A one-dimensional domain is discretized into N sub-domains of width Dx with node numbers i = 0, 1, 2, 3…………, N. If the time scale is discretized in steps of Dt, 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

Explanation:

\(\frac{{\partial v}}{{\partial t}} = \beta \frac{{{\partial ^2}v}}{{\partial {x^2}}}\)

\(\frac{{\partial v}}{{\partial t}} = \frac{{V_i^{\left( {n + 1} \right)}V_i^{\left( n \right)}}}{{{\rm{\Delta }}t}}\) (using forward time finite difference approximation)

Also, \({f^{11}}\left( x \right) = \frac{{{\partial ^2}f}}{{d{x^2}}} = \frac{{f\left( {x + h} \right) - 2f\left( x \right) + f\left( {x - h} \right)}}{{{h^2}}}\) 

(Using centered space finite difference approximation)

\(\Rightarrow \frac{{{\partial ^2}v}}{{\partial {x^2}}} = \frac{{V_{i + 1}^{\left( n \right)} - 2V_i^{\left( n \right)} + V_{i - 1}^{\left( n \right)}}}{{{{\left( {{\rm{\Delta }}x} \right)}^2}}}\)

\(\therefore \frac{{\partial v}}{{\partial t}} = \beta \frac{{{\partial ^2}v}}{{\partial {x^2}}}\) can be represented as

\(\frac{{V_i^{\left( {n + 1} \right)} - V_i^{\left( n \right)}}}{{{\rm{\Delta }}t}} = \beta \left[ {\frac{{V_{i + 1}^{\left( n \right)} - 2V_i^{\left( n \right)} + V_{i - 1}^{\left( n \right)}}}{{{{\left( {{\rm{\Delta }}x} \right)}^2}}}} \right]\)