An Improvement of a Nonclassical Numerical Method for the Computation of Fractional Derivatives

2009 ◽  
Vol 131 (1) ◽  
Author(s):  
Kai Diethelm
Keyword(s):  
Numerical Calculation ◽  
Numerical Method ◽  
Dynamic Systems ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Differential Operators ◽  
Standard Methods ◽  
Efficient Approach

Standard methods for the numerical calculation of fractional derivatives can be slow and memory consuming due to the nonlocality of the differential operators. Yuan and Agrawal (2002, “A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives,” ASME J. Vibr. Acoust., 124, pp. 321–324) have proposed a more efficient approach for operators whose order is between 0 and 1 that differs substantially from the traditional concepts. It seems, however, that the accuracy of the results can be poor. We modify the approach, adapting it better to the properties of the problem, and show that this leads to a significantly improved quality. Our idea also works for operators of order greater than 1.

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.

scholarly journals New insights on novel coronavirus 2019-nCoV/SARS-CoV-2 modelling in the aspect of fractional derivatives and fixed points

2021 ◽  
Vol 18 (6) ◽  
pp. 8683-8726
Author(s):  
Sumati Kumari Panda ◽  
◽  
Abdon Atangana ◽  
Juan J. Nieto ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Differential Operators ◽  
The Novel ◽  
Different Types ◽  
Novel Coronavirus

<abstract><p>Extended orthogonal spaces are introduced and proved pertinent fixed point results. Thereafter, we present an analysis of the existence and unique solutions of the novel coronavirus 2019-nCoV/SARS-CoV-2 model via fractional derivatives. To strengthen our paper, we apply an efficient numerical scheme to solve the coronavirus 2019-nCoV/SARS-CoV-2 model with different types of differential operators.</p></abstract>

An efficient numerical method for the optimal control of fractional-order dynamic systems

2018 ◽  
Vol 24 (22) ◽  
pp. 5312-5320 ◽  
Author(s):  
Ehsan Mohammadzadeh ◽  
Naser Pariz ◽  
Seyed Kamal Hosseini Sani ◽  
Amin Jajarmi
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Dynamic Systems ◽  
Internal Model ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Numerical Examples ◽  
Internal Model Principle ◽  
Comparative Results ◽  
Persistent Disturbances

This paper aims to investigate an efficient numerical scheme for the optimal control of fractional-order dynamic systems. By using the Grünwald–Letnikov approximation for the fractional derivatives and introducing a new transformation in the calculus of variations, the fractional optimal control problem under consideration is converted into a linear programming problem. Then, the internal model principle is employed in order to extend the new scheme for the fractional dynamic systems affected by the external persistent disturbances. Numerical examples and comparative results verify the validity and applicability of the new technique.

Aspects on viscoelasticity modeling of HDPE using fractional derivatives: Interpolation procedures and efficient numerical scheme

2021 ◽  
pp. 1-16
Author(s):  
T.C. da Costa-Haveroth ◽  
G.A. Haveroth ◽  
A. Kühl ◽  
J.L. Boldrini ◽  
M.L. Bittencourt ◽  
...  
Keyword(s):  
Numerical Scheme ◽  
Fractional Derivatives

A Numerical Scheme for a Class of Parametric Problem of Fractional Variational Calculus

2012 ◽  
Vol 7 (2) ◽  
Author(s):  
Om P. Agrawal ◽  
M. Mehedi Hasan ◽  
X. W. Tangpong
Keyword(s):  
Analytical Solutions ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimization Problems ◽  
Practical Interest ◽  
Variational Calculus ◽  
Functional Minimization ◽  
Orthogonal Functions ◽  
Order Problem ◽  
Parametric Problem

Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in the future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such properties as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered, and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.

scholarly journals Generalized fractional differential ring

2021 ◽  
Vol 5 (1) ◽  
pp. 279-287
Author(s):  
Zeinab Toghani ◽  
◽  
Luis Gaggero-Sager ◽  
Keyword(s):  
Fractional Derivative ◽  
Infinite Number ◽  
Fractional Derivatives ◽  
Differential Operators ◽  
Differential Ring ◽  
Fractional Differential

There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there is an infinite number of possible definitions of fractional derivatives, all are correct as differential operators each of them must be properly defined its algebra. We introduce a generalized version of fractional derivative that extends the existing ones in the literature. To those extensions it is associated a differentiable operator and a differential ring and applications that shows the advantages of the generalization. We also review the different definitions of fractional derivatives and it is shown how the generalized version contains the previous ones as a particular cases.

scholarly journals Experiments and numerical calculation to determine aerodynamic characteristics of flows around 3D wings

2014 ◽  
Vol 36 (2) ◽  
pp. 133-143 ◽  
Author(s):  
Nguyen Hong Son ◽  
Hoang Thi Bich Ngoc ◽  
Dinh Van Phong ◽  
Nguyen Manh Hung
Keyword(s):  
Numerical Calculation ◽  
Wind Tunnel ◽  
Pressure Distribution ◽  
Numerical Method ◽  
Aerodynamic Characteristics ◽  
Numerical Results ◽  
Computational Method ◽  
Wing Surface ◽  
Thickness Profile

The report presents method and results of experiments in wind tunnel to determine aerodynamic characteristics of 3D wings by measuring pressure distribution on the wing surfaces. Simultaneously, a numerical method by using sources and doublets distributed on panel elements of wing surface also is carried out to calculate flows around 3D wings. This computational method allows solving inviscid problems for wings with thickness profile. The experimental and numerical results are compared to each other to verify the built program that permits to extend the range of applications with the variation of wing profiles, wing planforms, and incidence angles.

A Universal Predictor-corrector Algorithm for Numerical Simulation of Generalized Fractional Differential Equations

2021 ◽  
Author(s):  
Zaid Odibat
Keyword(s):  
Numerical Simulation ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Differential Operators ◽  
Numerical Solutions ◽  
Fractional Operators ◽  
Initial Value ◽  
Generalized Fractional Integral ◽  
Fractional Differential ◽  
Predictor Corrector

Abstract This study introduces some remarks on generalized fractional integral and differential operators, that generalize some familiar fractional integral and derivative operators, with respect to a given function. We briefly explain how to formulate representations of generalized fractional operators. Then, mainly, we propose a predictor-corrector algorithm for the numerical simulation of initial value problems involving generalized Caputo-type fractional derivatives with respect to another function. Numerical solutions of some generalized Caputo-type fractional derivative models have been introduced to demonstrate the applicability and efficiency of the presented algorithm. The proposed algorithm is expected to be widely used and utilized in the field of simulating fractional-order models.

A Numerical Method for Mean Drift Forces Acting on a 3-D Body Under Wave and Current

1997 ◽  
Vol 41 (02) ◽  
pp. 108-117
Author(s):  
Tzung-hang Lee ◽  
Bo-qi Sun
Keyword(s):  
Numerical Calculation ◽  
Green Function ◽  
Numerical Method ◽  
Present Method ◽  
Small Amplitude ◽  
Correction Term ◽  
Current Speed ◽  
Forward Speed ◽  
The Green Function ◽  
Drift Forces

Under the assumptions of potential theory, small-amplitude motion, small steady disturbance and low current speed, the problem is simplified and then a form of Green function of the simplified equation is derived. For ease of numerical calculation, following the idea of Huijsmans & Hermans (1985), the Green function is expended into an oscillating source G0 at zero forward speed and a correction term G1 to take into account low forward speed. Then the expressions for G0 and G1 are derived. It is shown that these expressions have good numerical calculation performance. To test the present method, it is applied to a hemisphere and a half submerged ellipsoid.

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