Exact analytic solution of an unsolvable class of first Lane–Emden equation for polytropic gas sphere

Torres-Cordoba, Rafael; Martínez‐García, Edgar A.

doi:10.1016/j.newast.2020.101458

Cited by 7 publications

(5 citation statements)

References 7 publications

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“…Exact solutions are available for particular cases of Eq. (3) [7], but not for the general case according to the best of our knowledge. This is motivation enough for seeking approximate solutions.…”

Section: Introductionmentioning

confidence: 57%

A Modified Spectral Relaxation Method for Some Emden-Fowler Equations

Marewo¹

2021

Recent Advances in Numerical Simulations

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In this chapter, we present a modified version of the spectral relaxation method for solving singular initial value problems for some Emden-Fowler equations. This study was motivated by the several applications that these equations have in Science. The first step of the method of solution makes use of linearisation to solve the model problem on a small subinterval of the problem domain. This subinterval contains a singularity at the initial instant. The first step is combined with using the spectral relaxation method to recursively solve the model problem on the rest of the problem domain. We make use of examples to demonstrate that the method is reliable, accurate and computationally efficient. The numerical solutions that are obtained in this chapter are in good agreement with other solutions in the literature.

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Section: Introductionmentioning

confidence: 57%

A Modified Spectral Relaxation Method for Some Emden-Fowler Equations

Marewo¹

2021

Recent Advances in Numerical Simulations

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“…It is also shown in [15] that Adomian's method may be interpreted as a homotopy perturbation method. It seems that the transformation method in [16] gives solution differs from the numerical solutions for the polytropic gas sphere problem with polytropic indices m = 2 and 3. One of the shortcomings of the analytical methods is that many terms of complex forms which are hard to deduce are needed to be included into the series expansion to increase the accuracy and it is difficult to find approximated formulae for large domains.…”

Section: Introductionmentioning

confidence: 97%

“…type of equation, some methods including analytical ones, numerical ones and their combinations have been proposed in the literature. For analytical methods, Adomian decomposition method [8], [9], homotopy method [10]- [15], and transformation method [16] are developed. Adomian decomposition method was first proposed to solve polytropic gas sphere problem in [8] where the Lane-Emden equation is reformulated into a form of operator equation onto which Adomian decomposition is applied and a recursive relation of the components in the series expansion of the solution can be derived.…”

Section: Introductionmentioning

confidence: 99%

A Deep-Learning-Based Method for Solving Nonlinear Singular Lane-Emden Type Equation

Long

Wang

et al. 2020

IEEE Access

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The nonlinear Lane-Emden type equation can be used to describe many physical phenomena. To solve this type of equation, a method based on deep neural network is proposed. The output layer of this network has two layers, the last one of which scaling the outputs of their neighbors with an aim at coping with issues where the values of function to be approximated are much less or larger than the order of 1. The Lane-Emden equation and its initial conditions are employed to construct the loss function, and the problem of solving Lane-Emden equation is transformed into an optimization problem. A hybrid method combined Adam and L-BFGS-B methods is used to solve the optimization problem and consequently the Lane-Emden type equation is solved. To increase the accuracy, an adaptive strategy is incorporated into the training data sampling method. Especially, a strategy in coping with the issue about solving white-dwarf problem is proposed. Numerical experiments are conducted in which reference solutions including analytical solutions and numerical solutions given by Runge-Kutta method are used to verify the effectiveness of our proposed method. More importantly, the solutions over large domains are calculated by the use of the proposed method. The results show that the results given by our method are in good agreement with the reference solutions, and in cases where existent neural-network-based method fails, our method is able to produce convincible results.INDEX TERMS Deep learning, artificial neural network, Lane-Emden type equation, polytropic gas sphere model, isothermal gas sphere model, white-dwarf model.

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“…In fact, the vast majority of phenomena in natural or social sciences are nonlinear, but sometimes they are approximated as linear problems to reduce complexity. In physics (Seadawy et al , 2018; Zhou and Zhu, 2015; Zhou et al , 2014; Hosseini et al , 2021; Hosseini et al , 2020; Yildrim et al , 2021), thermal science (Nazir et al , 2022; Sohail et al , 2022; Imran et al , 2020; Sohail et al , 2021), vibration (He et al , 2022; Wang and Si, 2023a, 2023b), biological science (Kumar et al , 2019; Lü* et al , 2021; Yin et al , 2021a, 2021b), astronomy (Torres-Córdoba and Martínez-García, 2021; Yadav et al , 2021) and other disciplines, there are various nonlinear motion phenomena, which need to be described by NPDEs. Therefore, the solution of NPDEs naturally becomes an important part of nonlinear theory.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of breather, multi-wave, interaction and other wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

Wang

2023

HFF

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Purpose The purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves. Design/methodology/approach By means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions. Findings The breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before. Originality/value The solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.

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Exact analytic solution of an unsolvable class of first Lane–Emden equation for polytropic gas sphere

Cited by 7 publications

References 7 publications

A Modified Spectral Relaxation Method for Some Emden-Fowler Equations

A Modified Spectral Relaxation Method for Some Emden-Fowler Equations

A Deep-Learning-Based Method for Solving Nonlinear Singular Lane-Emden Type Equation

Dynamics of breather, multi-wave, interaction and other wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

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