Explicit solution of Lane-Emden type equations via a novel recurrence and Padé approximation approach

This article introduces a novel recursive algorithm for obtaining explicit solutions to initial value problems of Lane-Emden type equations. By combining the traditional power series method with Adomian polynomials, expressed in terms of solution coefficients, the algorithm achieves high accuracy and converges rapidly to the exact solution within only a few iterations. This formulation not only simplifies the solution process but also improves computational efficiency over several existing semi-analytical approaches by requiring fewer iterations to reach a desired level of accuracy. Additionally, the Padé approximation is applied to the power series solution to accelerate convergence and expand the convergence region, allowing the solution to remain accurate over a wider interval. Error analysis using absolute and residual errors confirms that the proposed method, both independently and in combination with Padé approximants, outperforms existing methods in terms of precision and applicability. Several examples illustrate the method’s accuracy, efficiency, and reliability in solving nonlinear singular initial value problems.