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Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed orders

I. Podlubny, T. Skovranek, B. M. Vinagre Jara, I. Petras, V. Verbitsky, Y. Chen
2013 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
techniques for fractional-and distributed-order differential equations.  ...  Numerical integration and differentiation on non-equidistant grids is introduced and illustrated by several examples of numerical solution of differential equations with fractional derivatives of constant  ...  The methods presented in this paper finally allow fractional-order differentiation and integration, and also distributed-order differentiation and integration, of non-uniformly sampled signals.  ... 
doi:10.1098/rsta.2012.0153 pmid:23547230 fatcat:4qllyzrvuvhd7fporvrdotsjsm

Analysis of Fractional Dynamic Systems

Fawang Liu, Richard Magin, Changpin Li, Alla Sikorskii, Santos Bravo Yuste
2014 The Scientific World Journal  
Zhou et al. discusses the monotonicity, the concavity, and the convexity of some functions arising in solutions of fractional differential equations.  ...  Mao et al. investigates the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient.  ...  Mao et al. investigates the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient.  ... 
doi:10.1155/2014/760634 pmid:24892087 pmcid:PMC4032648 fatcat:cvsc2lulzjarfgcfvjd7vyphmq

Numerical analysis for distributed-order differential equations

Kai Diethelm, Neville J. Ford
2009 Journal of Computational and Applied Mathematics  
Distributed order differential equation, multi-term fractional differential equation, numerical solution, convergence *  ...  In this paper we present and analyse a numerical method for the solution of a distributed order differential equation of the general form where the derivative D r * is taken to be a fractional derivative  ...  equations and which together motivate the theme of this paper: the development of a reliable, convergent scheme for the solution of basic distributed order differential equations.  ... 
doi:10.1016/j.cam.2008.07.018 fatcat:j3vqt53cs5fmnfknyytk3qhaiu

Recent Developments in Integral Transforms, Special Functions, and Their Extensions to Distributions Theory

Adem Kılıçman, Mustafa Bayram, Hassan Eltayeb
2013 Abstract and Applied Analysis  
Secer studied the numerical solution and simulation of second-order parabolic PDEs with Sinc-Galerkin method which covers to solve second-order PDEs by numerical method.  ...  ., numerical solutions of the second-order one-dimensional telegraph equation were studied which was based on the reproducing kernel 2 Abstract and Applied Analysis Hilbert space method; in an interesting  ...  Further we also acknowledge the assistance and help which were provided by the editorial board members of the journal during the preparation of this special issue as well as during the smooth publishing  ... 
doi:10.1155/2013/379856 fatcat:zwhwjclqkfdape4svsq4hmknoq

Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations

X. Wang, F. Liu, X. Chen
2015 Advances in Mathematical Physics  
Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature  ...  We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case.  ...  Diethelm and Ford [23] introduced and analyzed a numerical method for the solution of a distributed-order differential equations. Meerschaert et al.  ... 
doi:10.1155/2015/590435 fatcat:z5c25xxnzbethfrjtvu4vfg2r4

A Pseudo-spectral Method for Time Distributed Order Two-sided Space Fractional Differential Equations

Shina Daniel Oloniiju, Sicelo Praisegod Goqo, Precious Sibanda
2021 Taiwanese journal of mathematics  
In this study, we describe a scheme for obtaining the numerical solutions of time distributed order multidimensional two-sided space fractional differential equations.  ...  The time distributed order fractional differential operator is approximated using the composite Simpson's rule, and the solution of the resulting differential equation is expressed as a linear combination  ...  a distributed order fractional differential equation.  ... 
doi:10.11650/tjm/210501 fatcat:3wvht33cbnenfhyuqc46myur2y

Stability Analysis of Distributed Order Fractional Chen System

H. Aminikhah, A. Refahi Sheikhani, H. Rezazadeh
2013 The Scientific World Journal  
In addition, we have found that chaos exists in the double fractional order Chen system. Numerical solutions are used to verify the analytical results.  ...  Based on the asymptotic stability theory of nonlinear distributed order fractional systems, the stability of distributed order fractional Chen system is discussed.  ...  algorithm, we can obtain the numerical solution of the fractional differential equations (26) and multiterm fractional differential equations (27) , where = 1, 2, . . . , for = /ℎ and where is the total  ... 
doi:10.1155/2013/645080 pmid:24489508 pmcid:PMC3893869 fatcat:dmxaz7i5ojhglhnxmf2vgebghq

A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation

Q. Liu, F. Liu, Y.T. Gu, P. Zhuang, J. Chen, I. Turner
2015 Applied Mathematics and Computation  
This paper aims to develop a meshless approach based on the Point Interpolation Method (PIM) for numerical simulation of a space fractional diffusion equation.  ...  Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach.  ...  Fig. 3 . 3 Numerical solutions for κ = 1.6 with different fractional order α at t = 1.  ... 
doi:10.1016/j.amc.2015.01.092 fatcat:e3glwkw2a5h73miekjcjaj7dbq

Deterministic analysis of distributed order systems using operational matrix

Pham Luu Trung Duong, Ezra Kwok, Moonyong Lee
2016 Applied Mathematical Modelling  
Recently, distributed order systems as a generalized concept of fractional order have been a major focus in science and engineering areas, and have rapidly extended application across a wide range of disciplines  ...  This paper proposes a novel numerical scheme to analyze the behavior of single input single output linear systems in the time domain with a single distributed order differentiator/integrator by using operational  ...  Fig. 3 . 3 Solutions of distributed order relaxation equation with ρ(α) = 6α(1 − α) in Example 2. Fig. 4 . 4 Solutions and error of distributed order differential equation in Example 3.  ... 
doi:10.1016/j.apm.2015.09.035 fatcat:yow7dgolnjehrksddjkga4s6fi

Preface [chapter]

2020 Fractional Differential Equations  
This part is the important foundation of the following numerical solutions to fractional partial differential equations (FPDEs).  ...  The analytical solutions to two kinds of fractional ordinary differential equations (FODEs) are given, from which, readers can have a general idea on the behaviors of solutions to FODEs.  ...  Chapter 5 considers the finite difference methods for solving a class of the timespace fractional differential equations.  ... 
doi:10.1515/9783110616064-201 fatcat:hblgddtqnjgujoprmuzngvkiuq

Page 8134 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
Here, the numerical properties of partial differential equations of fractional order a, 1 <a < 2, are stud- ied.  ...  (l-ORNL-CEN; Oak Ridge, TN) Numerical methods for the solution of partial differential equations of fractional order. (English summary) J. Comput. Phys. 192 (2003), no. 2, 406-421.  ... 

Time-fractional heat conduction in a finite composite cylinder with heat source

Stanisław Kukla, Urszula Siedlecka
2020 Journal of Applied Mathematics and Computational Mechanics  
In this paper, the effect of the fractional order of the Caputo time-derivative occurring in heat conduction models on the temperature distribution in a finite cylinder consisting of an inner solid cylinder  ...  An analytical solution of the problem is derived in the form of the double series of eigenfunctions. Numerical examples are presented. MSC 2010: 26A33, 34A08, 80A20  ...  Numerical analysis shows the effect of the derivative fractional order on the temperature distribution in the cylinder.  ... 
doi:10.17512/jamcm.2020.2.07 fatcat:h7oepkig75e3fbm4tpukev3bsy

Fractional Dynamics

Carlo Cattani, Renato Spigler
2018 Fractal and Fractional  
Needless to say, these applications require the development of practical mathematical tools in order to extract quantitative information from models, newly reformulated in terms of fractional differential  ...  differential equations, and stochastic integral problems arising in all fields of science, engineering, and other applied fields.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/fractalfract2020019 fatcat:7ldhty2f3zgwxdbt4dyic2s7wm

Bayesian Derivative Order Estimation for a Fractional Logistic Model

Francisco J. Ariza-Hernandez, Martin P. Arciga-Alejandre, Jorge Sanchez-Ortiz, Alberto Fleitas-Imbert
2020 Mathematics  
In this paper, we consider the inverse problem of derivative order estimation in a fractional logistic model.  ...  By MCMC samples of the marginal posterior distributions, we estimate the order of the derivative and the growth rate parameter in the dynamic model, as well as the noise in the observations.  ...  In this paper, our interest is to estimate the order of the derivative in a differential equation based on measurements data, under uncertainty, of the solutions of a fractional-order logistic model.  ... 
doi:10.3390/math8010109 fatcat:zfs5yaai4zh2vlpvk5wkvuzxde

Fractional Calculus: Theory and Applications

Francesco Mainardi
2018 Mathematics  
Fractional calculus is allowing integrals and derivatives of any positive order (the term fractional is kept only for historical reasons).[...]  ...  Numerical Methods The paper "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial" by Roberto Garrappa [9] aims to provide a tutorial for the numerical solution of  ...  The paper "Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag-Leffler Functions" by Marina Popolizio [10] focuses on a numerical approach to solve Multiterm Fractional  ... 
doi:10.3390/math6090145 fatcat:bvduvpwutffmlmj2qwcaavolfa
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