On spectral methods for solving variable-order fractional integro-differential equations

Doha, E. H.; Abdelkawy, M. A.; Amin, A. Z. M.; Lopes, António M.

doi:10.1007/s40314-017-0551-9

Cited by 27 publications

(8 citation statements)

References 40 publications

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“…For instance, Haar wavelet collocation method for three-dimensional elliptic partial differential equations, 13 fractional-order Legendre-Laguerre collocation method for solving fractional partial differential equations, 14 spectral methods for solving variable-order fractional integro-differential equations. 15 Also, the proposed method was considered by many papers, for more details can refer to preveious studies. [16][17][18][19][20] In this paper, a new Genocchi-fractional Laguerre function and their properties are presented.…”

Section: Introductionmentioning

confidence: 99%

“…Presently, many papers have been devoted to the study of a spectral method to investigate various scientific models. For instance, Haar wavelet collocation method for three‐dimensional elliptic partial differential equations, fractional‐order Legendre‐Laguerre collocation method for solving fractional partial differential equations, spectral methods for solving variable‐order fractional integro‐differential equations . Also, the proposed method was considered by many papers, for more details can refer to preveious studies …”

Section: Introductionmentioning

confidence: 99%

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Application of the modified operational matrices in multiterm variable‐order time‐fractional partial differential equations

Dehestani

Ordokhani

Razzaghi

2019

Math Methods in App Sciences

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In this paper, we present a novel discrete scheme based on Genocchi polynomials and fractional Laguerre functions to solve multiterm variable-order time-fractional partial differential equations (M-V-TFPDEs) in the large interval. In this purpose, the accurate modified operational matrices are constructed to reduce the problems into a system of algebraic equations. Also, the computational algorithm based on the method and modified operational matrices in the large interval is easily implemented. Furthermore, we discuss the error estimation of the proposed method. Ultimately, to confirm our theoretical analysis and accuracy of numerical approach, several examples are presented. KEYWORDSerror estimation, Genocchi-fractional Laguerre functions, modified operational matrices, variable-order time-fractional partial differential equations MSC CLASSIFICATION

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Application of the modified operational matrices in multiterm variable‐order time‐fractional partial differential equations

Dehestani

Ordokhani

Razzaghi

2019

Math Methods in App Sciences

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“…On the other hand, spectral methods have been applied to different types of fractional differential equations. The spectral collocation method [27], as the best spectral method in terms of accuracy, has been applied recently for solving different types of fractional partial differential equations and fractional integro-differential equations [28].…”

Section: Introductionmentioning

confidence: 99%

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

Amin,

Abdelkawy,

Solouma

et al. 2023

Fractal Fract

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One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss–Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme’s accuracy and compare it to existing findings.

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“…On the top of this list, the spectral methods [2-5, 7, 10, 11, 39] have been improved recently. Spectral methods are exceedingly used to construct numerical algorithms for solving fractional differential equations [1,13,16,17,40,42]. In the spectral methods, the numerical solution is approximated as a truncated sum of assured basis functions.…”

Section: Introductionmentioning

confidence: 99%

Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

Doha

Abdelkawy

Amin

et al. 2019

NAMC

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In this manuscript, we introduce a spectral technique for approximating the variable-order fractional Riccati equation (VO-FRDEs). Firstly, the solution and its space fractional derivatives is expanded as shifted Chebyshev polynomials series. Then we determine the expansion coefficients by reducing the VO-FRDEs and its conditions to a system of algebraic equations. We show the accuracy and applicability of our numerical approach through four numerical examples.

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On spectral methods for solving variable-order fractional integro-differential equations

Cited by 27 publications

References 40 publications

Application of the modified operational matrices in multiterm variable‐order time‐fractional partial differential equations

Application of the modified operational matrices in multiterm variable‐order time‐fractional partial differential equations

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

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