New analytical method of solution to a nonlinear singular fractional Lane–Emden type equation

We consider a nonlinear singular fractional Lane–Emden type differential equation

$${}^{LC}{\mathcal{D}}_{a^{+}}^{\alpha }\phi (t)+\frac{\lambda }{t^{\alpha -\beta }}{}^{LC}{\mathcal{D}}_{a^{+}}^{\beta }\varpi (t,\phi (t))=0,0<\beta <\alpha <1,0<a<t\le T,$$

with an initial condition $\phi (a)=\nu$ assumed to be bounded and non-negative, $\varpi :[a,T]\times \mathbb{R}\to \mathbb{R}$ a Lipschitz continuous function, and ${}^{LC}{\mathcal{D}}_{a^{+}}^{\alpha },{}^{LC}{\mathcal{D}}_{a^{+}}^{\beta }$ are Liouville–Caputo derivatives of orders $0<\alpha ,\beta <1$. A new analytical method of solution to the nonlinear singular fractional Lane–Emden type equation using fractional product rule and fractional integration by parts formula is proposed. Furthermore, we prove the existence and uniqueness and the growth estimate of the solution. Examples are given to illustrate our results.