## 4 Numerical Experimentation and Results  
In this section, we implement the proposed overlapping hybrid block method to generate numerical solutions for problems that are typically difficult to solve numerically using standard methods. Before employing the hybrid block approach to integrate the nonlinear IVPs, they are first linearized. For a general nonlinear system of first-order IVPs, the linearisation method is detailed below. Consider the system of \(s\) nonlinear first-order IVPs of the form  
\[ \begin{split} y_{k}^{\prime}=f_{k}(t,y_{1},\dots,y_{s})& =\mathcal{L}_{k}(t,y_{1},\dots,y_{k-1},y_{k+1},\dots,y_{s})y_{k} \\ &+\mathcal{N}_{k}(t,y_{1},\dots,y_{k-1},y_{k},y_{k+1}\dots,y_{s} )\end{split}  \tag{25}\]  
where \(\mathcal{L}_{k}(\cdots)\) is the coefficient of the state variable \(y_{k}\), and \(\mathcal{N}_{k}(\cdots)\) is the component of the \(k\)-th equation that is not linear in \(y_{k}\), for \(k=1,2,\dots,s\). The hybrid block method is sequentially applied iteratively to each equation in the system (25), with the solution of the \(k\)-th equation used immediately when solving the \((k+1)\)th equation. Consequently, the iterative scheme is developed as  
\[ \begin{split} y_{k,r+1}^{\prime}&=\mathcal{L}_{k}(t,y_{1,r+1},\dots,y_{k-1,r+1},y_{k+1,r},\dots,y_{s,r})y_{k,r+1}\\ &+\mathcal{N}_{k}(t,y_{1,r+1},\dots,y_{k-1,r+1},y_{k,r},\dots,y_ {s,r}).\end{split}  \tag{26}\]  
This linearization method employs the Gauss-Seidel method for decoupling and solving large systems of nonlinear equations. The method has been adapted from the waveform relaxation method that has been reported in the literature (see, for example, [12; 13]). Noting that (26) is linear, it can be applied to the HBM method after setting  
\[f(t,y)=\phi(t)y+\psi(t)\]  
where \(\phi=\mathcal{L}(\cdots)\) and \(\psi=\mathcal{N}(\cdots)\) are known functions of \(t\). As a result, the matrix form of the HBM, (11), becomes  
\[A_{1}Y_{n+1}=A_{0}Y_{n}+h\hat{B}_{0}(\boldsymbol{\phi_{n}}^{(d)}Y_{n}+ \boldsymbol{\psi_{n}})+h\hat{B}_{1}(\boldsymbol{\phi_{n+1}}^{(d)}Y_{n+1}+ \boldsymbol{\psi_{n+1}}) \tag{27}\]  
or  
\[P_{n}Y_{n+1}=Q_{n} \tag{28}\]  
where  
\[ \begin{split} P_{n}&=A_{1}-h\hat{B}_{1} \boldsymbol{\phi_{n+1}}^{(d)},\\ Q_{n}&=\Big{(}A_{0}+h\hat{B}_{0}\boldsymbol{\phi_ {n}}^{(d)}\Big{)}Y_{n}+h\big{(}\hat{B}_{0}\boldsymbol{\psi_{n}}+\hat{B}_{1} \boldsymbol{\psi_{n+1}}\big{)}\\ \boldsymbol{\psi_{n}}&=\big{[}\boldsymbol{\psi_{n}}, \boldsymbol{\psi_{n-p_{1}}},\dots,\boldsymbol{\psi_{n-p_{n-1}}}\big{]}^{T}, \quad\boldsymbol{\psi_{n+1}}=\big{[}\boldsymbol{\psi_{n+p_{1}}},\boldsymbol{ \psi_{n+p_{2}}},\dots,\boldsymbol{\psi_{n+p_{m}}}\big{]}^{T},\\ \boldsymbol{\phi_{n}}^{(d)}&=\begin{bmatrix} \boldsymbol{\phi_{n}}&&\\ &\boldsymbol{\phi_{n-p_{1}}}&&\\ &&\boldsymbol{\phi_{n-p_{n-1}}}&&\\ &&\boldsymbol{\phi_{n+p_{m}}}&&\\ \end{bmatrix},\quad\boldsymbol{\phi}_{n+1}^{(d)}=\begin{bmatrix} \boldsymbol{\phi_{n+p_{1}}}&&\\ &&\boldsymbol{\phi_{n+p_{2}}}&&\\ &&\boldsymbol{\phi_{n+p_{m}}}&&\\ &&\boldsymbol{\phi_{n+p_{m}}}&&\\ \end{bmatrix}\end{split} \]  
All numerical examples in this study were solved using (28) applied iteratively over 20 iterations for each integration block \([t_{n},t_{n+1}]\). Through numerical experimentation, it was determined that sufficient convergence would have occurred prior to the 20 th iteration level. All numerical results presented in this study were generated by iterating the scheme (28) up to \(r=20\) for all \(n\) values. In light of the linearisation of nonlinear IVPs via partitioning, the hybrid block method defined by (28) is referred to as the linear partitioning overlapping hybrid block method. The method is computationally efficient and easy to implement. Due to the lack of derivatives and Jacobian matrices, it circumvents the challenges of Newton-based iterative techniques that rely on derivatives. In the examples, given below, the linear partitioning overlapping hybrid block method was applied using equally spaced and the shifted Legendre-based optimal intra-step points.