## 6 Numerical Results  
Here, we provide four numerical examples to show that the proposed algorithms are accurate, effective, and applicable. It is known that the current method, namely the generalized shifted Chebyshev collocation method (GSCCM), is more accurate when compared with the numerical methods using the radial basis function collocation method [54], the cubic B-spline collocation method [55], the finite difference method [56], and the Crank-Nicolson finite difference scheme [57]. The absolute error (AE) is the observed discrepancy between the measured and real values of the approximate solution, and it is defined as follows:  
\[E(z,\tau)=|\mathcal{U}(z,\tau)-\widetilde{\mathcal{U}}(z,\tau)|, \tag{73}\]  
where \(\mathcal{U}(z,\tau)\) and \(\widetilde{\mathcal{U}}(z,\tau)\) are the exact and numerical solutions, respectively. In addition, the maximum absolute errors (MAEs) are given by the following:  
\[\text{MAEs}=\text{Max}\{E(z,\tau):\forall(z,\tau)\in[0,\mathcal{L}]\times[0, \mathcal{T}]\}=\text{errors in }L^{\infty}. \tag{74}\]  
The errors in \(L^{2}\) can be computed using the following formula:  
\[\sqrt{\frac{\sum\limits_{i=0}^{M}\sum\limits_{j=0}^{M}(\text{Exact-approximate})^ {2}}{(N+1)(M+1)}}. \tag{75}\]  
**Example 1** ([51]). : _Consider the following second-order linear PDE:_  
\[\frac{\partial^{2}\mathcal{U}(z,\tau)}{\partial z^{2}}-3\frac{\partial^{2} \mathcal{U}(z,\tau)}{\partial z\partial\tau}+\frac{\partial^{2}\mathcal{U}(z,\tau)}{\partial\tau^{2}}=\mathcal{F}(z,\tau),\,0\leq z\leq 1,\,0<\tau\leq 1, \tag{76}\]  
_with the following conditions:_  
\[\mathcal{U}(z,0)=\sin(z),\,\mathcal{U}_{\tau}(z,0)=-\sin(z),\,0 \leq z\leq 1,\] \[\mathcal{U}(0,\tau)=0,\,\mathcal{U}_{z}(0,\tau)=e^{-\tau},\,0< \tau\leq 1,\]  
_whose exact solution is as follows:_  
\[\mathcal{U}(z,\tau)=e^{-\tau}\sin(z)\text{ and }\mathcal{F}(z,\tau)=3e^{-z}\cos(z).\]  
_The absolute values of the error function at the points \((z,\tau)\) are provided in Table 1 by taking \(\sigma=0,\,1,\,2,\,3,\,4\) and \(N=M=12\). From this table, one can see the high order of accuracy of the presented method. Figure 1 shows the space-time of the approximate solution (left) and the exact solution (right) with \(\sigma=5\) and \(N=M=12\). Figure 2 shows the space-time graphs of the AE functions at \(\sigma=5\) with \(N=M=8\) and \(N=M=12\), respectively. Figure 3 shows the  
**Example 2** ([52]). : _Consider the following telegraph equation with constant coefficients:_  
\[\frac{\partial^{2}\mathcal{U}(z,\tau)}{\partial\tau^{2}}-\frac{\partial^{2} \mathcal{U}(z,\tau)}{\partial z^{2}}+\frac{\partial\mathcal{U}(z,\tau)}{ \partial\tau}+\mathcal{U}(z,\tau)=\mathcal{F}(z,\tau),\,0\leq z\leq 1,\,0<\tau \leq 1, \tag{77}\]  
_governed by the following time initial and spatial boundary conditions:_  
\[\mathcal{U}(z,0)=\mathcal{U}_{\tau}(z,0)=0,\,0\leq z\leq 1,\] \[\mathcal{U}(0,\tau)=\mathcal{U}(1,\tau)=0,\,0<\tau\leq 1,\]  
_whose exact solution is as follows:_  
\[\text{MAEs}=\text{Max}\{E(z,\tau):\forall(z,\tau)\in[0,\mathcal{L}]\times[0, \mathcal{T}]\}=\text{errors in }L^{\infty}. \tag{74}\]0  
_In Tables 2 and 3, which compare the \(L^{\infty}\) and \(L^{2}\) errors for Example 2, the GSCCM method (\(\sigma=3\), \(N=M=14\)) demonstrates remarkable superiority in terms of accuracy compared to the other methods. The errors produced by this method are significantly lower, reaching magnitudes as small as \(10^{-10}\) in some cases, reflecting its high efficiency in solving the problem in Example 2. One of the most notable advantages of GSCCM is its ability to achieve unprecedented accuracy compared to the radial basis function collocation [54] and cubic B-spline collocation methods [55], even when the time step (\(\tau\)) is small. Although the CPU time for this method is relatively higher, this computational cost is justified by its exceptional precision, making it highly suitable for applications that require extremely accurate results.