An optimal decomposition method for analytical and numerical solution of third-order Emden–Fowler type equations

The current study formulated an optimal decomposition approach to solve different nonlinear third-order Emden–Fowler equations that arise in various scientific applications. To avoid singularity at x=0, the Emden–Fowler equation is converted into a Volterra integral equation. This allows the mathematical computation to be reduced due to single evaluations of the integral present in the problem. The optimal decomposition method is then applied to the resulting integral equations to compute closed-form and approximate numerical solutions. Some theorems are provided for the existence of a unique solution corresponding to integral equations. The mathematical formulation is further supported by conducting the convergence analysis of the proposed method. The accuracy and efficiency of the new approach is examined by working out several examples, and the obtained results demonstrate that the proposed method provides a reliable approach to compute the approximate series solutions and the exact solutions. The new approach resolves the issue of the existing method, which allows only to capture the solutions for smaller values of x (less than equal to 1).