Invariant Solutions for Nonhomogeneous Discrete Diffusion Equation

Table 3

The optimal systems for discrete centrifugal barrier and respective invariant solutions.

Optimal systems	Invariant solutions
𝑋1𝑒	𝑢(𝑥,𝑡)=𝑐1+𝑥^(1)𝑐2+𝑇^2𝑥4𝑥^(−1)−𝑥^(−2)−6
𝑋2𝑒	No invariant solution
𝑋3𝑒	Δ𝑦1𝑤+2𝑦𝑤(𝑦)=𝑇𝑥𝑥^(−2)𝑇𝑥 (reduced form); 𝑢(𝑥,𝑡)=𝑐∫^0−∞𝑒^((𝑡(2)/𝑥(2))−1)𝑑𝑡+2𝑡^(1)𝑥^(2)𝑇𝑥^−2𝑥^(−2)
𝑋2𝑒+𝑋3𝑒	Δ𝑦𝑣−𝐻(𝑥,𝑡)𝑣=−𝑇^2𝑥𝑥^(−2); Where 𝐻()=1𝑥,𝑡2𝑥^(2)𝑡^(2)𝜎𝑡+12𝑥^(2)𝑡^(1)+1