A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives

2002 ◽  
Vol 124 (2) ◽  
pp. 321-324 ◽  
Author(s):  
Lixia Yuan ◽  
Om P. Agrawal
Keyword(s):  
Differential Equations ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Integral Formula ◽  
Past History ◽  
Analytical Techniques ◽  
Numerical Studies ◽  
History Of ◽  
Fractional Differential ◽  
Derivatives Of

This paper presents a numerical scheme for dynamic analysis of mechanical systems subjected to damping forces that are proportional to fractional derivatives of displacements. These equations appear in the modeling of frequency dependent viscoelastic damping of materials. In the scheme presented, the fractional differential equation governing the dynamics 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. In contrast to other numerical techniques, this method does not require one to store the past history of the response. Numerical studies show that the solution converges as the number of Laguerre node points increase. Further, results obtained using this scheme agree well with those obtained using analytical techniques.

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.

Initialization Issues of the Caputo Fractional Derivative

2005 ◽  
Author(s):  
B. N. Narahari Achar ◽  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Past History ◽  
The Past ◽  
History Of ◽  
Fractional Differential ◽  
Generally True

The importance of proper initialization in taking into account the history of a system whose time evolution is governed by a differential equation of fractional order, has been established by Lorenzo and Hartley, who also gave the method of properly incorporating the effect of the past (history) by means of an initialization function for the Riemann-Liouville and the Grunwald formulations of fractional calculus. The present work addresses this issue for the Caputo fractional derivative and cautions that the commonly held belief that the Caputo formulation of fractional derivatives properly accounts for the initialization effects is not generally true when applied to the solution of fractional differential equations.

Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives

2021 ◽  
Vol 24 (2) ◽  
pp. 483-508
Author(s):  
Mohammed D. Kassim ◽  
Nasser-eddine Tatar
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Logarithmic Function ◽  
Fractional Differential ◽  
Derivatives Of ◽  
L’Hôpital’S Rule

Abstract The asymptotic behaviour of solutions in an appropriate space is discussed for a fractional problem involving Hadamard left-sided fractional derivatives of different orders. Reasonable sufficient conditions are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach a logarithmic function as time goes to infinity. This generalizes and extends earlier results on integer order differential equations to the fractional case. Our approach is based on appropriate desingularization techniques and generalized versions of Gronwall-Bellman inequality. It relies also on a kind of Hadamard fractional version of l'Hopital’s rule which we prove here.

scholarly journals Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2132
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Analytic Solutions ◽  
Fractional Integrals ◽  
Function Type ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

scholarly journals Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1675
Author(s):  
Nur Amirah Zabidi ◽  
Zanariah Abdul Majid ◽  
Adem Kilicman ◽  
Faranak Rabiei
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lagrange Interpolation ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Third Order ◽  
Fractional Differential ◽  
Derivatives Of

Differential equations of fractional order are believed to be more challenging to compute compared to the integer-order differential equations due to its arbitrary properties. This study proposes a multistep method to solve fractional differential equations. The method is derived based on the concept of a third-order Adam–Bashforth numerical scheme by implementing Lagrange interpolation for fractional case, where the fractional derivatives are defined in the Caputo sense. Furthermore, the study includes a discussion on stability and convergence analysis of the method. Several numerical examples are also provided in order to validate the reliability and efficiency of the proposed method. The examples in this study cover solving linear and nonlinear fractional differential equations for the case of both single order as α∈(0,1) and higher order, α∈1,2, where α denotes the order of fractional derivatives of Dαy(t). The comparison in terms of accuracy between the proposed method and other existing methods demonstrate that the proposed method gives competitive performance as the existing methods.

scholarly journals On existence–uniqueness results for proportional fractional differential equations and incomplete gamma functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Zaid Laadjal ◽  
Thabet Abdeljawad ◽  
Fahd Jarad
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Uniqueness Of Solutions ◽  
Gamma Functions ◽  
Uniqueness Results ◽  
Incomplete Gamma Functions ◽  
Fractional Differential ◽  
Derivatives Of

AbstractIn this article, we employ the lower regularized incomplete gamma functions to demonstrate the existence and uniqueness of solutions for fractional differential equations involving nonlocal fractional derivatives (GPF derivatives) generated by proportional derivatives of the form $$ D^{\rho }= (1-\rho )+ \rho D, \quad \rho \in [0,1], $$ D ρ = ( 1 − ρ ) + ρ D , ρ ∈ [ 0 , 1 ] , where D is the ordinary differential operator.

scholarly journals Solving Higher-Order Fractional Differential Equations by the Fuzzy Generalized Conformable Derivatives

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Atimad Harir ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Higher Order ◽  
Conformable Derivatives ◽  
Fuzzy Solution ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, the generalized concept of conformable fractional derivatives of order q ∈ n , n + 1 for fuzzy functions is introduced. We presented the definition and proved properties and theorems of these derivatives. The fuzzy conformable fractional differential equations and the properties of the fuzzy solution are investigated, developed, and proved. Some examples are provided for both the new solutions.

Matrix Approach to Discretization of Ordinary and Partial Differential Equations of Arbitrary Real Order: The Matlab Toolbox

2009 ◽  
Author(s):  
Igor Podlubny ◽  
Tomas Skovranek ◽  
Blas M. Vinagre Jara
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Matrix Approach ◽  
Matlab Toolbox ◽  
Partial Differential ◽  
The Matrix ◽  
Fractional Differential ◽  
Derivatives Of ◽  
Partial Fractional Differential Equations

The method developed recently by Podlubny et al. (I. Podlubny, Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, pp. 359–386; I. Podlubny et al., Journal of Computational Physics, vol. 228, no. 8, 1 May 2009, pp. 3137–3153) makes it possible to immediately obtain the discretization of ordinary and partial differential equations by replacing the derivatives with their discrete analogs in the form of triangular strip matrices. This article presents a Matlab toolbox that implements the matrix approach and allows easy and convenient discretization of ordinary and partial differential equations of arbitrary real order. The basic use of the functions implementing the matrix approach to discretization of derivatives of arbitrary real order (so-called fractional derivatives, or fractional-order derivatives), and to solution of ordinary and partial fractional differential equations, is illustrated by examples with explanations.

Initialization of Fractional Differential Equations: Background and Theory

2007 ◽  
Author(s):  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Companion Paper ◽  
Laplace Transforms ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
Higher Order Derivative ◽  
Fractional Differential ◽  
Derivatives Of

It has been known that the initialization of fractional operators requires time-varying functions, a complicating factor. This paper simplifies the process of initialization of fractional differential equations by deriving Laplace transforms for the initialized fractional integral and derivative that generalize those for the integer-order operators. This paper provides background on past work in the area and determines the Laplace transforms for initialized fractional integrals of any order and fractional derivatives of order less than one. A companion paper in this conference extends the theory to higher order derivative operators and provides application insight.

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