STOCHASTIC ANALYSIS OF DYNAMIC SYSTEMS CONTAINING FRACTIONAL DERIVATIVES

2001 ◽  
Vol 247 (5) ◽  
pp. 927-938 ◽  
Author(s):  
O.P. AGRAWAL
Keyword(s):  
Dynamic Systems ◽  
Stochastic Analysis ◽  
Fractional Derivatives

An Improvement of a Nonclassical Numerical Method for the Computation of Fractional Derivatives

2009 ◽  
Vol 131 (1) ◽  
Author(s):  
Kai Diethelm
Keyword(s):  
Numerical Calculation ◽  
Numerical Method ◽  
Dynamic Systems ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Differential Operators ◽  
Standard Methods ◽  
Efficient Approach

Standard methods for the numerical calculation of fractional derivatives can be slow and memory consuming due to the nonlocality of the differential operators. Yuan and Agrawal (2002, “A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives,” ASME J. Vibr. Acoust., 124, pp. 321–324) have proposed a more efficient approach for operators whose order is between 0 and 1 that differs substantially from the traditional concepts. It seems, however, that the accuracy of the results can be poor. We modify the approach, adapting it better to the properties of the problem, and show that this leads to a significantly improved quality. Our idea also works for operators of order greater than 1.

A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives

1998 ◽  
Author(s):  
Lixia Yuan ◽  
Om P. Agrawal
Keyword(s):  
Differential Equations ◽  
Dynamic Systems ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Integral Formula ◽  
Analytical Technique ◽  
First Order ◽  
Numerical Studies ◽  
Fractional Differential ◽  
Derivatives Of

Abstract This paper presents a numerical scheme for dynamic analysis of mechanical systems subjected to damping forces which are proportional to fractional derivatives of displacements. In this scheme, a fractional differential equation governing the dynamic of a system is transformed into a set of differential equations with no fractional derivative terms. Using Laguerre integral formula, this set is converted to a set of first order ordinary differential equations, which are integrated using a numerical scheme to obtain the response of the system. Numerical studies show that the solution converges as the number of Laguerre node points are increased. Further, results obtained using this scheme agree well with those obtained using an analytical technique.

scholarly journals On Dynamic Systems in the Frame of Singular Function Dependent Kernel Fractional Derivatives

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 946 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Fadila Madjidi ◽  
Fahd Jarad ◽  
Ndolane Sene
Keyword(s):  
Laplace Transform ◽  
Dynamic Systems ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Singular Function ◽  
Existence And Uniqueness Theorem ◽  
Lyapunov Direct Method ◽  
Caputo Fractional Derivatives ◽  
Special Case

In this article, we discuss the existence and uniqueness theorem for differential equations in the frame of Caputo fractional derivatives with a singular function dependent kernel. We discuss the Mittag-Leffler bounds of these solutions. Using successive approximation, we find a formula for the solution of a special case. Then, using a modified Laplace transform and the Lyapunov direct method, we prove the Mittag-Leffler stability of the considered system.

scholarly journals On Nonnegative Solutions of Fractionalq-Linear Time-Varying Dynamic Systems with Delayed Dynamics

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
M. De la Sen
Keyword(s):  
Dynamic Systems ◽  
Fractional Derivatives ◽  
Linear Time ◽  
Asymptotic Properties ◽  
Time Varying ◽  
Caputo Fractional Derivatives ◽  
Nonnegative Solutions ◽  
Time Varying Systems ◽  
Fractional Differential ◽  
The Stability

This paper is devoted to the investigation of nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic linear time-varying systems involving delayed dynamics with delays. The dynamic systems are described based onq-calculus and Caputo fractional derivatives on any order.

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