Lane-Emden type equation is a nonlinear differential equation appears in many fields such as stellar structure, radioactive cooling and modeling of clusters of galaxies. In this work, this equation is investigated usi...Lane-Emden type equation is a nonlinear differential equation appears in many fields such as stellar structure, radioactive cooling and modeling of clusters of galaxies. In this work, this equation is investigated using a semi-analytical method called the Variation of parameters method with an auxiliary parameter. In the applied technique, an unknown auxiliary parameter is inserted in Variation of Parameters Method to solve some special cases of these equations. The used algorithm is easy to implement and very effective. The obtained solutions are also fairly accurate.展开更多
We study a modified version of the Lane-Emden equation of the second kind modelling a thermal explosion in an infinite cylinder and a sphere. We first show that the solution to the relevant boundary value problem is b...We study a modified version of the Lane-Emden equation of the second kind modelling a thermal explosion in an infinite cylinder and a sphere. We first show that the solution to the relevant boundary value problem is bounded and that the solutions are monotone decreasing. The upper bound, the value of the solution at zero, can be approximated analytically in terms of the physical parameters. We obtain solutions to the boundary value problem, using both the Taylor series （which work well for weak nonlinearity） and the b-expansion method （valid for strong nonlinearity）. From here, we are able to deduce the qualitative behavior of the solution profiles with a change in any one of the physical parameters.展开更多
In the present paper, two new generating sets, of homology invariant functions will be established. Moreover, by the aid of two independent homology invariant functions of each set we established the transformed first...In the present paper, two new generating sets, of homology invariant functions will be established. Moreover, by the aid of two independent homology invariant functions of each set we established the transformed first order Lane-Emden equation. The first equation for polytropic index n ≠–1, ±∞ depends on five free parameters, while the other equation is for, n = ±∞ and depends on three free parameters.展开更多
Lane-Emden differential equations of order fractional has been studied.Numerical solution of this type is considered by collocation method. Some of examples are illustrated. The comparison between numerical and analyt...Lane-Emden differential equations of order fractional has been studied.Numerical solution of this type is considered by collocation method. Some of examples are illustrated. The comparison between numerical and analytic methods has been introduced.展开更多
In this work we apply the differential transformation method or DTM for solving some classes of Lane-Emden type equations as a model for the dimensionless density distribution in an isothermal gas sphere and as a stud...In this work we apply the differential transformation method or DTM for solving some classes of Lane-Emden type equations as a model for the dimensionless density distribution in an isothermal gas sphere and as a study of the gravitational potential of (white-dwarf) stars , which are nonlinear ordinary differential equations on the semi-infinite domain [1] [2]. The efficiency of the DTM is illustrated by investigating the convergence results for this type of the Lane-Emden equations. The numerical results show the reliability and accuracy of this method.展开更多
We study the anti-symmetric solutions to the Lane-Emden type system involving fractional Laplacian(-△)^(s)(0<s<1).First,we obtain a Liouville type theorem in the often-used defining space L_(2s).An interesting ...We study the anti-symmetric solutions to the Lane-Emden type system involving fractional Laplacian(-△)^(s)(0<s<1).First,we obtain a Liouville type theorem in the often-used defining space L_(2s).An interesting lower bound on the solutions is derived to estimate the Lipschitz coefficient in the sub-linear cases.Considering the anti-symmetric property,one can naturally extend the defining space from L_(2s) to L_(2s+1).Surprisingly,with this extension,we show the existence of non-trivial solutions.This is very different from the previous results of the Lane-Emden system.展开更多
The three-dimensional spherical polytropic Lane-Emden problem is yr r+(2/r)y_(r)+y^(m)=0,y(0)=1,y_(r)(0)=0 where m∈[0,5]is a constant parameter.The domain is r∈[0,ξ]whereξis the first root of y(r).We recast this a...The three-dimensional spherical polytropic Lane-Emden problem is yr r+(2/r)y_(r)+y^(m)=0,y(0)=1,y_(r)(0)=0 where m∈[0,5]is a constant parameter.The domain is r∈[0,ξ]whereξis the first root of y(r).We recast this as a nonlinear eigenproblem,with three boundary conditions andξas the eigenvalue allowing imposition of the extra boundary condition,by making the change of coordinate x≡r/ξ:y_(xx)+(2/x)y_(x)+ξ^(2) y^(m)=0,y(0)=1,y_(x)(0)=0,y(1)=0.We find that a Newton-Kantorovich iteration always converges from an m-independent starting point y^((0))(x)=cos([π/2]x),ξ^((0))=3.We apply a Chebyshev pseudospectral method to discretize x.The Lane-Emden equation has branch point singularities at the endpoint x=1 whenever m is not an integer;we show that the Chebyshev coefficients are a_(n)∼constant/n^(2m+5) as n→∞.However,a Chebyshev truncation of N=100 always gives at least ten decimal places of accuracy—much more accuracy when m is an integer.The numerical algorithm is so simple that the complete code(in Maple)is given as a one page table.展开更多
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 li...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 dis-cussed through graphs.The auxiliary parameter¯h offers a suitable mode of handling the region of con-vergence.The outcomes reveal that the q-HATM is attractive,reliable,efficient and very effective.展开更多
This paper is devoted to studying the existence of positive solutions for the following integral system {u(x)=∫_(R^n)|x-y|~λv-~q(y)dy, ∫_(R^n)|x-y|~λv-~p(y)dy,p,q>0,λ∈(0,∞),n≥1.It is shown that if(u,v) is a...This paper is devoted to studying the existence of positive solutions for the following integral system {u(x)=∫_(R^n)|x-y|~λv-~q(y)dy, ∫_(R^n)|x-y|~λv-~p(y)dy,p,q>0,λ∈(0,∞),n≥1.It is shown that if(u,v) is a pair of positive Lebesgue measurable solutions of this integral system, then 1/(p-1)+1/(q-1)=λ/n, which is different from the well-known case of the Lane-Emden system and its natural extension, the Hardy-Littlewood-Sobolev type integral equations.展开更多
In this paper we propose a collocation method for solving Lane-Emden type equation which is nonlinear or-dinary differential equation on the semi-infinite domain. This equation is categorized as singular initial value...In this paper we propose a collocation method for solving Lane-Emden type equation which is nonlinear or-dinary differential equation on the semi-infinite domain. This equation is categorized as singular initial value problems. We solve this equation by the generalized Laguerre polynomial collocation method based on Her-mite-Gauss nodes. This method solves the problem on the semi-infinite domain without truncating it to a fi-nite domain and transforming domain of the problem to a finite domain. In addition, this method reduces so-lution of the problem to solution of a system of algebraic equations.展开更多
In the following black hole model, electrons and positrons form a neutral gas which is confined by gravitation. The smaller masses are supported against gravity by electron degeneracy pressure. Larger masses are suppo...In the following black hole model, electrons and positrons form a neutral gas which is confined by gravitation. The smaller masses are supported against gravity by electron degeneracy pressure. Larger masses are supported by ideal gas and radiation pressure. In each case, the gas is a polytrope which satisfies the Lane-Emden equation. Solutions are found that yield the physical properties of black holes, for the range 1000 to 100 billion solar masses.展开更多
文摘Lane-Emden type equation is a nonlinear differential equation appears in many fields such as stellar structure, radioactive cooling and modeling of clusters of galaxies. In this work, this equation is investigated using a semi-analytical method called the Variation of parameters method with an auxiliary parameter. In the applied technique, an unknown auxiliary parameter is inserted in Variation of Parameters Method to solve some special cases of these equations. The used algorithm is easy to implement and very effective. The obtained solutions are also fairly accurate.
基金supported by the National Science Foundation of U.S.A.(No.1144246)
文摘We study a modified version of the Lane-Emden equation of the second kind modelling a thermal explosion in an infinite cylinder and a sphere. We first show that the solution to the relevant boundary value problem is bounded and that the solutions are monotone decreasing. The upper bound, the value of the solution at zero, can be approximated analytically in terms of the physical parameters. We obtain solutions to the boundary value problem, using both the Taylor series （which work well for weak nonlinearity） and the b-expansion method （valid for strong nonlinearity）. From here, we are able to deduce the qualitative behavior of the solution profiles with a change in any one of the physical parameters.
文摘In the present paper, two new generating sets, of homology invariant functions will be established. Moreover, by the aid of two independent homology invariant functions of each set we established the transformed first order Lane-Emden equation. The first equation for polytropic index n ≠–1, ±∞ depends on five free parameters, while the other equation is for, n = ±∞ and depends on three free parameters.
文摘Lane-Emden differential equations of order fractional has been studied.Numerical solution of this type is considered by collocation method. Some of examples are illustrated. The comparison between numerical and analytic methods has been introduced.
文摘In this work we apply the differential transformation method or DTM for solving some classes of Lane-Emden type equations as a model for the dimensionless density distribution in an isothermal gas sphere and as a study of the gravitational potential of (white-dwarf) stars , which are nonlinear ordinary differential equations on the semi-infinite domain [1] [2]. The efficiency of the DTM is illustrated by investigating the convergence results for this type of the Lane-Emden equations. The numerical results show the reliability and accuracy of this method.
基金supported by National Natural Science Foundation of China(Grant Nos.12031012,11831003 and 11701207)Natural Science Foundation of Henan Province of China(Grant No.222300420499)。
文摘We study the anti-symmetric solutions to the Lane-Emden type system involving fractional Laplacian(-△)^(s)(0<s<1).First,we obtain a Liouville type theorem in the often-used defining space L_(2s).An interesting lower bound on the solutions is derived to estimate the Lipschitz coefficient in the sub-linear cases.Considering the anti-symmetric property,one can naturally extend the defining space from L_(2s) to L_(2s+1).Surprisingly,with this extension,we show the existence of non-trivial solutions.This is very different from the previous results of the Lane-Emden system.
基金supported by the National Science Foundation through grants OCE0451951 and ATM 0723440.
文摘The three-dimensional spherical polytropic Lane-Emden problem is yr r+(2/r)y_(r)+y^(m)=0,y(0)=1,y_(r)(0)=0 where m∈[0,5]is a constant parameter.The domain is r∈[0,ξ]whereξis the first root of y(r).We recast this as a nonlinear eigenproblem,with three boundary conditions andξas the eigenvalue allowing imposition of the extra boundary condition,by making the change of coordinate x≡r/ξ:y_(xx)+(2/x)y_(x)+ξ^(2) y^(m)=0,y(0)=1,y_(x)(0)=0,y(1)=0.We find that a Newton-Kantorovich iteration always converges from an m-independent starting point y^((0))(x)=cos([π/2]x),ξ^((0))=3.We apply a Chebyshev pseudospectral method to discretize x.The Lane-Emden equation has branch point singularities at the endpoint x=1 whenever m is not an integer;we show that the Chebyshev coefficients are a_(n)∼constant/n^(2m+5) as n→∞.However,a Chebyshev truncation of N=100 always gives at least ten decimal places of accuracy—much more accuracy when m is an integer.The numerical algorithm is so simple that the complete code(in Maple)is given as a one page table.
文摘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 dis-cussed through graphs.The auxiliary parameter¯h offers a suitable mode of handling the region of con-vergence.The outcomes reveal that the q-HATM is attractive,reliable,efficient and very effective.
基金Supported by National Natural Science Foundation of China(11126148,11501116,11671086,11871208)Natural Science Foundation of Hunan Province of China(2018JJ2159)+1 种基金the Project Supported by Scientific Research Fund of Hunan Provincial Education Department(16C0763)the Education Department of Fujian Province(JA15063)
文摘This paper is devoted to studying the existence of positive solutions for the following integral system {u(x)=∫_(R^n)|x-y|~λv-~q(y)dy, ∫_(R^n)|x-y|~λv-~p(y)dy,p,q>0,λ∈(0,∞),n≥1.It is shown that if(u,v) is a pair of positive Lebesgue measurable solutions of this integral system, then 1/(p-1)+1/(q-1)=λ/n, which is different from the well-known case of the Lane-Emden system and its natural extension, the Hardy-Littlewood-Sobolev type integral equations.
文摘In this paper we propose a collocation method for solving Lane-Emden type equation which is nonlinear or-dinary differential equation on the semi-infinite domain. This equation is categorized as singular initial value problems. We solve this equation by the generalized Laguerre polynomial collocation method based on Her-mite-Gauss nodes. This method solves the problem on the semi-infinite domain without truncating it to a fi-nite domain and transforming domain of the problem to a finite domain. In addition, this method reduces so-lution of the problem to solution of a system of algebraic equations.
文摘In the following black hole model, electrons and positrons form a neutral gas which is confined by gravitation. The smaller masses are supported against gravity by electron degeneracy pressure. Larger masses are supported by ideal gas and radiation pressure. In each case, the gas is a polytrope which satisfies the Lane-Emden equation. Solutions are found that yield the physical properties of black holes, for the range 1000 to 100 billion solar masses.
