Discretization of fractional differential equations by a piecewise constant approximation

Angstmann, Christopher N.; Henry, Bryan R.; Jacobs, Byron A.; McGann, Anna V.

doi:10.1051/mmnp/2017063

Cited by 8 publications

(7 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…According to Angstmann et al [35], the generalized Euler method (GEM) and piece wise continuous argument (PCWA) methods do not give the proper results, which is why we also used two other techniques (the Adams-Bashforth-Moulton method and the Grunwald-Letnikov method) to show the efficiency of the measles model in Sects. 4.3 and 4.4.…”

Section: Piece Wise Continuous Argument Methodsmentioning

confidence: 99%

Comparison of fractional order techniques for measles dynamics

et al. 2019

View full text Add to dashboard Cite

A mathematical model which is non-linear in nature with non-integer order φ, 0 < φ ≤ 1 is presented for exploring the SIRV model with the rate of vaccination μ 1 and rate of treatment μ 2 to describe a measles model. Both the disease free F 0 and the endemic F * points have been calculated. The stability has also been argued for using the theorem of stability of non-integer order differential equations. R 0 , the basic reproduction number exhibits an imperative role in the stability of the model. The disease free equilibrium point F 0 is an attractor when R 0 < 1. For R 0 > 1, F 0 is unstable, the endemic equilibrium F * subsists and it is an attractor. Numerical simulations of considerable model are also supported to study the behavior of the system.

show abstract

Section: Piece Wise Continuous Argument Methodsmentioning

confidence: 99%

Comparison of fractional order techniques for measles dynamics

et al. 2019

View full text Add to dashboard Cite

show abstract

“…At present, the expressions of fractional differential mainly include Riemann-Liouville, Grünwald-Letnikov, and Caputo [16][17][18], and the most commonly used expressions are Grünwald-Letnikov (G-L) expressions. The G-L differential is defined by:…”

Section: Fractional Derivativementioning

confidence: 99%

Fractional Modeling for Quantitative Inversion of Soil-Available Phosphorus Content

Xiong

Tian

2018

Mathematics

View full text Add to dashboard Cite

The study of field spectra based on fractional-order differentials has rarely been reported, and traditional integer-order differentials only perform the derivative calculation for 1st-order or 2nd-order spectrum signals, ignoring the spectral transformation details between 0th-order to 1st-order and 1st-order to 2nd-order, resulting in the problem of low-prediction accuracy. In this paper, a spectral quantitative analysis model of soil-available phosphorus content based on a fractional-order differential is proposed. Firstly, a fractional-order differential was used to perform a derivative calculation of original spectral data from 0th-order to 2nd-order using 0.2-order intervals, to obtain 11 fractional-order spectrum data. Afterwards, seven bands with absolute correlation coefficient greater than 0.5 were selected as sensitive bands. Finally, a stepwise multiple linear regression algorithm was used to establish a spectral estimation model of soil-available phosphorus content under different orders, then the prediction effect of the model under different orders was compared and analyzed. Simulation results show that the best order for a soil-available phosphorus content regression model is a 0.6 fractional-order, the coefficient of determination (), root mean square error (RMSE), and ratio of performance to deviation (RPD) of the best model are 0.7888, 3.348878, and 2.001142, respectively. Since the RPD value is greater than 2, the optimal fractional model established in this study has good quantitative predictive ability for soil-available phosphorus content.

show abstract

“…FDEs are a special type of Volterra equations with weakly singular kernels [13]. To treat the singularity, different strategies can be employed such as graded meshes or transformations [7,14,27] Other numerical methods include non-polynomial spline methods [17], spectral methods [11,33], piecewise constant approximation or integrabilization methods [3,4] and the discrete time random walk approach [1,2,5]. Yet, another strategy, which we use in this article, is integration by parts.…”

Section: Introductionmentioning

confidence: 99%

On the order reduction of approximations of fractional derivatives: an explanation and a cure

2023

View full text Add to dashboard Cite

Finite-difference based approaches are common for approximating the Caputo fractional derivative. Often, these methods lead to a reduction of the convergence rate that depends on the fractional order. In this note, we approximate the expressions in the fractional derivative components using a separate quadrature rule for the integral and a separate discretization of the derivative in the integrand. By this approach, the error terms from the Newton–Cotes quadrature and the differentiation are isolated and it is possible to conclude that the order dependent error is inevitable when the end points of the interval are included in the quadrature. Furthermore, we show experimentally that the theoretical findings carries over to quadrature rules without the end points included. Finally we show how to increase accuracy for smooth functions, and compensate for the order dependent loss.

show abstract

Discretization of fractional differential equations by a piecewise constant approximation

Cited by 8 publications

References 34 publications

Comparison of fractional order techniques for measles dynamics

Comparison of fractional order techniques for measles dynamics

Fractional Modeling for Quantitative Inversion of Soil-Available Phosphorus Content

On the order reduction of approximations of fractional derivatives: an explanation and a cure

Contact Info

Product

Resources

About