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Junsheng Duan - One of the best experts on this subject based on the ideXlab platform.
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the volterra integral form of the lane Emden equation new derivations and solution by the adomian decomposition method
Journal of Applied Mathematics and Computing, 2015Co-Authors: Randolph Rach, Abdulmajid Wazwaz, Junsheng DuanAbstract:In this paper, we present new alternate derivations for the Volterra integral forms of the Lane-Emden equation that we derived in our last work (Wazwaz et al. in Appl Math Comput 219:5004–5019, 2013). The main focus will be on Lane-Emden equations for the shape factor of \(k=1\), where an alternate derivation for L’Hospital’s formula will be developed. Following our approach, the Adomian decomposition method, which provides an efficient algorithm for analytic approximate solutions of the Lane-Emden equation, will be used. Our results are supported by investigating several numerical examples that include linear and nonlinear initial value problems.
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solving coupled lane Emden boundary value problems in catalytic diffusion reactions by the adomian decomposition method
Journal of Mathematical Chemistry, 2014Co-Authors: Randolph Rach, Junsheng DuanAbstract:In this paper, we consider the coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. First, we utilize systems of Volterra integral forms of the Lane–Emden equations and derive the modified recursion scheme for the components of the decomposition series solutions. The numerical results display that the Adomian decomposition method gives reliable algorithm for analytic approximate solutions of these systems. The error analysis of the sequence of the analytic approximate solutions can be performed by using the error remainder functions and the maximal error remainder parameters, which demonstrate an approximate exponential rate of convergence.
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adomian decomposition method for solving the volterra integral form of the lane Emden equations with initial values and boundary conditions
Applied Mathematics and Computation, 2013Co-Authors: Randolph Rach, Junsheng DuanAbstract:In this paper, we use the systematic Adomian decomposition method to handle the integral form of the Lane-Emden equations with initial values and boundary conditions. The Volterra integral form of the Lane-Emden equation overcomes the singular behavior at the origin x=0. We confirm our belief that the Adomian decomposition method provides efficient algorithm for analytic approximate solutions of the equation. Our results are supported by investigating several numerical examples that include initial value problems and boundary value problems as well. Finally we consider the modified decomposition method of Rach, Adomian and Meyers for the Volterra integral form.
Huihui Zeng - One of the best experts on this subject based on the ideXlab platform.
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on nonlinear asymptotic stability of the lane Emden solutions for the viscous gaseous star problem
Advances in Mathematics, 2016Co-Authors: Tao Luo, Zhouping Xin, Huihui ZengAbstract:Abstract This paper proves the nonlinear asymptotic stability of the Lane–Emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant γ lies in the stability range ( 4 / 3 , 2 ) . It is shown that for small perturbations of a Lane–Emden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier–Stokes–Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is C 1 / 2 -Holder continuous across the vacuum boundary provided that γ lies in ( 4 / 3 , 2 ) . The key is to establish the global-in-time regularity uniformly up to the vacuum boundary, which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of the Lane–Emden solution with detailed convergence rates, and detailed large time behaviors of solutions near the vacuum boundary. In particular, it is shown that every spherical surface moving with the fluid converges to the sphere enclosing the same mass inside the domain of the Lane–Emden solution with a uniform convergence rate and the large time asymptotic states for the vacuum free boundary problem (1.1.2a) , (1.1.2b) , (1.1.2c) , (1.1.2d) , (1.1.2e) , (1.1.2f) are determined by the initial mass distribution and the total mass. To overcome the difficulty caused by the degeneracy and singular behavior near the vacuum free boundary and coordinates singularity at the symmetry center, the main ingredients of the analysis consist of combinations of some new weighted nonlinear functionals (involving both lower-order and higher-order derivatives) and space–time weighted energy estimates. The constructions of these weighted nonlinear functionals and space–time weights depend crucially on the structures of the Lane–Emden solution, the balance of pressure and gravitation, and the dissipation. Finally, the uniform boundedness of the acceleration of the vacuum boundary is also proved.
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on nonlinear asymptotic stability of the lane Emden solutions for the viscous gaseous star problem
arXiv: Analysis of PDEs, 2015Co-Authors: Tao Luo, Zhouping Xin, Huihui ZengAbstract:This paper proves the nonlinear asymptotic stability of the Lane-Emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant $\gamma$ lies in the stability range $(4/3, 2)$. It is shown that for small perturbations of a Lane-Emden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier-Stokes-Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary provided that $\gamma$ lies in $(4/3, 2)$. The key is to establish the global-in-time regularity uniformly up to the vacuum boundary, which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of the Lane-Emden solution with detailed convergence rates, and detailed large time behaviors of solutions near the vacuum boundary.
Randolph Rach - One of the best experts on this subject based on the ideXlab platform.
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the volterra integral form of the lane Emden equation new derivations and solution by the adomian decomposition method
Journal of Applied Mathematics and Computing, 2015Co-Authors: Randolph Rach, Abdulmajid Wazwaz, Junsheng DuanAbstract:In this paper, we present new alternate derivations for the Volterra integral forms of the Lane-Emden equation that we derived in our last work (Wazwaz et al. in Appl Math Comput 219:5004–5019, 2013). The main focus will be on Lane-Emden equations for the shape factor of \(k=1\), where an alternate derivation for L’Hospital’s formula will be developed. Following our approach, the Adomian decomposition method, which provides an efficient algorithm for analytic approximate solutions of the Lane-Emden equation, will be used. Our results are supported by investigating several numerical examples that include linear and nonlinear initial value problems.
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solving coupled lane Emden boundary value problems in catalytic diffusion reactions by the adomian decomposition method
Journal of Mathematical Chemistry, 2014Co-Authors: Randolph Rach, Junsheng DuanAbstract:In this paper, we consider the coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. First, we utilize systems of Volterra integral forms of the Lane–Emden equations and derive the modified recursion scheme for the components of the decomposition series solutions. The numerical results display that the Adomian decomposition method gives reliable algorithm for analytic approximate solutions of these systems. The error analysis of the sequence of the analytic approximate solutions can be performed by using the error remainder functions and the maximal error remainder parameters, which demonstrate an approximate exponential rate of convergence.
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adomian decomposition method for solving the volterra integral form of the lane Emden equations with initial values and boundary conditions
Applied Mathematics and Computation, 2013Co-Authors: Randolph Rach, Junsheng DuanAbstract:In this paper, we use the systematic Adomian decomposition method to handle the integral form of the Lane-Emden equations with initial values and boundary conditions. The Volterra integral form of the Lane-Emden equation overcomes the singular behavior at the origin x=0. We confirm our belief that the Adomian decomposition method provides efficient algorithm for analytic approximate solutions of the equation. Our results are supported by investigating several numerical examples that include initial value problems and boundary value problems as well. Finally we consider the modified decomposition method of Rach, Adomian and Meyers for the Volterra integral form.
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comparison of the adomian decomposition method and the variational iteration method for solving the lane Emden equations of the first and second kinds
Kybernetes, 2011Co-Authors: Randolph RachAbstract:Purpose – The purpose of this paper is to provide a comparison of the Adomian decomposition method (ADM) with the variational iteration method (VIM) for solving the Lane‐Emden equations of the first and second kinds.Design/methodology/approach – The paper examines the theoretical framework of the Adomian decomposition method and compares it with the variational iteration method. The paper seeks to determine the relative merits and computational benefits of both the Adomian decomposition method and the variational iteration method in the context of the important physical models of the Lane‐Emden equations of the first and second kinds.Findings – The Adomian decomposition method is shown to readily solve the Lane‐Emden equations of both the first and second kinds for all positive real values of the system coefficient α and for all positive real values of the nonlinear exponent m. The decomposition series solution of these nonlinear differential equations requires the calculation of the Adomian polynomials a...
Vineet Kumar Singh - One of the best experts on this subject based on the ideXlab platform.
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an analytic algorithm of lane Emden type equations arising in astrophysics using modified homotopy analysis method
Computer Physics Communications, 2009Co-Authors: Om P Singh, Rajesh K Pandey, Vineet Kumar SinghAbstract:Abstract Lane–Emden type equation models many phenomena in mathematical physics and astrophysics. It is a nonlinear differential equation which describes the equilibrium density distribution in self-gravitating sphere of polytropic isothermal gas, has a singularity at the origin, and is of fundamental importance in the field of stellar structure, radiative cooling, modeling of clusters of galaxies. An efficient analytic algorithm is provided for Lane–Emden type equations using modified homotopy analysis method, which is different from other analytic techniques as it itself provides us with a convenient way to adjust convergence regions even without Pade technique. Some examples are given to show its validity.
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an analytic algorithm of lane Emden type equations arising in astrophysics using modified homotopy analysis method
Computer Physics Communications, 2009Co-Authors: Om P Singh, Rajesh K Pandey, Vineet Kumar SinghAbstract:Abstract Lane–Emden type equation models many phenomena in mathematical physics and astrophysics. It is a nonlinear differential equation which describes the equilibrium density distribution in self-gravitating sphere of polytropic isothermal gas, has a singularity at the origin, and is of fundamental importance in the field of stellar structure, radiative cooling, modeling of clusters of galaxies. An efficient analytic algorithm is provided for Lane–Emden type equations using modified homotopy analysis method, which is different from other analytic techniques as it itself provides us with a convenient way to adjust convergence regions even without Pade technique. Some examples are given to show its validity.
S Chakraverty - One of the best experts on this subject based on the ideXlab platform.
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numerical solution of nonlinear singular initial value problems of Emden fowler type using chebyshev neural network method
Neurocomputing, 2015Co-Authors: Susmita Mall, S ChakravertyAbstract:In this investigation, a new algorithm has been proposed to solve singular initial value problems of Emden-Fowler type equations. Approximate solutions of these types of equations have been obtained by applying Chebyshev Neural Network (ChNN) model for the first time. The Emden-Fowler type equations are singular in nature. Here, we have considered single layer Chebyshev Neural Network model to overcome the difficulty of singularity. The computations become efficient because the procedure does not need to have hidden layer. A feed forward neural network model with error back propagation principle is used for modifying the network parameters and to minimize the computed error function. We have compared analytical and numerical solutions of linear and nonlinear Emden-Fowler equations respectively with the approximate solutions obtained by proposed ChNN method. Their good agreements and less CPU time in computations than the traditional artificial neural network (ANN) show the efficiency of the present methodology.
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chebyshev neural network based model for solving lane Emden type equations
Applied Mathematics and Computation, 2014Co-Authors: Susmita Mall, S ChakravertyAbstract:The objective of this paper is to solve second order non-linear ordinary differential equations of Lane-Emden type using Chebyshev Neural Network (ChNN) model. These equations are categorized as singular initial value problems. Artificial Neural Network (ANN) model is used here to overcome the difficulty of the singularity. A single layer neural network is used and the hidden layer is eliminated by expanding the input pattern by Chebyshev polynomials. Here we have used feed forward neural network model and principle of error back propagation. Homogeneous and non-homogeneous Lane-Emden equations are considered to show effectiveness of Chebyshev Neural Network model.
