An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications

Analytic approximate solutions of Volterra’s population and some scientific models by power series method

In this paper, we have implement an analytic approximate method based on power series method (PSM) to obtain asolutions for Volterra’s population model of population growth of a species in a closed system. The numerical solution isobtained by combining the PSM and Pad´e technique. The Pad´e approximation that often show superior performance overseries approximation are effectively used in the analysis to capture essential behavior of the population u(t) of identicalindividuals. The results demonstrate that the method has many merits such as being derivative-free, overcome the difficultyarising in calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM).It does not require to calculate Lagrange multiplier as in Variational Iteration Method (VIM) and no needs to construct ahomotopy and solve the corresponding algebraic equations as in Homotopy Perturbation Method (HPM). Moreover, weused this method to solve some scientific models, namely, the hybrid selection model, the Riccati model and the logisticmodel to provide the analytic solutions. The obtained analytic approximate solutions of applying the PSM is in fullagreement with the results obtained with those methods available in the literature. The software used for the calculationsin this study was MATHEMATICAr 8.0

A Multi-Domain Spectral Collocation Approach for Solving Lane-Emden Type Equations

In this work, we explore the application of a novel multi-domain spectral collocation method for solving general non-linear singular initial value differential equations of the Lane-Emden type. The proposed solution approach is a simple iterative approach that does not employ linearisation of the differential equations. Spectral collocation is used to discretise the iterative scheme to form matrix equations that are solved over a sequence of non-overlapping sub-intervals of the domain. Continuity conditions are used to advance the solution across the non-overlapping sub-intervals. Different Lane-Emden equations that have been reported in the literature have been used for numerical experimentation. The results indicate that the method is very effective in solving Lane-Emden type equations. Computational error analysis is presented to demonstrate the fast convergence and high accuracy of the method of solution