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A one-dimensional domain is discretized into $N$ sub-domains of width $\Delta x$ with node numbers $i=0,1,2,3, \dots , N$. If the time scale is discretized in steps of $\Delta t$, the forward-time and centered-space finite difference approximation at i th node and n th time step, for the partial differential equation $\frac{\partial v}{\partial t} = \beta \frac{\partial ^2 v}{\partial x^2}$ is

  1. $\frac{v_i^{(n+1)}-v_i^{(n)}}{\Delta t} = \beta \bigg[ \frac{v_{i+1}^{(n)} – 2v_i^{(n)} + v_{i-1}^{(n)}}{(\Delta x)^2} \bigg]$
  2. $\frac{v_i^{(n)}-v_i^{(n-1)}}{\Delta t} = \beta \bigg[ \frac{v_{i+1}^{(n)} – 2v_i^{(n)} + v_{i-1}^{(n)}}{2 \Delta x} \bigg]$
  3. $\frac{v_i^{(n)}-v_i^{(n-1)}}{\Delta t} = \beta \bigg[ \frac{v_{i+1}^{(n)} – 2v_i^{(n)} + v_{i-1}^{(n)}}{(\Delta x)^2} \bigg]$
  4. $\frac{v_i^{(n)}-v_i^{(n-1)}}{2 \Delta t} = \beta \bigg[ \frac{v_{i+1}^{(n)} – 2v_i^{(n)} + v_{i-1}^{(n)}}{2 \Delta x} \bigg]$
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