Invariant Solutions for Nonhomogeneous Discrete Diffusion Equation

Table 2

The optimal systems for discrete harmonic oscillator and respective invariant solutions.

Optimal systems	Invariant solutions
𝑋1𝑒	𝑢(𝑥,𝑡)=𝑐1+𝑥^(1)𝑐2−𝑘^2𝑇𝑥^−2𝑥^(2)+6𝑥^(1)+6
𝑋2𝑒	No invariant solution
𝑋3𝑒	𝑢(𝑥,𝑡)=𝑐(𝑘+1)^2−𝑘^2𝑇𝑥^−2𝑥𝑘+1^(2)+2𝑥^(1)+2𝐾+1(𝑘+1)^2
𝑋4𝑒	𝑢(𝑥,𝑡)=𝑐+𝑘^2𝑇𝑥^−2𝑥^(2)+2𝑥^(1)−1
𝑋5𝑒	𝑢(𝑥,𝑡)=𝑐1+𝑥^(1)𝑐2+4𝑡^(1)+2+𝑘^2𝑇𝑥^−2𝑥^(2)+8+4𝑘^2𝑇𝑥^−2𝑥^(1)+(2𝜎𝑥−1+6𝑘^2𝑇𝑥^−2)