## 4 Numerical Treatment of Telegraph Equation  
In this section, we utilize a numerical method for treating the following telegraph equation [52]:  
\[\frac{\partial^{2}\mathcal{U}(z,\tau)}{\partial\tau^{2}}+2\zeta(z,\tau)\frac{ \partial\mathcal{U}(z,\tau)}{\partial\tau}+\eta^{2}(z,\tau)\mathcal{U}(z,\tau )=\theta(z,\tau)\frac{\partial^{2}\mathcal{U}(z,\tau)}{\partial z^{2}}+ \mathcal{F}(z,\tau),\quad 0<z<\mathcal{L},\quad 0<\tau\leq\mathcal{T}. \tag{37}\]  
governed by the following conditions:  
\[\mathcal{U}(z,0)= \mathcal{G}_{1}(z),\,\mathcal{U}_{\tau}(z,0)=\mathcal{G}_{2}(z), \quad z\in(0,\mathcal{L}), \tag{38}\] \[\mathcal{U}(0,\tau)= \mathcal{H}_{1}(\tau),\,\mathcal{U}(\mathcal{L},\tau)=\mathcal{H} _{2}(\tau),\quad\tau\in(0,\mathcal{T}]. \tag{39}\]  
For \(\zeta>0,\eta=0\) Equation (37) represents a damped wave equation, and if \(\zeta>\eta>0\), it is the telegraph equation. By substituting (21), (22), (24), and (26) in (37), we obtain the following approximate formulas:  
\[\Delta^{(2)}\Psi_{N}(\tau)\mathbf{U}\Phi_{M}(z) +2\zeta(z,\tau)\Delta\Psi_{N}(\tau)\mathbf{U}\Phi_{M}(z)+\eta^{2} (z,\tau)\Psi_{N}(\tau)\mathbf{U}\Phi_{M}(z) \tag{40}\] \[\approx\theta(z,\tau)\Psi_{N}(\tau)\mathbf{U}\Delta^{(2)}\Phi_{M}( z)+\mathcal{F}(z,\tau),\,0<z<\mathcal{L},\,0<\tau\leq\mathcal{T}.\]  
Moreover, the conditions in (38) and (39) imply the following equations:  
\[\Psi_{N}(0)\mathbf{U}\Phi_{M}(z) \approx\mathcal{G}_{1}(z), \tag{41}\] \[\Psi_{N}^{{}^{\prime}}(0)\mathbf{U}\Phi_{M}(z) \approx\mathcal{G}_{2}(z),\] (42) \[\Psi_{N}(\tau)\mathbf{U}\Phi_{M}(0) \approx\mathcal{H}_{1}(\tau),\] (43) \[\Psi_{N}(\tau)\mathbf{U}\Phi_{M}(\mathcal{L}) \approx\mathcal{H}_{2}(\tau), \tag{44}\]Similarly, as in the previous section, We insert the collocation nodes in (40)-(44). Accordingly, we can write the following:  
\[\Delta^{(2)}\Psi_{N}(\tau^{(\sigma)}_{\mathcal{I},j})\mathbf{U}\Phi_ {M}(z^{(\sigma)}_{\mathcal{I},j})+2\zeta(z^{(\sigma)}_{\mathcal{I},j}\tau^{( \sigma)}_{\mathcal{I},j})\Delta\Psi_{N}(\tau^{(\sigma)}_{\mathcal{I},j}) \mathbf{U}\Phi_{M}(z^{(\sigma)}_{\mathcal{I},j})+\eta^{2}(z^{(\sigma)}_{ \mathcal{I},j},\tau^{(\sigma)}_{\mathcal{I},j})\Psi_{N}(\tau^{(\sigma)}_{ \mathcal{I},j})\mathbf{U}\Phi_{M}(z^{(\sigma)}_{\mathcal{I},j}) \tag{45}\] \[=\theta(z^{(\sigma)}_{\mathcal{I},j},\tau^{(\sigma)}_{\mathcal{I},j})\mathbf{V}_{N}(\tau^{(\sigma)}_{\mathcal{I},j})\mathbf{U}\Delta^{(2)} \Phi_{M}(z^{(\sigma)}_{\mathcal{I},j})+\mathcal{F}(z^{(\sigma)}_{\mathcal{I}, j},\tau^{(\sigma)}_{\mathcal{I},j})\quad 1\leq i\leq M-1,\;1\leq j\leq N-1,\] (46) \[\Psi_{N}(0)\mathbf{U}\Phi_{M}(z^{(\sigma)}_{\mathcal{I},j})= \mathcal{G}_{1}(z^{(\sigma)}_{\mathcal{I},j}),\;0\leq i\leq M,\] (47) \[\Psi_{N}(\tau^{(\sigma)}_{\mathcal{I},j})\mathbf{U}\Phi_{M}(0)= \mathcal{H}_{1}(\tau^{(\sigma)}_{\mathcal{I},j}),\;1\leq j\leq N-1,\] (48) \[\Psi_{N}(\tau^{(\sigma)}_{\mathcal{I},j})\mathbf{U}\Phi_{M}( \mathcal{L})=\mathcal{H}_{2}(\tau^{(\sigma)}_{\mathcal{I},j}),\;1\leq j\leq N -1. \tag{49}\]  
An iterative approach developed by [PERSON] may be used to solve the \((N+1)\times(M+1)\) algebraic equations obtained. This allows us to compute \(\widetilde{\mathcal{U}}(z,\tau)\) given in (21).