A New Approximation Method Based on Boubaker Polynomials to Solve a Type of Lane-Emden Differential Equations Arising in Astrophysics

Abstract

In this essay, a new numerical method for solving linear Lane-Emden differential equations arising in astrophysics is proposed. The method begins by approximating the second derivative in the equation using a finite series of the Boubaker polynomials, which is then expressed in matrix form. By integrating this approximation and applying the derivative condition, the expression for the first derivative term is obtained. A second integration, along with the other boundary condition, yields the Boubaker series representation of the unknown function. Using this matrix formulation and associated operations, the problem is reduced to a system of linear algebraic equations. Furthermore, an error estimation and improvement technique are introduced and applied, along with the proposed method, to numerical examples. The results demonstrate the effectiveness of the new method, showing improved accuracy compared with other standard polynomial approaches. This is evident from the tables and graphs presented in the numerical examples.