## 5 Application  
Now, we illustrate all of that mentioned above with the following mildly nonlinear elliptic equation:  
\[u_{xx}+u_{yy}=u^{5/3}. \tag{21}\]  
This type of equation is of interest in the theory of gas dynamics [7]. An associated Dirichlet problem can be formulated as follows. Suppose that the equation is satisfied in the interior of the square \(0\leq x,y\leq 1\) in \(\mathbb{R}^{2}\) and that \(u(x,y)>0\) is given and continuous on the boundary of the square ([8]):  
Figure 2: Domains of parameters of [PERSON]’s method associated with Corollary 4 (magenta region) and Theorem 1 (gray region) for \(p=\frac{1}{10},\frac{1}{5},\frac{2}{5},\frac{4}{5}\). \[u(x,0)=2x^{2}-x+1,\ \ u(x,1)=2,\ \ 0\leq x\leq 1, \tag{22}\] \[u(0,y)=2y^{2}-y+1,\ \ u(1,y)=2,\ \ 0\leq y\leq 1.\]  
Our discussion focuses on the formulation of the finite difference equation for the elliptic boundary value Problems (21)-(22). The method of finite differences applied to this problem yields a finite system of equations. For general use, iterative techniques often represent the best approach to the solution of such finite systems of equations. Specifically, central difference approximations for (21) are used, so that Problems (21)-(22) are reduced to the problem of finding a real zero of a function \(F:\Omega\subseteq\mathbb{R}^{m}\longrightarrow\mathbb{R}^{m}\), namely a real solution \(\mathbf{x}^{*}\) of a nonlinear system \(F(\mathbf{x})=0\) with \(m\) equations and \(m\) unknowns. The common technique used to approximate \(\mathbf{x}^{*}\) is the application of iterative methods. In this case, [PERSON]'s method goes on being the most used iterative method for approximating \(\mathbf{x}^{*}\), since this method is one of the most efficient. For Problems (21)-(22) in \(\mathbb{R}^{2}\), Equation (21) can be approximated using central difference approximations for the spacial derivatives. Consider a grid with step size \(h=\frac{1}{N+1}\) in \(x\) and \(k=\frac{1}{M+1}\) in \(y\) defined over the region \(D\), so that \(D\) is partitioned into a grid consisting of \((N+1)\times(M+1)\) rectangles with sides \(h=\frac{1}{N+1}\) and \(k=\frac{1}{M+1}\). The mesh points \((x_{i},y_{j})\) are given by:  
\[x_{i}=ih,\ \ \ y_{i}=jk,\ \ \ i=0,1,\ldots,N+1,\ \ \ j=0,1,\ldots,M+1.\]  
Considering the following finite difference expressions to approximate the partial differentials:  
\[u_{xx}(x_{i},y_{j})=\frac{u(x_{i-1},y_{j})-2u(x_{i},y_{j})+u(x_{i+1},y_{j})}{ h^{2}}+\mathcal{O}(h^{2}),\]  
\[u_{yy}(x_{i},y_{j})=\frac{u(x_{i},y_{j-1})-2u(x_{i},y_{j})+u(x_{i},y_{j+1})}{ k^{2}}+\mathcal{O}(k^{2}),\]  
Equation (21) is approximated at each interior grid point \((x_{i},y_{j})\) by the difference equation:  
\[\frac{u(x_{i+1},y_{j})-2u(x_{i},y_{j})+u(x_{i-1},y_{j})}{h^{2}}+\frac{u(x_{i}, y_{j+1})-2u(x_{i},y_{j})+u(x_{i},y_{j-1})}{k^{2}}=u(x_{i},y_{j})^{5/3},\]  
for \(i=1,2,\ldots,N\) and \(j=1,2,\ldots,M\). The boundary conditions are:  
\[u(x_{i},y_{0})=2x_{i}^{2}-x_{i}+1,\ \ u(x_{i},y_{M})=2,\ \ i=1,2, \ldots,N,\] \[u(x_{0},y_{j})=2y_{j}^{2}-y_{j}+1,\ \ u(x_{N},y_{j})=2,\ \ j=0,1, \ldots,M+1.\]  
If we now denote the approximate value of \(u(x_{i},y_{j})\) as \(u_{i,j}\), we obtain the difference equation:  
Figure 2:0  
for \(i=1,2,\ldots,N\) and \(j=1,2,\ldots,M\), with:  
Figure 2:1 Figure 2:2Equation (21) with the boundary conditions given in (22) forms an \(NM\times NM\) nonlinear system of equations. To set up the nonlinear system, the \(NM=m\) interior grid points are labeled row-by-row from \(x_{1}\) to \(x_{m}\) starting from the left-bottom corner point. The resulting system is:  
Figure 2:3  
where:  
\[A=\left( \begin{array}{cccccc}B&C&0&\cdots&\cdots&0\\ C&B&C&\ddots&&\vdots\\ 0&C&B&C&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&\ddots&0\\ \vdots&&\ddots&\ddots&\ddots&C\\ 0&\cdots&\cdots&0&C&B\end{array} \right)_{M\times M},\]  
\[B=\left(