Lane Emden Problems Research Articles

UNIFORM AND NON-UNIFORM HAAR WAVELETS SOLUTIONS OF FRACTIONAL REGULAR α-SINGULAR LANE-EMDEN EQUATIONS

In this paper, we propose nonlinear fractional Lane-Emden equations, Dαy(x) +λ|x Dβy(x) + f(y) = 0, 1 < α ≤ 2, 0 < β ≤ 1, 0 < x, with initial conditions, y(0) = A, y (0) = B, where Dα,Dβ are Caputo fractional derivatives, λ = 1,2 and A,B are constants, 0 < β ≤ 1, 1 < α ≤ 2 and f(y) is a nonlinear function of y. We developed two collocation methods, namely the uniform fractional Haar wavelet collocation method and the nonuniform fractional Haar wavelet collocation method, and computed the solutions. In this procedure, fractional Haar integrations are used to find out the nonlinear system, which, when solved, gives the required solution. Our findings indicate that when (α, β) approach (2, 1), the solutions of the fractional and classical Lane-Emden problems become arbitrarily close to each other.

New sign-changing solutions for the 2D Lane-Emden problem with large exponents

New sign-changing solutions for the 2D Lane-Emden problem with large exponents

Uniqueness, multiplicity and nondegeneracy of positive solutions to the Lane-Emden problem

In this paper, we study the nearly critical Lane-Emden equations(⁎){−Δu=up−εinΩ,u>0inΩ,u=0on∂Ω, where Ω⊂RN with N≥3, p=N+2N−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 a1,⋯,ak 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 (⁎).

A NUMERICAL STUDY OF 2D-LANE-EMDEN PROBLEM USING 2D-BOUBAKER POLYNOMIALS

The paper presents a numerical solution for the two-dimensional Lane-Emden problem using two-dimensional Boubaker polynomials. The method involves utilizing the operational matrix of differentiation and collocation method to convert the problem into a system of algebraic equations. The proposed approach, based on two-dimensional Boubaker polynomials operational matrices, is shown to be straightforward and effective. The validity and applicability of the method are demonstrated through illustrative examples.

Asymptotic behavior of least energy nodal solutions for biharmonic Lane–Emden problems in dimension four

Asymptotic behavior of least energy nodal solutions for biharmonic Lane–Emden problems in dimension four

An effective approach based on Smooth Composite Chebyshev Finite Difference Method and its applications to Bratu-type and higher order Lane–Emden problems

The Smooth Composite Chebyshev Finite Difference method is generalized for higher order initial and boundary value problems. Round-off and truncation error analyses and convergence analysis of the method are also extended to higher order. The proposed method is applied to obtain the highly precise numerical solutions of boundary or initial value problems of the Bratu and higher order Lane Emden types. To visualize the competency of the presented method, the obtained results are compared with nine different methods, namely, Bezier curve method, Adomian decomposition method, Operational matrix collocation method, Direct collocation method, Haar Wavelet Collocation, Bernstein Collocation Method, Improved decomposition method, Quartic B-Spline method and New Cubic B-spline method. The comparisons show that the presented method is highly accurate than the other numerical methods and also gets rid of the singularity of the given problems.

Asymptotic behavior of least energy solutions to the Finsler Lane-Emden problem with large exponents

<p style='text-indent:20px;'>In this paper we are concerned with the least energy solutions to the Lane-Emden problem driven by an anisotropic operator, so-called the Finsler <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>-Laplacian, on a bounded domain in <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb{R}}^N $\end{document}</tex-math></inline-formula>. We prove several asymptotic formulae as the nonlinear exponent gets large.</p>

Morse index computation for radial solutions of the Hénon problem in the disk

We compute the Morse index m(up) of any radial solution up of the semilinear problem: (P)−Δu=|x|α|u|p−1uinBu=0on∂Bwhere B is the unit ball of R2 centered at the origin, α≥0 is fixed and p>1 is sufficiently large. In the case α=0, i.e. for the Lane–Emden problem, this leads to the following Morse index formula m(up)=4m2−m−2,for p large enough, where m is the number of nodal domains of u.

Non-degeneracy and local uniqueness of positive solutions to the Lane-Emden problem in dimension two

We are concerned with the Lane-Emden problem{−Δu=upinΩ,u>0inΩ,u=0on∂Ω, where Ω⊂R2 is a smooth bounded domain and p>1 is sufficiently large.Improving some known asymptotic estimates on the solutions, we prove the non-degeneracy and local uniqueness of the multi-spikes positive solutions for general domains. Our methods mainly use ODE's theory, various local Pohozaev identities, blow-up analysis and the properties of Green's function.

Approximate solution of Lane-Emden problem via modified Hermite operation matrix method


 
 
 
 Lane-Emden equations are singular initial value problems and they are important in mathematical physics and astrophysics. The aim of this present paper is presenting a new numerical method for finding approximate solution to Lane-Emden type equations arising in astrophysics based on modified Hermite operational matrix of integration. The proposed technique is based on taking the truncated modified Hermite series of the solution function in the Lane-Emden equation and then transferred into a matrix equation together with the given conditions. The obtained result is system of linear algebraic equation using collection points. The suggested algorithm is applied on some relevant physical problems as Lane-Emden type equations. 
 
 
 

Approximate nonradial solutions for the Lane-Emden problem in the ball

Abstract In this paper we provide a numerical approximation of bifurcation branches from nodal radial solutions of the Lane Emden Dirichlet problem in the unit ball in ℝ2, as the exponent of the nonlinearity varies. We consider solutions with two or three nodal regions. In the first case our numerical results complement the analytical ones recently obtained in [11]. In the case of solutions with three nodal regions, for which no analytical results are available, our analysis gives numerical evidence of the existence of bifurcation branches. We also compute additional approximations indicating presence of an unexpected branch of solutions with six nodal regions. In all cases the numerical results allow to formulate interesting conjectures.

Computational Method for Reaction Diffusion-Model Arising in a Spherical Catalyst

In this paper, we consider Lane–Emden problems which have many applications in sciences. Mainly we focus on two special cases of Lane–Emden boundary value problems which models reaction–diffusion equations in a spherical catalyst and spherical biocatalyst. Here we propose a method to obtain approximate solution of these models. The main reason for using this technique is high accuracy and low computational cost compared to some other methods. Numerical results are shown using tables and figures. Accuracy of the computational method is shown by comparing numerical results by analytical methods.

First order smooth composite Chebyshev finite difference method for solving coupled Lane-Emden problem in catalytic diffusion reactions

A new effective technique based on Chebyshev Finite Difference Method is introduced. First order smoothness of the approximation polynomial at the end points of each sub-interval is imposed in addition to the continuity condition. Both round-off and truncation error analyses are given besides the convergence analysis. Coupled Lane Emden boundary value problem in Catalytic Diffusion Reactions is investigated by using presented method. The obtained results are compared with the existing methods in the literature and it is observed that the proposed method gives more reliable results than the others.

A novel algorithm based on the Legendre wavelets spectral technique for solving the Lane–Emden equations

In this research, we present an iterative spectral method for the approximate solution of a class of Lane–Emden equations. In this procedure, we initially extend the Legendre wavelet which is appropriate for any time interval. Thereafter, the Guass-Legendre collection points of the Legendre wavelet are acquired. Employing this new approach, the iterative spectral technique converts the differential equation to a set of algebraic equations which diminishes the computational costs effectively. By solving the obtained algebraic equations, an accurate approximate solution for the assumed Lane–Emden equation is achieved. The present technique is validated by solving a number of Lane–Emden problems and are compared with other existing methods. The numerical simulations demonstrate that the new algorithm is simple and it has highly accuracy.

Quasi-radial solutions for the Lane–Emden problem in the ball

We consider the semilinear elliptic problem where B is the unit ball of $${\mathbb {R}}^2$$ centered at the origin and $$p\in (1,+\infty )$$. We prove the existence of sign-changing solutions to ($${{\mathcal {E}}}_p$$) having 2 nodal domains, whose nodal line does not touch $$\partial B$$ and which are non-radial. We call these solutions quasi-radial. The result is obtained for any p sufficiently large, considering least energy nodal solutions in spaces of functions invariant under suitable dihedral groups of symmetry and proving that they fulfill the required qualitative properties. We also show that these symmetric least energy solutions are instead radial for p close enough to 1, thus displaying a breaking of symmetry phenomenon in dependence on the exponent p. We then investigate the nonradial bifurcation at certain values of p from the sign-changing radial least energy solution of ($${{\mathcal {E}}}_p$$). The bifurcation result gives again, with a different approach and for values of p close to the ones at which the bifurcations appear, the existence of non-radial but quasi-radial nodal solutions.

Application of a new hybrid method for solving singular fractional Lane–Emden-type equations in astrophysics

In this research, a hybrid numerical method combining cosine and sine (CAS) wavelets and Green’s function approach is created to acquire the arrangements of fractional Lane–Emden Problem. The suggested methodology for detecting the solution of nonlinear equations dependent on variations of germinal algorithms is applied on nonlinear fractional Lane–Emden Problem under some smooth conditions and results in an iterative scheme of nonlinear equations Because of its efficiency, this technique is applied on a large variety of equations from the boundary value problems to the optimization. This paper is extending the suggested methodology technique for fractional Lane–Emden Problem. Moreover, the feature of the present novel method is utilized to convert the problem under observance into a system of algebraic equations that can be illuminated by suitable algorithms. A rapprochement of results has likewise been obtained using the present strategy and those reported using other techniques seem to indicate the precision and computational efficiency to establish the suitability of the Green-CAS wavelet method.

Sharp quantization for Lane–Emden problems in dimension two

In this short note, we prove a sharp quantization for positive solutions of Lane-Emden problems in a bounded planar domain. This result has been conjectured by De Marchis, Ianni and Pacella [6, Remark 1.2].

Explicit Runge–Kutta methods for starting integration of Lane–Emden problem

Traditionally, when constructing explicit Runge–Kutta methods we demand the satisfaction of the trivial simplifying assumption. Thus, f1=f(x0,y0) is always used as the first stage of these methods when applied to the Initial Value Problem (IVP): y′(x)=f(x,y),y(x0)=y0. Here we examine the case with f1=f(x0+c1h,y0),(h: the step) and c1 ≠ 0. We derive the order conditions for arbitrary order and construct a 5th order method at the standard cost of six stages per step. This method is found to outperform other classical Runge–Kutta pairs with orders 5(4) when applied to problems with singularity at the beginning (e.g. Lane–Emden problem).

Morse index and uniqueness of positive solutions of the Lane-Emden problem in planar domains

We compute the Morse index of 1-spike solutions of the semilinear elliptic problem(Pp){−Δu=up in Ωu=0 on ∂Ωu>0 in Ω where Ω⊂R2 is a smooth bounded domain and p>1 is sufficiently large.When Ω is convex, our result, combined with the characterization in [21], a result in [40] and with recent uniform estimates in [37], gives the uniqueness of the solution to (Pp), for p large. This proves, in dimension two and for p large, a longstanding conjecture.

Solutions of the Singular IVPs of Lane-Emden type equations by combining Laplace transformation and perturbation technique

AbstractIn this paper, we propose an efficient method to solve linear and nonlinear singular initial value problems of Lane-Emden type equations by combining Laplace transformation and homotopy perturbation methods. The method is based upon Laplace transform, polynomial series and perturbation technique. Several examples, including some well-known Lane-Emden problems, are presented to show the ability and accuracy of the modify method.