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

TL;DR

This work addresses nonlinear fractional Lane-Emden boundary-value problems with Caputo derivatives $D^\alpha$ and $D^\beta$, formulated as $D^\alpha y(x) + \frac{\lambda}{x^\beta}D^\beta y(x) + f(y)=0$ on $0<x<1$ with mixed BCs, modeling heat-conduction-type phenomena. The authors develop the Uniform Fractional Haar Wavelet Collocation Method (UFHWCM), combining quasilinearization with uniform Haar-wavelet collocation and fractional Haar integration to transform the nonlinear problem into a sequence of linear systems that are efficiently solved for Haar coefficients. They provide convergence and stability analyses, showing the error decays with mesh refinement and that the linear system remains well-conditioned (with $\|T^{-1}\|_2$ near $0.7071$ across cases). Numerical experiments on three Test Cases demonstrate consistent residual reduction as the mesh is refined and confirm that the fractional solutions converge to the classical Lane-Emden solution when $(\alpha,\beta)\to(2,1)$. The method offers a robust, high-accuracy approach for fractional β-singular BVPs and related heat-conduction models.

Abstract

This paper introduces nonlinear fractional Lane-Emden equations of the form, $$ D^α y(x) + \fracλ{x^β}~ D^β y(x) + f(y) =0, ~ ~1 < α\leq 2, ~~ 0< β\leq 1, ~~ 0 < x < 1,$$ subject to boundary conditions, $$ y'(0) =\mathbf{a} , ~~~ \mathbf{c}~ y'(1) + \mathbf{d}~ y(1) = \mathbf{b},$$ where, $D^α, D^β$ represent Caputo fractional derivative, $\mathbf{a, b,c,d} \in \mathbb{R}$, $ λ= 1, 2$, and $f(y)$ is non linear function of $y.$ 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 $(α, β)$ approach $(2,1),$ the solutions of the fractional and classical Lane-Emden become identical.

Figures (10)

• Figure 1: The demonstration of the method design of the proposed work.
• Figure 2:
• Figure 4:
• Figure 6: Residual error for problem \ref{['P3Test1']} at $\alpha =1.99, \beta =0.99.$
• Figure 7:

Theorems & Definitions (8)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3