## 4 Validation of the Proposed Numerical Algorithm for the Inversion of Laplace Transformation  
### Example 1  
To confirm the accuracy of the proposed method and the computational analysis code, consider the initial value problem as follows  
\[f^{\prime\prime}(t)+9f(t)=2\sin 3t, \tag{36}\]  
Figure 3: Flowchart of numerical inversion of Laplace transformation. given the initial conditions are:  
\[f(0)=1,\ f^{\prime}(0)=0. \tag{37}\]  
Using Laplace transform to find \(f(t)\), take Laplace transform of both sides of Equation (36):  
\[L\{f^{\prime\prime}(t)\}+L\{9f(t)\}=L\{2\sin 3t\}, \tag{38}\]  
\[s^{2}F(s)-sf(0)-f^{\prime}(0)+9F(s)=\frac{6}{s^{2}+9}. \tag{39}\]  
Substituting Equation (37) into Equation (39), we obtain \(F(s)\) as follows:  
\[F(s)=\frac{6}{(s^{2}+9)^{2}}+\frac{s}{s^{2}+9}. \tag{40}\]  
Take the inverse Laplace transform Equation (40) to find \(f(t)\) as follows:  
\[f(t)=L^{-1}\{F(s)\}=L^{-1}\Bigg{\{}\frac{6}{(s^{2}+9)^{2}}\Bigg{\}}+L^{-1} \Bigg{\{}\frac{s}{s^{2}+9}\Bigg{\}}. \tag{41}\]  
The inverse Laplace transform of the second term is:  
\[L^{-1}\Bigg{\{}\frac{s}{s^{2}+9}\Bigg{\}}=\cos 3t. \tag{42}\]  
Since the inverse Laplace transform of the first term is more complicated, we can use the convolution theorem [19] to find the Laplace transform of the first term:  
\[L^{-1}\Bigg{\{}\frac{6}{(s^{2}+9)^{2}}\Bigg{\}}=L^{-1}\Bigg{\{}\frac{2}{3} \frac{3}{(s^{2}+9)}\frac{3}{(s^{2}+9)}\Bigg{\}}=\frac{2}{3}(f*g)(t), \tag{43}\]  
where \(f(t)=g(t)=\sin 3t\), the convolution of Equation (43) is determined as:  
\[L^{-1}\Bigg{\{}\frac{6}{(s^{2}+9)^{2}}\Bigg{\}}=\frac{2}{3}\!\int_{0}^{t}\sin 3 t\sin 3(t-\tau)d\tau. \tag{44}\]  
Evaluating the integral of Equation (44), we obtain:  
Figure 3:0  
Thus, the exact solution of Equation (36) is:  
Figure 3:1  
Instead of using the convolution theorem [19], we can apply the numerical inversion of the Laplace transform for \(F(s)\) as presented in Section 3. The results between the two methods are shown in Figure 4, which demonstrates a high degree of agreement between the exact and numerical solutions, thereby validating the effectiveness and accuracy of the numerical approach for the numerical inversion of the Laplace transform. Figure 3:2 Comparison of stress wave history between the proposed numerical solution and [PERSON]’s analytical solution at various locations in the rods. Figure 3:3 Comparison of velocity history between the proposed numerical solution and [PERSON]’s analytical solution at various locations in the rods. Investigation of Wave Propagation in a Rayleigh-Love Rod under the Collinear Impact of a Striker Rod with Different Impedances  
This section verifies stress wave propagation in a rod under the collinear impact of a striker rod with different impedances. The proposed numerical solution is compared to the results provided by [PERSON] et al. [11], which do not include [PERSON]'s effect, the stress of the striker rod in the \(k\)th cycle is:  
Figure 3:4  
The stress of the semi-infinite rod in the \(k\)th cycle is:  
Figure 3:5  
where \(v_{1,k}\) is the particle velocity of the striker rod in the cycle \(k\)th and \(v_{2,k}\) is the particle velocity of the semi-infinite rod in the cycle \(k\)th. Investigation of Stress Wave Propagation in a Semi-Infinite Rod and a Striker Rod at the Interface \(x=0\) m  
Based on the stress equation with cyclically varying values without considering the effects of lateral inertia and Poisson's ratio on the striker rod (Equation (48)) and the semi-infinite rod (Equation (49)), we will now compare the results with those of the proposed numerical solution. This comparison aims to evaluate the results obtained from modifications in the cross-sectional area and material properties of the rods, as indicated in Figure 3:6 In the first case, we investigate the position at \(x=0\), where the interaction between the two rods occurs upon impact. The results obtained are illustrated in Figure 9 and Figure 3:7