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The Emden-Fowler equation has important applications in some mathematical and physical problems. In this paper, a smoothing transformation is given for the initial value problem of the Emden-Fowler equation, and then the equation is further transformed into an equivalent Volterra integral equation of the second kind. The series expansion of the solution to the integral equation about the origin is obtained by Picard iteration, which shows that the solution of the transformed equation is sufficiently smooth at the origin. The series solution and its Padé approximation are usually only accurate near the origin. Hence, a high-precision Chebyshev collocation method is designed to obtain the numerical solution on a finite interval. The convergence of the scheme is analysed, and the error with the maximum norm is estimated. Numerical examples illustrate the high accuracy of the proposed method for solving the initial value problem of the Emden-Fowler equation.

Series solution and Chebyshev collocation method for the initial value problem of Emden-Fowler equation