# Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter-Saxton Equation
## 3 Method of Solution  
In this section, we shall employ the operational matrices of integration based on Fibonacci wavelets as discussed in Section 2 for the numerical treatment of the non-linear Hunter-Saxton Equation (2). To begin with, we recall the non-linear Hunter-Saxton equation:  
\[\frac{\partial^{2}u(\eta,\zeta)}{\partial\eta\partial t}+u(\eta,\zeta)\frac{ \partial^{2}u(\eta,\zeta)}{\partial\eta^{2}}+\frac{1}{2}\bigg{(}\frac{ \partial u(\eta,\zeta)}{\partial\eta}\bigg{)}^{2}=0, \tag{27}\]  
with initial condition and boundary conditions:  
\[u(\eta,0)=f_{0}(\eta),\qquad\frac{\partial u(\eta,0)}{\partial \zeta}=f_{1}(\eta)\qquad 0\leq\eta\leq 1 \tag{28}\] \[u(0,\zeta)=g_{0}(\zeta),\qquad\frac{\partial u(0,\zeta)}{ \partial\eta}=g_{1}(\zeta),\qquad 0<t\leq 1,\]where \(f_{0},f_{1},g_{0}\), and \(g_{1}\) possess second-order continuous derivative and belong to \(L^{2}[0,1)\). Following the quasi-linearization technique as discussed in the previous section, Relation (27) is transformed to a linear differential equation as  
\[\frac{\partial^{2}u_{r+1}(\eta,\zeta)}{\partial\eta\partial\zeta} +u_{r+1}(\eta,\zeta)\frac{\partial^{2}u_{r}(\eta,\zeta)}{\partial \eta^{2}}+\frac{\partial u_{r}(\eta,\zeta)}{\partial\eta}\frac{\partial u_{r+ 1}(\eta,\zeta)}{\partial\eta}+\frac{\partial f_{r}(\eta,\zeta)}{\partial\eta} \frac{\partial^{2}f_{r+1}(\eta,\zeta)}{\partial\eta^{2}}\] \[-\frac{1}{2}\bigg{(}\frac{\partial f_{r}(\eta,\zeta)}{\partial \eta}\bigg{)}^{2}-\frac{\partial^{2}f_{r}(\eta,\zeta)}{\partial\eta^{2}}f_{r }(\eta,\zeta)=0. \tag{29}\]  
with initial guess \(u_{0}(\eta,\zeta)=f_{0}(\eta)+\zeta f_{1}(\eta)\). In order to solve (29), we approximate the highest partial derivative appearing in (29) by the Fibonacci wavelet basis as  
\[\frac{\partial^{3}u(\eta,\zeta)}{\partial\eta^{2}\partial\zeta}=\sum_{\ell=0 }^{2^{k-1}M}h_{\ell}\phi(\eta)=\mathbf{H}^{T}\Phi(\eta), \tag{30}\]  
where \(\mathbf{H}\) is the unknown wavelet coefficient vector given by (17) to be determined. Integrating (30) with respect to \(\zeta\) between limits \(\zeta_{s}\) and \(\zeta\) yields  
\[\frac{\partial^{2}u(\eta,\zeta)}{\partial\eta^{2}} =(\zeta-\zeta_{s})\sum_{i=0}^{2^{k-1}M}h_{\ell}\phi(\eta)+\frac{ \partial^{2}u(\eta,\zeta_{s})}{\partial\eta^{2}}\] \[=(\zeta-\zeta_{s})\mathbf{H}^{T}\Phi_{\ell}+\frac{\partial^{2}u( \eta,\zeta_{s})}{\partial\eta^{2}}. \tag{31}\]  
Integrating (31) with respect to \(\eta\) between the limits \(0\) and \(\eta\), we obtain  
\[\frac{\partial u(\eta,\zeta)}{\partial\eta}=(\zeta-\zeta_{s})\sum_{i=0}^{2^{k -1}M}h_{\ell}\mathbf{S}_{\ell,1}(\eta)+\frac{\partial u(\eta,\zeta_{s})}{ \partial\eta}-\frac{\partial u(0,\zeta_{s})}{\partial\eta}+\frac{\partial u( 0,\zeta)}{\partial\eta}. \tag{32}\]  
Again, integrating (32) with respect to \(\eta\) between the limits \(0\) and \(\eta\) yields  
\[u(\eta,\zeta)=(\zeta-\zeta_{s})\sum_{i=0}^{2^{k-1}M}h_{\ell}\mathbf{S}_{\ell,2 }(\eta)+u(\eta,\zeta_{s})-u(0,\zeta_{s})+\eta\bigg{(}\frac{\partial u(0,\zeta )}{\partial\eta}-\frac{\partial u(0,\zeta_{s})}{\partial\eta}\bigg{)}+u(0, \zeta). \tag{33}\]  
Differentiating (33) with respect to \(\zeta\), we have  
\[u(\eta,0)=f_{0}(\eta),\qquad\frac{\partial u(\eta,0)}{\partial \zeta}=f_{1}(\eta)\qquad 0\leq\eta\leq 1 \tag{28}\]0  
Next, we differentiate (34) with respect to \(\eta\) to obtain  
\[u(\eta,0)=f_{0}(\eta),\qquad\frac{\partial u(\eta,0)}{\partial \zeta}=f_{1}(\eta)\qquad 0\leq\eta\leq 1 \tag{28}\]1By employing the initial and boundary conditions (28) and \(\zeta\rightarrow\zeta_{s+1}\) with \(\Delta\zeta=\zeta_{s+1}-\zeta_{s}\), Equations (31)-(34) take the form  
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