Analytical solution of time-fractional Emden-Fowler equations arising in astronomy and mathematical physics.

The time-fractional Emden-Fowler model is an interesting mathematical model because it is widely used and important in mathematics and applied sciences. The current investigation compiles the Elzaki transform homotopy perturbation method (ET-HPM) and the Laplace transform homotopy perturbation method (LT-HPM) for the analytical solution of the considered model. In their developments, we show that fractional orders are simple to compute, and the resulting series lead to accurate results after a few iterations. We consider the fractional orders in the context of the Caputo form. The primary aim of these approaches is to compare the analytical results of the models under consideration. These schemes do not impose any constraints on the parameters or make any assumptions that could complicate the process of obtaining the analytical results. We show that the derived outcomes are in convergent sequence with a rapid iterative formula. The iterative series and the graphical illustrations in various fractional orders confirm the efficiency of the proposed approaches. The derived results are also compared to the precise solutions, which show that our suggested approaches are simple, effective, and accurate.