Convergence Analysis of the Short-Memory Principle for Fractional Models

2021 ◽  
Author(s):  
Oliver Stark ◽  
Felix Strehle ◽  
Soren Hohmann
Keyword(s):  
Convergence Analysis ◽  
Fractional Models ◽  
Short Memory ◽  
Short Memory Principle

scholarly journals A Numerical Algorithm for the Solutions of ABC Singular Lane–Emden Type Models Arising in Astrophysics Using Reproducing Kernel Discretization Method

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 923 ◽  
Author(s):  
Omar Abu Arqub ◽  
Mohamed S. Osman ◽  
Abdel-Haleem Abdel-Aty ◽  
Abdel-Baset A. Mohamed ◽  
Shaher Momani
Keyword(s):  
Convergence Analysis ◽  
Fractional Order ◽  
Numerical Algorithm ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Discretization Method ◽  
Fractional Models ◽  
Discretization Technique ◽  
And Mathematics

This paper deals with the numerical solutions and convergence analysis for general singular Lane–Emden type models of fractional order, with appropriate constraint initial conditions. A modified reproducing kernel discretization technique is used for dealing with the fractional Atangana–Baleanu–Caputo operator. In this tendency, novel operational algorithms are built and discussed for covering such singular models in spite of the operator optimality used. Several numerical applications using the well-known fractional Lane–Emden type models are examined, to expound the feasibility and suitability of the approach. From a numerical viewpoint, the obtained results indicate that the method is intelligent and has several features stability for dealing with many fractional models emerging in physics and mathematics, using the new presented derivative.

A note on short memory principle of fractional calculus

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Yiheng Wei ◽  
Yuquan Chen ◽  
Songsong Cheng ◽  
Yong Wang
Keyword(s):  
Fractional Calculus ◽  
Continuous Time ◽  
Special Kind ◽  
The Other ◽  
Memory Length ◽  
Other Hand ◽  
Classical Principle ◽  
Short Memory ◽  
Short Memory Principle ◽  
Discrete Argument

AbstractIn this paper, from the classical short memory principle under Grünwald-Letnikov definition, several novel short memory principles are presented and investigated. On one hand, the classical principle is extended to Riemann-Liouville and Caputo cases. On the other hand, a special kind of principles are formulated by introducing a discrete argument instead of the continuous time, resulting in principles with fixed memory length or fixed memory step. Apart from these, several interesting properties of the proposed principles are revealed profoundly.

Bifurcation Analysis of the Fractional Duffing System Based on the Improved Short Memory Principle Method

2021 ◽  
Author(s):  
Ruiqun Ma ◽  
Bo Zhang ◽  
Haiwei Yun ◽  
Jinglong Han
Keyword(s):  
System Dynamics ◽  
Bifurcation Diagram ◽  
Fractional Order ◽  
Chaotic Motion ◽  
Excitation Frequency ◽  
Nonlinear Dynamic Analysis ◽  
Frequency Change ◽  
Duffing System ◽  
Short Memory ◽  
Short Memory Principle

Abstract In this study, the improved short memory principle method is introduced to the analysis of the dynamic characteristics of the fractional Duffing system, and the basis for the improvement of the short memory principle method is provided. The influence of frequency change on the dynamic performance of the fractional Duffing system is studied using nonlinear dynamic analysis methods, such as phase portrait, Poincare map and bifurcation diagram. Moreover, the dynamic behaviour of the fractional Duffing system when the fractional order and excitation amplitude change is investigated. The analysis shows that when the excitation frequency changes from 0.43 to 1.22, the bifurcation diagram contains four periodic and three chaotic motion regions. Periodic motion windows are found in the three chaotic motion regions. Results confirm that the frequency and amplitude of the external excitation and the fractional order of damping have a greater impact on system dynamics. Thus, attention should be paid to the design and analysis of system dynamics.

scholarly journals Improved Short Memory Principle Method for Solving Fractional Damped Vibration Equations

2020 ◽  
Vol 10 (21) ◽  
pp. 7566
Author(s):  
Ruiqun Ma ◽  
Jinglong Han ◽  
Xiaoxuan Yan
Keyword(s):  
Initial Step ◽  
Binomial Coefficient ◽  
Degree Of Freedom ◽  
Step Size ◽  
Single Degree Of Freedom ◽  
Memory Time ◽  
Short Memory ◽  
Short Memory Principle ◽  
Initial Step Size ◽  
Two Degree Of Freedom

In this paper, an improved short memory principle based on the Grünwald–Letnikov definition is proposed and applied in solving fractional vibration differential equations. The improved idea is to adjust the truncation of memory time in short memory principle (SMP) to the truncation of binomial coefficient terms, and the finite coefficients are repeatedly applied to the step size gradually enlarged. In this method, a very small initial step size is used to meet the accuracy requirements, and then the step size is gradually enlarged to prolong the memory time and reduce the amount of calculation. Examples of free vibration, forced vibration with a single-degree-of-freedom and a vehicle suspension two-degree-of-freedom vibration reduction model verify the method’s accuracy and effectiveness.

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