## 6 Applications  
### Bratu Problem in One Dimension  
Consider the one-dimensional Bratu problem stated as [18]  
\[\frac{d^{2}U}{dx^{2}}+\lambda\exp U(x)=0,\,\lambda>0,\,0<x<1, \tag{24}\]  
subject to the conditions \(U(0)=U(1)=0\). 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. Here \(\lambda_{c}\) is known as critical value and its value is \(8(\eta^{2}-1)\), where \(\eta\) denotes the fixed point of \(\coth\left(x\right)\). The analytic solution for Equation (24) is found to be  
\[U(x)=-2\log_{c}\left[\frac{\cosh\left(x-\frac{1}{2}\right)\frac{\theta}{2}}{ \cosh\left(\frac{\theta}{4}\right)}\right]. \tag{25}\]  
The constant \(\theta\) has to be found such that it satisfies the boundary conditions. The critical value of \(\lambda\) is obtained by using a similar procedure found in [19]. Substitute Equation (25) in (24) and convert it by using collocation at \(x=\dfrac{1}{2}\) as it is the middle point of \([0,1]\). Instead, one can choose a different value for \(x\), although it may give higher-order approximation but collocating at \(x=\dfrac{1}{2}\) will give equal distribution of the interval throughout the region and hence we get  
\[\theta^{2}=2\lambda\cosh^{2}\left(\frac{\theta}{4}\right). \tag{26}\]  
Differentiate the above Equation (26) with respect to \(\theta\) and by taking \(\dfrac{d\lambda}{d\theta}=0\), it is seen that \(\lambda_{c}\) satisfies  
\[\theta=\dfrac{1}{2}\lambda_{c}\cosh\left(\frac{\theta}{4}\right)\sinh\left( \frac{\theta}{4}\right). \tag{27}\]  
Substituting the value of \(\lambda_{c}\) from (27) in (26) one gets the value of \(\theta_{c}\),  
\[\frac{\theta_{c}}{4}=\coth\left(\frac{\theta_{c}}{4}\right), \tag{28}\]  
for which \(\theta_{c}=4.798714560\) can be obtained using an iterative method. Then the value of \(\lambda_{c}=3.513830720\) is obtained from Equation (26). Figure 1 display this critical value of \(\lambda_{c}\). Equation (24) is discretized using the finite-difference scheme to obtain  
\[F_{j}(U_{j})=\frac{U_{j+1}-2U_{j}+U_{j-1}}{h^{2}}+\lambda\exp U_{j}=0,\,j=1(1 )M-1, \tag{29}\]  
subject to the conditions \(U_{0}=U_{M}=0\) with mesh length \(h=1/M\). There are \(M-1\) difference equations. The Jacobian matrix obtained is sparse and it has three non-zero entries on the main and two sub-diagonals. The solution of the above finite-difference equations converges to the solution of 1-D Bratu problem with the help of initial vector \(U^{(0)}=(0,0,..,0)^{T}\).  
We take \(M=100\) and experiment for \(350\)\(\lambda\)'s in the interval \(\lambda\in(0,\,3.5]\) (mesh length = 0.01). For every \(\lambda\), take \(M_{\lambda}\) to be the minimum number of iterations so that \(\|U_{j}^{(r+1)}-U_{j}^{(r)}\|_{2}<10^{-13}\), where, the computed value \(U_{j}^{(r)}\) is calculated for 13-decimal-place accuracy. Let \(\overline{M_{\lambda}}\) represents the average of iteration number for the \(350\)\(\lambda\)'s. Table 3 presents the numerical results for one-dimensional Bratu problem, where \(N\) denotes the number of iterations for convergence. It is observed from all the methodsfound in Table 3, whenever the methods have higher-order convergence, a greater number of points in \(\lambda\) converge with fewer iterations. For the proposed \(5^{th}PJ\) and \(8^{th}PJ\) methods, a greater number of points in \(\lambda\) converge within four iterations. However, the least average iteration number (\(\overline{M_{\lambda}}\)) reduces as the order of method increases. Therefore, it is found that \(5^{th}PJ\) and \(8^{th}PJ\) methods are better among the methods taken for comparison since they have the least average iteration number (\(\overline{M_{\lambda}}\)).