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

Adomian Decomposition Method for Solving Higher Order Boundary Value Problems

In this paper, we present efficient numerical algorithms for the approximate solution of linear and non-linear higher order boundary value problems. Algorithms are, based on Adomian decomposition. Also, the Laplace Transformation with Adomian decomposition technique is proposed to solve the problems when Adomian series diverges. Three examples are given to illustrate the performance of each technique. Keyword: Higher order Singular boundary value problems, Adomian decomposition techniques, Laplace transformations

SOLVING NONLINEAR BOUNDARY VALUE PROBLEMS USING THE HOMOTOPY ANALYSIS METHOD

Analytical solutions of differential equations are very important for all researchers from different discipline. Obtaining such solutions is difficult in most cases, especially if the differential equation is nonlinear. One of the mostly used methods are the series methods, where the solution is represented as an infinite series. Different methods are available to evaluate the terms of this series. These methods include the well-known Taylor series method, the Adomian decomposition method, the Homotopy iteration method, and the Homotopy analysis method. In this thesis we give a survey of the different series methods available to solve initial and boundary value problems. The methods to be presented are the Taylor series method, the Adomina decomposition method, and the Homotopy analysis method. The main features of each method will be presented and the error analysis will be discussed as well. For the Homotopy analysis method, the error is controlled by introducing the parameter known as ℏ, then the error is controlled by monitoring the value of the solution at a specific point for different values of ℏ. This produces what is known as the ℏ curve. The mathematical foundation of this method is not very well established, and the method will not work at all times. The error for the Taylor series and the Adomian decomposition method is controlled by adding more terms to the series solution which might be costly and difficult to calculate especially if the differential equation is nonlinear. In this study we will show that the error can be controlled by other means. A modified Taylor series method has been developed and will be discussed. The method is based on controlling the error through different choices of the point of expansion. The mathematical foundation of the method and application of the method to differential equations with singularities and eigenvalue problems will be presented

Some Problems on Variational Iteration Method

In this research project paper, I introduce some basic idea of Variational iteration method and its algorithm to solve the equations ODE & PDE, fractional differential equation, fractal differential equation and differential-difference equations. Also, some linear and nonlinear differential equations like Burger’s equation, Fisher’s equation,Wave equation and Schrodinger equation are solved by using Variational iteration method. Then I compare this method with Adomian decomposition method (ADM) and modified Variational iteration method (MVIM). The advantage of VIM, it does not require a small parameter in an equation as perturbation technique needs. The VIM is used to solve effectively, easily, and accurately a large class of non-linear problems with approximations which converge rapidly to accurate solutions. For linear problems, its exact solution can be obtained by only one iteration step due to the fact that the Lagrange multiplier can be exactly identified