A free fractional viscous oscillator as a forced standard damped vibration

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
Giuseppe Devillanova ◽  
Giuseppe Carlo Marano
Keyword(s):  
Fractional Derivative ◽  
Initial Conditions ◽  
Free Response ◽  
Resolution Method ◽  
Single Degree Of Freedom ◽  
Caputo’S Fractional Derivative ◽  
First Order ◽  
Fractional Differential ◽  
Free Dynamic ◽  
Fractional Displacement

AbstractThis paper contains a survey on one of the mathematical approaches used to solve a fractional differential equation whose solution gives the free dynamic response of viscoelastic single degree of freedom systems (viscosity is actually modelled by a fractional displacement derivative instead of first order one). The paper shall deal with Caputo’s fractional derivative since its Laplace Transform (on which the resolution method is based) only depends by lower integer order (and therefore measurable and physically meaningful) derivatives given as initial conditions. The paper provides a deep mathematical analysis of the properties of the solution expressed in terms of the mechanical parameters meaning. Additionally, important physical implications are reported exhibiting a richer dynamic behavior if compared to the standard damping case (velocity linear dependence). Some important consequences in the use of Caputo’s fractional derivative are reported, and some limitations to possible viscous parameters values are obtained. Finally, it is shown that free response of fractional derivative equation solution is mathematically equivalent to a suitably forced solution of the integer model.

scholarly journals A Study of Fractional Differential Equation with a Positive Constant Coefficient via Hilfer Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Hoa Ngo Van ◽  
Vu Ho
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Positive Constant ◽  
Constant Coefficient ◽  
Initial Conditions ◽  
Hilfer Fractional Derivative ◽  
Fractional Differential ◽  
The Fixed Point Theorem ◽  
Rassias Stability

The aim of the paper is to consider the existence and uniqueness of solution of the fractional differential equation with a positive constant coefficient under Hilfer fractional derivative by using the fixed-point theorem. We also prove the bounded and continuous dependence on the initial conditions of solution. Besides, Hyers–Ulam stability and Hyers–Ulam–Rassias stability are discussed. Finally, we provide an example to demonstrate our main results.

scholarly journals Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2231
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Power Law ◽  
Dynamic Processes ◽  
Nonlinear Fractional Differential Equation ◽  
First Order ◽  
Fractional Differential ◽  
Special Case

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

Functions of bounded fractional differential variation – A new concept

2016 ◽  
Vol 23 (3) ◽  
pp. 417-427 ◽  
Author(s):  
Jyotindra C. Prajapati ◽  
Krunal B. Kachhia
Keyword(s):  
Fractional Derivative ◽  
Order Derivative ◽  
Normed Algebra ◽  
Classical Analysis ◽  
First Order ◽  
Fractional Differential

AbstractThe idea of functions of bounded differential variation was introduced by Bhatt, Dabhi and Kachhia in [2]. In the present paper, we introduce functions of bounded fractional differential variation using the Caputo-type fractional derivative instead of the commonly used first-order derivative. Various properties and relation with some known results of classical analysis are also studied. We prove that the space ${\mathrm{BFDV}[a,b]}$ of all functions of bounded fractional differential variation on ${[a,b]}$ is a normed algebra under certain type of norms.

scholarly journals The Extremal Solution To Conformable Fractional Differential Equations Involving Integral Boundary Condition

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 186 ◽  
Author(s):  
Shuman Meng ◽  
Yujun Cui
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Integral Boundary Conditions ◽  
Lower Solution ◽  
Stieltjes Integral ◽  
Integral Boundary ◽  
First Order ◽  
Monotone Iterative ◽  
Fractional Differential

In this article, by using the monotone iterative technique coupled with the method of upper and lower solution, we obtain the existence of extremal iteration solutions to conformable fractional differential equations involving Riemann-Stieltjes integral boundary conditions. At the same time, the comparison principle of solving such problems is investigated. Finally, an example is given to illustrate our main results. It should be noted that the conformal fractional derivative is essentially a modified version of the first-order derivative. Our results show that such known results can be translated and stated in the setting of the so-called conformal fractional derivative.

scholarly journals Existence and Integral Representation of Scalar Riemann-Liouville Fractional Differential Equations with Delays and Impulses

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 607
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Lower Limit ◽  
Fractional Differential Equations ◽  
Initial Conditions ◽  
Integral Representations ◽  
Time Integral ◽  
Fractional Differential ◽  
Set Up ◽  
Banach Principle

Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and impulses are studied and initial conditions and impulsive conditions are set up in an appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence of the delay in the equation. We study the case of a fixed lower limit of the fractional derivative and the case of a changeable lower limit at each impulsive time. Integral representations of the solutions in all considered cases are obtained. Existence results on finite time intervals are proved using the Banach principle.

scholarly journals Analytic Approach for Solving System of Fractional Differential Equations

2021 ◽  
Vol 32 (1) ◽  
pp. 14
Author(s):  
Nabaa N Hasan ◽  
Zainab John
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Initial Conditions ◽  
Analytic Approach ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential

In this paper, Sumudu transformation (ST) of Caputo fractional derivative formulae are derived for linear fractional differential systems. This formula is applied with Mittage-Leffler function for certain homogenous and nonhomogenous fractional differential systems with nonzero initial conditions. Stability is discussed by means of the system's distinctive equation.

A uniqueness result for a fractional differential equation

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Rui Ferreira
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Uniqueness Result ◽  
Caputo’S Fractional Derivative ◽  
Fractional Differential

AbstractWe obtain a uniqueness result for a differential equation depending on Caputo’s fractional derivative. An example is included to illustrate our result.

scholarly journals The fractional nonlinear $${\mathcal{PT}}$$ dimer

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Mario I. Molina
Keyword(s):  
Fractional Derivative ◽  
Laplace Transformation ◽  
Initial Conditions ◽  
Critical Value ◽  
Order Derivative ◽  
Marked Contrast ◽  
Transformation Technique ◽  
First Order ◽  
Main Effect ◽  
Gain Loss

AbstractWe examine a fractional discrete nonlinear Schrodinger dimer, where the usual first-order derivative in the time evolution is replaced by a non integer-order derivative. The dimer is nonlinear (Kerr) and $${\mathcal{{PT}}}$$ PT -symmetric, and for localized initial conditions we examine the exchange dynamics between both sites. By means of the Laplace transformation technique, the linear $${{\mathcal{{PT}}}}$$ PT dimer is solved in closed form in terms of Mittag–Leffler functions, while for the nonlinear regime, we resort to numerical computations using the direct explicit Grunwald algorithm. In general, we see that the main effect of the fractional derivative is to produce a monotonically decreasing time envelope for the amplitude of the oscillatory exchange. In the presence of $${\mathcal{{PT}}}$$ PT symmetry, the oscillations experience some amplification for gain/loss values below some threshold, while beyond threshold, the amplitudes of both sites grow unbounded. The presence of nonlinearity can arrest the unbounded growth and lead to a selftrapped state. The trapped fraction decreases as the nonlinearity is increased past a critical value, in marked contrast with the standard (non-fractional) case.

scholarly journals Practical stability for Riemann–Liouville delay fractional differential equations

2021 ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Practical Stability ◽  
Razumikhin Technique ◽  
Fractional Differential ◽  
Derivatives Of

AbstractIn this paper, we study a system of nonlinear Riemann–Liouville fractional differential equations with delays. First, we define in an appropriate way initial conditions which are deeply connected with the fractional derivative used. We introduce an appropriate generalization of practical stability which we call practical stability in time. Several sufficient conditions for practical stability in time are obtained using Lyapunov functions and the modified Razumikhin technique. Two types of derivatives of Lyapunov functions are used. Some examples are given to illustrate the introduced definitions and results.

Symmetry properties of fractional order transport equations

2012 ◽  
Vol 9 (1) ◽  
pp. 59-64
Author(s):  
R.K. Gazizov ◽  
A.A. Kasatkin ◽  
S.Yu. Lukashchuk
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Group ◽  
Initial Conditions ◽  
Group Analysis ◽  
Equivalence Transformation ◽  
Point Change ◽  
Infinitesimal Operator ◽  
Symmetry Properties ◽  
Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

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