Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials

Rad, Jamal Amani; Kazem, Saeed; Shaban, M.; Parand, Kourosh; Yıldırım, Ahmet

doi:10.1002/mma.2794

Cited by 55 publications

(30 citation statements)

References 44 publications

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“…For example, the nonlinear oscillation of earthquake [8], the fractional optimal control problems for dynamic systems [9,10,11,12], and the fluid-dynamic models with fractional derivatives can eliminate the deficiency arising from the assumption of continuous traffic flow [13,14,15]. During the last decades, several methods have been used to solve fractional differential equations, fractional partial differential equations, fractional integro-differential equations, the initial and boundary value problems, and dynamic systems containing fractional derivatives, such as Adomian's decomposition method [16,17], fractional-order Legendre functions [18], fractional-order Chebyshev functions of the second kind [19], Homotopy analysis method [20], Bessel functions and spectral methods [21], Legendre and Bernstein polynomials [22], finite element methods [23], Legendre collocation [24], modified spline collocation [25], multiquadratic radial basis functions [26], and other methods [27,28,29,30,31,32,33]. …”

Section: Summary Of Fractional Calculus Historymentioning

confidence: 99%

Novel orthogonal functions for solving differential equations of arbitrary order

2017

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Fractional calculus and the fractional differential equations have appeared in many physical and engineering processes. Therefore, an efficient and suitable method to solve them is very important. In this paper, novel numerical methods are introduced based on the fractional order of the Chebyshev orthogonal functions (FCF) with Tau and collocation methods to solve differential equations of the arbitrary (integer or fractional) order. The FCFs are obtained from the classical Chebyshev polynomials of the first kind. Also, the operational matrices of the fractional derivative and the product for the FCFs have been constructed. To show the efficiency and capability of these methods we have solved some well-known problems: the momentum, the Bagley-Torvik, and the Lane-Emden differential equations, then have compared our results with the famous methods in other papers.

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Section: Summary Of Fractional Calculus Historymentioning

confidence: 99%

Novel orthogonal functions for solving differential equations of arbitrary order

2017

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“…In another paper, Wang et al [13], used the Grünwald-Letnikov definition to approximate the solution of the FDDE's and also discussed the convergence of the method. In [14], Rad et al have discussed the operational matrices of the Bernstein's and Legendre polynomials and they have applied it to solve the FDDE's with the Tau method.…”

Section: Introductionmentioning

confidence: 99%

An Approximate Method for Solving Fractional Delay Differential Equations

Pandey

Kumar

Mohaptra

2016

Int. J. Appl. Comput. Math

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This paper presents an approximate method for solving a kind of fractional delay differential equations defined in terms of Caputo fractional derivatives. The approximate method is based on the application of the Bernstein's operational matrix of fractional differentiation. First, Bernstein operational matrix of fractional differentiation is presented generalizing the idea of Bernstein's operational matrix of derivative for integer orders, and then applied to solve the nonlinear fractional delay differential equations. The operational matrix method combined with the typical tau method reduces the fractional delay differential equation into system of nonlinear equations. Solving these nonlinear equations the desired solution is achieved. Two different cases of the fractional delay differential equations are illustrated and solved using the presented method. Numerical results and discussions demonstrate the applicability of the proposed method.

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“…Since no exact solution exists for FDE, most efforts have supplied numerical and analytical methods to solve these equations. Indeed, many powerful methods have been recently developed, such as the Adomian decomposition method, homotopy analysis method, homotopy perturbation method, collocation method, finite difference method and Tau method [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

A New Variational Iteration Method for a Class of Fractional Convection-Diffusion Equations in Large Domains

2017

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Abstract:In this paper, we introduced a new generalization method to solve fractional convection-diffusion equations based on the well-known variational iteration method (VIM) improved by an auxiliary parameter. The suggested method was highly effective in controlling the convergence region of the approximate solution. By solving some fractional convection-diffusion equations with a propounded method and comparing it with standard VIM, it was concluded that complete reliability, efficiency, and accuracy of this method are guaranteed. Additionally, we studied and investigated the convergence of the proposed method, namely the VIM with an auxiliary parameter. We also offered the optimal choice of the auxiliary parameter in the proposed method. It was noticed that the approach could be applied to other models of physics.

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Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials

Cited by 55 publications

References 44 publications

Novel orthogonal functions for solving differential equations of arbitrary order

Novel orthogonal functions for solving differential equations of arbitrary order

An Approximate Method for Solving Fractional Delay Differential Equations

A New Variational Iteration Method for a Class of Fractional Convection-Diffusion Equations in Large Domains

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