Numerical solutions of fractional advection–diffusion equations with a kind of new generalized fractional derivative

Xu, Yufeng; He, Zhen; Xu, Qinwu

doi:10.1080/00207160.2013.799277

Cited by 26 publications

(36 citation statements)

References 22 publications

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“…Many authors in different fields such as chemical physics, fluid flows, electrical networks, viscoelasticity, try to present a model of these phenomena by boundary value problems of fractional differential equations [1][2][3][4]. In order to achieve extra information in fractional calculus, interested readers can refer to more valuable books and papers that are written by other authors [5][6][7][8][9][10][11][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

On the existence and uniqueness of solutions to a new class of fractional boundary value problems

Neamaty

Yadollahzadeh

Darzi

et al. 2015

International Journal of Computer Mathematics

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In this article, we verify the existence and uniqueness of solution for a class of nonlinear boundary value problem for fractional differential equation of order 3 < α ≤ 4, with the standard fractional derivative in the sense of Riemann-Liouville. By using a variety of tools including the Schauder fixed point theorem, the Krasnoseĺskii fixed point theorem and contraction mapping principle, existence and uniqueness results are obtained. Some examples are given to expression the results.

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Section: Introductionmentioning

confidence: 99%

On the existence and uniqueness of solutions to a new class of fractional boundary value problems

Neamaty

Yadollahzadeh

Darzi

et al. 2015

International Journal of Computer Mathematics

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“…However, the analytical form solutions are very complicated with infinity serials or integrations, which makes them not suitable for fast computing, while numerical methods are more practical for these equations in applications. In [19], Xu proposes a finite difference scheme for timefractional advection-diffusion equations with generalized 2 International Journal of Differential Equations fractional derivative [19]. Later, a finite difference scheme and an analytical solution are studied for generalized timefractional diffusion equation by Xu and Argrawal [20,21].…”

Section: Introductionmentioning

confidence: 99%

Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator

Zheng

2019

International Journal of Differential Equations

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Generalized fractional operators are generalization of the Riemann-Liouville and Caputo fractional derivatives, which include Erdélyi-Kober and Hadamard operators as their special cases. Due to the complicated form of the kernel and weight function in the convolution, it is even harder to design high order numerical methods for differential equations with generalized fractional operators. In this paper, we first derive analytical formulas for − ℎ ( > 0) order fractional derivative of Jacobi polynomials. Spectral approximation method is proposed for generalized fractional operators through a variable transform technique. Then, operational matrices for generalized fractional operators are derived and spectral collocation methods are proposed for differential and integral equations with different fractional operators. At last, the method is applied to generalized fractional ordinary differential equation and Hadamard-type integral equations, and exponential convergence of the method is confirmed. Further, based on the proposed method, a kind of generalized grey Brownian motion is simulated and properties of the model are analyzed.

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“…There are some other fractional derivatives such as distributed order fractional derivatives [19], Risez fractional derivatives [137] and new generalized fractional derivatives [6]. We are not going further on these topics but encouraging readers to the excellent papers [14,31,32,39,51,63,100,128,130,131,137] for approximation methods.…”

Section: Other Fractional Derivativesmentioning

confidence: 99%

“…There has been some pioneer works on numerical treatment for various generalized fractional problems covering fractional Burgers equation [128], fractional sub-di↵usion equation [130], fractional advection-di↵usion equation [131], generalized van der Pol equation [127] as well as oscillator equation [129]. To my best knowledge, these topics are all new and first discussed therein since the new generalized fractional operator is proposed in [6].…”

mentioning

confidence: 99%

“…To my best knowledge, these topics are all new and first discussed therein since the new generalized fractional operator is proposed in [6]. To get these equations, one can simply replace the original fractional operator by this new generalized fractional operator, like in [128,130,131] or an even more general operator involving a kernel function, e.g. [5,127,129].…”

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confidence: 99%

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Numerical methods for fractional differential equations

Li¹

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Fractional di↵erential equations have received much attention in recent decades likely due to its powerful ability in modeling 'memory' processes, which are mostly observed in real world. Fractional models have been widely investigated and applied in many fields such as physics, biochemistry, electrical engineering, continuum and statistical mechanics. It is shown by many researchers that fractional derivatives can provide more accurate models than integer order derivatives do. However, for most models involving fractional operators, it is very challenging if not impossible to get analytical solutions. This motivates us to propose numerical techniques, especially those that can obtain numerical approximations both e ciently and e↵ectively. In this thesis, we aim to construct high order accuracy numerical scheme for two categories: (i) four classes of typical fractional problems, and (ii) some classes of generalized fractional problems. In tackling the first category, we focus on the improvement of the theoretical spatial convergence order mainly using parametric quintic spline. The parametric spline method has been used in previous work and has been numerically shown to get satisfying performance. Unfortunately, no strict theoretical result is established and we note that the analysis is not trivial. Due to this reason, we are greatly motivated to further investigate this method and provide strict analysis which is extremely important for a numerical method. Our contributions for topics in the first category mainly lies in two aspects: 1) we develop new numerical methods for fractional problems that improve the spatial convergence order achieved so far in the literature; 2) we establish the solvability, stability and convergence results of the proposed methods that are not given in previous work relating to parametric spline method. The other category is about numerical treatment on some classes of generalized fractional problems. As the name suggests, the generalized fractional problems include typical fractional problems as special cases and may represent even more: v Contents vi both existing and potential new cases. In fact, it is shown that many integral equations can be solved in a much elegant way using generalized fractional operators and it can somehow blur the distinction between the di↵erential and integral equations. For this topic, the numerical investigations are very scarce and the temporal convergence order of the existing work is not so satisfying and may not meet the demand in reality. Motivated by these two observations, we shall construct some new approximations of generalized fractional derivatives through generalizations of some typical techniques, e.g. L2 1 method and weighted shifted Grünwald-Letnikov method to improve the temporal convergence order. This will help to solve this kind of problems more e ciently and e↵ectively.

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Numerical solutions of fractional advection–diffusion equations with a kind of new generalized fractional derivative

Cited by 26 publications

References 22 publications

On the existence and uniqueness of solutions to a new class of fractional boundary value problems

On the existence and uniqueness of solutions to a new class of fractional boundary value problems

Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator

Numerical methods for fractional differential equations

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