Lane-Emden Type Equations Research Articles

A hybrid quantum-spectral-successive linearization method for general Lane–Emden type equations

A hybrid quantum-spectral-successive linearization method for general Lane–Emden type equations

An efficient recursive technique with Padé approximation for a kind of Lane–Emden type equations emerging in various physical phenomena

The study numerically examined a class of nonlinear singular differential problems known as the Lane–Emden differential equation, which emerges in numerous real-world situations. The primary goal of this work is to formulate a computationally efficient iterative technique for solving the nonlinear Lane–Emden initial value problems. The proposed approach is a hybrid of the homotopy perturbation method and the Padé approximation. The nonlinear singular Lane–Emden initial value problem (SLEIVP) is transformed into an equivalent recursive integral employing the Picard’s approach. To resolve the singularity and nonlinearity, the recursive integral equation is transformed into a system of integral equations by using the homotopy notion. Furthermore, to enhance the convergence rate of the technique, Padé approximation is taken into account. The convergence analysis for the proposed approach is also conducted. The present technique is tested on SLEIVPs and numerical findings are compared with the existing techniques, to demonstrate the accuracy, effectiveness and ease of use.

Combining Laplace transform and Variational iteration method for solving singular IVPs and BVPs of Lane–Emden type equation

This study uses the combined Laplace transform and variational iteration method to analytically solve initial value problems (IVPs) and boundary value problems (BVPs) of Lane- Emden- equation with singular behaviour at . Initially, this method requires calculating the lagrange multiplayer, and after its value was calculated, it was found to be equal , this method is applying on several problems and the results were improved using Padé approximations, the results were compared with the exact solution by calculating the absolute error value between them.

Analytical method for systems of nonlinear singular boundary value problems

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.

GEPINN: An innovative hybrid method for a symbolic solution to the Lane–Emden type equation based on grammatical evolution and physics-informed neural networks

In this paper, we present an innovative and powerful combination of grammatical evolution and a physics-informed neural network approach for symbolically solving the Lane–Emden type equation, which is a nonlinear ordinary differential equation. We employ a grammatical evolution algorithm based on a context-free grammar to construct a mathematical expression comprising some parameters. Subsequently, these parameters are determined using the physics-informed neural networks approach. To achieve this, the computational graph of the mathematical expression generated in each iteration of the grammatical evolution is treated as a network. To assess the proposed method, we consider the Lane–Emden type equation. The proposed method demonstrated that it is a capable method for symbolically solving nonlinear ordinary differential equations accurately.

A Physics-Informed Neural Network model combined Pell–Lucas polynomials for solving the Lane–Emden type equation

A Physics-Informed Neural Network model combined Pell–Lucas polynomials for solving the Lane–Emden type equation

A novel numerical solution to lane-emden type equations using monte carlo technique

Monte Carlo (MC) methods are increasingly recognized as severe in many computational scientific fields and have diverse applications in many branches of science. This paper systematically provides two computational algorithms based on MC methods to solve different forms of Lane-Emden (LE) type equations. The proposed algorithms introduce solutions to 11 LE equations under various complex conditions. The performance and comparative study of numerical solutions based on the MC algorithms were computationally analyzed using other numerical/analytical methods available in the literature. We find that the MC solutions agree with the exact or Runge–Kutta solutions and different numerical methods applied to solve these equations.

An effective QLM-based Legendre matrix algorithm to solve the coupled system of fractional-order Lane-Emden equations

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.

Numerical algorithm for solving real-life application problems of Lane–Emden type equation

This paper presents an advanced variational iteration method for finding the approximate-exact solution for initial value problems of the Lane–Emden type equation that arise in several applications by employing the quasilinearization approach to the variational iteration method. Also, the convergence analysis of the method is studied under the Lipschitz condition. To demonstrate the applicability of the method, several application problems arise in isothermal gas spheres, stellar structures, and physics with different shape factors and are successfully solved by means of this method. The comparison is made with the available exact solution and the existing methods (Mall and Chakraverty, 2014; Umesh and Kumar, 2021; El-Essawy et al., 2023; Ahmad et al., 2016). We see that the method meets our expectations and reaches an exact solution in a few iterations. The optimality of the method is investigated by the absolute errors and absolute residual errors. The present method is time-efficient, requires less computational work, and converges to an exact solution in a few iterations. We also provided the CPU time to obtain an approximate solution to all the problems for the numerical investigation.

Existence and Uniqueness of Positive Solutions for Semipositone Lane-Emden Equations on the Half-Axis

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.

Modelling Lane–Emden type equations using Physics-Informed Neural Networks

The solutions of Lane–Emden (LE) type equations arising in astrophysics for the polytropic gas spheres, the isothermal gas sphere, and the white dwarf stars are revisited. We explore the potentiality of recent methods of deep learning using neural networks constrained by the physics and called Physics-Informed Neural Networks (PINNs). The basics of PINNs is introduced for solving each equation individually. The method consists in constraining the equation residual at some collocation dataset in addition to the boundary data via a minimization procedure. When the training process is complete, a learned differentiable function is obtained that can generate solution at any value of the variable. The novelty of this study is the additional possibility of learning solutions for several equations collectively with the same network, e.g. for the polytropic equations family for all the indices. We demonstrate the performances of PINNs in comparison with classical numerical methods. Advantages and drawbacks are highlighted. Interestingly, PINNs are meshless methods that can quasi-instantaneously generate the solution and its derivative once trained. However, the training procedure and accuracy of the method remain two future points of improvement.

A numerical method based on Legendre wavelet and quasilinearization technique for fractional Lane-Emden type equations

A numerical method based on Legendre wavelet and quasilinearization technique for fractional Lane-Emden type equations

Numerical Solutions for Singular Lane-Emden Equations Using Shifted Chebyshev Polynomials of the First Kind

This paper describes an algorithm for obtaining approximate solutions to a variety of well-known Lane-Emden type equations. The algorithm expands the desired solution y(x) ≃ yN(x), in terms of shifted Chebyshev polynomials of first kind such that yN(i)(0) = y(i)(0) (i = 0, 1, ..., N). The derivative values y(j)(0) for j = 2, 3, ..., are computed by using the given differential equation and its initial conditions. This makes approximate solutions more consistent with the exact solutions of given differential equations. The explicit expressions of the expansion coefficients of yN(x) are obtained. The suggested method is much simpler compared to any other method for solving this initial value problem. An excellent agreement between the exact and the approximate solutions is found in the given examples. In addition, the error analysis is presented.

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>

Fourier Ayrıştırma Metodu ile Lane-Emden Denkleminin Çözümü

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.

Monte Carlo simulation of Lane–Emden type equations arising in astrophysics

Monte Carlo (MC) method played an essential role in many areas of human activity and has found application in many branches of physical science. This paper proposes a computational technique based on MC algorithms to solve Lane–Emden (LE) type equations. We analyze four LE equations arising in astrophysics: the positive and negative indices of the polytropic gas spheres, the isothermal gas sphere, and the white dwarf equation. We calculated eleven models (i.e., eleven LE equations) of the positive index polytropes, nine for the negative index polytrope, the isothermal gas sphere, and the white dwarf equation. Comparing the MC and numerical/analytical models gives good agreement for the four LE equations under study.

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.

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.

A novel conformable Laplace transform for conformable fractional Lane–Emden type equations

In the current work, we introduce a relatively new integral operator, a modification of conformable Laplace transform that can handle some conformable fractional differential equations with singularities of a given kind. Some essential properties and theorems are discussed to show the efficiency and applicability of the new transform. We use this transform in conjunction with the Adomian decomposition method to solve various classes of conformable fractional Lane–Emden type equations.

New analytical method of solution to a nonlinear singular fractional Lane–Emden type equation

<abstract><p>We consider a nonlinear singular fractional Lane–Emden type differential equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ {}^{LC}\mathcal{D}^\alpha_{a^+}\varphi(t)+\frac{\lambda}{t^{\alpha-\beta}}\, \, {}^{LC}\mathcal{D}^\beta_{a^+}\varpi(t, \varphi(t)) = 0, \, \, 0<\beta<\alpha<1, \, \, 0< a<t\leq T, $\end{document} </tex-math></disp-formula></p> <p>with an initial condition $ \varphi(a) = \nu $ assumed to be bounded and non-negative, $ \varpi:[a, T]\times\mathbb{R}\rightarrow \mathbb{R} $ a Lipschitz continuous function, and $ {}^{LC}\mathcal{D}^\alpha_{a^+}, {}^{LC}\mathcal{D}^\beta_{a^+} $ are Liouville–Caputo derivatives of orders $ 0 < \alpha, \beta < 1 $. A new analytical method of solution to the nonlinear singular fractional Lane–Emden type equation using fractional product rule and fractional integration by parts formula is proposed. Furthermore, we prove the existence and uniqueness and the growth estimate of the solution. Examples are given to illustrate our results.</p></abstract>