Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas

2018 ◽  
Vol 21 (6) ◽  
pp. 1493-1505
Author(s):  
Takahiro Yajima ◽  
Shunya Oiwa ◽  
Kazuhito Yamasaki
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Tangent Vector ◽  
Plane Curve ◽  
Initial Time ◽  
Composite Function ◽  
Long Period ◽  
Fractional Differential ◽  
Order Tangent ◽  
Over Time

Abstract This paper discusses a construction of fractional differential geometry of curves (curvature of curve and Frenet-Serret formulas). A tangent vector of plane curve is defined by the Caputo fractional derivative. Under a simplification for the fractional derivative of composite function, a fractional expression of Frenet frame of curve is obtained. Then, the Frenet-Serret formulas and the curvature are derived for the fractional ordered frame. The different property from the ordinary theory of curve is given by the explicit expression of arclength in the fractional-order curvature. The arclength part of the curvature takes a large value around an initial time and converges to zero for a long period of time. This trend of curvature may reflect the memory effect of fractional derivative which is progressively weaken for a long period of time. Indeed, for a circle and a parabola, the curvature decreases over time. These results suggest that the basic property of fractional derivative is included in the fractional-order curvature appropriately.

scholarly journals Modeling the Dependence of Barometric Pressure with Altitude Using Caputo and Caputo–Fabrizio Fractional Derivatives

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Muath Awadalla ◽  
Yves Yameni Noupoue Yannick ◽  
Kinda Abu Asbeh
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Error Rate ◽  
Fractional Derivatives ◽  
Classical Method ◽  
Main Tool ◽  
Barometric Pressure ◽  
Proposed Model ◽  
Fractional Differential ◽  
The Relationship

This work is dedicated to the study of the relationship between altitude and barometric atmospheric pressure. There is a consistent literature on this relationship, out of which an ordinary differential equation with initial value problems is often used for modeling. Here, we proposed a new modeling technique of the relationship using Caputo and Caputo–Fabrizio fractional differential equations. First, the proposed model is proven well-defined through existence and uniqueness of its solution. Caputo–Fabrizio fractional derivative is the main tool used throughout the proof. Then, follow experimental study using real world dataset. The experiment has revealed that the Caputo fractional derivative is the most appropriate tool for fitting the model, since it has produced the smallest error rate of 1.74% corresponding to the fractional order of derivative α  = 1.005. Caputo–Fabrizio was the second best since it yielded an error rate value of 1.97% for a fractional order of derivative α  = 1.042, and finally the classical method produced an error rate of 4.36%.

Multistage Adomian Decomposition Method for Solving NLP Problems Over a Nonlinear Fractional Dynamical System

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Fırat Evirgen ◽  
Necati Özdemir
Keyword(s):  
Dynamical System ◽  
Fractional Order ◽  
Decomposition Method ◽  
Optimization Problem ◽  
Adomian Decomposition Method ◽  
Optimal Solution ◽  
Initial Time ◽  
Adomian Decomposition ◽  
Fractional Differential ◽  
Fractional Dynamical System

This paper deals with implementation of the multistage Adomian decomposition method (MADM) to solve a class of nonlinear programming (NLP) problems, which are reformulated with a nonlinear system of fractional differential equations. The multistage strategy is used to investigate the relation between an equilibrium point of the fractional order dynamical system and an optimal solution of the NLP problem. The preference of the method lies in the fact that the multistage strategy gives this relation in an arbitrary longtime interval, while the Adomian decomposition method (ADM) gives the optimal solution just only in the neighborhood of the initial time. The numerical results taken by the fractional order MADM show that these results are compatible with the solution of NLP problem rather than the ADM. Furthermore, in some cases the fractional order MADM can perform more rapid convergency to the optimal solution of optimization problem than the integer order ones.

scholarly journals p-Moment Mittag–Leffler Stability of Riemann–Liouville Fractional Differential Equations with Random Impulses

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1379
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan ◽  
Peter Kopanov
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Initial Time ◽  
Dini Derivative ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Type Of Stability

Fractional differential equations with impulses arise in modeling real world phenomena where the state changes instantaneously at some moments. Often, these instantaneous changes occur at random moments. In this situation the theory of Differential equations has to be combined with Probability theory to set up the problem correctly and to study the properties of the solutions. We study the case when the time between two consecutive moments of impulses is exponentially distributed. In connection with the application of the Riemann–Liouville fractional derivative in the equation, we define in an appropriate way both the initial condition and the impulsive conditions. We consider the case when the lower limit of the Riemann–Liouville fractional derivative is fixed at the initial time. We define the so called p-moment Mittag–Leffler stability in time of the model. In the case of integer order derivative the introduced type of stability reduces to the p–moment exponential stability. Sufficient conditions for p–moment Mittag–Leffler stability in time are obtained. The argument is based on Lyapunov functions with the help of the defined fractional Dini derivative. The main contributions of the suggested model is connected with the implementation of impulses occurring at random times and the application of the Riemann–Liouville fractional derivative of order between 0 and 1. For this model the p-moment Mittag–Leffler stability in time of the model is defined and studied by Lyapunov functions once one defines in an appropriate way their Dini fractional derivative.

scholarly journals On the Stability of Fractional Differential Equations Involving Generalized Caputo Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Minh Duc Tran ◽  
Vu Ho ◽  
Hoa Ngo Van
Keyword(s):  
Global Existence ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Caputo Derivative ◽  
Fractional Differential ◽  
The Stability ◽  
The Given

This work presents the results of the global existence for fractional differential equations involving generalized Caputo derivative with the case of the fractional order derivative α∈1,2. In addition, the Ulam–Hyers–Mittag-Leffler stability of the given problems is also established.

scholarly journals Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 216
Author(s):  
Rafail K. Gazizov ◽  
Stanislav Yu. Lukashchuk
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Anomalous Diffusion ◽  
Higher Order ◽  
Recursion Operators ◽  
Fractional Differential ◽  
Time Fractional Derivative ◽  
Linear Fractional Differential Equations ◽  
Infinite Sequences

Higher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order α∈(0,1)∪(1,2). It is proved that the equation in question has infinite sequences of nontrivial higher-order symmetries that are generated by two local recursion operators. It is also shown that some of the obtained higher-order symmetries can be rewritten as fractional-order symmetries, and corresponding fractional-order recursion operators are presented. The proposed approach for finding higher-order symmetries is applicable for a wide class of linear fractional differential equations.

scholarly journals Oscillation Behavior for a Class of Differential Equation with Fractional-Order Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Shouxian Xiang ◽  
Zhenlai Han ◽  
Ping Zhao ◽  
Ying Sun
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Generalized Riccati Transformation ◽  
Fractional Differential ◽  
Oscillation Theorems ◽  
Positive Integers

By using a generalized Riccati transformation technique and an inequality, we establish some oscillation theorems for the fractional differential equation[atpt+qtD-αxt)γ′ − b(t)f∫t∞‍(s-t)-αx(s)ds = 0, fort⩾t0>0, whereD-αxis the Liouville right-sided fractional derivative of orderα∈(0,1)ofxandγis a quotient of odd positive integers. The results in this paper extend and improve the results given in the literatures (Chen, 2012).

scholarly journals Analytic Solution for theRLElectric Circuit Model in Fractional Order

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
P. V. Shah ◽  
A. D. Patel ◽  
I. A. Salehbhai ◽  
A. K. Shukla
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Analytic Solution ◽  
Electrical Circuit ◽  
Special Cases ◽  
The Laplace Transform ◽  
Fractional Differential

This paper provides an analytic solution ofRLelectrical circuit described by a fractional differential equation of the order0<α≤1. We use the Laplace transform of the fractional derivative in the Caputo sense. Some special cases for the different source terms have also been discussed.

scholarly journals Integral presentations of the solution of a boundary value problem for impulsive fractional integro-differential equations with Riemann-Liouville derivatives

2021 ◽  
Vol 7 (2) ◽  
pp. 2973-2988
Author(s):  
Ravi Agarwal ◽  
◽  
Snezhana Hristova ◽  
Donal O'Regan ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Initial Time ◽  
Qualitative Properties ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Time Point

<abstract><p>Riemann-Liouville fractional differential equations with impulses are useful in modeling the dynamics of many real world problems. It is very important that there are good and consistent theoretical proofs and meaningful results for appropriate problems. In this paper we consider a boundary value problem for integro-differential equations with Riemann-Liouville fractional derivative of orders from $ (1, 2) $. We consider both interpretations in the literature on the presence of impulses in fractional differential equations: With fixed lower limit of the fractional derivative at the initial time point and with lower limits changeable at each impulsive time point. In both cases we set up in an appropriate way impulsive conditions which are dependent on the Riemann-Liouville fractional derivative. We establish integral presentations of the solutions in both cases and we note that these presentations are useful for furure studies of existence, stability and other qualitative properties of the solutions.</p></abstract>

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