MDPI - Publisher of Open Access Journals

6 pages, 203 KiB

Open AccessArticle

Noether Symmetries of a Generalized Coupled Lane-Emden-Klein-Gordon-Fock System with Central Symmetry

by B. Muatjetjeja, S. O. Mbusi and A. R. Adem

Symmetry 2020, 12(4), 566; https://doi.org/10.3390/sym12040566 - 5 Apr 2020

Cited by 13 | Viewed by 2017

In this paper we carry out a complete Noether symmetry analysis of a generalized coupled Lane-Emden-Klein-Gordon-Fock system with central symmetry. It is shown that several cases transpire for which the Noether symmetries exist. Moreover, we derive conservation laws connected with the admitted Noether [...] Read more.

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

9 pages, 264 KiB

Open AccessArticle

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

by Muhammad Umar Farooq, Chaudry Masood Khalique and Fazal M. Mahomed

Symmetry 2019, 11(10), 1244; https://doi.org/10.3390/sym11101244 - 4 Oct 2019

Cited by 2 | Viewed by 2270

Abstract

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. Full article

(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)

16 pages, 402 KiB

Open AccessArticle

Solving the Systems of Equations of Lane-Emden Type by Differential Transform Method Coupled with Adomian Polynomials

by Lie-jun Xie, Cai-lian Zhou and Song Xu

Mathematics 2019, 7(4), 377; https://doi.org/10.3390/math7040377 - 25 Apr 2019

Cited by 14 | Viewed by 4344

Abstract

In this work, we applied the improved differential transform method to find the solutions of the systems of equations of Lane-Emden type arising in various physical models. With our proposed scheme, the desired solutions take the form of a convergent series with easily computable components. The results disclosing the relation between the differential transforms of multi-variables and the corresponding Adomian polynomials are proven. One can see that both the differential transforms and the Adomian polynomials of those nonlinearities have the same mathematical structure merely with constants instead of variable components. By using this relation, we computed the differential transforms of nonlinear functions given in the systems. The validity and applicability of the proposed method are illustrated through several homogeneous and nonhomogeneous nonlinear systems. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

► Show Figures

Figure 1

9 pages, 725 KiB

Open AccessArticle

Noether-Like Operators and First Integrals for Generalized Systems of Lane-Emden Equations

by M. Umar Farooq

Symmetry 2019, 11(2), 162; https://doi.org/10.3390/sym11020162 - 1 Feb 2019

Cited by 4 | Viewed by 2492

Abstract

Coupled systems of Lane–Emden equations are of considerable interest as they model several physical phenomena, for instance population evolution, pattern formation, and chemical reactions. Assuming a complex variational structure, we classify the generalized system of Lane–Emden type equations in relation to Noether-like operators and associated first integrals. Various forms of functions appearing in the considered system are taken, and it is observed that the Noether-like operators form an Abelian algebra for the corresponding Euler–Lagrange-type systems. Interestingly, we find that in many cases, the Noether-like operators satisfy the classical Noether symmetry condition and become the Noether symmetries. Moreover, we observe that the classical Noetherian integrals and the first integrals we determine using the complex Lagrangian approach turn out to be the same for the underlying system of Lane–Emden equations. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

9 pages, 148 KiB

Open AccessArticle

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

by Ben Muatjetjeja and Chaudry Masood Khalique

Math. Comput. Appl. 2010, 15(3), 325-333; https://doi.org/10.3390/mca15030325 - 1 Dec 2010

Cited by 20 | Viewed by 1611

Abstract

Systems of Lane-Emden equations arise in the modelling of several physical phenomena, such as pattern formation, population evolution and chemical reactions. In this paper we construct Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a coupled Lane-Emden system. Then the first integrals with respect to Noether and partial Noether operators are obtained for the Lane-Emden system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. However, the gauge function is different in certain cases. Full article

Search Results (5)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (5)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI