Invariant Solutions for Nonhomogeneous Discrete Diffusion Equation

Table 4

Generating solutions for free heat equation.

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)+𝜀,𝑡^(1),𝑢)	𝑢(𝑥,𝑡)=𝑓𝑥^(1)−𝜀,𝑡^(1)
𝑋3𝑒	𝐺3(𝑥^(1),𝑡^(1),𝑢𝑒^𝜀)	𝑢(𝑥,𝑡)=𝑒^𝜀𝑓𝑥^(1),𝑡^(1)
𝑋4𝑒	𝐺4𝑥^(1)𝑒^𝜀,𝑡^(1)𝑒^2𝜀,𝑢	𝑢(𝑥,𝑡)=𝑓𝑥^(1)𝑒^−𝜀,𝑡^(1)𝑒^−2𝜀
𝑋5𝑒	𝐺5𝑥^(1)+2𝑡^(1)𝜀,𝑡^(1),𝑢exp−𝑥^(1)𝜀+𝜀^2𝑡^(1)	𝑢(𝑥,𝑡)=exp𝜀^2𝑡^(1)−𝑥^(1)𝜀𝑓𝑥^(1)−2𝑡^(1)𝑒^𝜀,𝑡^(1)
𝑋6𝑒	𝐺6𝑥^(1)1−4𝜀𝑡^(1),𝑡^(1)1−4𝜀𝑡^(1),𝑢1−4𝜀𝑡^(1)exp−𝑥^(2)𝜀+𝑥^(1)𝜎𝑥𝜀1−4𝜀𝑡^(1)	𝑢()=1𝑥,𝑡4𝜋𝑡^(1)exp−𝑥^(2)−𝑥^(1)𝜎𝑥𝜀4𝑡^(1)