Soft Numerical Computing in Uncertain Dynamic Systems

${\partial }_{x_{\mathit{ii}-\mathit{gH}},x_{i-\mathit{gH}}}u{i,j}_{}\oplus {\partial }_{y_{i-\mathit{gH}},y_{\mathit{ii}-\mathit{gH}}}u{i,j}_{}=f\left(x_i,y_j\right)$

$\frac{1}{h^2}\left(\left(-1\right)u_{i,j}{\ominus }_H\left(-1\right)u_{i+1,j}{\ominus }_H{u}_{i,j}\oplus u_{i-1,j}\right)$

$\oplus \frac{1}{h^2}\left(u_{i,j+1}{\ominus }_H{u}_{i,j}{\ominus }_H\left(-1\right)u_{i,j-1}\oplus \left(-1\right)u_{i,j}\right)=f\left(x_i,y_j\right)$

${\ominus }_H\left(-1\right)u_{i+1,j}{\ominus }_{H}2\odot u_{i,j}\oplus \left(-2\right)u_{i,j}\oplus u_{i-1,j}\oplus u_{i,j+1}{\ominus }_H\left(-1\right)u_{i,j-1}=h^2{f}_{\mathit{ij}},$

$i,j=0,1,\dots ,N.$

The level-wise form is:

$u_{i+1,j,u}-2u_{i,j,l}-2u_{i,j,u}+u_{i-1,}$