An operational matrix based on Chelyshkov polynomials for solving multi-order fractional differential equations

Talaei, Younes; Asgari, M.

doi:10.1007/s00521-017-3118-1

Cited by 39 publications

(36 citation statements)

References 18 publications

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“…It can be seen that the new method performs better than the method in [31] and much better than the technique in [41]. The error achieved by the presented method becomes smaller and smaller with the increment of n and converge to zero at n = 20.…”

Section: Illustrative Examplesmentioning

confidence: 80%

“…Symbol "-" means that the result for n is unavailable for that method. From Table 1, it can be seen that the errors achieved by the presented method are less than those in [31] for all values of n. In addition to that, when n increases, the errors are reduced until they become zero at n = 10, η = 0.25 and n = 13, η = 0.25. This means that the presented method is more accurate than that in [31] for this problem.…”

Section: Illustrative Examplesmentioning

confidence: 85%

“…Table 1 shows a comparison of the results obtained by the present method and those in [31] in terms of L ∞ ω η and L 2 ω η errors for different values of n and η. Note that the results for the method that we compare have been executed by the method in [31] and we use these results to make a direct comparison with the presented method. Symbol "-" means that the result for n is unavailable for that method.…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…The exact solution of this problem is y(x) = x 4 − 1 2 x 3 . Table 2 describes the results obtained by the present method and those in [31,41] in terms of L 2 ω η errors for different values of n with α = 1 4 .…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…Using these polynomials, a solution has been given of weakly singular integral equations in [27], Volterra-Fredholm integral equations in [28], linear functional integro-differential equations with variable coefficients in [29] and Volterra-Fredholm-Hammerstein integral equations in [30]. Furthermore, the operational matrix of fractional derivatives based on Chelyshkov polynomials for solving multi-order fractional differential equations has been presented in [31] and the operational matrix of fractional integration to solve a class of nonlinear fractional differential equations has been introduced in [32]. Recently, Talaei [33] has proposed a numerical algorithm based on FCHFs for solving linear weakly singular Volterra integral equation.…”

Section: Introductionmentioning

confidence: 99%

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An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

2020

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Fractional differential equations have been applied to model physical and engineering processes in many fields of science and engineering. This paper adopts the fractional-order Chelyshkov functions (FCHFs) for solving the fractional differential equations. The operational matrices of fractional integral and product for FCHFs are derived. These matrices, together with the spectral collocation method, are used to reduce the fractional differential equation into a system of algebraic equations. The error estimation of the presented method is also studied. Furthermore, numerical examples and comparison with existing results are given to demonstrate the accuracy and applicability of the presented method.

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Section: Illustrative Examplesmentioning

confidence: 80%

Section: Illustrative Examplesmentioning

confidence: 85%

Section: Illustrative Examplesmentioning

confidence: 99%

Section: Illustrative Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

2020

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A numerical method for variable‐order fractional version of the coupled 2D Burgers equations by the 2D Chelyshkov polynomials

Hosseininia

Heydari

Ghaini

2021

Math Methods in App Sciences

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This paper represents a system of variable-order (VO) time fractional 2D Burgers equations and expresses a semidiscrete approach by applying the 2D Chelyshkov polynomials (CPs) for solving this system. In this model, the fractional derivative of the Caputo type is considered. To solve this system, we first discretize the VO time fractional derivatives. Next, we obtain a recurrent algorithm by using the weighted finite difference method with parameter . Then, utilizing the 2D CPs, we expand the unknown solution and replace it in the main system. In the sequel, we use the differentiation operational matrices and the collocation method to extract an algebraic system of equations which can be easily solved.The validity of the formulated method is investigated through three numerical examples. KEYWORDS 2D burgers equation, Chelyshkov polynomials (CPs), variable-order (VO) time fractional derivative MSC CLASSIFICATION 35R11 INTRODUCTIONThe subject of variable-order (VO) fractional calculus (integration/differentiation of VO fractional order ) is a useful mathematical tool for deeper analysis of dynamical phenomena. 1 In fact, the mathematical systems modelled by this novel concept in science and engineering show more accuracy and sensitivity. 2 We remind that obtaining an analytical solution for the problem that includes VO fractional operator is often very complex and usually impossible. So, using approximate methods are inevitable for solving such problems. The interested reader can find some numerical methods for solving such VO fractional problems in previous works. [3][4][5][6][7] By using the time fractional model of Burgers' equation, many problems in physics can be modelled such as electromagnetic wave, weak shock propagation, flow systems, electromagnetic waves and shock waves in a viscous medium. [8][9][10][11] So, solving different types of this equation is highly demanded. An approximate technique for solving a system of fractional Burgers' equations has been proposed in Korpinar et al. 12 Li et al. 13 suggested a numerical scheme for the time fractional model of Burgers' equation by employing the separation of variables method and Cole-Hopf transformation. The method of differential transformation has been applied in Abazari and Borhanifar 14 for solving some categories of the Burgers equation. In Veereshaa and Prakashab, 15 a numerical method for solving coupled time fractional Burgers' equation has been proposed.The Chelyshkov polynomials (CPs) as a most important sequence of polynomials have many privileges, such as the orthogonality and spectral accuracy. During recent years, the CPs have been extensively employed for solving diverse

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Obtaining Controllable Pseudo-Upper and Lower Triangular Multi-Order State-Space Realizations from a Special Case of Incommensurate Fractional-Order Transfer Functions

Tabatabaei

2024

Circuits Syst Signal Process

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An operational matrix based on Chelyshkov polynomials for solving multi-order fractional differential equations

Cited by 39 publications

References 18 publications

An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

A numerical method for variable‐order fractional version of the coupled 2D Burgers equations by the 2D Chelyshkov polynomials

Obtaining Controllable Pseudo-Upper and Lower Triangular Multi-Order State-Space Realizations from a Special Case of Incommensurate Fractional-Order Transfer Functions

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