# Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter-Saxton Equation
## 4 Error Estimation and Convergence Analysis  
This section is entirely devoted to study the error bounds and convergence results associated with the Fibonacci wavelet method. **Theorem 1**. : _Let \(u^{*}(\eta)=\sum_{n=1}^{2^{k-1}}\sum_{m=0}^{M-1}h_{n,m}\,\phi_{n,m}(\eta)\) be the Fibonacci wavelet expansion of any sufficiently smooth real-valued function \(u(\eta)\) belonging to the space \(C^{M}[0,1)\). Then, the norm truncated error \(e(\eta)\) is given by_  
\[\left\|e(\eta)\right\|_{2}=\left\|u-u^{*}\right\|_{2}\leq\frac{R}{  
From the above expansion, one can obtain  
\[\left|u(\eta)-\hat{u}(\eta)\right|=\frac{u^{(M)}(s)\eta^{M}}{M!},\quad s\in[0,1). \tag{40}\]  
As the interval \([0,1)\) can be divided into sub-intervals of the form \(I_{n,k}=\left[\frac{(n-1)}{2^{k-1}},\frac{n}{2^{k-1}}\right]\), therefore, by virtue of (40), we obtain  
\[\left\|u-u^{*}\right\|_{L^{2}[0,1]}^{2} =\left\|u-C^{T}\Phi\right\|_{L^{2}[0,1]}^{2}\] \[=\sum_{n=1}^{2^{k-1}}\left\|u-A^{T}\hat{F}\right\|_{I_{n,k}}^{2}\] \[\leq\sum_{n=1}^{2^{k-1}}\left\|u-u^{*}\right\|_{I_{n,k}}^{2}\] \[\leq\sum_{n=1}^{2^{k-1}}\int_{I_{n,k}}\left[u(\eta)-u^{*}(\eta) \right]^{2}d\eta\] \[\leq\sum_{n=1}^{2^{k-1}}\int_{I_{n,k}}\left(\frac{\eta^{M}max_{x \in I_{n,k}}|u^{M}(s)|}{M!}\right)^{2}d\eta\] \[\leq\int_{0}^{1}\left(\frac{\eta^{M}max_{x\in I_{n,k}}|u^{M}(s)| }{M!}\right)^{2}d\eta\] \[=\frac{R^{2}}{M!^{2}\sqrt{2M+1}},\]  
where \(\hat{F}(\eta)=\left[F_{0}(\eta),F_{1}(\eta),\ldots,F_{M-2}(\eta),F_{M-1}(\eta )\right]^{T}\) and \(R=\max_{s\in[0,1)}|u^{M}(\eta)|\). Hence, \(\|e(\eta)\|\to 0\) as \(M\to\infty\). This completes the proof of Theorem 1. Theorem 2: Let \(u(\eta)\) be a bounded continuous function in \(L^{2}[0,1)\) with bound \(\tilde{M}\). Then, the Fibonacci wavelet expansion given by (14) converges uniformly to \(u(\eta)\) with  
\[\left|h_{n,m}\right|<\frac{\tilde{M}\,\delta\,r}{2^{k-1/2}\,2^{m}},\quad\delta =\frac{1}{\sqrt{\mathbb{S}_{m}}}. \tag{41}\]  
Proof: \(u(\eta)\in L^{2}[0,1)\) can be approximated by the Fibonacci wavelets as  
\[u(\eta)\approx\sum_{n=1}^{2^{k-1}}\sum_{m=0}^{M-1}h_{n,m}\,\phi_{n,m}(\eta),\]  
where  
\[\left\|u-u^{*}\right\|_{L^{2}[0,1]}^{2} =\left\|u-C^{T}\Phi\right\|_{L^{2}[0,1]}^{2}\]0 \[\left\|u-u^{*}\right\|_{L^{2}[0,1]}^{2} =\left\|u-C^{T}\Phi\right\|_{L^{2}[0,1]}^{2}\]1 \[\left\|u-u^{*}\right\|_{L^{2}[0,1]}^{2} =\left\|u-C^{T}\Phi\right\|_{L^{2}[0,1]}^{2}\]2 \[\left\|u-u^{*}\right\|_{L^{2}[0,1]}^{2} =\left\|u-C^{T}\Phi\right\|_{L^{2}[0,1]}^{2}\]3By taking \(2^{k-1}\eta-n+1=t\), we obtain  
\[\left\|u-u^{*}\right\|_{L^{2}[0,1]}^{2} =\left\|u-C^{T}\Phi\right\|_{L^{2}[0,1]}^{2}\]4  
Consequently, we obtain  
\[\left\|u-u^{*}\right\|_{L^{2}[0,1]}^{2} =\left\|u-C^{T}\Phi\right\|_{L^{2}[0,1]}^{2}\]5 \[\left\|u-u^{*}\right\|_{L^{2}[0,1]}^{2} =\left\|u-C^{T}\Phi\right\|_{L^{2}[0,1]}^{2}\]6  
Applying the property of Fibonacci polynomials, we have  
\[\left\|u-u^{*}\right\|_{L^{2}[0,1]}^{2} =\left\|u-C^{T}\Phi\right\|_{L^{2}[0,1]}^{2}\]7  
where \(r\) is a constant. Therefore, (42) becomes  
\[\left\|u-u^{*}\right\|_{L^{2}[0,1]}^{2} =\left\|u-C^{T}\Phi\right\|_{L^{2}[0,1]}^{2}\]8  
Thus, we conclude that \(\sum_{n=1}^{2^{k}-1}\sum_{m=0}^{M-1}h_{n,m}\) converges absolutely provided \(k,m\to\infty\), and hence, the Fibonacci wavelet series given by (14) converges uniformly to \(u(\eta)\). Towards the culmination, we provide an Algorithm 1 of the Fibonacci wavelet method to obtain a numerical solution of the Hunter-Saxton Equation (27). ```
\(\bullet\) Input: The level of resolution \(k=1,2,\dots\) and \(m=0,1,\dots,M-1\), \(0\leq\eta\leq 1\). \(\bullet\) Define the Fibonacci wavelet \(\phi_{n,m}(\eta)\) by Equation (10). \(\bullet\) Construct the vector \(\Phi(\eta)\) from (18). \(\bullet\) Construct the Fibonacci wavelet basis matrix \(\Phi(\eta)\) using collocation points (20). \(\bullet\) Compute the operational matrices of integration \(\mathbf{S}_{1}\) and \(\mathbf{S}_{2}\).