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We construct a new family of sign-changing solutions for a two-dimensional Lane-Emden problem with large exponent whose shape resembles a tower with alternating sign of bubbles solving different singular Liouville equations on the whole plane. [ABSTRACT FROM AUTHOR]

This paper presents new relativistic composite polytropic models for compact stars by simultaneously solving Einstein field equations with the polytropic state equation to simulate the spherically symmetric, static matter distribution. Using a non-uniform polytropic index, we get the Tolman–Oppenheimer–Volkoff equation for the relativistic composite polytrope (CTOV). To analyze the star's structure, we numerically solve the CTOV equation and compute the Emden and mass functions for various relativistic parameters and polytropic indices appropriate for neutron stars. The calculation results show that, as the relativistic parameter approaches zero, we recover the well-known Lane-Emden equation from the Newtonian theory of polytropic stars; thus, testing the computational code by comparing composite Newtonian models to those in the literature yields good agreement. We compute composite relativistic models for the neutron star candidates Cen X-3, SAXJ1808.4-3658, and PSR J1614-22304. We compare the findings with various existing models in the literature. Based on the accepted models for PSR J1614-22304 and Cen X-3, the star's core radius is predicted to be between 50 and 60% percent of its total radius, while we found that the radius of the core of star SAXJ1808.4-3658 is around 30% of the total radius. Our findings show that the neutron star structure may be approximated by a composite relativistic polytrope, resulting in masses and radii that are quite consistent with observation. [ABSTRACT FROM AUTHOR]

In this paper, the gravity with a deviation is considered. Modification of the Lane-Emden equation and Jeans’ instability condition is performed based on the gravity with a deviation. Some exact and numerical solutions are given for the modified Lane-Emden equation. [ABSTRACT FROM AUTHOR]

The purpose of this study is to propose a computationally effective algorithm for the numerical evaluation of a fractional-order system of singular Lane-Emden type equations arising in physical problems. The fractional operator considered is in the sense of the Liouville-Caputo derivative. The presented matrix collocation method is based upon a combination of the quasilinearization method (QLM) and the shifted Legendre functions (SLFs) and is called QLM-SLFs method. By applying first the QLM to the nonlinear underlying system, we get a family of linear equations. Hence, a spectral matrix collocation scheme relied on the SLFs is designed to solve the resulting sequence of linear system of equations at very few iterations. The uniform convergence of the shifted Legendre expansion series solution is established. To illustrate the effectiveness of the proposed QLM-SLFs technique in the present paper, three test examples are carried out. The applicability and validity of the proposed method are testified through comparisons with the outcomes of other existing procedures in the literature. The proposed QLM-SLFs method is efficient and easy to implement. The approximation obtained by the method also converges quickly to the solutions of the underlying model problem. In comparison with available existing computational procedures, the QLM-SLFs approach shows that the use of Legendre functions together with QLM provides solutions with high accuracy and exponential convergence rate. [ABSTRACT FROM AUTHOR]

A high-order numerical scheme based on collocation of a quintic B-spline over finite element is proposed for the numerical solution of a class of nonlinear singular boundary value problems (SBVPs) arising in various physical models in engineering and applied sciences. Five illustrative examples are presented to illustrate the applicability and accuracy of the method. In order to justify the advantage of the proposed numerical scheme, the computed results are compared with the results obtained by two other fourth-order numerical methods, namely the finite difference method (Chawla et al. in BIT 28(1):88–97, 1988) and B-spline collocation method (Goh et al. in Comput Math Appl 64:115–120, 2012). [ABSTRACT FROM AUTHOR]

In this paper, modulus of continuity, second kind Chebyshev wavelet and Hölder class are studied. The moduli of continuity and approximations of functions whose second derivative belonging to Hölder class have been determined by second kind Chebyshev wavelet. The operational matrix of integration for second kind Chebyshev wavelet has been framed. Using this, numerical solutions of non-linear singular differential equations have been obtained. The modulus of continuity, approximations, solution of Lane-Emden equation of index p=1 as well as the comparison with the exact solution and applicability of second kind Chebyshev wavelet method in finding numerical solution of Chandrasekhar's white dwarf equation are significant achievements of this paper. [ABSTRACT FROM AUTHOR]

The positive solutions to the critical Lane–Emden equation in R n have been classified completely by the moving plane method. Afterwards, with additional conditions such as limited energy or volume, there is proof that relies entirely on integration by parts and inequalities techniques. In this paper, the authors provide a new approach to obtain the same classification results without further assumptions. [ABSTRACT FROM AUTHOR]

In this paper we derive the best constant for the following L^{\infty }-type Gagliardo-Nirenberg interpolation inequality \begin{equation*} \|u\|_{L^{\infty }}\leq C_{q,\infty,p} \|u\|^{1-\theta }_{L^{q+1}}\|\nabla u\|^{\theta }_{L^p},\quad \theta =\frac {pd}{dp+(p-d)(q+1)}, \end{equation*} where parameters q and p satisfy the conditions p>d\geq 1, q\geq 0. The best constant C_{q,\infty,p} is given by \begin{equation*} C_{q,\infty,p}=\theta ^{-\frac {\theta }{p}}(1-\theta)^{\frac {\theta }{p}}M_c^{-\frac {\theta }{d}},\quad M_c≔\int _{\mathbb {R}^d}u_{c,\infty }^{q+1} dx, \end{equation*} where u_{c,\infty } is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when u=Au_{c,\infty }(\lambda (x-x_0)) for any real numbers A, \lambda >0 and x_{0}\in \mathbb {R}^d. In fact, the generalized Lane-Emden equation in \mathbb {R}^d contains a delta function as a source and it is a Thomas-Fermi type equation. For q=0 or d=1, u_{c,\infty } have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that u_{c,m}\to u_{c,\infty } and C_{q,m,p}\to C_{q,\infty,p} as m\to +\infty for d=1, where u_{c,m} and C_{q,m,p} are the function achieving equality and the best constant of L^m-type Gagliardo-Nirenberg interpolation inequality, respectively. [ABSTRACT FROM AUTHOR]

We present an algorithm for the result of differential equations (DEs) by using linearly independent Hosoya polynomials of trees. With the newly adopted strategy, the desired outcome is expanded in the form of a collection of continuous polynomials over an interval. Nevertheless, compared to other methods for solving differential equations, this method's precision and effectiveness relies on the size of the collection of Hosoya polynomials, and the process is easier. Excellent agreement between the exact and approximate solutions is obtained when the current scheme is used to crack linear and nonlinear equations. Potentially, this method could be used in more intricate systems for which there are no exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, the clique polynomial method (CPM) is proposed for the numerical solution of Singular Lane-Emden type equations. A new operational matrix of integration concerning clique polynomials of the complete graph has been generated in its generalized representation. This operational matrix is applied to differential equations and is transformed into a system of algebraic equations that can be solved efficiently with the help of Newton's iterative solver. The efficiency of the developed method is revealed by considering illustrative examples, and the obtained results are compared favorably with the corresponding exact solution and errors. [ABSTRACT FROM AUTHOR]

The standard Lane–Emden equations model several physical phenomena such as isotropic continuous media, thermal behaviour of a spherical cloud of gas, and isothermal gas spheres. Systems of Lane–Emden-type equations appear in the modelling of the concentration of carbon substrate and oxygen, catalytic diffusion reactions, the steady state concentration of carbon dioxide, dusty fluid models, and pattern formation. In solving singular boundary value problems, one faces challenges resulting from the divergence of the associated variable coefficients at the singular points. This research article explores the power series approach to analytically approximate solutions to a class of systems of strongly nonlinear singular boundary value problems of Lane–Emden-type. The nonlinear terms in the proposed problems are transformed into power series using the generalised Cauchy product before establishing explicit recursion formulae for the expansion coefficients of the system of series solutions. The initial conditions required in the proposed boundary value problems are assumed and determined from a set of nonlinear algebraic equations resulting from the given right boundary conditions. Three special systems of nonlinear singular boundary value problems of Lane–Emden type are presented to demonstrate the proposed method’s reliability, effectiveness, and accuracy. The obtained approximate solutions are compared with the exact solutions (where they are available), or with other existing results (where the exact solution is not readily available). The series solution of the first example has a slow convergent rate, while the results of the other two examples are in excellent agreement with the exact solutions and other published results.

This article is concerned with the structure of solutions to the elliptic problem for a Hénon-type equation with a forcing term: −Δu=α(x)up+κμ,inRN,u>0,inRN,(Pκ)\hspace{11.3em}-\Delta u=\alpha \left(x){u}^{p}+\kappa \mu ,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}u\gt 0,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{13.0em}\left({{\rm{P}}}_{\kappa }) where N≥3N\ge 3, p>1p\gt 1, κ>0\kappa \gt 0, and α\alpha is a positive continuous function in RN\{0}{{\mathbb{R}}}^{N}\setminus \left\{0\right\}, and μ\mu is a nonnegative Radon measure in RN{{\mathbb{R}}}^{N}. Under suitable assumptions on the exponent pp, the coefficient α\alpha , and the forcing term μ\mu , we give a complete classification of the existence/nonexistence of solutions to problem (Pκ{{\rm{P}}}_{\kappa }) with respect to κ\kappa .

The Majorana transformation makes it possible to reduce the Thomas–Fermi equation to a first-order differential equation. This reduction is possible due to the special scaling property of the Thomas–Fermi equation under homology transformations. Such reductions are well known in the context of stellar astrophysics, where the use of homology-invariant variables has long proved useful. We use homology-invariant variables in the context of the Thomas–Fermi equation to demystify the origin of the otherwise mysterious Majorana transformation. [ABSTRACT FROM AUTHOR]

We analyze the Lane–Emden equations in the cylindrical framework. Although the explicit forms of the solutions (which are also called polytropes) are not known, we identify some of their qualitative properties. In particular, possible critical points and zeros of the polytropes are investigated and discussed, leading to possible improvements in the approximation methods which are currently employed. The cases when the critical parameter is odd and even are separately analyzed. Furthermore, we propose a technique to evaluate the distance between a pair of polytropes in small intervals. [ABSTRACT FROM AUTHOR]

In the present study, we introduce a collocation approach utilizing quintic B-spline functions as bases for solving systems of Lane Emden equations which have various applications in theoretical physics and astrophysics. The method derives a solution for the provided system by converting it into a set of algebraic equations with unknown coefficients, which can be easily solved to determine these coefficients. Examining the convergence theory of the proposed method reveals that it yields a fourth-order convergent approximation. It is confirmed that the outcomes are consistent with the theoretical investigation. Tables and graphs illustrate the proficiency and consistency of the proposed method. Findings validate that the newly employed method is more accurate and effective than other approaches found in the literature. All calculations have been performed using Mathematica software. [ABSTRACT FROM AUTHOR]

By applying the gluing method, we construct infinitely many axial symmetric singular positive solutions to the Lane-Emden equation: ΔHu+up=0,inHn\{0}{\Delta }_{{\mathbb{H}}}u+{u}^{p}=0\left,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{H}}}^{n}\backslash \left\{0\right\} on the Heisenberg group Hn{{\mathbb{H}}}^{n}, where n>1,Q⁄(Q−4)

The equations of Lane-Emden (LE) can be visualized in various phenomena of astrophysics, fluid mechanics, polymer science and material science, thus the main concern of the present study is to put a novel effort to resolve these equations by utilizing the artificial neural networking approach incorporation with Vieta-Lucas wavelets called as VLW-ANN method. This unique combination of neural networking and Vieta-Lucas wavelets has been prepared to reduce the computational challenges as well as to overcome the obstacles while dealing with singularity. Many examples of the LE variety are solved by this approach. The effectiveness, accuracy and simplicity of the VLW-ANN scheme are demonstrated by a comparative study between the VLW-ANN results and existing results. Additionally, the results are shown in tables and figures, which give a more favorable impression of the scheme’s dependability. VLW-ANN scheme will provide interesting results for other non-linear models. [ABSTRACT FROM AUTHOR]

In this research, we study the numerical solution of fractional Lane-Emden type equations, which emerge mainly in astrophysics applications. We propose a numerical approach making use of Legendre wavelets and the quasilinearization technique. The nonlinear term in fractional Lane-Emden type equations is iteratively linearized using the quasilinearization technique. The linearized equations are then solved using the Legendre wavelet collocation method. The proposed method is quite effective to overcome the singularity in fractional Lane-Emden type equations. Convergence and error analysis of the proposed method are given. We solve some test problems to compare the effectiveness of the proposed method with some other numerical methods in the literature. [ABSTRACT FROM AUTHOR]

In this study, a high-order compact finite difference method is used to solve Lane-Emden equations with various boundary conditions. The norm is to use a first-order finite difference scheme to approximate Neumann and Robin boundary conditions, but that compromises the accuracy of the entire scheme. As a result, new higher-order finite difference schemes for approximating Robin boundary conditions are developed in this work. We test the applicability and performance of the method using different examples of Lane-Emden equations. Convergence analysis is provided, and it is consistent with the numerical results. The results are compared with the exact solutions and published results from other methods. The method produces highly accurate results, which are displayed in tables and graphs. [ABSTRACT FROM AUTHOR]

In this study, we investigate a system of coupled nonlinear Lane-Emden equations that arise from catalytic diffusion reactions. A highly effective algorithm, which heavily relies on the differential transform method, is proposed to solve this system. The algorithm produces a convergent series solution with components that can be easily computed. The Adomian polynomials corresponding to the given system are utilized for calculating the differential transforms of its nonlinearities with multiple variables. A practical numerical example is provided to validate the effectiveness and accuracy of the present scheme. The numerical results obtained by our developed approach show a significantly lower error rate compared to other existing approaches.—In this study, we investigate a system of coupled nonlinear Lane-Emden equations that arise from catalytic diffusion reactions. A highly effective algorithm, which heavily relies on the differential transform method, is proposed to solve this system. The algorithm produces a convergent series solution with components that can be easily computed. The Adomian polynomials corresponding to the given system are utilized for calculating the differential transforms of its nonlinearities with multiple variables. A practical numerical example is provided to validate the effectiveness and accuracy of the present scheme. The numerical results obtained by our developed approach show a significantly lower error rate compared to other existing approaches. [ABSTRACT FROM AUTHOR]

This study is concerned by the Collocation method for solving Volterra integral equations form of the Lane-Emden type numerically through the so-called Touchard polynomials, which are essentially binomial polynomials. Where the system of miniature linear equations is solved numerically using the MATALEB program. Then some examples are presented to verify the reliability and effectiveness of the method in addition to the speed of its convergence. [ABSTRACT FROM AUTHOR]

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space D 0 1 , p \mathcal{D}^{{1,p}}_{0} into L q L^{q} and the summability properties of the distance function. We prove that, in the superconformal case (i.e. when 푝 is larger than the dimension), these two facts are equivalent, while in the subconformal and conformal cases (i.e. when 푝 is less than or equal to the dimension), we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behavior of the positive solution of the Lane–Emden equation for the 푝-Laplacian with sub-homogeneous right-hand side, as the exponent 푝 diverges to ∞. The case of first eigenfunctions of the 푝-Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies. [ABSTRACT FROM AUTHOR]

In this paper, we study the nearly critical Lane-Emden equations (⁎) { − Δ u = u p − ε in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N with N ≥ 3 , p = N + 2 N − 2 and ε > 0 is small. Our main result is that when Ω is a smooth bounded convex domain and the Robin function on Ω is a Morse function, then for small ε the equation (⁎) has a unique solution, which is also nondegenerate. As for non-convex domain, we also obtain exact number of solutions to (⁎) under some conditions. In general, the solutions of (⁎) may blow-up at multiple points a 1 , ⋯ , a k of Ω as ε → 0. In particular, when Ω is convex, there must be a unique blow-up point (i.e., k = 1). In this paper, by using the local Pohozaev identities and blow-up techniques, even having multiple blow-up points (non-convex domain), we can prove that such blow-up solution is unique and nondegenerate. Combining these conclusions, we finally obtain the uniqueness, multiplicity and nondegeneracy of solutions to (⁎). [ABSTRACT FROM AUTHOR]

Purpose: The numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present the biorthogonal flatlet multiwavelet collocation method (BFMCM) as a numerical scheme for a class of system of Lane–Emden equations with initial or boundary or four-point boundary conditions. Design/methodology/approach: The approach is involved in combining the biorthogonal flatlet multiwavelet (BFM) with the collocation method. The authors investigate the properties and procedure of the BFMCM for first time on this class of equations. By using the BFM and the collocation points, the method is constructed and it transforms the nonlinear differential equations problem into a system of nonlinear algebraic equations. The unknown coefficients of the assuming solution are determined by solving the obtained system. Additionally, convergence analysis and numerical stability of the suggested method are provided. Findings: According to the attained results, the proposed BFMCM has more accurate results in comparison with results of other methods. The maximum absolute errors are calculated by using the BFMCM for comparison purposes provided. Originality/value: The key desirable properties of BFMCM are its efficiency, simple applicability and minimizes errors. Therefore, the proposed method can be used to solve nonlinear problems or problems with singular points. [ABSTRACT FROM AUTHOR]

Semipositone Lane–Emden type equations are considered on the half-axis. Such equations have been used in modelling several phenomena in astrophysics and mathematical physics and are often difficult to solve analytically. We provide sufficient conditions for the existence of a positive continuous solution and we describe its global behavior. Our approach is based on a perturbed operator technique and fixed point theorems. Some examples are presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

We propose a new class of SBVPs which deals with exothermic reactions. We also propose four computationally stable methods to solve singular nonlinear BVPs by using Hermite wavelet collocation which are coupled with Newton's quasilinearization and Newton-Raphson method. We compare the results which are obtained by using Hermite wavelets with the results obtained by using Haar wavelets. The efficiency of these methods are verified by applying these four methods on Lane-Emden equations. Convergence analysis is also presented. [ABSTRACT FROM AUTHOR]

In this work, we investigate astrophysical systems in a Newtonian regime using anisotropic matter. For this purpose, we considered that both radial and tangential pressures satisfy a generalized Chaplygin-type equation of state. Using this model, we found the Lane–Emden equation for this system and solved it numerically for several sets of parameters. Finally, we explored the mass supported by this physical system and compared it with the Chandrasekhar mass. [ABSTRACT FROM AUTHOR]

We derive a generic spatial eigenvalue problem governing stars in hydrostatic equilibrium. Our approach generalizes the various Lane–Emden equations finding use over the past century, allowing for more general equations of state (EoS) while ensuring a stellar structure with finite size (without the need for artificial truncation of the radius). We show that the resulting stellar structure is encoded into two quantities: the eigenvalue, which determines the total size or mass of the star, and the density distribution, which encodes the internal structure. While our formalism recovers known results for polytrope and white dwarf EoS, we also study additional EoS, such as those incorporating excluded volumes or those calibrated through viral expansions. We obtain numerical values for the stellar structure under a variety of frameworks, comparing and contrasting stellar structure under different EoS. Interestingly, we show how different EoS can be calibrated to give solutions with the same stellar structure, highlighting the arbitrariness of a particular EoS for replicating observations. This leads us to comment on general properties EoS should obey to describe physically realistic stars. We also consider hydrostatic gas clouds immersed in larger regions having non-zero ambient density. We compare three analytical methods for finding solutions of these eigenvalue problems, including Taylor series solutions, the variational approximation, and the non-perturbative delta-expansion method. Although each method has benefits and drawbacks, we show that the delta-expansion method provides the most accuracy in replicating stellar structure. [ABSTRACT FROM AUTHOR]

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.

The pseudo-spectral method is used as a technique to employ the first derivative of the well-known Legendre polynomials (FDLs) as novel basis functions. Then, the FDLs Gauss- Lobatto quadrature weights (FDLs-GLQWs) and zeros (FDLs-GLQZs) have been calculated. Consequently, a matrix for differentiation (D-matrix) and another for integration (B-matrix) have been created. To solve several types of ordinary differential problems (ODPs), BVPs, integro–differential equations (IDEs), optimal control problems (OCPs), we designed three algorithms that relied on those matrices. Each algorithm for each type. The convergence of the designed algorithms has been verified theoretically by the error analysis. Finally, the proposed algorithms are applied to several numerical examples.

We analyze the Lane–Emden equations in the cylindrical framework. Although the explicit forms of the solutions (which are also called polytropes) are not known, we identify some of their qualitative properties. In particular, possible critical points and zeros of the polytropes are investigated and discussed, leading to possible improvements in the approximation methods which are currently employed. The cases when the critical parameter is odd and even are separately analyzed. Furthermore, we propose a technique to evaluate the distance between a pair of polytropes in small intervals.

We concern a family \{(u_{\varepsilon },v_{\varepsilon })\}_{\varepsilon > 0} of solutions of the Lane-Emden system on a smooth bounded convex domain \Omega in \mathbb {R}^N \[ \begin {cases} -\Delta u_{\varepsilon } = v_{\varepsilon }^p & \text {in } \Omega, \\ -\Delta v_{\varepsilon } = u_{\varepsilon }^{q_{\varepsilon }} & \text {in } \Omega, \\ u_{\varepsilon },\, v_{\varepsilon } > 0 & \text {in } \Omega, \\ u_{\varepsilon } = v_{\varepsilon } =0 & \text {on } \partial \Omega, \end {cases} \] for N \ge 4, \max \{1,\frac {3}{N-2}\} < p < q_{\varepsilon } and small \[ \varepsilon ≔\frac {N}{p+1} + \frac {N}{q_{\varepsilon }+1} - (N-2) > 0. \] This system appears as the extremal equation of the Sobolev embedding W^{2,(p+1)/p}(\Omega) \hookrightarrow L^{q_{\varepsilon }+1}(\Omega), and is also closely related to the Calderón-Zygmund estimate. Under the natural energy condition, we prove that the multiple bubbling phenomena may arise for the family \{(u_{\varepsilon },v_{\varepsilon })\}_{\varepsilon > 0}, and establish a detailed qualitative and quantitative description. If p < \frac {N}{N-2}, the nonlinear structure of the system makes the interaction between bubbles so strong, so the determination process of the blow-up rates and locations is completely different from that of the classical Lane-Emden equation. If p \ge \frac {N}{N-2}, the blow-up scenario is relatively close to that of the classical Lane-Emden equation, and only single-bubble solutions can exist. Even in the latter case, we have to devise a new method to cover all p near \frac {N}{N-2}. We also deduce a general existence theorem that holds on any smooth bounded domains. [ABSTRACT FROM AUTHOR]

Analytic approximations are presented for the exact solutions of the Volterra type nonlinear integral equation of the second kind for classical gaseous polytropes in closed form. This equation is considered as the integral equivalent of the Lane-Emden differential equation with boundary conditions which describes the standard polytropic models in terms of a Cauchy problem. With the aid of a linear approximation of this equation and general heuristic considerations of a physical character, as well as with the aid of a graphical model and variation of the parameters of the approximating functions, approximate expressions for the Emden functions and the dimensionless density are obtained in closed form with a mean square accuracy from ~10-4 to a few percent for a series of values of the polytropic index n of practical interest (n = 0.5, 3, 4, 6, ∞). Our previous approximation for the spatial density of the isothermal model is compared with a pseudo-isothermal law describing the distribution of the density of dark matter surrounding spiral galaxies and used by various authors for studying their rotation curves. [ABSTRACT FROM AUTHOR]

In this research, exponential approximation is used to solve a class of nonlinear Emden-Fowler equations. This method is based on the matrix forms of exponential functions and their derivatives using collocation points. To demonstrate the usefulness of the method, we apply it to some different problems. The numerical approximate solutions are compared with available (existing) exact (analytical) solutions to show the accuracy of the proposed method. The method has been checked with several examples to show its validity and reliability. The reported examples illustrate that the method is reasonably efficient and accurate. [ABSTRACT FROM AUTHOR]

This paper suggests the Chebyshev pseudo-spectral approach to solve the variable-order fractional Lane–Emden differential equations (VOFLEDE). The variable-order fractional derivative (VOFD) is defined in the Caputo sense. The proposed method transforms the problem into a set of algebraic equations that can be solved for unknowns. Few examples are discussed to exhibit the viability and effectiveness of the approach. The present study indicates the accuracy, efficiency, and powerfulness of the Chebyshev collocation method in solving the VOFD Lane–Emden equation. Error bound and convergence analysis of the method is also discussed. It is worth noticing that using lesser collocation nodes in computation is another advantage of the technique, which eventually reduces the computational cost. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach for tackling the Lane-Emden equation, a significant nonlinear differential equation of paramount importance in the realms of physics and astrophysics. We employ the Chimp optimization algorithm in conjunction with Chebyshev polynomials to devise an innovative solution strategy. Inspired by the behavioral patterns of chimpanzees, the Chimp algorithm is harnessed to optimize the Chebyshev polynomial approximations, thereby transforming the Lane-Emden equation into an unconstrained optimization problem. Our method’s effectiveness is demonstrated through a series of numerical experiments, showcasing its capability to precisely solve the Lane-Emden equation across various polytropic indices. [ABSTRACT FROM AUTHOR]

The basic idea of the present study is to apply the local fractional Sumudu decomposition method (LFSDM) presented in [31] to solve Lane-Emden of index m and Emden-Fowler equation of index m with local fractional derivative, in order to obtain non differentiable analytical solutions. The results of the solved local differential fractional equations show the effectiveness of this method. [ABSTRACT FROM AUTHOR]

The temperature in a self-gravitating star has been widely described in astrophysics by the Lane-Emden differential equation. To solve the non-linear equations, the Lane-Emden differential equation is used as a prototype for testing new mathematical and numerical techniques. The problems of singular initial value of Lane-Emden type was solved by the Differential Transform Method in this study. This method is applicable to solve various linear and nonlinear problems which decrease size of computational work. Here, we introduced some numerical problems to describe the high accuracy and efficiency of our proposed method. [ABSTRACT FROM AUTHOR]

We study the sharp constant for the embedding of W^{1,p}_0(\Omega) into L^q(\Omega), in the case 2p and q is sufficiently close to p, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side. The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of p-Laplace–type equations by L. Damascelli and B. Sciunzi. [ABSTRACT FROM AUTHOR]

This study is concerned by the Collocation method for solving Volterra integral equations form of the Lane-Emden type numerically through the so-called Touchard polynomials, which are essentially binomial polynomials. Where the system of miniature linear equations is solved numerically using the MATALEB program. Then some examples are presented to verify the reliability and effectiveness of the method in addition to the speed of its convergence.

In this article, we tried to get the solution of a class of Lane Emden type equations by using the Fourier Decomposition Method. This method is obtained by using the Fourier transform and the Adomian Decomposition method (FADM) together.

In this paper, a new construction machine learning method is proposed based on Rational Chebyshev polynomials to solve Lane–Emden type differential equations. The method is used for training, and the parameters are obtained by the error back propagation principle. By comparing with Chebyshev neural network proposed in [Appl. Math. Comp. 247:100-114(2014)][Appl. Soft. Comp. 43:347-356(2016)], our results are more accurate. [ABSTRACT FROM AUTHOR]

The well known results of the theory of classical gaseous polytropes are presented in the framework of an integral approach where the standard Lane-Emden differential equation for a spherically symmetric gravitating mass is examined via its equivalent in the form of a nonlinear integral Volterra equation of the 2nd kind. It is shown that the inverse Laplace transform of the Lane-Emden equation for polytropes with an index of n=5 (Schuster model) is a recurrence relation for Bessel functions of the first kind. The invariance of the nonlinear integral Volterra equation with respect to homological transformations is shown, as well as the possibility of obtaining singular solutions under certain conditions. It is also shown that for whole integral and half integral polytrope indices, this equation is equivalent to a multidimensional integral equation, and finding the expansion of the Emden function in a power series of the dimensionless distance ξ from the center of the polytrope is equivalent to finding the Neumann series and the iterated nuclei in the Fredholm theory. Approximations of the Emden functions in closed form and their applicability to various astrophysical objects will be presented and discussed in the second part of this paper. Polytropes of other geometries and dimensionalities are not considered here. [ABSTRACT FROM AUTHOR]

The pseudo-spectral method is used as a technique to employ the first derivative of the well-known Legendre polynomials (FDLs) as novel basis functions. Then, the FDLs Gauss- Lobatto quadrature weights (FDLs-GLQWs) and zeros (FDLs-GLQZs) have been calculated. Consequently, a matrix for differentiation (D-matrix) and another for integration (B-matrix) have been created. To solve several types of ordinary differential problems (ODPs), BVPs, integro–differential equations (IDEs), optimal control problems (OCPs), we designed three algorithms that relied on those matrices. Each algorithm for each type. The convergence of the designed algorithms has been verified theoretically by the error analysis. Finally, the proposed algorithms are applied to several numerical examples. [ABSTRACT FROM AUTHOR]

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 L 2 . 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. [ABSTRACT FROM AUTHOR]

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 L s 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. [ABSTRACT FROM AUTHOR]