Invariant Solutions for Nonhomogeneous Discrete Diffusion Equation

Table 1

Optimal systems for free heat equation and respective invariant solutions.

Optimal systems	Invariant solutions
𝑋1𝑒	𝑢(𝑥,𝑡)=𝑐1+𝑥^(1)𝑐2
𝑋2𝑒	𝑢(𝑥,𝑡)=𝑐
𝑋3𝑒	No invariant solution
𝑋2𝑒−𝑋1𝑒	𝑢(𝑥,𝑡)=𝑐1+𝑐22^𝑥−𝑡
𝑋4𝑒	𝑤+1𝑦+12𝑦𝑤𝑦=0 (reduced form); 𝑢=𝑐∫^0−∞𝑡^((𝑥(2)−𝑥(1)𝜎𝑥)𝑡(−1)−1)ln𝑡^(1)(2𝑥^(1)−𝜎𝑥)𝑡^(−1)𝑒^2𝑡𝑑𝑡
𝑋4𝑒+𝑋1𝑒	𝑢(𝑥,𝑡)=𝑐1+𝑐2𝑥^(2)+𝑥^(1)𝜎𝑥