Uniform Haar Wavelet Solutions for Fractional Regular $\beta$-Singular BVPs Modeling Human Head Heat Conduction under Febrifuge Effects

This paper introduces nonlinear fractional Lane-Emden equations of the form, $$D^{\alpha }y(x)+\frac{\lambda }{x^{\beta }}{D}^{\beta }y(x)+f(y)=0,1<\alpha \le 2,0<\beta \le 1,0<x<1,$$ subject to boundary conditions, $$y^{\mathrm{\prime }}(0)=\mathbf{a},\mathbf{c}{y}^{\mathrm{\prime }}(1)+\mathbf{d}y(1)=\mathbf{b},$$ where, $D^{\alpha },D^{\beta }$ represent Caputo fractional derivative, $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}\in \mathbb{R}$, $\lambda =1,2$, and $f(y)$ is non linear function of $y\mathrm{.}$ We have developed collocation method namely, uniform fractional Haar wavelet collocation method and used it to compute solutions. The proposed method combines the quasilinearization method with the Haar wavelet collocation method. In this approach, fractional Haar integrations is used to determine the linear system, which, upon solving, produces the required solution. Our findings suggest that as the values of $(\alpha ,\beta )$ approach $(2,1),$ the solutions of the fractional and classical Lane-Emden become identical.