scholarly journals A new general fractional derivative Goldstein-Kac-type telegraph equation

2020 ◽  
Vol 24 (6 Part B) ◽  
pp. 3893-3898
Author(s):  
Ping Cui ◽  
Yi-Ying Feng ◽  
Jian-Gen Liu ◽  
Lu-Lu Geng
Keyword(s):  
Fractional Order ◽  
Telegraph Equation ◽  
Order Derivative ◽  
Mathematical Tool ◽  
Fractional Order Derivative ◽  
Derivative Formula ◽  
The Laplace Transform ◽  
The One ◽  
First Time ◽  
Derivatives Of

In this paper, we consider the Riemann-Liouville-type general fractional derivatives of the non-singular kernel of the one-parametric Lorenzo-Hartley function. A new general fractional-order-derivative Goldstein-Kac-type telegraph equation is proposed for the first time. The analytical solution of the considered model with the graphs is obtained with the aid of the Laplace transform. The general fractional-order-derivative formula is as a new mathematical tool proposed to model the anomalous behaviors in complex and power-law phenomena.

Digital Fractional Order Savitzky-Golay Differentiator and Its Application

2011 ◽  
Author(s):  
Dali Chen ◽  
Dingyu Xue ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Least Square ◽  
One Dimension ◽  
Order Derivative ◽  
Noisy Signal ◽  
Fractional Order Derivative ◽  
Least Square Fitting ◽  
Image Enhancing ◽  
The One ◽  
Derivatives Of

Firstly the one-dimension digital fractional order Savitzky-Golay differentiator (1-D DFOSGD), which generalizes the Savitzky-Golay filter from the integer order to the fractional order, is proposed to estimate the fractional order derivative of the noisy signal. The polynomial least square fitting technology and the Riemann-Liouville fractional order derivative definition are used to ensure robust and accuracy. Experiments demonstrate that 1-D DFOSGD can estimate the fractional order derivatives of both ideal signal and noisy signal accurately. Secondly, the two-dimension DFOSGD is obtained from 1-D DFOSGD by defining a group of direction operators, and a new image enhancing method based on 2-D DFOSGD is presented. Experiments demonstrate that 2-D DFOSGD has very good performance on image enhancement.

Geometric interpretation of fractional-order derivative

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Geometry ◽  
Fractional Order ◽  
Infinite Series ◽  
Fractional Derivatives ◽  
Geometric Interpretation ◽  
Order Derivative ◽  
Jet Bundles ◽  
Fractional Order Derivative ◽  
Integer Order ◽  
Derivatives Of

AbstractA new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

scholarly journals General fractional calculus operators containing the generalized Mittag-Leffler functions applied to anomalous relaxation

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 317-326 ◽  
Author(s):  
Xiaojun Yang
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Integral Transforms ◽  
Kernel Functions ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Laplace Integral ◽  
Fractional Calculus Operators ◽  
Anomalous Relaxation ◽  
First Time

In this paper, we address a family of the general fractional calculus operators of Wiman and Prabhakar types for the first time. The general Mittag-Leffler function to structure the kernel functions of the fractional order derivative operators and their Laplace integral transforms are considered in detail. The formulations are as the mathematical tools proposed to investigate the anomalous relaxation.

scholarly journals A comparison study of steady-state vibrations with single fractional-order and distributed-order derivatives

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 809-818 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Cui-Ping Cheng ◽  
Lian Chen
Keyword(s):  
Steady State ◽  
Weight Function ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Order Derivative ◽  
Similar Response ◽  
Driving Frequency ◽  
Fractional Order Derivative ◽  
The One ◽  
Distributed Order

AbstractWe conduct a detailed study and comparison for the one-degree-of-freedom steady-state vibrations under harmonic driving with a single fractional-order derivative and a distributed-order derivative. For each of the two vibration systems, we consider the stiffness contribution factor and damping contribution factor of the term of fractional derivatives, the amplitude and the phase difference for the response. The effects of driving frequency on these response quantities are discussed. Also the influences of the orderαof the fractional derivative and the parameterγparameterizing the weight function in the distributed-order derivative are analyzed. Two cases display similar response behaviors, but the stiffness contribution factor and damping contribution factor of the distributed-order derivative are almost monotonic change with the parameterγ, not exactly like the case of single fractional-order derivative for the orderα. The case of the distributed-order derivative provides us more options for the weight function and parameters.

scholarly journals A new general fractional-order derivative with Rabotnov fractional-exponential kernel

2019 ◽  
Vol 23 (6 Part B) ◽  
pp. 3711-3718 ◽  
Author(s):  
Xiao-Jun Yang ◽  
Minvydas Ragulskis ◽  
Thiab Taha
Keyword(s):  
Fractional Order ◽  
Physical Models ◽  
Order Derivative ◽  
Mathematical Approach ◽  
Liouville Type ◽  
Derivative Operator ◽  
Fractional Order Derivative ◽  
Fractional Exponential Function ◽  
Complex Phenomena ◽  
First Time

In this article, a general fractional-order derivative of the Riemann-Liouville type with the non-singular kernel involving the Rabotnov fractional-exponential function is addressed for the first time. A new general fractional-order derivative model for the anomalous diffusion is discussed in detail. The general fractional-order derivative operator formula is as a novel and mathematical approach proposed to give the generalized presentation of the physical models in complex phenomena with power law.

scholarly journals Fractional Model for a Class of Diffusion-Reaction Equation Represented by the Fractional-Order Derivative

2020 ◽  
Vol 4 (2) ◽  
pp. 15 ◽  
Author(s):  
Ndolane Sene
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Fractional Order ◽  
Integration Method ◽  
Diffusion Reaction ◽  
Order Derivative ◽  
Reaction Equation ◽  
Fractional Model ◽  
Fractional Order Derivative ◽  
The Laplace Transform

This paper proposes the analytical solution for a class of the fractional diffusion equation represented by the fractional-order derivative. We mainly use the Grunwald–Letnikov derivative in this paper. We are particularly interested in the application of the Laplace transform proposed for this fractional operator. We offer the analytical solution of the fractional model as the diffusion equation with a reaction term expressed by the Grunwald–Letnikov derivative by using a double integration method. To illustrate our findings in this paper, we represent the analytical solutions for different values of the used fractional-order derivative.

scholarly journals A new general fractional-order wave model involving Miller-Ross kernel

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 953-957
Author(s):  
Linming Dou ◽  
Xiao-Jun Yang ◽  
Jiangen Liu
Keyword(s):  
Analytical Solution ◽  
Fractional Order ◽  
Wave Model ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Complex Processes ◽  
First Time

In the paper we consider a general fractional-order wave model with the general fractional-order derivative involving the Miller-Ross kernel for the first time. The analytical solution for the general fractional-order wave model is investigated in detail. The obtained result is given to explore the complex processes in the mining rock.

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