Transient Heat Diffusion with Temperature-Dependent Conductivity and Time-Dependent Heat Transfer Coefficient

Table 4

Reductions by elements of the optimal systems.

Symmetry	Reductions
Γ1+𝛼Γ3	𝜌=𝑟𝑒^−𝛼𝑡;𝜃=𝑒^2𝛼𝑡/𝑛𝐺(𝜌), where 𝐺 satisfies 2𝛼𝑛𝐺−𝑛𝐺^𝑛−1(𝐺)^2𝑚=(𝜌+𝛼𝜌)𝐺^′+𝐺^𝑛𝐺^′′.
Γ3+Γ2	𝜌=𝑟exp(𝑒^−𝑛𝑠𝑡);𝜃=exp{(−𝑠𝑡−2𝑒^−𝑛𝑠𝑡𝑛)𝑡}𝐺(𝜌), where 𝐺 satisfies 2𝑠𝐺=𝑛𝑠𝜌𝐺^′+𝑛𝐺^𝑛−1(𝐺^′)^2+𝑚𝜌𝐺^𝑛𝐺^′+𝐺^𝑛𝐺^′′.
Γ3	𝜃=𝑟^2/𝑛𝐺(𝑡); with 𝐺 satisfying 𝐺^′2(𝑡)=(𝑚+𝑛+1)𝐺^𝑛+1−𝑠𝐺.