Noether, Partial Noether Operators and First Integrals for the Coupled Lane-Emden System

Muatjetjeja, Ben; Khalique, Chaudry Masood

doi:10.3390/mca15030325

Cited by 6 publications

(4 citation statements)

References 30 publications

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“…Lane–Emden equations also arise in the modelling of the several physical phenomena as population evolution, chemical reactions, pattern formation, etc. (Muatjetjeja and Khalique, 2010). The main difficulty in solving the singular differential equations is the singularity behaviour that occurs at r = 0.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions

Umesh

Kumar

2020

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Purpose The purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size partitions. Design/methodology/approach First, the authors describe the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. After that, for the fast calculation of the Adomian polynomials, an algorithm is presented based on Duan’s corollary and Rach’s rule. Then, MADM is discussed for the unequal step-size partitions of the domain, to obtain the numerical solution of Lane–Emden-type equations. Moreover, convergence analysis and an error bound for the approximate solution are discussed. Findings The proposed method removes the singular behaviour of the problems and provides the high precision numerical solution in the large effective region of convergence in comparison to the other existing methods, as shown in the tested examples. Originality/value Unlike the other methods, the proposed method does not require linearization or perturbation to obtain an analytical and numerical solution of singular differential equations, and the obtained results are more physically realistic.

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Section: Introductionmentioning

confidence: 99%

Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions

Umesh

Kumar

2020

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“…We next consider the following system of Lane-Emden equations [1][2][3][4][5][6][7] a particular case of (1.1) with p 1 (x) = x k 1 and p 2 (x) = x k 2 as…”

Section: Introductionmentioning

confidence: 99%

“…In [3,4,33], authors considered the system of Lane-Emden equations (1.2) with boundary conditions y ′ 1 (0) = y ′ 2 (0) = 0, y 1 (1) = 1, y 2 (1) = 2 and shape factors k 1 = k 2 = 2 and f 1 = a y 2 1 + b y 1 y 2 , f 2 = c y 2 1 − d y 1 y 2 , occurs in catalytic diffusion reactions with the parameters a, b, c, d are actual chemical reactions.…”

Section: Introductionmentioning

confidence: 99%

Analytic solution of system of singular nonlinear differential equations with Neumann-Robin boundary conditions arising in astrophysics

Singh

2020

Preprint

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In this paper, we propose a new approach for the approximate analytic solution of system of Lane-Emden-Fowler type equations with Neumann-Robin boundary conditions. The algorithm is based on Green's function and the homotopy analysis method. This approach depends on constructing Green's function before establishing the recursive scheme for the approximate analytic solution of the equivalent system of integral equations. Unlike Adomian decomposition method (ADM) [1], the present method contains adjustable parameters to control the convergence of the approximate series solution. Convergence and error estimation of the present is provided under quite general conditions. Several examples are considered to demonstrate the accuracy of the current algorithm. Computational results reveal that the proposed approach produces better results as compared to some existing iterative methods.

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“…Similarly, for the above stated forms of P(x), F and G along with few more, ample literature on the classification of the above system with respect to Lie symmetries, Noether symmetries and production of associated first integrals is available. In this regard, the interested reader is referred to [18][19][20][21] (and references in these papers for more details). Seeking the symmetry properties and conservation laws of LE-system (2) with respect to different cases which we present here is novel.…”

Section: Introductionmentioning

confidence: 99%

First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach

2019

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The aim of the present work is to classify the Noether-like operators of two-dimensional physical systems whose dynamics is governed by a pair of Lane-Emden equations. Considering first-order Lagrangians for these systems, we construct corresponding first integrals. It is seen that for a number of forms of arbitrary functions appearing in the set of equations, the Noether-like operators also fulfill the classical Noether symmetry condition for the pairs of real Lagrangians and the generated first integrals are reminiscent of those we obtain from the complex Lagrangian approach. We also investigate the cases in which the underlying systems are reducible via quadrature. We derive some interesting results about the nonlinear systems under consideration and also find that the algebra of Noether-like operators is Abelian in a few cases.

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Noether, Partial Noether Operators and First Integrals for the Coupled Lane-Emden System

Cited by 6 publications

References 30 publications

Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions

Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions

Analytic solution of system of singular nonlinear differential equations with Neumann-Robin boundary conditions arising in astrophysics

First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach

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