Numerical Solutions for Singular Lane-Emden Equations Using Shifted Chebyshev Polynomials of the First Kind

Abstract

This paper describes an algorithm for obtaining approximate solutions to a variety of well-known Lane-Emden type equations. The algorithm expands the desired solution y(x) ≃ y_N(x), in terms of shifted Chebyshev polynomials of first kind such that y_N⁽ⁱ⁾(0) = y⁽ⁱ⁾(0) (i = 0, 1, ..., N). The derivative values y^(j)(0) for j = 2, 3, ..., are computed by using the given differential equation and its initial conditions. This makes approximate solutions more consistent with the exact solutions of given differential equations. The explicit expressions of the expansion coefficients of y_N(x) are obtained. The suggested method is much simpler compared to any other method for solving this initial value problem. An excellent agreement between the exact and the approximate solutions is found in the given examples. In addition, the error analysis is presented.