# Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter-Saxton Equation
## 5 Numerical Examples and Discussion  
This section constitutes the centerpiece of this article: the operational matrices of integration as constructed in Section 2 based on the Fibonacci wavelets are utilized for obtaining the approximate solution of the non-linear Hunter-Saxton Equation (27) under certain initial and boundary conditions. To facilitate the narrative, we take into account some examples whose exact solutions are available in the open literature. This allows us to compare the obtained results with exact solutions and the solutions acquired through other numerical techniques. The accuracy of the proposed method is determined by the absolute error:  
\[E_{abs}=\big{|}u(\eta,\zeta)-u_{n,m}(\eta,\zeta)\big{|}, \tag{44}\]  
where \(u(\eta,\zeta)\) and \(u_{n,m}(\eta,\zeta)\), respectively, denote the exact and approximate solutions. **Example 1**. : _Consider Equation (29) subject to the initial condition:_  
\[u(\eta,0)=2\eta,\quad\frac{\partial u(\eta,0)}{\partial\zeta}=-2\eta,\quad 0 \leq\eta\leq 1, \tag{45}\]  
_and the boundary condition:_  
\[u(0,\zeta)=0,\quad\frac{\partial u(0,\zeta)}{\partial\eta}=\frac{2}{1+\zeta}, \quad 0<\zeta\leq 1. \tag{46}\]  
_The analytic solution of the problem (29) with respect to the initial and boundary conditions (45) and (46) is_  
\[u(\eta,\zeta)=\frac{2\eta}{1+\zeta}. \tag{47}\]  
_The initial guess is given by_  
\[f_{0}(\eta,\zeta)=2\eta-2\zeta\eta. \tag{48}\]  
_The comparison of the exact solution and the approximate solution for \(k=3,M=4\) and \(k=4,M=5\) is depicted in Figures 1 and 2, respectively. From Figure 3, the absolute error for \(k=2,M=3\) and \(k=4,M=5\) is represented via three-dimensional plots, from which we infer that as we increase the wavelet parameters \(k\) and \(M\), the absolute error decreases. To show the legitimacy of the proposed method, we compare the obtained absolute error with the absolute error of the trigonometric \(B\)-spline [9] and the Haar wavelet methods [23] in Tables 1-3 for different values of \(\zeta\), from which one can easily infer that the results achieved through the Fibonacci wavelets are reasonably effective in comparison to the other existing numerical methods._  
Figure 1: Comparison of analytic and FWM solutions of Example 1 at \(k=3,M=4\) for different \(\zeta\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\eta\) & **Present Method** & **Haar Wavelet Method**[23] & _B-Spline_[9] \\ \hline
0.1 & \(1.1052\times 10^{-11}\) & \(1.00\times 10^{-11}\) & \(5.0871\times 10^{-8}\) \\
0.2 & \(4.5815\times 10^{-11}\) & \(5.00\times 10^{-11}\) & \(5.8416\times 10^{-8}\) \\
0.3 & \(4.5253\times 10^{-11}\) & \(7.00\times 10^{-11}\) & \(7.1641\times 10^{-8}\) \\
0.4 & \(6.6008\times 10^{-11}\) & \(1.10\times 10^{-9}\) & \(5.9301\times 10^{-8}\) \\
0.5 & \(1.6400\times 10^{-11}\) & \(2.00\times 10^{-10}\) & \(5.8624\times 10^{-8}\) \\
0.6 & \(1.4383\times 10^{-11}\) & \(4.00\times 10^{-10}\) & \(5.7901\times 10^{-8}\) \\
0.7 & \(5.5616\times 10^{-12}\) & \(4.00\times 10^{-10}\) & \(7.4068\times 10^{-9}\) \\
0.8 & \(1.2655\times 10^{-11}\) & \(4.00\times 10^{-10}\) & \(4.5148\times 10^{-9}\) \\
0.9 & \(3.9552\times 10^{-11}\) & \(4.00\times 10^{-10}\) & \(1.5287\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the proposed method’s absolute error with the HWM and _B_-spline methods of Example 1 for \(\zeta=0.1\).