# Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter-Saxton Equation
## 5 Numerical Examples and Discussion
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\eta\) & **Present Method** & **Haar Wavelet Method**[23] & _B-Spline_[9] \\ \hline
0.1 & \(4.0590\times 10^{-13}\) & \(7.36\times 10^{-9}\) & \(7.3711\times 10^{-10}\) \\
0.2 & \(2.0500\times 10^{-12}\) & \(5.21\times 10^{-8}\) & \(5.3433\times 10^{-9}\) \\
0.3 & \(1.6620\times 10^{-12}\) & \(6.91\times 10^{-8}\) & \(9.6802\times 10^{-9}\) \\
0.4 & \(2.4243\times 10^{-12}\) & \(8.88\times 10^{-7}\) & \(8.2694\times 10^{-9}\) \\
0.5 & \(6.0234\times 10^{-12}\) & \(9.16\times 10^{-7}\) & \(8.1740\times 10^{-9}\) \\
0.6 & \(5.2824\times 10^{-13}\) & \(9.46\times 10^{-7}\) & \(8.0728\times 10^{-9}\) \\
0.7 & \(1.3101\times 10^{-13}\) & \(1.85\times 10^{-6}\) & \(1.0326\times 10^{-9}\) \\
0.8 & \(4.6474\times 10^{-13}\) & \(1.88\times 10^{-6}\) & \(6.2946\times 10^{-10}\) \\
0.9 & \(4.4524\times 10^{-12}\) & \(1.91\times 10^{-6}\) & \(2.1314\times 10^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the proposed method’s absolute error with the HWM and _B_-spline methods of Example 1 for \(\zeta=0.01\). Figure 3: (a) Absolute error at \(k=2\), \(M=3\); (b) absolute error at \(k=4\), \(M=5\) of Example 1. Figure 2: Three-dimensional comparison of the approximate and exact solution at \(k=4\), \(M=5\) of Example 1. **Example 2**. : _Consider Equation (29) subject to the initial conditions:_  
\[u(\eta,0)=(2+3\eta)^{2/3}+2\eta+2,\quad\frac{\partial u(\eta,0)}{\partial\zeta}=2 \eta+\frac{2\eta}{\left(3\eta+2\right)^{1/3}},\quad 0\leq\eta\leq 1, \tag{49}\]  
_and the boundary conditions:_  
\[u(0,\zeta)=\frac{2^{2/3}+2}{\left(1+\zeta\right)^{2}},\quad\frac{\partial u(0, \zeta)}{\partial\eta}=\frac{2^{2/3}\Big{(}\zeta+2^{1/3}(\zeta+1)+1\Big{)}}{(1+ \zeta)^{2}},\quad 0<\zeta\leq 1. \tag{50}\]  
_The analytic solution of the problem (29) together with the initial and boundary conditions (49) and (50) is_  
\[u(\eta,\zeta)=\frac{(2+3\eta(1+\zeta))^{2/3}+2\eta(1+\zeta)+2}{\left(1+\zeta \right)^{2}}. \tag{51}\]  
_The initial guess is given by_  
\[u_{0}(\eta,\zeta)=(2+3\eta)^{2/3}+2\eta+2+\zeta\Bigg{(}2\eta+\frac{2\eta}{ \left(3\eta+2\right)^{1/3}}\Bigg{)}. \tag{52}\]  
_Figures 4 and 5 depict the graphical comparison of the proposed method with the exact solution for \(k=3\), \(M=4\) and \(k=4\), \(M=5\), respectively, which suggests that the Fibonacci wavelets are in good agreement with the analytic solution of the problem. Moreover, in Figure 6, the absolute error for different \(k=2,M=3\) and \(k=4\), \(M=5\) is shown via a three-dimensional plot, which elucidates that, upon increasing the wavelet parameters \(k\) and \(M\), the absolute error decreases. Finally, a comparison of the absolute error of the proposed method with the trigonometric \(B\)-spline [9] and the Laguerre wavelet methods [24] is given in Tables 4-6 for different values of \(\zeta\). From these tables, we conclude that the proposed method is more accurate than the other existing numerical methods._ \begin{table}
\begin{tabular}{c c c c} \hline \(\eta\) & **Present Method** & **Laguerre Wavelet Method**[24] & _B-Spline_[9] \\ \hline
0.1 & \(1.0628\times 10^{-10}\) & \(1.8010\times 10^{-10}\) & \(1.3312\times 10^{-9}\) \\
0.2 & \(4.0301\times 10^{-10}\) & \(1.5112\times 10^{-10}\) & \(5.7990\times 10^{-9}\) \\
0.3 & \(2.5753\times 10^{-10}\) & \(3.9270\times 10^{-9}\) & \(1.0590\times 10^{-8}\) \\
0.4 & \(2.5521\times 10^{-10}\) & \(1.4702\times 10^{-8}\) & \(5.6427\times 10^{-7}\) \\
0.5 & \(1.1001\times 10^{-9}\) & \(1.6749\times 10^{-8}\) & \(5.4550\times 10^{-7}\) \\
0.6 & \(2.2703\times 10^{-11}\) & \(1.9008\times 10^{-8}\) & \(5.3513\times 10^{-7}\) \\
0.7 & \(1.9953\times 10^{-10}\) & \(3.5397\times 10^{-8}\) & \(5.8207\times 10^{-8}\) \\
0.8 & \(2.1422\times 10^{-10}\) & \(8.8317\times 10^{-8}\) & \(3.5256\times 10^{-8}\) \\
0.9 & \(1.0988\times 10^{-9}\) & \(2.4388\times 10^{-8}\) & \(1.1877\times 10^{-8}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison of the proposed method’s absolute error with the HWM and _B_-spline methods of Example 2 for \(\zeta=0.1\).