A review of operational matrices and spectral techniques for fractional calculus

Bhrawy, A. H.; Taha, Taha M.; Machado, J. A. Tenreiro

doi:10.1007/s11071-015-2087-0

Cited by 164 publications

(78 citation statements)

References 70 publications

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“…By (26) together with the constants a = 0, b = 1 2 , it is easy to get the first RPS approximate solutions for the time fractional ALW equations as:…”

Section: Applicationsmentioning

confidence: 99%

“…Therefore, an immediate and natural question arises if we can get the explicit approximate solutions of time fractional WBK equations. In recent years, many powerful techniques have been extended and developed to obtain numerical and analytical solutions of fractional differential equations, such as the tau spectral method [20], the spectral collocation method [21][22][23][24], the Jacobi-Gauss-Lobatto collocation method [25], the operational matrices and spectral techniques [26] and the mesh-less boundary collocation methods [27,28].…”

Section: Introductionmentioning

confidence: 99%

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Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method

Wang

Chen

2015

Entropy

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In this paper, a new analytic iterative technique, called the residual power series method (RPSM), is applied to time fractional Whitham-Broer-Kaup equations. The explicit approximate traveling solutions are obtained by using this method. The efficiency and accuracy of the present method is demonstrated by two aspects. One is analyzing the approximate solutions graphically. The other is comparing the results with those of the Adomian decomposition method (ADM), the variational iteration method (VIM) and the optimal homotopy asymptotic method (OHAM). Illustrative examples reveal that the present technique outperforms the aforementioned methods and can be used as an alternative for solving fractional equations.

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“…By (26) together with the constants a = 0, b = 1 2 , it is easy to get the first RPS approximate solutions for the time fractional ALW equations as:…”

Section: Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method

Wang

Chen

2015

Entropy

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“…By definition of fractional derivative, to compute the solution at the current time level one needs to save all the previous solutions, which makes the storage expensive if low-order methods are employed for spatial discretization. One of the main advantage of the spectral method is the fact that it can relax this storage limit since it needs fewer grid points to produce a highly accurate solution [11,12].…”

Section: Introductionmentioning

confidence: 99%

Galerkin spectral method for the fractional nonlocal thermistor problem

Ammi¹,

Torres

2017

Computers & Mathematics with Applications

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“…For example in [3,4], the authors have achieved the necessary conditions for optimization of FOCPs with the Caputo fractional derivative. The interested reader can refer to [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] for some recent works on FOCPs. In this paper, we propose a new efficient and accurate computational method based on HFs for solving the following FOCP [6]:…”

Section: Introductionmentioning

confidence: 99%

An efficient computational method based on the hat functions for solving fractional optimal control problems

et al. 2016

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In this paper, an efficient and accurate computational method based on the hat functions (HFs) is proposed for solving a class of fractional optimal control problems (FOCPs). In the proposed method, the fractional optimal control problem under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved. To this end, the fractional state and control variables are expanded by the HFs with unknown coefficients. Then, the operational matrix of fractional integration of the HFs with some properties of these basis functions are employed to achieve a nonlinear algebraic equation, replacing the performance index and a nonlinear system of algebraic equations, replacing the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extremum is applied, which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined Lagrange multipliers. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable and Lagrange multipliers. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

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A review of operational matrices and spectral techniques for fractional calculus

Cited by 164 publications

References 70 publications

Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method

Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method

Galerkin spectral method for the fractional nonlocal thermistor problem

An efficient computational method based on the hat functions for solving fractional optimal control problems

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