Nonlinear Lane-Emden Equations 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.

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

Taylor wavelet solution of linear and nonlinear Lane-Emden equations

This study aims to use the Taylor wavelet method to solve linear and nonlinear Lane-Emden equations. An advantage of the method is the orthonormality property of the polynomials which reduce the computational cost. Another advantage is that the nonlinear terms do not need to be approximated. The application of the method reduces the differential equations to a system of algebraic equations. Six differential equations that model different physical problems with initial and boundary conditions are solved to illustrate the efficiency and accuracy of the Taylor wavelet method. The results obtained from the method are compared with other numerical results and exact solutions and presented in terms of absolute error tables and graphics. We observe from these results that the method is highly accurate and capable of obtaining the exact solution when it is in the form of a polynomial.

Modified Hermite Operational Matrix Method for Nonlinear Lane-Emden Problem

Nonlinear Lane –Emden equations are significant in astrophysics and mathematical physics. The aim of this paper is submitted a new approximate method for finding solution to nonlinear Lane-Emden type equations appearing in astrophysics based on modified Hermite integration operational matrix. The suggest technique is based on taking the truncated modified Hermite series of the solution function in the nonlinear Lane-Emden equation and then changed into a matrix equation with the given conditions. Nonlinear system of algebraic equation using collection points is obtained. The present method is applied on some relevant physical problems as nonlinear Lane-Emden type equations.

Two Efficient Computational Algorithms to Solve the Nonlinear Singular Lane-Emden Equations

In this paper, two efficient computational algorithms based on Rational and Exponential Bessel (RB and EB) functions are compared to solve several well-known classes of nonlinear Lane-Emden type models. The problems, which are define in some models of non-Newtonian fluid mechanics and mathematical physics, are nonlinear ordinary differential equations of second-order over the semi-infinite interval and have a singularity at x = 0. The nonlinear Lane-Emden equations are converted to a sequence of linear differential equations by utilizing the quasilinearization method (QLM), and then these linear equations are solved by RB and EB collocation methods. Afterwards, the obtained results are compared with the solution of other methods to demonstrate the efficiency and applicability of the proposed methods.

A Modified Homotopy Perturbation Method for Nonlinear Singular Lane–Emden Equations Arising in Various Physical Models

A modified homotopy perturbation method for solving a class of nonlinear Lane–Emden equations with boundary conditions arising in various physical models is proposed. The proposed algorithm is based on the homotopy perturbation method and integral form of the Lane–Emden equation. The integral form of the problem overcomes the singular behavior at the origin. The accuracy and applicability of our algorithm is examined by solving two singular models: (i) the second kind Lane–Emden equation used to model a thermal explosion in an infinite cylinder or a sphere and (ii) the nonlinear singular problem with Neumann boundary conditions.

A new algorithm for solving the nonlinear Lane–Emden equations arising in astrophysics

Hybrid orthonormal Bernstein and block-pulse function wavelet method for many phenomena in mathematical physics and astrophysics is considered. We use a fundamental administrator and convert Lane–Emden conditions into necessary conditions. The upsides of utilizing the proposed strategy are introduced. At that point, an effective error estimation for the proposed technique is additionally presented lastly a few examinations and their numerical arrangements are given; and looking at between the numerical outcomes acquired from alternate strategies, we demonstrate the high exactness and productivity of the proposed strategy. Our method is characterized then the singular equations are transformed to Volterra integro-differential equations. We modify this equations to an algebraic system of equations. The solution to this application is achieved by solving this system and the constructed solutions are on approximation form. The acquired results guarantee the method provides a truncate solution to the Lane–Emden equations.

A Smart Amalgamation of Spectral Neural Algorithm for Nonlinear Lane-Emden Equations with Simulated Annealing

Abstract The actual motivation of this paper is to develop a functional link between artificial neural network (ANN) with Legendre polynomials and simulated annealing termed as Legendre simulated annealing neural network (LSANN). To demonstrate the applicability, it is employed to study the nonlinear Lane-Emden singular initial value problem that governs the polytropic and isothermal gas spheres. In LSANN, minimization of error is performed by simulated annealing method while Legendre polynomials are used in hidden layer to control the singularity problem. Many illustrative examples of Lane-Emden type are discussed and results are compared with the formerly used algorithms. As well as with accuracy of results and tranquil implementation it provides the numerical solution over the entire finite domain.

A numerical approach for a class of astrophysics equations using piecewise spectral-variational iteration method

PurposeThe main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical physics and astrophysics.Design/methodology/approachThis method is based on a combination of Chebyshev interpolation and standard variational iteration method (VIM) and matching it to a sequence of subintervals. Unlike the spectral method and the VIM, the proposed PSVIM does not require the solution of any linear or nonlinear system of equations and analytical integration.FindingsSome well-known classes of Lane-Emden type equations are solved as examples to demonstrate the accuracy and easy implementation of this technique.Originality/valueIn this paper, a new and efficient technique is proposed to solve the nonlinear Lane-Emden equations. The proposed method overcomes the difficulties arising in calculating complicated and time-consuming integrals and terms that are not needed in the standard VIM.

Jacobi rational–Gauss collocation method for Lane–Emden equations of astrophysical significance

In this paper, a new spectral collocation method is applied to solve Lane–Emden equations on a semi-infinite domain. The method allows us to overcome difficulty in both the nonlinearity and the singularity inherent in such problems. This Jacobi rational–Gauss method, based on Jacobi rational functions and Gauss quadrature integration, is implemented for the nonlinear Lane–Emden equation. Once we have developed the method, numerical results are provided to demonstrate the method. Physically interesting examples include Lane–Emden equations of both first and second kind. In the examples given, by selecting relatively few Jacobi rational–Gauss collocation points, we are able to get very accurate approximations, and we are thus able to demonstrate the utility of our approach over other analytical or numerical methods. In this way, the numerical examples provided demonstrate the accuracy, efficiency, and versatility of the method.

The Taylor Matrix Method for Approximate Solution of Lane-Emden Equation with index-n

Many problems in mathematical physics can be formulated as an equation of Lane-Emden type. There are many methods for the solution of this equation. One of these methods is the Taylor matrix method. The only types of nonlinear equations that this method has been applied so far are the Riccati and Abel equations. In this study, an algorithm based on the Taylor matrix method is proposed and applied to the nonlinear Lane-Emden equation with index-n. An example is also given.

An extension of the Tau numerical algorithm for the solution of linear and nonlinear Lane-Emden equations

The purpose of this paper is to present a numerical algorithm for solving the Lane–Emden equations as singular initial value problems. The proposed algorithm is based on an operational Tau method (OTM). The main idea behind the OTM is to convert the desired problem to some operational matrices. Firstly, we use a special integral operator and convert the Lane–Emden equations to integral equations. Then, we use OTM to linearize the integral equations to some operational matrices and convert the problem to an algebraic system. The concepts, properties, and advantages of OTM and its application for solving Lane–Emden equations are presented. Some orthogonal polynomials are also used to reduce the volume of computations. Finally, several experiments of Lane–Emden equations including linear and nonlinear terms are given to illustrate the validity and efficiency of the proposed algorithm. Copyright © 2012 John Wiley & Sons, Ltd.

Existence of nonlinear Lane-Emden equation offractional order

Existence of nonlinear Lane-Emden equation offractional order

The modified decomposition method for analytic treatment of differential equations

The modified decomposition method is applied for analytic treatment of nonlinear differential equations. The modified method accelerates the rapid convergence of the series solution, dramatically reduces the size of work, and provides the exact solution by using few iterations only without any need to the so-called Adomian polynomials. Numerical illustrations that include nonlinear PDEs, nonlinear Klein–Gordon equations, and nonlinear Lane–Emden equation, are investigated to show the pertinent features of the technique.

Linearization techniques for singular initial-value problems of ordinary differential equations

Linearization methods for singular initial-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus yielding piecewise analytical solutions and globally smooth solutions. The accuracy of these methods is assessed by comparisons with exact and asymptotic solutions of homogeneous and non-homogeneous, linear and nonlinear Lane–Emden equations. It is shown that linearization methods provide accurate solutions even near the singularity or the zeros of the solution. In fact, it is shown that linearization methods provide more accurate solutions than methods based on perturbation methods. It is also shown that the accuracy of these techniques depends on the nonlinearity of the ordinary differential equations and may not be a monotonic function of the step size.