Numerical resolution of Emden's equation using Adomian polynomials

Purpose: In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium. Design/methodology/approach: In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known. Findings: Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal. Originality/value: The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.This work has been supported by the Ministerio de Ciencia e Innovación, project TIN2009-10581

NUMERICAL SOLUTIONS OF SINGULAR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS USING SAID-BALL POLYNOMIALS

In this article, the collocation method based on Said-Ball polynomials have been used to solve the singular nonlinear ordinary differential equations of various orders numerically. An operational matrix forms of these ordinary differential equations are obtained from Said-Ball polynomial with variated relations of solution and different derivatives. The presented method reduces the given problem to a system of nonlinear algebraic equations, which removed the singularity of ordinary differential equations. Resulting system is solved using Newton\u27s iteration method to get the coefficients of Said-Ball polynomials. We obtained approximate solutions of the problem under study. Numerical results have been obtained and compared with exact and other works. The presented method gives impressive solutions, that show the accuracy and reliability of the proposed method

Group Classification of a family of second-order differential equations

We find the group of equivalence transformations for equations of the form $y''= A(x)y' + F(y),$ where $A$ and $F$ are arbitrary functions. We then give a complete group classification of these families of equations, using a direct method of analysis, together with the equivalence transformations.Comment: 13 page

Modified differential transform method for singular lane-emden equations in integer and fractional order

In the present work the modified differential transform method, incorporating the Adomian polynomials into the differential transform method(DTM), is used to solve the nonlinear and singular Lane-Emden equations in integer and fractional order. Numerical examples with different types are solved. The results show that this method is very effective and simple.Publisher's Versio

A technique for solving system of generalized Emden-Fowler equation using Legendre wavelet

This article is concerned with the development of an efficient numerical algorithm for the solution of a system of generalized nonlinear Emden-Fowler equation. The proposed algorithm is based on the Legendre wavelet operational matrix of integration technique. This method decreases the storage and computational complexity due to its calculation on the subinterval [ (n-1)/2^(k-1) , n/2^(k-1)) of [0,1]. The main highlight of this method is to converts the system of the differential equation into an equivalent system of nonlinear algebraic equations, which greatly simplifies approximation. Some numerical example shows that the proposed scheme is very efficient and reliable.Publisher's Versio

A Successive Linearization Method Approach to Solve Lane-Emden Type of Equations

We propose a new application of the successive linearization method for solving singular initial and boundary value problems of Lane-Emden type. To demonstrate the reliability of the proposed method, a comparison is made with results from existing methods in the literature and with exact analytical solutions. It was found that the method is easy to implement, yields accurate results, and performs better than some numerical methods