scholarly journals Applications of fractional calculus in equiaffine geometry: plane curves with fractional order

2021 ◽  
Author(s):  
Muhittin Evren Aydin ◽  
Adela Mihai ◽  
Asif Yokus
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Plane Curves

scholarly journals APPLICATIONS OF FRACTIONAL CALCULUS IN EQUIAFFINE GEOMETRY: PLANE CURVES WITH FRACTIONAL ORDER

2020 ◽  
Author(s):  
Muhittin Aydin ◽  
Adela Mihai ◽  
Asıf Yokus
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Fundamental Theorem ◽  
Plane Curve ◽  
Plane Curves

In this paper, we introduce the notions of equiaffine arclength and curvature with fractional order for a plane curve and compare them with the standard ones. In terms of the equiaffine curvature with fractional order we obtain an equiaffine Frenet formula and then construct an analogue of the Fundamental Theorem. The plane curves of constant equiaffine curvature with fractional-order are classified. Several examples are also illustrated.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

The bouncing ball and the Grünwald-Letnikov definition of fractional derivative

2021 ◽  
Vol 24 (4) ◽  
pp. 1003-1014
Author(s):  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Restitution Coefficient ◽  
Classical Physics ◽  
Bouncing Ball ◽  
Mechanical Experiment ◽  
Fractional Models ◽  
Definition Of

Abstract This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

scholarly journals Nonlinear Viscoelastic-Plastic Creep Model of Rock Based on Fractional Calculus

2022 ◽  
Vol 2022 ◽  
pp. 1-7
Author(s):  
Erjian Wei ◽  
Bin Hu ◽  
Jing Li ◽  
Kai Cui ◽  
Zhen Zhang ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Engineering Application ◽  
Creep Model ◽  
Nonlinear Viscoelastic ◽  
Rock Creep ◽  
Order Change ◽  
Creep Constitutive Model ◽  
Plastic Creep

A rock creep constitutive model is the core content of rock rheological mechanics theory and is of great significance for studying the long-term stability of engineering. Most of the creep models constructed in previous studies have complex types and many parameters. Based on fractional calculus theory, this paper explores the creep curve characteristics of the creep elements with the fractional order change, constructs a nonlinear viscoelastic-plastic creep model of rock based on fractional calculus, and deduces the creep constitutive equation. By using a user-defined function fitting tool of the Origin software and the Levenberg–Marquardt optimization algorithm, the creep test data are fitted and compared. The fitting curve is in good agreement with the experimental data, which shows the rationality and applicability of the proposed nonlinear viscoelastic-plastic creep model. Through sensitivity analysis of the fractional order β2 and viscoelastic coefficient ξ2, the influence of these creep parameters on rock creep is clarified. The research results show that the nonlinear viscoelastic-plastic creep model of rock based on fractional calculus constructed in this paper can well describe the creep characteristics of rock, and this model has certain theoretical significance and engineering application value for long-term engineering stability research.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

scholarly journals The Asymptotic Behaviours of a Class of Neutral Delay Fractional-Order Pantograph Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Baojun Miao ◽  
Xuechen Li
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Asymptotic Estimates ◽  
Stability Of Solutions ◽  
Neutral Delay ◽  
Estimates Of Solutions ◽  
Fractional Delay ◽  
Asymptotic Behaviours

By using fractional calculus and the summation by parts formula in this paper, the asymptotic behaviours of solutions of nonlinear neutral fractional delay pantograph equations with continuous arguments are investigated. The asymptotic estimates of solutions for the equation are obtained, which may imply asymptotic stability of solutions. In the end, a particular case is provided to illustrate the main result and the speed of the convergence of the obtained solutions.

Fractional Order Dynamics in the Trajectory Planning of Redundant and Hyper-Redundant Manipulators

2003 ◽  
Author(s):  
Fernando B. M. Duarte ◽  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Trajectory Planning ◽  
Free Space ◽  
Trajectory Optimization ◽  
Generalized Inverse ◽  
Resolution Of Singularities ◽  
Control Algorithms ◽  
Redundant Manipulators ◽  
Kinematic Control

Redundant manipulators have some advantages when compared with classical arms because they allow the trajectory optimization, both on the free space and on the presence of obstacles, and the resolution of singularities. For this type of arms the proposed kinematic control algorithms adopt generalized inverse matrices but, in general, the corresponding trajectory planning schemes show important limitations. Motivated by these problems this paper studies the pseudoinverse-based trajectory planning algorithms, using the theory of fractional calculus.

scholarly journals Stability of Fractional Order Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Margarita Rivero ◽  
Sergei V. Rogosin ◽  
José A. Tenreiro Machado ◽  
Juan J. Trujillo
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
State Of The Art ◽  
Point Of View ◽  
Fractional Order Systems ◽  
Short Period ◽  
Systems And Control ◽  
Different Types ◽  
The Stability ◽  
And Control

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled.

scholarly journals The Fractional Complex Step Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Rabha W. Ibrahim ◽  
Hamid A. Jalab
Keyword(s):  
Differential Operator ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Liouville Operator ◽  
Strict Sense ◽  
Step Method ◽  
Test Function

It is well known that the complex step method is a tool that calculates derivatives by imposing a complex step in a strict sense. We extended the method by employing the fractional calculus differential operator in this paper. The fractional calculus can be taken in the sense of the Caputo operator, Riemann-Liouville operator, and so forth. Furthermore, we derived several approximations for computing the fractional order derivatives. Stability of the generalized fractional complex step approximations is demonstrated for an analytic test function.

Formulation of a State Equation Including Fractional-Order State-Vectors

2007 ◽  
Author(s):  
Masaharu Kuroda
Keyword(s):  
Fractional Calculus ◽  
Control Theory ◽  
Fractional Order ◽  
State Equation ◽  
The State ◽  
Space Representation ◽  
Fractional Dynamics ◽  
Modern Control Theory ◽  
Order State ◽  
Modern Control

In recent years, applications of fractional calculus have flourished in various science and engineering fields. Particularly in engineering, control engineering appears to be expanding aggressively in its applications. Exemplary are the CRONE controller and the PIλDμ controller, which is categorizable into applications of fractional calculus in classical control theory. A state equation can be called the foundation of modern control theory. However, the relationship between fractional derivatives and the state equation has not been examined sufficiently. Consequently, a systematic procedure referred to by every researcher on the fractional-calculus side or control-theory side has not yet been established. For this study, therefore, involvement of fractional-order derivatives into a state equation is demonstrated here for ready comprehension by researchers. First, the procedures are explained generally; then the technique to incorporate the fractional-order state-vector into a conventional state equation is given as an example of the applications. The state-space representation in this study is useful not only for modeling a controlled system with fractional dynamics, but also for design and implementation of a controller to control fractional-order states. After we complete installation of the basic parts, we can apply the benefits of modern control theory, including robust control theories such as H-infinity and μ-analysis and synthesis in their integrities, to this fractional-order state-equation.

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