Emden Equation Research Articles

Spectral Collocation Approach via Normalized Shifted Jacobi Polynomials for the Nonlinear Lane-Emden Equation with Fractal-Fractional Derivative

Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations with generalized Caputo derivatives that appear in the study of astronomical objects. The offered solution is approximated as a truncated series of the normalized shifted Jacobi polynomials under the assumption that the exact solution is an element in L2. The spectral collocation method is used as a solver to obtain the unknown expansion coefficients. The Jacobi roots are used as collocation nodes. Our solutions can easily be a generalization of the solutions of the classical LE equation, by obtaining a numerical solution based on new parameters, by fixing these parameters to the classical case, we obtain the solution of the classical equation. We provide a meticulous convergence analysis and demonstrate rapid convergence of the truncation error concerning the number of retained modes. Numerical examples show the effectiveness and applicability of the method. The primary benefits of the suggested approach are that we significantly reduce the complexity of the underlying differential equation by solving a nonlinear system of algebraic equations that can be done quickly and accurately using Newton’s method and vanishing initial guesses.

Numerical solution of Lane-Emden pantograph delay differential equation: stability and convergence analysis

In this work, a collocation approach-based Bernstein operational matrix of differentiation method is used for obtaining the numerical solution of a class of modified Lane-Emden equation with delay in pantograph sense. The proposed numerical algorithm provides numerical solution by discretising the Lane-Emden pantograph delay differential equation into a system of algebraic equations which can be solved directly using any mathematical software. The consistency of the proposed numerical technique is verified with the convergence analysis of the proposed algorithm. The stability analysis of the model is also given using the Lyapunov function. Test examples and graphical representations of their solutions are included to illustrate the applicability and superiority of the proposed method over existing methods.

Numerical solution for the system of Lane-Emden type equations using cubic B-spline method arising in engineering

<abstract><p>In this study, we develop a collocation method based on cubic B-spline functions for effectively solving the system of Lane-Emden type equations arising in physics, star structure, and astrophysics. To overcome the singularity behavior of the considered system at τ = 0, we apply the L'Hôpital rule. Furthermore, we have carried out a convergence analysis of the proposed method and have demonstrated that it has a second-order convergence. To demonstrate the effectiveness, accuracy, simplicity, and practicality of the method, five test problems are solved numerically and the maximum absolute errors of the proposed method are compared with those of some existing methods.</p></abstract>

Approximation of functions belonging to Hölder’s class and solution of Lane-Emden differential equation using Gegenbauer wavelets

In this paper, a very new technique based on the Gegenbauer wavelet series is introduced to solve the Lane-Emden differential equation. The Gegenbauer wavelets are derived by dilation and translation of an orthogonal Gegenbauer polynomial. The orthonormality of Gegenbauer wavelets is verified by the orthogonality of classical Gegenbauer polynomials. The convergence analysis of Gegenbauer wavelet series is studied in H?lder?s class. H?lder?s class H?[0,1) and H?[0,1) of functions are considered, H?[0,1) class consides with classical H?lder?s class H?[0, 1) if ?(t) = t?, 0 < ? ? 1. The Gegenbauer wavelet approximations of solution functions of the Lane-Emden differential equation in these classes are determined by partial sums of their wavelet series. In briefly, four approximations E(1) 2k?1,0, E(1) 2k?1,M, E(2) 2k?1,0, E(2) 2k?1,M of solution functions of classes H?[0, 1), H?[0, 1) by (2k?1, 0)th and (2k?1,M)th partial sums of their Gegenbauer wavelet expansions have been estimated. The solution of the Lane-Emden differential equation obtained by the Gegenbauer wavelets is compared to its solution derived by using Legendre wavelets and Chebyshev wavelets. It is observed that the solutions obtained by Gegenbauer wavelets are better than those obtained by using Legendre wavelets and Chebyshev wavelets, and they coincide almost exactly with their exact solutions. This is an accomplishment of this research paper in wavelet analysis.

A Novel Approach to Simulate Lane-Emden and Emden-Fowler Equations Using Curriculum Learning-Based Unsupervised Symplectic Artificial Neural Network

A Novel Approach to Simulate Lane-Emden and Emden-Fowler Equations Using Curriculum Learning-Based Unsupervised Symplectic Artificial Neural Network

Numerical solution of Lane-Emden Pantograph Delay Differential Equation: Stability and Convergence Analysis.

Numerical solution of Lane-Emden Pantograph Delay Differential Equation: Stability and Convergence Analysis.

On Semilinear Elliptic Equations with Hardy-Leray Potentials

This paper is concerned with a semilinear elliptic equation with the Hardy-Leray potential. We employ the method of moving planes to prove the radial symmetry of positive solutions. Based on this result, we obtain the Liouville theorem in subcritical case. In addition, we find special radial solutions in critical case. All the properties above are similar to the corresponding results of the Lane-Emden equation.

Pointwise A Priori Estimates for Solutions to Some p-Laplacian Equations

In this article, we apply blow-up analysis to study pointwise a priori estimates for some p-Laplacian equations based on Liouville type theorems. With newly developed analysis techniques, we first extend the classical results of interior gradient estimates for the harmonic function to that for the p-harmonic function, i.e., the solution of Δpu = 0, x ∈ Ω. We then obtain singularity and decay estimates of the sign-changing solution of Lane-Emden-Fowler type p-Laplacian equation −Δpu = |u|λ − 1u, x ∈ Ω, which are then extended to the equation with general right hand term f(x, u) with certain asymptotic properties. In addition, pointwise estimates for higher order derivatives of the solution to Lane-Emden type p-Laplacian equation, in a case of p = 2, are also discussed.

Stellar Structure Model in the Post-Newtonian Approximation

In this work the influence of the post-Newtonian corrections to the equations of stellar structure are analyzed. The post-Newtonian Lane–Emden equation follows from the corresponding momentum density balance equation. From a polytropic equation of state the solutions of the Lane–Emden equations in the Newtonian and post-Newtonian theories are determined and the physical quantities for the Sun, the white dwarf Sirius B and neutron stars with masses M ≃ 1.4 M ⊙, 1.8 M ⊙ and 2.0 M ⊙ are calculated. It is shown that the post-Newtonian corrections to the fields of mass density, pressure and temperature are negligible for the Sun and Sirius B, but for stars with strong fields the differences become important. For the neutron stars analyzed here the central pressure and the central temperature which follow from the post-Newtonian Lane–Emden equation are about fifty to sixty percent greater than those of the Newtonian theory and the central mass density is about three to four percent smaller.

A non-local semilinear eigenvalue problem

We prove that positive solutions of the fractional Lane–Emden equation with homogeneous Dirichlet boundary conditions satisfy pointwise estimates in terms of the best constant in Poincaré’s inequality on all open sets, and are isolated in L^1 on smooth bounded ones, whence we deduce the isolation of the first non-local semilinear eigenvalue.

An efficient technique based on higher order Haar wavelet method for Lane–Emden equations

This paper aims at introducing an effective computational technique for finding the solution to well-known Lane–Emden type equations by employing the higher order numerical method based on Haar wavelet expansions. A modified version of Haar wavelet method, known as Higher order Haar wavelet method has been discussed to enhance the accuracy and rate of convergence. This numerical method successfully tackles the singularity at x=0 and gives the approximated solution of the given Lane–Emden type equations in terms of higher order Haar wavelet expansions. Various numerical examples of Lane–Emden type equations have been discussed to demonstrate the validity and efficiency of the proposed algorithm. The numerical findings obtained by employing the present method have been presented with the aid of tables and graphs. Also, the comparisons made with well-known existing techniques, including finite difference schemes, various spline and decomposition methods, help to manifest the better accuracy and high efficiency of the present approach.

Stability of 2D steady Euler flows related to least energy solutions of the Lane-Emden equation

In this paper, we investigate nonlinear stability of planar steady Euler flows related to least energy solutions of the Lane-Emden equation in a smooth bounded domain. We prove the orbital stability of these flows in terms of both the Ls norm of the vorticity for any s∈(1,+∞) and the energy norm. As a consequence, nonlinear stability is obtained when the least energy solution is unique, which actually holds for a large class of domains and exponents. The proofs are based on a new variational characterization of least energy solutions in terms of the vorticity, a compactness argument, and proper use of conserved quantities of the Euler equation.

A convergence-preserving non-standard finite difference scheme for the solutions of singular Lane-Emden equations

The derivation of numerical schemes for the solution of Lane-Emden equations requires meticulous consideration because they are highly nonlinear in nature; have singularity behaviors at the origin and in some cases, their exact solutions are known for only a few parameter ranges. This explains the reason for the failure of some existing methods in approximating such problems. Thus, in this research, a convergence-preserving Non-Standard Finite Difference Scheme (NSFDS) shall be derived for the solution of Lane-Emden equations. The main idea in this work is the approximation of both the linear and nonlinear terms non-locally and the renormalization/reformulation of the numerator and denominator functions of the Lane-Emden equations. The need for this approach came up due to some setbacks of existing methods where the exact solutions’ qualitative properties are not routinely transferred to the approximate solutions. The analysis of some properties of the NSFDS like convergence, dynamical consistency, monotone dependence on the initial value, and monotonicity of solutions as well as perturbation solution analysis shall be carried out. The NSFDS was then employed in solving some classes of singular initial value problems of the Lane-Emden type and the results obtained show that the proposed method is efficient and computationally cheap.

Schrödinger–Poisson systems under gradient fields

A singularity-free generalisation of Newtonian gravity can be constructed (Lazar in Phys Rev D 102:096002, 2020) within the framework of gradient field theory. This procedure offers a straightforward regularisation of Newtonian gravity and remains equally well applicable to other fields, such as electromagnetic fields. Here, with the aim of finding potentially measurable effects of gradient fields on the dispersion properties of various media, we present a quantum kinetic treatment of matter under gradient fields. The method is based on the application of the Wigner–Moyal procedure to the modified Schrödinger–Poisson equation emerging in the framework of gradient field theory. This allows us to treat, on equal footing, three different scenarios, namely self-gravitating systems, plasmas, and cold atoms in magneto-optical traps. We address the signature of gradient fields in the elementary excitations of these media. In particular, we estimate this effect to be accessible in state-of-the-art plasma-based experiments. We discuss in detail the classical kinetic and hydrodynamic limits of our approach and obtain a class of generalised Lane–Emden equations, in the context of gradient field theory, which remain valid in the three scenarios discussed here.

Chebyshev-quasilinearization method for solving fractional singular nonlinear Lane-Emden equations

In this paper, we propose a method for solving some classes of the singular fractional nonlinear Lane-Emden type equations. The method is proposed by utilizing the second-kind Chebyshev wavelets in conjunction with the quasilinearization technique. The operational matrices for the second-kind Chebyshev wavelets are used. The method is tested on the fractional standard Lane-Emden equation, the fractional isothermal gas spheres equation, and some other examples. We compare the results produced by the present method with some well-known results to show the accuracy and efficiency of the method.

Wavelet solution of a strongly nonlinear Lane–Emden equation

Wavelet solution of a strongly nonlinear Lane–Emden equation

Frameworks for generalized anisotropic conformally flat polytropes in f(R) gravity

The purpose of this paper is to analyze the conformally flat spherically symmetric fluid distribution with generalized polytropic equations of state. We have developed two different framework for two different definitions of generalized polytropes. The frameworks for development of modified Lane–Emden equation are presented for both cases. The conformally flat condition is used to calculate anisotropy factor which transform considered systems into consistent systems. Tolman mass function is used in the polytropic models to check their stability.

Construct a realistic stellar model with polytropic relation

The aim of this study is to construct a simple stellar model with non-uniform polytropic index. We find that the Emden equation cannot deal with the polytrope gas sphere with non-uniform polytropic index in a real star, and then we construct a realistic stellar model. The key point is that we should solve the two independent equations for density and pressure due to the essence of polytropic relation, but not the Emden equation which combines the hydrostatic balance and polytropic relation. We take the Sun for a computational example to find that this simple model yields quite a good result compared to the MESA code. The advantage of this simple model lies in its much simpler equation of state than that in the standard stellar model.

Neutrino bound states and bound systems

Yukawa interactions of neutrinos with a new light scalar boson ϕ can lead to formation of stable bound states and bound systems of many neutrinos (ν-clusters). For allowed values of the coupling y and the scalar mass mϕ, the bound state of two neutrinos would have the size larger than 1012 cm. Bound states with sub-cm sizes are possible for keV scale sterile neutrinos with coupling y > 10−4. For ν-clusters we study in detail the properties of final stable configurations. If there is an efficient cooling mechanism, these configurations are in the state of degenerate Fermi gas. We formulate and solve equations of the density distributions in ν-clusters. In the non-relativistic case, they are reduced to the Lane-Emden equation. We find that (i) stable configurations exist for any number of neutrinos, N; (ii) there is a maximal central density ∼ 109 cm−3 determined by the neutrino mass; (iii) for a given mϕ there is a minimal value of Ny3 for which stable configurations can be formed; (iv) for a given strength of interaction, Sϕ = (ymν/mϕ)2, the minimal radius of ν-clusters exists. We discuss the formation of ν-clusters from relic neutrino background in the process of expansion and cooling of the Universe. One possibility realized for Sϕ > 700 is the development of instabilities in the ν-background at T < mν which leads to its fragmentation. For Sϕ ∈ [70, 700]) they might be formed via the growth of initial density perturbations in the ν-background and virialiazation, in analogy with the formation of Dark Matter halos. For allowed values of y, cooling of ν-clusters due to ϕ-bremsstrahlung and neutrino annihilation is negligible. The sizes of ν-clusters may range from ∼ km to ∼ 5 Mpc.

Integration of the Lane–Emden equation for relativistic anisotropic polytropes through gravitational decoupling: a novel approach

In this work we propose a novel approach to integrate the Lane–Emden equations for relativistic anisotropic polytropes. We take advantage of the fact that Gravitational Decoupling allows to decrease the number of degrees of freedom once a well known solution of the Einstein field equations is provided as a seed so after demanding the polytropic equation for the radial pressure the system is automatically closed. The approach not only allows to extend both isotropic or anisotropic known solutions but simplifies the computation of the Tolman mass whenever the Minimal Geometric Deformation is considered given that the g_{tt} component of the metric remains unchanged. We illustrate how the the method works by analyzing the solutions obtained from Tolman IV, Durgapal IV and Wymann IIa isotropic systems as a seed for the integration.