An efficient numerical method for variable order fractional functional differential equation

Yang, Jiansheng; Yao, Huanmin; Wu, Boying

doi:10.1016/j.aml.2017.08.020

Cited by 49 publications

(29 citation statements)

References 17 publications

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“…Moreover, Jia et al [51] relied on the simplified reproducing kernel method to solve the efficient numerical scheme for VO fractional equation and verified its convergence. Yang et al [128] proposed the VO-FDEs depending on the reproducing kernel splines method. Most importantly, this method is able to successfully reduce computational cost and provided accurate approximate solutions.…”

Section: Numerical Methods For Time Fdesmentioning

confidence: 99%

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Sun

Chang

Zhang

et al. 2019

FCAA

284

138

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Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

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Section: Numerical Methods For Time Fdesmentioning

confidence: 99%

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Sun

Chang

Zhang

et al. 2019

FCAA

284

138

View full text Add to dashboard Cite

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“…where α(t), β(t) are defined in (13) and (14). In order to obtain our main results, we start off by carrying on essential analysis to the equation of (26).…”

Section: Existence Of Approximate Solutionmentioning

confidence: 99%

“…Recently, some authors have considered the applications of derivatives of variable order in various sciences such as anomalous diffusion modeling, mechanical applications, multi-fractional Gaussian noises. Among these, there have been many works dealing with numerical methods for some class of variable order fractional differential equations, for instance, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Unique Existence Result of Approximate Solution to Initial Value Problem for Fractional Differential Equation of Variable Order Involving the Derivative Arguments on the Half-Axis

Zhang

2019

Mathematics

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The semigroup properties of the Riemann–Liouville fractional integral have played a key role in dealing with the existence of solutions to differential equations of fractional order. Based on some results of some experts’, we know that the Riemann–Liouville variable order fractional integral does not have semigroup property, thus the transform between the variable order fractional integral and derivative is not clear. These judgments bring us extreme difficulties in considering the existence of solutions of variable order fractional differential equations. In this work, we will introduce the concept of approximate solution to an initial value problem for differential equations of variable order involving the derivative argument on half-axis. Then, by our discussion and analysis, we investigate the unique existence of approximate solution to this initial value problem for differential equation of variable order involving the derivative argument on half-axis. Finally, we give examples to illustrate our results.

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“…Jafari et al [19] obtained the approximate solution of differential equations with variable order using operational matices. In [20,21], the functional boundary value problems with variable order are solved by reproducing kernel method. Ganji and Jafari [22] applied Jacobi polynomials to obtain solution of multi VODE.…”

Section: Introductionmentioning

confidence: 99%

A numerical scheme to solve variable order diffusion-wave equations

2019

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Original scientific paper https://doi.org/10.2298/TSCI190729371MIn this work, we consider variable order diffusion-wave equations. We choose variable order derivative in the Caputo sense. First, we approximate the unknown functions and its derivatives using Bernstein basis. Then, we obtain operational matrices based on Bernstein polynomials. Finally, with the help of these operational matrices and collocation method, we can convert variable order diffusion-wave equations to an algebraic system. Few examples are given to demonstrate the accuracy and the competence of the presented technique.

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An efficient numerical method for variable order fractional functional differential equation

Cited by 49 publications

References 17 publications

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Unique Existence Result of Approximate Solution to Initial Value Problem for Fractional Differential Equation of Variable Order Involving the Derivative Arguments on the Half-Axis

A numerical scheme to solve variable order diffusion-wave equations

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