# Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter-Saxton Equation
## 1 Introduction
[PERSON] [11] formulated a stable and convergent finite difference method for the modified Hunter-Saxton equation on a periodic domain. Nonetheless, very recently, [PERSON] et al. [12] demonstrated a highly applicable aspect of the Hunter-Saxton equation while undertaking the numerical treatment of the liquid crystal model described through the Hunter-Saxton equation by employing the Sinc collocation method via the theta weighted scheme. Over the last couple of decades, wavelet theory has seen unprecedented progress in the form of innovative and efficient numerical algorithms. Wavelet-based numerical schemes have defhroned the usual numerical methods and have received much attention from researchers working in various disciplines of science and engineering [13; 14]. In particular, the wavelet-based matrix methods have acquired a respectable status in numerical analysis due to their easy implementation, simple procedure, and rapid convergence. Moreover, they are computer-oriented and do not require calculating the inverse of the wavelet matrix,and less CPU time is needed as the major blocks of the wavelet operational matrix are computed once and reused in subsequent computations. As of now, numerous linear and non-linear differential equations have been solved by employing various wavelet families, including the Haar wavelets, Euler wavelets, Bernoulli wavelets, Chebyshev wavelets, Legendre wavelets, Hermite wavelets, Gegenbauer wavelets, and ultraspherical wavelets [15; 16; 17; 18; 19; 20; 21; 22]. To harness the advantages of wavelet-based numerical methods in the context of the non-linear Hunter-Saxton equation, many wavelet-based methods have been employed for obtaining the numerical solution. To mention a few, [PERSON] et al. [23] utilized the Haar wavelet quasi-linearization approach to obtain the approximate solution of the Hunter-Saxton equation. Recently, [PERSON] et al. [24] developed a collocation technique based on the Laguerre wavelet for the numerical solution of the Hunter-Saxton equation and exhibited the uniform convergence of the employed method. The origin of the Fibonacci numbers dates back to the early 1200s with the seminal work of Italian mathematician [PERSON] studying the famous rabbit problem. The Fibonacci numbers \(F_{m}=F_{m-1}+F_{m-2},m\geq 2\) with initial values \(F_{0}=0,F_{1}=1\) are the terms of the sequence \(\{0,1,1,2,3,5,\dots\}\). Using the Fibonacci-like recurrence relations, [PERSON] and [PERSON] introduced a large class of polynomials commonly known as Fibonacci polynomials. The polynomials \(F_{m}(\eta)\) introduced by [PERSON] are defined by the recurrence relation \(F_{m}(\eta)=\eta F_{m-1}(\eta)+F_{m-2}(\eta),\ \eta\in\mathbb{R},m\geq 3\) with \(F_{1}(\eta)=1,F_{2}(\eta)=\eta\), whereas the polynomials introduced by [PERSON] have the general form \(F_{m}(\eta)=F_{m-1}(\eta)+\eta F_{m-2}(\eta)\), \(m\geq 3\) with \(F_{1}(\eta)=F_{2}(\eta)=1\). Later on, [PERSON] defined another class of Fibonacci polynomials of the form \(F_{m}(\eta)=2\eta F_{m-1}(\eta)+F_{m-2}(\eta),m\geq 2\), where \(F_{0}(\eta)=0,F_{1}(\eta)=1\). Over a couple of decades, some prominent offshoots of the Fibonacci polynomials have appeared in the open literature. For instance, [PERSON] and [PERSON] [25] introduced the notion of \(h\)-Fibonacci polynomials and studied the derivatives of these polynomials in terms of the convolution of \(h\)-Fibonacci polynomials. Mathematically, the \(h(\eta)\)-Fibonacci polynomials \(\{F_{h,m}(\eta)\}_{m=0}^{\infty}\) are defined by the recurrence relation \(F_{h,m+1}(\eta)=h(\eta)F_{h,m}(\eta)+F_{h,m-1},m\geq 1\), where \(F_{h,0}=0\), \(F_{h,1}=1\), and \(h(\eta)\) is a polynomial with real coefficients. It is pertinent to mention that for \(h(\eta)=\eta\) and \(h(\eta)=2\eta\), one can obtain the Fibonacci polynomials of the Catalan and Byrd type, respectively [26].