Invariant Solutions for Nonhomogeneous Discrete Diffusion Equation

Table 5

Generating solutions for discrete harmonic oscillator.

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),𝑢𝑒^𝜀)	𝑢(𝑥,𝑡)=𝑒^𝜀𝑓𝑥^(1),𝑡^(1)
𝑋3𝑒	𝐺3𝑥^(1)+𝜀,𝑡^(1),𝑢exp𝑥^(1)𝑘𝜀+𝑘𝜀^22	𝑢(𝑥,𝑡)=exp𝑥^(1)𝑘𝜀−𝑘𝜀^22𝑓𝑥^(1)−𝜀,𝑡^(1)
𝑋4𝑒	𝐺4𝑥^(1)+𝜀,𝑡^(1),𝑢exp−𝑥^(1)𝑘𝜀−𝑘𝜀^22	𝑢(𝑥,𝑡)=exp−𝑥^(1)𝑘𝜀+𝑘𝜀^22𝑓𝑥^(1)−𝜀,𝑡^(1)
𝑋5𝑒	𝐺5𝑥^(1)𝑒^𝜀,𝑡^(1)1+𝜀,𝑢expexp2(𝑥^(2)−𝑥^(1)𝜎𝑥)𝑒^4𝜀+𝜀,𝑘=1	𝑢(𝑥,𝑡); 1=expexp2(𝑥^(2)−𝑥^(1)𝜎𝑥)+𝜀𝑓𝑥^(1)𝑒^2𝜀,𝑡^(1)−𝜀
𝑋6𝑒	𝐺5𝑥^(1)𝑒^−2𝜀,𝑡^(1)1+𝜀,𝑢expexp2(𝑥^(2)−𝑥^(1)𝜎𝑥)𝑒^4𝜀−𝜀,𝑘=1	𝑢(𝑥,𝑡); 1=expexp−2𝑥^(2)−𝑥^(1)𝜎𝑥𝑒^8𝜀−𝜀𝑓𝑥^(1)𝑒^2𝜀,𝑡^(1)−𝜀