## 4 Numerical Examples
### 1-D Bratu Problem
The finite dimensional problem using standard finite difference scheme is given by  
\[F_{j}(U_{j})=\frac{U_{j+1}-2U_{j}+U_{j-1}}{h^{2}}+\lambda\exp U_{j}=0,\,j=1..N-1 \tag{31}\]  
with discrete boundary conditions \(U_{0}=U_{N}=0\) and the step size \(h=1/N\). There are \(N-1\) unknowns (\(n=N-1\)). The Jacobian is a sparse matrix and its typical number of nonzero per row is three. It  
Figure 1: Variation of \(\theta\) for different values of \(\lambda\). is known that the finite difference scheme converges to the lower solution of the 1-D Bratu using the starting vector \(U^{(0)}=(0,0,..,0)^{T}\). We use \(N=101\) (\(n=100\)) and test for \(350\)\(\lambda\)'s in the interval \((0,3.5]\) (interval width = 0.01). For each \(\lambda\), we let \(M_{\lambda}\) be the minimum number of iterations for which \(\|U_{j}^{(k+1)}-U_{j}^{(k)}\|_{2}<1e-13\), where the approximation \(U_{j}^{(k)}\) is calculated correct to 14 decimal places. Let \(\overline{M_{\lambda}}\) be the mean of iteration number for the 350 \(\lambda\)'s. Figure 2 and Table 3 give the results for the 1-D Bratu problem, where \(M\) represents number of iterations for convergence. It can be observed from the six methods considered in Table 3 that as \(\lambda\) increases to its critical value, the number of iterations required for convergence increase. However, as the order of method increases, the mean of iteration number decreases. The \(6^{th}HM\) is the most efficient method among the six methods because it has the lowest mean iteration number and the highest number of \(\lambda\) converging in 2 iterations. For each \(\lambda\), we find the minimum order of the \((2r+4)^{th}\)_HM_ family so that we reach convergence in 2 iterations and the results are shown in Figure 3. It can be observed that as the value of \(\lambda\) increases, the value of \(p\) required for convergence in 2 iterations also increases. For \(\lambda\in[0.01,0.04]\), we require \(p=4\) (\(4^{th}HM\)). For \(\lambda\in[0.05,0.35]\), we require \(p=6\) (\(6^{th}HM\)). For \(\lambda\in[0.36,0.83]\), we require \(p=8\) (\(8^{th}HM\)). For \(\lambda\in[0.84,1.29]\), we require \(p=10\) (\(10^{th}HM\)). For \(\lambda\in[1.30,1.66]\), we require \(p=12\) (\(12^{th}HM\)). For \(\lambda\in[1.66,1.95]\), we require \(p=14\) (\(14^{th}HM\)). For \(\lambda\in[1.96,2.19]\), we require \(p=16\) (\(16^{th}HM\)). For \(\lambda\in[2.20,2.37]\), we require \(p=18\) (\(18^{th}HM\)). For \(\lambda\in[2.38,2.52]\), we require \(p=20\) \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \(M=2\) & \(M=3\) & \(M=4\) & \(M=5\) & \(M>5\) & \(\overline{M_{\lambda}}\) \\ \hline \(2^{nd}NM\) & 0 & 12 & 114 & 143 & 81 & 4.92 \\ \(3^{rd}HM\) & 0 & 140 & 206 & 2 & 2 & 3.62 \\ \(4^{th}SGS\) & 4 & 237 & 100 & 8 & 1 & 3.33 \\ \(4^{th}HM\) & 4 & 234 & 103 & 7 & 2 & 3.35 \\ \(6^{th}CHMT\) & 3 & 213 & 124 & 8 & 2 & 3.42 \\ \(6^{th}HM\) & 35 & 281 & 32 & 1 & 1 & 3.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of number of \(\lambda\)’s in different methods for 1-D Bratu problem. Figure 2: Variation of number of iteration with \(\lambda\) for the \(2^{nd}NM\), \(3^{rd}HM\), \(4^{th}HM\) and \(6^{th}HM\) methods. (\(20^{th}HM\)). For \(\lambda\in[2.53,2.64]\), we require \(p=22\) (\(22^{th}HM\)) and so on. We notice that the width of the interval decrease and the order of the family is very high as \(\lambda\) tends to its critical value. Finally, for \(\lambda=3.5\), we require \(p=260\) to reach convergence in 2 iterations.