# Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter-Saxton Equation
## 1 Introduction  
Non-linear partial differential equations play a vital role in modeling real-life phenomena, especially in the field of applied sciences such as meteorology, aerospace engineering, oceanography, geology, astronomy, and many other disciplines. For the realistic treatment of these non-linear models, several numerical techniques such as the barycentric interpolation collocation method [1], Fourier spectral method [2], and reproducing kernel method [3] have been adopted for the numerical treatment of these non-linear partial differential equations, such as the Klein-Gordon-Zakharov equations, Schrodinger equation, Duffing systems, and many more. The Hunter-Saxton equation is one among those non-linear partial differential equations that describe waves of a nematic liquid crystal in an enormous director field and arise at the short-wave limit of the Camassa-Holm equation--an integrable model of shallow-water waves propagating uni-directionally across a flat bottom [4]. This equation has both the bi-Hamiltonian and bi-variational structure of integrable partial differential equations, often referred to as the Hunter-Saxton equation, which re-expresses the geodesic flow in the diffeomorphism group of a circle. Formally, the non-linear Hunter-Saxton equation is given by\[\frac{1}{\partial\eta}\bigg{(}\frac{\partial u(\eta,\zeta)}{\partial \zeta}+u(\eta,\zeta)\frac{\partial u(\eta,\zeta)}{\partial\eta}\bigg{)}=\frac{1} {2}\bigg{(}\frac{\partial u(\eta,\zeta)}{\partial\eta}\bigg{)}^{2}. \tag{1}\]  
Equation (1) can also be redrafted as  
\[\frac{\partial^{2}u(\eta,\zeta)}{\partial\eta\partial\zeta}+u(\eta,\zeta) \frac{\partial^{2}u(\eta,\zeta)}{\partial\eta^{2}}+\bigg{(}\frac{\partial u( \eta,\zeta)}{\partial\eta}\bigg{)}^{2}, \tag{2}\]  
where \(\eta\) and \(\zeta\) denote the space and time coordinates, respectively, with initial and boundary conditions:  
\[u(\eta,0)=g_{0}(\eta)\quad\text{and}\lim_{\eta\to\infty}u(\eta,0)=0. \tag{3}\]  
In practice, it is more appropriate to impose a boundary condition of the form:  
\[\lim_{\eta\to\infty}u(\eta,0)=0, \tag{4}\]  
or even the initial and boundary conditions:  
\[ \begin{split} u(\eta,0)=f_{0}(\eta),\qquad\frac{\partial u(\eta, 0)}{\partial\zeta}=f_{1}(\eta)\qquad 0\leq\eta\leq 1\\ u(0,\zeta)=g_{0}(\zeta),\qquad\frac{\partial u(0,\zeta)}{ \partial\eta}=g_{1}(\zeta),\qquad 0<\zeta\leq 1,\end{split}  \tag{5}\]  
where \(f_{0}(\eta),f_{1}(\eta),g_{0}(t)\), and \(g_{1}(t)\) are sufficiently smooth functions. Keeping in view the prolificacy of the Hunter-Saxton equation governed by the differential Equation (2), a substantial amount of research has been carried out in the quest to obtain numerical solutions via several analytical and numerical techniques. For instance, [PERSON] [5] employed the homotopy analysis method (HAM) for the approximate solution of the non-linear Hunter-Saxton equation and showed that the HAM is more rapidly convergent than certain other iterative methods such as the variational iteration method, Adomian decomposition method, modified variational iteration method, and modified Adomian decomposition method, whereas [PERSON] et al. [6] established the global solutions of the periodic Hunter-Saxton equation with weak dissipation. [PERSON] et al. [7] obtained exact rational solutions of the generalized Hunter-Saxton equation by using the Pade approximation. Besides, [PERSON] [8] proposed the reduced differential transform method for obtaining the solution of the non-linear Hunter-Saxton equation, which does not require linearization, discretization, nor perturbation of the underlying non-linear differential equation. [PERSON] et al. [9] examined the efficiency of the cubic trigonometric \(B\)-spline method for obtaining the numerical solution of the non-linear Hunter-Saxton equation. [PERSON] et al. [10] applied the finite element method by reducing the governing non-linear partial differential equation to a system of ordinary differential equations in terms of shape functions.