Nonlinear Lane-Emden Type Equations Research Articles

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.

Laguerre collocation method for solving Lane-Emden type equations

In this paper, a Laguerre collocation method is presented in order to obtain numerical solutions for linear and nonlinear Lane-Emden type equations and their initial conditions. The basis of the present method is operational matrices with respect to modified generalized Laguerre polynomials(MGLPs) that transforms the solution of main equation and its initial conditions to the solution of a matrix equation corresponding to the system of algebraic equations with the unknown Laguerre coefficients. By solving this system, coefficients of approximate solution of the main problem will be determined. Implementation of the method is easy and has more accurate results in comparison with results of other methods.

An efficient hybrid computational technique for the time dependent Lane-Emden equation of arbitrary order

The study of dynamic behaviour of nonlinear models that arise in ocean engineering play a vital role in our daily life. There are many examples of ocean water waves which are nonlinear in nature. In shallow water, the linearization of the equations imposes severe conditions on wave amplitude than it does in deep water, and the strong nonlinear effects are observed. In this paper, q-homotopy analysis Laplace transform scheme is used to inspect time dependent nonlinear Lane-Emden type equation of arbitrary order. It offers the solution in a fast converging series. The uniqueness and convergence analysis of the considered model is presented. The given examples confirm the competency as well as accuracy of the presented scheme. The behavior of obtained solution for distinct orders of fractional derivative is discussed through graphs. The auxiliary parameter ħ offers a suitable mode of handling the region of convergence. The outcomes reveal that the q-HATM is attractive, reliable, efficient and very effective.

Singular curve and critical curve for doubly nonlinear Lane–Emden type equations

This paper is concerned with singular curve and critical curve for the periodic doubly nonlinear Lane–Emden type equation . In 2010, under a convex assumption on the domain Ω, Wang et al. (2010) considered a partial case of . While, there are no results about the cases of and before the present work. In this paper, we fill the gap for these two cases and give a total classification for the exponents. Furthermore, by constructing a special blow‐up sequence, we remove the convex assumption on the domain Ω, not only for the partial case considered in Wang et al. (2010) but also for other remaining cases.

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.

A Deep-Learning-Based Method for Solving Nonlinear Singular Lane-Emden Type Equation

The nonlinear Lane-Emden type equation can be used to describe many physical phenomena. To solve this type of equation, a method based on deep neural network is proposed. The output layer of this network has two layers, the last one of which scaling the outputs of their neighbors with an aim at coping with issues where the values of function to be approximated are much less or larger than the order of 1. The Lane-Emden equation and its initial conditions are employed to construct the loss function, and the problem of solving Lane-Emden equation is transformed into an optimization problem. A hybrid method combined Adam and L-BFGS-B methods is used to solve the optimization problem and consequently the Lane-Emden type equation is solved. To increase the accuracy, an adaptive strategy is incorporated into the training data sampling method. Especially, a strategy in coping with the issue about solving white-dwarf problem is proposed. Numerical experiments are conducted in which reference solutions including analytical solutions and numerical solutions given by Runge-Kutta method are used to verify the effectiveness of our proposed method. More importantly, the solutions over large domains are calculated by the use of the proposed method. The results show that the results given by our method are in good agreement with the reference solutions, and in cases where existent neural-network-based method fails, our method is able to produce convincible results.

Sine–cosine wavelets operational matrix method for fractional nonlinear differential equation

In this paper, we present a solution method for fractional nonlinear ordinary differential equations. We propose a method by utilizing the sine–cosine wavelets (SCWs) in conjunction with quasilinearization technique. The fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by quasilinearization technique. The operational matrices of fractional order integration for SCW are derived and utilized to transform the obtained discrete system into systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear differential equations. Convergence analysis and procedure of implementation for the proposed method are also considered. To illustrate the reliability and accuracy of the method, we tested the method on fractional nonlinear Lane–Emden type equation and temperature distribution equation.

A high-order numerical method for solving nonlinear Lane-Emden type equations arising in astrophysics

In this paper, some nonlinear Lane-Emden type equations arising in astrophysics are solved by Chebyshev Finite Difference Method. Convergence and error analysis of the method are examined. To show the applicability and efficiency, some astrophysics problems such as the isothermal gas spheres, standard Lane-Emden equation and white-dwarf equation are realized. Besides, the method carried out for some boundary value problems, and it is shown that the method also works with boundary conditions as well as with initial conditions without any modification. The results demonstrated that the proposed method is rather efficient and more accurate than the many methods given in the literature.

Chebyshev Operational Matrix Method for Lane-Emden Problem

AbstractIn the this paper, a new modified method is proposed for solving linear and nonlinear Lane-Emden type equations using first kind Chebyshev operational matrix of differentiation. The properties of first kind Chebyshev polynomial and their shifted polynomial are first presented. These properties together with the operation matrix of differentiation of first kind Chebyshev polynomial are utilized to obtain numerical solutions of a class of linear and nonlinear LaneEmden type singular initial value problems (IVPs). The absolute error of this method is graphically presented. The proposed framework is different from other numerical methods and can be used in differential equations of the same type. Several examples are illuminated to reveal the accuracy and validity of the proposed method.

An efficient computational method for the approximate solution of nonlinear Lane-Emden type equations arising in astrophysics

The key purpose of this article is to introduce an efficient computational method for the approximate solution of the homogeneous as well as non-homogeneous nonlinear Lane-Emden type equations. Using proposed computational method given nonlinear equation is converted into a set of nonlinear algebraic equations whose solution gives the approximate solution to the Lane-Emden type equation. Various nonlinear cases of Lane-Emden type equations like standard Lane-Emden equation, the isothermal gas spheres equation and white-dwarf equation are discussed. Results are compared with some well-known numerical methods and it is observed that our results are more accurate.

Haar Adomian Method for the Solution of Fractional Nonlinear Lane-Emden Type Equations Arising in Astrophysics

In this paper, we propose a method for solving some well-known classes of fractional Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. The method is proposed by utilizing Haar wavelets in conjunction with Adomian's decomposition method. The operational matrices for the Haar wavelets are derived and constructed. Procedure of implementation and convergence analysis of the method are presented. The method is tested on the fractional standard Lane-Emden equation and the fractional isothermal gas spheres equation. We compare the results produce by present method with some well-known results to show the accuracy and applicability of the method.

A Third-degree B-spline Collocation Scheme for Solving a Class of the Nonlinear Lane–-Emden Type Equations

A Third-degree B-spline Collocation Scheme for Solving a Class of the Nonlinear Lane–-Emden Type Equations

Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane–Emden type equations

In this article, a new method is generated to solve nonlinear Lane–Emden type equations using Legendre, Hermite and Laguerre wavelets. We are interested to note that these wavelets will give same solutions with good accuracy. Theorems on convergence analysis are stated and proved on the spaces, which are created by Legendre, Hermite and Laguerre wavelets bases and justified these spaces are equivalent to polynomial linear space generated by general polynomial basis. The main idea for obtaining numerical solutions depends on converting the differential equation with initial and boundary conditions into a system of linear or nonlinear algebraic equations with unknown coefficients. A very high level of accuracy reflects the reliability of this scheme for such problems.

Non-Standard Finite Difference Schemes for Solving Singular Lane-Emden Equation

In this paper we construct Non-Standard finite difference schemes (NSFD) for numerical solution of nonlinear Lane-Emden type equations which are nonlinear ordinary dierential equations on semi-infinite domain. They are categorized as singular initial value problems. This equation describes a variety of phenomena in theoretical physics and astrophysics. The presented schemes are obtained by using the Non-Standard finite difference method. The use of NSFD method and its approximations play an important role for the formation of stable numerical methods. The main advantage of the schemes is that the algorithm is very simple and very easy to implement. Thus, this method may be applied as a simple and accurate solver for ODEs and PDEs and it can also be utilized as an accurate algorithm to solve linear and nonlinear equations arising in physics and other fields of applied mathematics. Illustrative examples have been discussed to demonstrate validity and applicability of the technique and the results have been compared with the exact solutions.

A variational iteration method for solving nonlinear Lane–Emden problems

In this paper, an explicit analytical method called the variational iteration method is presented for solving the second-order singular initial value problems of the Lane–Emden type. In addition, the local convergence of the method is discussed. It is often useful to have an approximate analytical solution to describe the Lane–Emden type equations, especially in the case that the closed-form solutions do not exist at all. This convince us that an effective improvement of the method will be useful to obtain a better approximate analytical solution. The improved method is then treated as a local algorithm in a sequence of intervals. Besides, an adaptive version is suggested for finding accurate approximate solutions of the nonlinear Lane–Emden type equations. Some examples are given to demonstrate the efficiency and accuracy of the proposed method.

On the Numerical Solution of Singular Lane–Emden Type Equations Using Cubic B-spline Approximation

In this article, a numerical approximation is presented for solving the nonlinear Lane–Emden type equations. The method is based on cubic B-spline approximations that leads to a system of nonlinear equations and the unknowns have been obtained by using the optimization. Truncation error of this method is obtained. Some numerical examples are presented to demonstrate the validity and applicability of the method.

An Elegant Perturbation Iteration Algorithm for the Lane-Emden Equation

An all new technique has been devised to solve non-linear LaneEmden type equations. This novel technique is based on the Perturbation Iteration Algorithm. In this paper, a few examples are presented for the illustration of the power and wide usability of the proposed method. Moreover, a compare and contrast with the actual solution is provided. It has been evaluated that by employment of this method, the construction of perturbation solutions converging swiftly to the true solutions usually becomes easy, by giving us room to exactly demonstrate how -terms influence linearized equations. This swift convergence of the method gives solutions that are accurate quantitatively through relatively little iteration.

PICARD-REPRODUCING KERNEL HILBERT SPACE METHOD FOR SOLVING GENERALIZED SINGULAR NONLINEAR LANE-EMDEN TYPE EQUATIONS

An iterative method is discussed with respect to its effectiveness and capability of solving singular nonlinear Lane-Emden type equations using reproducing kernel Hilbert space method combined with the Picard iteration. Some new error estimates for application of the method are established. We prove the convergence of the combined method. The numerical examples demonstrates a good agreement between numerical results and analytical predictions.

A numerical approach for solving singular nonlinear Lane–Emden type equations arising in astrophysics

In this paper, we suggest a numerical method based upon hybrid of Chebyshev wavelets and finite difference methods for solving well-known nonlinear initial-value problems of Lane–Emden type. The useful properties of the Chebyshev wavelets and finite difference method are utilized to reduce the computation of the problem to a set of nonlinear algebraic equations. Making a comparison among the obtained results using the present method with those ones reported in literature by some other well-known methods confirms the accuracy and computational efficiency of the present technique.

Numerical Solution of Singular Lane-Emden Equation

A new approach for solving the nonlinear Lane-Emden type equations has been proposed. The method is based on Legendre wavelets approximations. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution.