## 6 Numerical Results
In Tables 4 and 5, detailed comparisons of the \(L^{\infty}\) and  
\[\text{MAEs}=\text{Max}\{E(z,\tau):\forall(z,\tau)\in[0,\mathcal{L}]\times[0, \mathcal{T}]\}=\text{errors in }L^{\infty}. \tag{74}\]1 The AE curves of the \(z\)-direction (**left**) and \(\tau\)-direction (**right**) for Example 1 with \(\sigma=5\) and \(N=M=12\). \[\text{MAEs}=\text{Max}\{E(z,\tau):\forall(z,\tau)\in[0,\mathcal{L}]\times[0, \mathcal{T}]\}=\text{errors in }L^{\infty}. \tag{74}\]2 Comparison of the curves of the analytical solutions and the approximate solutions at \(\tau=0\), \(0.5\), \(1\) (**left**) and \(z=0.0\), \(0.2\), \(0.4\) (**right**) for Example 1 with \(\sigma=5\) and \(N=M=12\). **Example 3** ([56]). : _Consider the following telegraph equation with variable coefficients:_  
\[ \begin{split}\frac{\partial^{2}\mathcal{U}(z,\tau)}{\partial\tau^{2} }+2e^{(z+\tau)}\frac{\partial\mathcal{U}(z,\tau)}{\partial\tau}-(1+z^{2}) \frac{\partial^{2}\mathcal{U}(z,\tau)}{\partial z^{2}}+\sin^{2}(z+\tau) \mathcal{U}(z,\tau)=\mathcal{F}(z,\tau),\\ 0\leq z\leq 1,\,0<\tau\leq 1,\end{split}  \tag{78}\]  
_governed by the following time initial and spatial boundary conditions:_  
\[ \begin{split}\mathcal{U}(z,0)=\sinh(z),\;\mathcal{U}_{\tau}(z,0) =-2\sinh(z),\;0\leq z\leq 1,\\ \mathcal{U}(0,\tau)=0,\;\mathcal{U}(1,\tau)=e^{-2\tau}\sinh(1), \;0<\tau\leq 1,\end{split} \]  
_whose exact solution is_  
\[\mathcal{U}(z,\tau)=e^{-2\tau}\sinh(z)\;\text{and}\;\mathcal{F}(z,\tau)=(3-z^{ 2}+\sin^{2}(z+\tau)-4e^{(z+\tau)})e^{-2\tau}\sinh(z).\]  
_In Table 6, we depict the MAEs using the proposed method with those obtained in [56] and [57], respectively. Table 7 presents the \(L^{\infty}\) and \(L^{2}\) errors along with the CPU time for Example 3 at \(\sigma=1.5\) and \(N=M=16\). The results indicate that the errors are extremely small but gradually increase as \(\tau\) increases, reflecting the impact of a larger time step on the accuracy. Furthermore, the CPU time rises with \(\tau\) due to the higher computational complexity. This demonstrates the efficiency of the method in Example 3, achieving high accuracy at the expense of the increased computational cost. Moreover, Figure 7 shows the space-time graphs of the approximate solution (left) and its AE function (right) with \(\sigma=2,\;\mathcal{T}=5\) and \(N=M=16\):_  
Figure 5: Space–time graphs of the approximate solution (**left**) and its AE function (**right**) for Example 2 with \(\sigma=2,\;\mathcal{T}=10\) and \(N=M=16\). Figure 6: Comparison of the curves of the analytical solutions and the approximate solutions at \(\tau=4,\;7,\;10\) (**left**) and \(z=0.3,\;0.6,\;0.9\) (**right**) for Example 2 with \(\sigma=2\) and \(N=M=16\). **Example 4** ([56; 57]). : _Consider the following singular telegraph equation:_  
\[ \begin{split}\frac{\partial^{2}\mathcal{U}(z,\tau)}{\partial\tau^{2 }}+\frac{2}{z^{2}}\frac{\partial\mathcal{U}(z,\tau)}{\partial\tau}&-( 1+z^{2})\frac{\partial^{2}\mathcal{U}(z,\tau)}{\partial z^{2}}+\frac{1}{z^{2} }\mathcal{U}(z,\tau)=\mathcal{F}(z,\tau),\\ & 0\leq z\leq 1,\,0<\tau\leq 1,\end{split}  \tag{79}\]  
_governed by the following initial time and spatial boundary conditions:_  
\[ \begin{split}\mathcal{U}(z,0)&=\sinh(z),\,\, \mathcal{U}_{\tau}(z,0)=-2\sinh(z),\,\,0\leq z\leq 1,\\ \mathcal{U}(0,\tau)&=0,\,\,\mathcal{U}(1,\tau)=e^{ -2\tau}\sinh(1),\,\,0<\tau\leq 1,\end{split} \]  
_whose exact solution is as follows:_  
\[\mathcal{U}(z,\tau)=e^{-2\tau}\sinh(z),\]  
_and_  
\[\mathcal{F}(z,\tau)=-\frac{e^{-2\tau}(z^{4}-3z^{2}+3)\,\sinh(z)}{z^{2}}.\]  
_Table 8 lists the MAEs at various choices of \(\mathcal{T}\). Moreover, Figure 8 presents the AE curves obtained by the present method at \(\tau=0.5\) and \(z=0.5\) with \(N=M=16\), and \(\sigma=3\), respectively._