# Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter-Saxton Equation
## 2 Fibonacci Wavelets and Operational Matrices
### Fibonacci Wavelets and Function Approximation  
For any positive real number \(\eta\), the Fibonacci polynomials are defined by the recurrence formula:  
\[F_{m+2}(\eta)=\eta F_{m+1}(\eta)+F_{m}(\eta),\;\;m\in\mathbb{N}, \tag{6}\]  
with initial conditions \(F_{0}(\eta)=0,F_{1}(\eta)=1\). Generally, the Fibonacci polynomials are defined by  
\[F_{m+1}(\eta)= \begin{cases}1,&m=0\\ \eta,&m=1\\ \eta F_{m}(\eta)+F_{m-1}(\eta),&m\geq 2,\end{cases} , \tag{7}\]  
Wavelets constitute an orthonormal family of functions generated from dilations and translations of a fixed function called the \"mother wavelet\" \(\psi(\eta)\). When the dilation \(\alpha\) and translation parameter \(\beta\) vary continuously, one can obtain the following family of continuous wavelets [13]:  
\[\phi_{\alpha,\beta}(\eta)=|\alpha|^{-1/2}\phi\bigg{(}\frac{\eta-\beta}{\alpha }\bigg{)}\quad\alpha,\beta\in\mathbb{R},\alpha\
eq 0 \tag{8}\]  
If we restrict the parameters \(\alpha\) and \(\beta\) to discrete values, say \(\alpha=\alpha_{0}^{-k},\beta=n\beta_{0}\alpha_{0}^{-k}\), \(\alpha_{0}>1,\beta_{0}>0\), for positive integers \(n\) and \(k\), we obtain the following family of discrete wavelets:  
\[\phi_{k,n}(\eta)=|\alpha_{0}|^{k/2}\phi\Big{(}\alpha_{0}^{k}\eta-n\beta_{0} \Big{)}. \tag{9}\]The discrete wavelet family \(\psi_{k,n}(\eta)\) forms a wavelet basis for \(L^{2}(\mathbb{R})\). Motivated by the above construction, we define the Fibonacci wavelets on the interval \([0,1]\) as [22]:  
\[\phi_{n,m}(\eta)=\left\{ \begin{array}{ll}\dfrac{2^{(k-1)/2}}{\sqrt{S_{m}}}F_{m }\big{(}2^{k-1}\eta-n+1\big{)},&\qquad\left(\dfrac{n-1}{2^{k-1}}\leq\eta<\dfrac {n}{2^{k-1}}\right)\\ 0,&\qquad\text{otherwise},\end{array} \right. \tag{10}\]  
where \(k\) and \(n\), respectively, denote the level of resolution and translation with \(k=1,2,\ldots,\)\(n=1,2,\ldots,2^{k-1}\). The Fibonacci polynomial \(F_{m}(\eta)\) of degree \(m\) can also be computed as [35]  
\[F_{m}(\eta)=\sum_{i=0}^{\lfloor\eta/2\rfloor}\binom{m-i}{i}\eta^{m-2i},\quad m \geq 0, \tag{11}\]  
where \(\lfloor\cdot\rfloor\) is the usual floor function with normalization factor  
\[S_{m} =\int_{0}^{1}\big{(}F_{m}\big{)}^{2}\,d\eta,\quad m=0,1,\ldots,M-1.\] \[=\sum_{i=0}^{\lfloor\eta/2\rfloor}\sum_{j=0}^{\lfloor\eta/2 \rfloor}\binom{m-i}{i}\binom{m-j}{j}\Big{(}2(m-i-j)+1\Big{)}^{-1}. \tag{12}\]  
If we take \(k=2,M=3\), we obtain the following family of Fibonacci wavelets:  
\[\phi_{1,0}(\eta) =\sqrt{2},\] \[\phi_{1,1}(\eta) =2\sqrt{6}\,\eta,\] \[\phi_{1,2}(\eta) =\sqrt{\frac{15}{14}}\,(1+4\eta^{2}),\] \[\phi_{2,0}(\eta) =\sqrt{2},\] \[\phi_{2,1}(\eta) =\sqrt{6}\,(2\eta-1),\] \[\phi_{2,2}(\eta) =\sqrt{\frac{30}{7}}\,\big{(}2\eta^{2}-2\eta+1\big{)}\]  
Any square integrable function \(u(\eta)\) can be approximated via Fibonacci wavelets as  
\[u(\eta)\approx\sum_{n=1}^{2^{k-1}}\sum_{m=0}^{M-1}h_{n,m}\,\phi_{n,m}(\eta) \tag{14}\]  
where \(h_{n,m}\) represents the Fibonacci wavelet coefficients, given by  
\[S_{m} =\int_{0}^{1}\big{(}F_{m}\big{)}^{2}\,d\eta,\quad m=0,1,\ldots,M-1.\]0  
The corresponding matrix form of (14) is given by  
\[S_{m} =\int_{0}^{1}\big{(}F_{m}\big{)}^{2}\,d\eta,\quad m=0,1,\ldots,M-1.\]1  
where \(\mathbf{H}\) represents the row vector:  
\[\mathbf{H}= \begin{bmatrix}h_{1,0},\ldots,h_{1,M-1},h_{2,0},\ldots,h_{2,M-1}, \ldots,h_{2^{k-1},0},\ldots,h_{2^{k-1},M-1}\end{bmatrix} ^{T}. \tag{17}\]  
The matrix \(\Phi(\eta)\) appearing in Equation (16) represents the Fibonacci wavelet matrix of order \(1\times 2^{k-1}M\) and is given by  
\[\Phi(\eta)= \begin{bmatrix}\phi_{1,0},\ldots,\phi_{1,M-1},\phi_{2,0},\ldots, \phi_{2,M-1},\ldots,\phi_{2^{k-1},0},\ldots,\phi_{2^{k-1},M-1}\end{bmatrix} ^{ T}.