## 6 Applications
### Bratu Problem in Two Dimension  
The following partial differential equations representing two-dimensional Bratu problem [19] is considered:  
\[\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial y^{2}}+ \lambda exp(U)=0,\;x,y\in D=[0,1]\times[0,1] \tag{30}\]  
satisfying the conditions  
\[U(x,y)=0,\;x,y\in\Gamma, \tag{31}\]  
where \(\Gamma\) represents the boundary of D. The two known bifurcated exact solutions of the Planar Bratu problem are obtained for the values of \(\lambda<\lambda_{c}\), one solution for \(\lambda=\lambda_{c}\) and for \(\lambda>\lambda_{c}\) solution does not exist. The analytic solution for Equation (30) is found to be  
\[U(x,y)=2\log_{\epsilon}\left[\frac{\cosh\left(\frac{\theta}{4}\right)\cosh \left((x-\frac{1}{2})(y-\frac{1}{2})\theta\right)}{\cosh\left((x-\frac{1}{2}) \left(\frac{\theta}{2}\right)\right)\cosh\left((y-\frac{1}{2})\left(\frac{ \theta}{2}\right)\right)}\right]. \tag{32}\]  
The constant \(\theta\) must be found such that it satisfies the boundary conditions. Substitute Equation (32) in (30) and convert it by using collocation at \((x,y)=\left(\frac{1}{2},\frac{1}{2}\right)\) as it is the center of the domain D. Instead,  
Figure 1: Variation of \(\theta\) for 1-D Bratu Problem \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \(\mathbf{N=2}\) & \(\mathbf{N=3}\) & \(\mathbf{N=4}\) & \(\mathbf{N=5}\) & \(\mathbf{N>5}\) & \(\overline{M_{\lambda}}\) \\ \hline \(2^{nd}NM\) & 0 & 12 & 115 & 142 & 81 & 4.93 \\ \(4^{th}NR\) & 12 & 257 & 76 & 4 & 1 & 3.21 \\ \(5^{th}ACT\) & 1 & 136 & 171 & 37 & 5 & 3.71 \\ \(5^{th}MBJ\) & 23 & 263 & 60 & 3 & 1 & 3.14 \\ \(5^{th}PJ\) & 23 & 276 & 48 & 2 & 1 & 3.10 \\ \(8^{th}SA\) & 23 & 263 & 60 & 3 & 1 & 3.13 \\ \(8^{th}MBJ\) & 23 & 262 & 61 & 2 & 2 & 3.14 \\ \(8^{th}PJ\) & 23 & 276 & 48 & 1 & 2 & 3.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of number of \(\lambda^{\prime}s\) requiring N iterations to converge in 1-D Bratu problem. one can choose a different value for \((x,y)\), although it may give higher-order approximation but collocating at \((x,y)=\left(\frac{1}{2},\frac{1}{2}\right)\) will give equal distribution of the interval throughout the region and hence we get  
\[\theta^{2}=\lambda\cosh^{2}\left(\frac{\theta}{4}\right). \tag{33}\]  
Differentiate the above Equation (33) with respect to \(\theta\) and by taking \(\frac{d\lambda}{d\theta}=0\), it is seen that \(\lambda_{c}\) satisfies  
\[\theta=\frac{1}{4}\lambda_{c}\cosh\left(\frac{\theta}{4}\right)\sinh\left( \frac{\theta}{4}\right). \tag{34}\]  
Substituting the value of \(\lambda_{c}\) from (34) in (33) one gets the value of \(\theta_{c}\),  
\[\frac{\theta_{c}}{4}=\coth\left(\frac{\theta_{c}}{4}\right), \tag{35}\]  
for which \(\theta_{c}=4.798714561\) can be obtained using an iterative method. Then the value of \(\lambda_{c}=7.027661438\) is obtained from Equation (34). Figure 2 display this critical value of \(\lambda_{c}\). Equation (30) is discretized using the finite-difference five-point formula with mesh length \(h\) on both the axes to obtain the nonlinear equations  
\[F(U_{i,j})=-(4U_{i,j}-\lambda h^{2}exp(U_{i,j}))+U_{i+1,j}+U_{i-1,j}+U_{i,j+1 }+U_{i,j-1}, \tag{36}\]  
where \(U_{i,j}\) is \(U\) at \((x_{i},y_{j})\), \(x_{i}=ih\), \(y_{j}=jh\), \(i,j=1(1)M\), subject to the conditions \(U_{i,0}=U_{i,M+1}=U_{0,j}=U_{M+1,j}=0\) for \(i,j=0(1)M+1\). Equation (36) represents a set of \(M\times M\) nonlinear equations in \(U_{i,j}\) which are then solved by using iterative methods. By taking \(M=10\) and \(M=20\), we experiment for 700 \(\lambda\)'s, in the interval \(\lambda\in(0,7]\) (mesh length = 0.01). For every \(\lambda\), take \(M_{\lambda}\) to be the minimum number of iterations so that \(\|U_{i,j}^{(r+1)}-U_{i,j}^{(r)}\|_{2}<10^{-11}\), where, the computed value \(U_{i,j}^{(r)}\) is calculated for 13-decimal-place accuracy. Let \(\overline{M_{\lambda}}\) represents the average of iteration number for the 700 \(\lambda\)'s.