Abstract
This paper presents an advanced variational iteration method for finding the approximate-exact solution for initial value problems of the Lane–Emden type equation that arise in several applications by employing the quasilinearization approach to the variational iteration method. Also, the convergence analysis of the method is studied under the Lipschitz condition. To demonstrate the applicability of the method, several application problems arise in isothermal gas spheres, stellar structures, and physics with different shape factors and are successfully solved by means of this method. The comparison is made with the available exact solution and the existing methods (Mall and Chakraverty, 2014; Umesh and Kumar, 2021; El-Essawy et al., 2023; Ahmad et al., 2016). We see that the method meets our expectations and reaches an exact solution in a few iterations. The optimality of the method is investigated by the absolute errors and absolute residual errors. The present method is time-efficient, requires less computational work, and converges to an exact solution in a few iterations. We also provided the CPU time to obtain an approximate solution to all the problems for the numerical investigation.
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