Boundary conditions ( $i=0,n$ , $j=1,2,\cdots ,m$ )

$u\left(0,j\right)=0$ , $u\left(n,j\right)=0$

Initial Conditions ( $i=1,2,\cdots ,n-1$ , $j=0$ )

$u_{i,1}=\frac{1}{2}\left(r{f}_{i-1}+\left(2-2r\right)f_i+r{f}_{i+1}+2k{g}_i\right)$ (centered differences (CD)) (19)

or

$u_{i,1}=r{f}_{i-1}+\left(1-2r\right)f_i+r{f}_{i+1}+k{g}_i-\frac{1}{2}{k}^2{c}^2{f^{″}}_i$ (backward differences (BD)) (20)

or

$u_{i,1}=f_i+k{g}_i+\frac{1}{2}{k}^2{c}^2{f^{″}}_i$ (forward differences (FD)) (21)

Rest Values ( $i=1,2,\cdots ,n-1$ , $j=1,2,\cdots ,m$ )

$u_{i,j+1}=r{u}_{i-1,j}+\left(2-2r\right)u_{i,j}+r{u}_{i+1,j}-u_{i,j-1}$ (22)