Highly efficient optimal decomposition approach and its mathematical analysis for solving fourth-order Lane–Emden–Fowler equations

In this paper, an optimal decomposition algorithm is introduced to solve a kind of nonlinear fourth-order Emden–Fowler equations (EFEs) that appear in many applied fields. Transforming the Emden–Fowler equation into a Volterra integral equivalent equation allows us to deal with the singularity at the endpoint $x=0$. This conversion also helps to reduce the computational cost of solving the problem. The existence and uniqueness of the solution of each integral equation obtained are established in the corresponding theorems. The convergence analysis further supports the theoretical findings. The accuracy and efficiency of the new method are tested against the existing method (Wazwaz et al., 2014) using numerous cases, and the results show that the presented scheme is a reliable method for computing approximate series solutions and even exact solutions. In addition, the new technique overcomes the drawback of the existing method, that provides only an approximation within a limited interval.