Numerical Study of Fractional Differential Equations of Lane-Emden Type by Method of Collocation

Mechee, Mohammed S.; Senu, Norazak

doi:10.4236/am.2012.38126

Cited by 42 publications

(36 citation statements)

References 23 publications

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“…The Lane-Emden differential equations are very important in astrophysics, for this reason, writing articles to review them [61,62,63,64,65]. Now, we consider the linear Lane-Emden equation of fractional order as follows with the initial conditions…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…Table 6 shows the comparison of the absolute error obtained by the present Tau method, the reproducing kernel method (RKM) [61] and the collocation method in Ref. [63].…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…Graphs of the residual errors for example 6 with m = 10 and α = 0.50, by using the Tau and collocation methods Table 6. Comparison of the present Tau method, the reproducing kernel method [61], and the collocation method [63] of the absolute errors for example 6 where S is the squeeze number, M is the Hartman number, a is a constant. a > 0 corresponds to suction and a < 0 corresponds to injection at the lower stationary disk in momentum problem [67] .…”

Section: Illustrative Examplesmentioning

confidence: 99%

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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: Illustrative Examplesmentioning

confidence: 99%

“…Table 6 shows the comparison of the absolute error obtained by the present Tau method, the reproducing kernel method (RKM) [61] and the collocation method in Ref. [63].…”

Section: Illustrative Examplesmentioning

confidence: 99%

Section: Illustrative Examplesmentioning

confidence: 99%

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Novel orthogonal functions for solving differential equations of arbitrary order

2017

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“…The most common spectral method from the strong form of the equations is known as collocation. In collocation techniques, the partial differential equation must be satisfied at a set of grid, or more precisely, collocation points (see, for instance, [6][7][8][9][10]). Spectral methods also have become increasingly popular for solving fractional differential equations [11][12][13][14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Approximate Solutions of Fisher's Type Equations with Variable Coefficients

Bhrawy

Alghamdi

2013

Abstract and Applied Analysis

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The spectral collocation approximations based on Legendre polynomials are used to compute the numerical solution of timedependent Fisher's type problems. The spatial derivatives are collocated at a Legendre-Gauss-Lobatto interpolation nodes. The proposed method has the advantage of reducing the problem to a system of ordinary differential equations in time. The four-stage A-stable implicit Runge-Kutta scheme is applied to solve the resulted system of first order in time. Numerical results show that the Legendre-Gauss-Lobatto collocation method is of high accuracy and is efficient for solving the Fisher's type equations. Also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations.

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“…Recently, an analytical and numerical solution to the Lane-Emden equation presented by Gorder and Vajravelu [14] by using the traditional power series approach and the homotopy anlaysis method. The collocation method has been used by Mechee and Senu [10] to obtain the numerical solution of the Lane-Emden type. Recently, Chebyshev Neural Network based model has been used by Mall and Chakraverty [7] to solve the Lane-Emden type equations where the artificial neutral network used to solve the singularity in Lane-Emden type equations.…”

Section: Introductionmentioning

confidence: 99%

Chebyshev neural network model with linear and nonlinear active functions

Chaharborj

Mahmoudi²

2016

IJBAS

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In this paper the second order non-linear ordinary differential equations of Lane-Emden type as singular initial value problems using Chebyshev Neural Network (ChNN) with linear and nonlinear active functions has been studied. Active functions as, F(z)=z, sinh(x), tanh(z) are considered to find the numerical results with high accuracy. Numerical results from Chebyshev Neural Network shows that linear active function has more accuracy and is more convenient compare to other functions.

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Numerical Study of Fractional Differential Equations of Lane-Emden Type by Method of Collocation

Abstract: Lane-Emden differential equations of order fractional has been studied. Numerical solution of this type is considered by collocation method. Some of examples are illustrated. The comparison between numerical and analytic methods has been introduced.

Cited by 42 publications

References 23 publications

Novel orthogonal functions for solving differential equations of arbitrary order

Novel orthogonal functions for solving differential equations of arbitrary order

Approximate Solutions of Fisher's Type Equations with Variable Coefficients

Chebyshev neural network model with linear and nonlinear active functions

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