## 5 The Error Bound  
In this section, the convergence of our spectral collocation method is examined in both the first- and two-dimensional generalized Chebyshev-weighted Sobolev spaces. Consider the following generalized Chebyshev-weighted Sobolev space:  
\[\mathbf{H}^{m}_{w(z)}(I)=\{u:D^{k}_{z}\,u\in L^{2}_{w(z)}(I),\,0\leq k\leq m\}, \tag{50}\]  
where \(I=(0,\mathcal{L})\), equipped with the following inner product, norm, and semi-norm:  
\[(u,v)_{\mathbf{H}^{m}_{w(z)}}=\sum_{k=0}^{m}(D^{k}_{z}\,u,D^{k}_ {z}\,v)_{L^{2}_{w(z)}}, \tag{51}\] \[||u||^{2}_{\mathbf{H}^{m}_{w(z)}}=(u,u)_{\mathbf{H}^{m}_{w(z)}}, \quad|u|_{\mathbf{H}^{m}_{w(z)}}=||D^{m}_{z}\,u||_{L^{2}_{w(z)}},\]  
where \(m\in N\). Also, assume the following two-dimensional Gegenbauer-weighted Sobolev space:  
\[\mathbf{H}^{r,s}_{\mathcal{O}(z,\tau)}(\Omega)=\{u:\frac{\partial^{p+q}\,u}{ \partial\,z^{p}\,\partial\,\tau^{q}}\in L^{2}_{\tilde{\omega}(z,\tau)}(\Omega ),\,r\geq p\geq 0,\,s\geq q\geq 0\}, \tag{52}\]  
equipped with the following norm and semi-norm:  
\[||u||_{\mathbf{H}^{s}_{\mathcal{O}(z,\tau)}}=\left(\sum_{p=0}^{r}\sum_{q=0}^{s }\left|\left|\frac{\partial^{p+q}\,u}{\partial\,z^{p}\,\partial\,\tau^{q}} \right|\right|^{2}_{L^{2}_{\tilde{\omega}(z,\tau)}}\right)^{\frac{1}{2}},\quad |u|_{\mathbf{H}^{r,s}_{\mathcal{O}(z,\tau)}}=\left|\left|\frac{\partial^{r+s} \,u}{\partial\,z^{r}\,\partial\,\tau^{s}}\right|\right|_{L^{2}_{\tilde{\omega}( z,\tau)}}, \tag{53}\]  
where \(\Omega=(0,\mathcal{L})\times(0,\mathcal{T})\), \(\tilde{\omega}(z,\tau)=w(z)\,w(\tau)\) and \(r,s\in N\). Lemma 1 ([53]): _For \(n\geq 1\), \(n+r>1\) and \(n+s>1\), where \(r,s\), are constants, we have the following:_  
\[\frac{\Gamma(n+r)}{\Gamma(n+s)}\leq\mathbf{o}^{rs}_{n}\,n^{r-s}, \tag{54}\]  
_where_  
\[\mathbf{o}^{rs}_{n}=exp\bigg{(}\frac{r-s}{2\,(n+s-1)}+\frac{1}{12\,(n+r-1)}+ \frac{(r-s)^{2}}{n}\bigg{)}. \tag{55}\]  
Remark 2: \(\mathbf{o}^{rs}_{n}\) _can be expressed as follows for the fixed \(r,s\):_  
\[\mathbf{o}^{rs}_{n}=1+\mathcal{O}(n^{-1}).\]  
**Theorem 2**. : _Suppose \(\widetilde{\mathcal{U}}(z)=\sum\limits_{i=0}^{M}u_{i}G_{i}^{(\sigma)}(z)\), is the approximate solution of \(\mathcal{U}(z)\in\mathbf{H}_{w(z)}^{m}(I)\). Then, for \(0\leq k\leq m\leq M+1\), we obtain the following:_  
\[||D_{z}^{k}\left(\mathcal{U}(z)-\widetilde{\mathcal{U}}(z)\right)||_{L^{2}_{w( z)}}\lesssim\mathcal{L}^{m-k}\,M^{\frac{1}{2}(m-k)\,\left(\frac{2\pi+2}{\pi+2} \right)}\,||D_{z}^{m}\,u(z)||_{L^{2}_{w(z)}}, \tag{56}\]  
_where \(\mathcal{A}\lesssim\mathcal{B}\) refers to the presence of a constant \(\
u\) with the following: \(\mathcal{A}\leq\
u\,\mathcal{B}\)._  
Proof. : The two expansions of \(\mathcal{U}(z)\) and \(\widetilde{\mathcal{U}}(z)\) allow us to write the following:  
\[D_{z}^{k}\left(\mathcal{U}(z)-\widetilde{\mathcal{U}}(z)\right)=\sum\limits_ {n=M+1}^{\infty}u_{n}\,D_{z}^{k}\,G_{n}^{(\sigma)}(z)=\sum\limits_{n=M+1}^{ \infty}u_{n}\,\sum\limits_{r=k}^{n}\frac{A_{r,n}^{\sigma}\,\Gamma(r+1)}{ \Gamma(r-k+1)}\,z^{r-k}. \tag{57}\]  
Taking \(||.||_{L^{2}_{w(z)}}^{2}\), for each side of the previous equation, we obtain the following:  
\[(u,v)_{\mathbf{H}^{m}_{w(z)}}=\sum_{k=0}^{m}(D^{k}_{z}\,u,D^{k}_ {z}\,v)_{L^{2}_{w(z)}}, \tag{51}\]0  
Also, we can write the following:  
\[(u,v)_{\mathbf{H}^{m}_{w(z)}}=\sum_{k=0}^{m}(D^{k}_{z}\,u,D^{k}_ {z}\,v)_{L^{2}_{w(z)}}, \tag{51}\]1  
Now, Equation (58) can be rewritten as follows:  
\[(u,v)_{\mathbf{H}^{m}_{w(z)}}=\sum_{k=0}^{m}(D^{k}_{z}\,u,D^{k}_ {z}\,v)_{L^{2}_{w(z)}}, \tag{51}\]2  
In virtue of the Stirling formula, it is possible to obtain the following inequalities:  
\[ \begin{split}&\frac{\Gamma^{2}(r+1)\,\Gamma\left(2\left(r-m \right)+\frac{1}{\sigma+2}\right)}{\Gamma^{2}(r-m+1)\,\Gamma(1-2\,m+2\,r)} \lesssim r^{2\,m}\left(2\left(r-m\right)\right)^{-\left(\frac{\sigma+1}{ \sigma+2}\right)},\\ &\frac{\Gamma^{2}(r+1)\,\Gamma\left(2\left(r-k\right)+\frac{1}{ \sigma+2}\right)}{\Gamma^{2}(r-k+1)\,\Gamma(1-2\,k+2\,r)}\lesssim r^{2\,k} \left(2\left(r-k\right)\right)^{-\left(\frac{\sigma+1}{\sigma+2}\right)}. \end{split}