A new algorithm for solving differential equations of Lane–Emden type

Wazwaz, Abdul‐Majid

doi:10.1016/s0096-3003(99)00223-4

Cited by 287 publications

(318 citation statements)

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“…Numerical and perturbation approaches to solve eqn (1.2) have been considered in Horedt [20,21], Bender [5] and Lema [29]. Other methods have also been invoked for the solution of eqn (1.2) (see Roxbough and Stocken [36], Adomian et al [2], Shawagfeh [38], Burt [8], Wazwaz [41] and Liao [27]). It is known that for r = 0, 1 and 5, eqn (1.2) has exact solutions (Chandrasekhar [9], Davis [11], Datta [10] and Wrubel [43]).…”

Section: Introductionmentioning

confidence: 99%

“…The transformation properties in these instances are known (see [15,32,40]). For n = 2, Wazwaz [41] considered eqn (1.7) for some particular functions f (y) using an algorithm based on the Adomian decomposition method. Our approach is completely different and provides a complete Noether point symmetry classification of eqn (1.7) for different forms of f (y).…”

Section: Introductionmentioning

confidence: 99%

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Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Khalique¹,

Mahomed²,

Muatjetjeja³

2008

JNMP

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We classify the Noether point symmetries of the generalized Lane-Emden equation y ′′ + ny ′ /x + f (y) = 0 with respect to the standard Lagrangian L = x n y ′2 /2 − x n f (y)dy for various functions f (y). We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f (y). One of them is f (y) = αy r , where r = 0, 1. The case r = 5 was considered previously and only a one-parameter family of solutions was presented. Here we provide a complete integration not only for r = 5 but for other r values. We also give the Lie point symmetries for each case. In two of the new cases, the single Noether symmetry is also the only Lie point symmetry.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Khalique¹,

Mahomed²,

Muatjetjeja³

2008

JNMP

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“…The first method is to transform the equation into an integrodifferential equation and then iterate this equation [2]. The second method is developed by [1], [3] and [4]. This method uses the Adomian decomposition method [5] and introduces the solution in the form of a power series.…”

Section: Shakeri (2008)mentioning

confidence: 99%

Approximate Solution to the Fractional Lane–Emden Type Equations

Nouh

Abdel-Salam

2017

Iran J Sci Technol Trans Sci

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Abstract.In this paper, approximate solutions for a class of fractional Lane-Emden type equations based on the series expansion method are presented. Various examples are introduced and discussed.The recurrence relation for the components of the approximate solution is constructed. In the standard integer order derivative, it is shown that the results are the same as those obtained by Adomian decomposition method, homotopy perturbation method, modified Laplace decomposition method and variational iteration method.

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“…The approximate solutions of the Lane-Emden equation were given by Adomian decomposition method [4]- [6], homotopy perturbation method [7]- [9], variational iteration method [10]- [12], differential transformation method [13], Wavelets-Collocation method [14], [15] and so on.…”

Section:  mentioning

confidence: 99%