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

$u (0, j) = 0$ , $u (n, j) = 0$

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

$\begin{array}{l} - 2 ω r u_{i - 1,1} + 2 (1 + 2 ω r) u_{i,1} - 2 ω r u_{i + 1,1} \\ = (1 - 2 ω) r u_{i - 1,0} + (2 + 4 ω r - 2 r) u_{i,0} + (1 - 2 ω) r u_{i + 1,0} \\ - 2 ω k r g_{i - 1} + 2 (1 - 2 ω r) k g_{i} - 2 ω k r g_{i + 1} \end{array}$ (CD) (32)

$\begin{array}{l} - ω r u_{i - 1,1} + (1 - 2 ω r) u_{i,1} - ω r u_{i + 1,1} \\ = (1 - ω) r u_{i - 1,0} + (1 + 2 ω r - 2 r) u_{i,0} + (1 - ω) r u_{i + 1,0} \\ + ω k r (- g_{i - 1} + 2 g_{i} - g_{i + 1}) + \frac{1}{2} ω c^{2} k^{2} r ({f^{″}}_{i - 1} - 2 {f^{″}}_{i} + {f^{″}}_{i}) + k g_{i} - \frac{1}{2} c^{2} k^{2} {f^{″}}_{i} \end{array}$ (BD) (33)

$u_{i, 1} = f_{i} + k g_{i} + \frac{1}{2} k^{2} c^{2} {f^{″}}_{i}$ (FD) (34)

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

$\begin{array}{l} - ω r u_{i - 1, j + 1} + (1 + 2 ω r) u_{i, j + 1} - ω r u_{i + 1, j + 1} \\ = (1 - 2 ω) r u_{i - 1, j} + (2 + 4 ω r - 2 r) u_{i, j} + (1 - 2 ω) r u_{i + 1, j} \\ + ω r u_{i - 1, j - 1} + (- 1 - 2 ω r) u_{i, j - 1} + ω r u_{i + 1, j - 1} \end{array}$ (35)