Invariant Solutions for Nonhomogeneous Discrete Diffusion Equation

Table 6

Generating solutions for discrete centrifugal barrier.

Generators	Infinitesimal transformations	Generating solution
𝑋1𝑒	𝐺1(𝑥^(1),𝑡^(1)+𝜀,𝑢)	𝑢(𝑥,𝑡)=𝑓𝑥^(1),𝑡^(1)−𝜀
𝑋2𝑒	𝐺2(𝑥^(1),𝑡^(1)𝑡,𝑢𝑒^𝜀)	𝑢(𝑥,𝑡)=𝑒^𝜀𝑓𝑥^(1),𝑡^(1)
𝑋3𝑒	𝐺3𝑥^(1)𝑒^𝜀,𝑡^(1)𝑒^2𝜀,𝑢	𝑢(𝑥,𝑡)=𝑓𝑥^(1)𝑒^−𝜀,𝑡^(1)𝑒^−2𝜀
𝑋4𝑒	𝐺6𝑥^(1)1−4𝜀𝑡^(1),𝑡^(1)1−4𝜀𝑡^(1),𝑢1−4𝜀𝑡^(1)𝑥exp^(2)−𝑥^(1)𝜎𝑥𝜀1−4𝜀𝑡^(1)	𝑢(𝑥,𝑡)=1+4𝜀𝑡^(1)𝑥exp^(2)−𝑥^(1)𝜎𝑥𝜀1+4𝜀𝑡^(1)𝑓𝑥1+4𝜀𝑡^(1),𝑡1+4𝜀𝑡^(1)