-difference scheme (ENFDS) and the Adams–Bashforth–Moulton (ABM) method are considered. An analysis of the errors of the methods is carried out on specific test examples. It is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method. Forced oscillations of linear fractional oscillators (LFO) are investigated. Using the ABM method, the amplitude–frequency characteristics (AFC) were constructed, which were compared with the AFC obtained by the analytical formula. The Q-factor of the LFO is investigated. It is shown that the orders of fractional derivatives are responsible for the intensity of energy dissipation in fractional vibrational systems. Specific mathematical models of LFOs are considered: a fractional analogue of the harmonic oscillator, fractional oscillators of Mathieu and Airy. Oscillograms and phase trajectories were constructed using the ABM method for various values of the parameters included in the model equation. The interpretation of the simulation results is carried out.

scholarly journals On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1387 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Characteristic Polynomial ◽  
Initial Conditions ◽  
Pell Numbers ◽  
Families Of Subsets

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about the roots of the characteristic polynomial of this sequence, but we will consider general initial conditions. Since there are currently several types of generalizations of the Pell sequence, it is very difficult for anyone to realize what type of sequence an author really means. Thus, we will call this sequence the generalized k-distance Tribonacci sequence (Tn(k))n≥0.

scholarly journals Novel Fractional Operators with Three Orders and Power-Law, Exponential Decay and Mittag–Leffler Memories Involving the Truncated M-Derivative

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 626 ◽  
Author(s):  
Jesús Emmanuel Solís-Pérez ◽  
José Francisco Gómez-Aguilar
Keyword(s):  
Numerical Analysis ◽  
Exponential Decay ◽  
Power Law ◽  
Chaotic Attractor ◽  
Lagrange Interpolation ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Fractional Operators ◽  
Conformable Derivative ◽  
Sensitive Dependence

In this research, novel M-truncated fractional derivatives with three orders have been proposed. These operators involve truncated Mittag–Leffler function to generalize the Khalil conformable derivative as well as the M-derivative. The new operators proposed are the convolution of truncated M-derivative with a power law, exponential decay and the complete Mittag–Leffler function. Numerical schemes based on Lagrange interpolation to predict chaotic behaviors of Rucklidge, Shimizu–Morioka and a hybrid strange attractors were considered. Additionally, numerical analysis based on 0–1 test and sensitive dependence on initial conditions were carried out to verify and show the existence of chaos in the chaotic attractor. These results showed that these novel operators involving three orders, two for the truncated M-derivative and one for the fractional term, depict complex chaotic behaviors.

scholarly journals Response of Fractionally Damped Beams with General Boundary Conditions Subjected to Moving Loads

2012 ◽  
Vol 19 (3) ◽  
pp. 333-347 ◽  
Author(s):  
R. Abu-Mallouh ◽  
I. Abu-Alshaikh ◽  
H.S. Zibdeh ◽  
Khaled Ramadan
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Damping Characteristics ◽  
The Laplace Transform ◽  
The Right

This paper presents the transverse vibration of Bernoulli-Euler homogeneous isotropic damped beams with general boundary conditions. The beams are assumed to be subjected to a load moving at a uniform velocity. The damping characteristics of the beams are described in terms of fractional derivatives of arbitrary orders. In the analysis where initial conditions are assumed to be homogeneous, the Laplace transform cooperates with the decomposition method to obtain the analytical solution of the investigated problems. Subsequently, curves are plotted to show the dynamic response of different beams under different sets of parameters including different orders of fractional derivatives. The curves reveal that the dynamic response increases as the order of fractional derivative increases. Furthermore, as the order of the fractional derivative increases the peak of the dynamic deflection shifts to the right, this yields that the smaller the order of the fractional derivative, the more oscillations the beam suffers. The results obtained in this paper closely match the results of papers in the literature review.

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