Conditional Lie-Bäcklund Symmetries and Reductions of the Nonlinear Diffusion Equations with Source

Table 1

CLBS (15) and IS (19) with 𝑙=2 of (14) for general 𝑚.

Number	Equation (14)	CLBS (15)	IS (19)
1	𝑣𝑡=𝑒^(𝑝−𝑞)𝑥𝑚𝑚𝑣𝑣𝑚+𝑘−1^𝑥𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝑚+𝑘−1𝑚𝑚𝑣𝑚+𝑘−1^𝑥𝑚+1+𝑚+𝑘−1𝑚𝛼𝑣+𝛽+𝛾𝑓1^(1)(𝑥)	𝜂=𝑣𝑥𝑥+𝑝−𝑞𝑚𝑣𝑥	𝑊𝑓1^(1)(𝑥),1
2	𝑣𝑡=𝑒^(𝑝−𝑞)𝑥𝑚𝑣𝑣𝑥^𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝛼𝑣+𝛽+𝛾𝑓1^(1)(𝑥)	𝜂=𝑣𝑥𝑥+𝑝−𝑞𝑚𝑣𝑥	𝑊𝑓1^(1)(𝑥),1
3	𝑣𝑡=𝑒^(𝑝−𝑞)𝑥𝑚𝑣𝑥^𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝛼𝑣+𝛽+𝛾𝑓1^(1)(𝑥)	𝜂=𝑣𝑥𝑥+𝑝−𝑞𝑚𝑣𝑥	𝑊𝑓1^(1)(𝑥),1
4	𝑣𝑡=𝑒^(𝑝−𝑞)𝑥𝑚𝑣^2−𝑝−(𝑚+1)𝑞𝑣𝑝−𝑞^𝑥𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝑝−𝑞𝑚𝑞(𝑝−(𝑚+1)𝑞)^2𝑒^(𝑝−𝑞)𝑥−()𝑞×𝑣𝑝−𝑚+1𝑣𝑝−𝑞^𝑥𝑚+1−𝑝−𝑞𝑝−(𝑚+1)𝑞𝛼𝑣+𝛽+𝛾𝑒^−((𝑝−𝑞)/𝑚)𝑥𝑝,𝑞≠𝑝,𝑚+1	𝜂=𝑣𝑥𝑥+𝑝−𝑞𝑚𝑣𝑥	𝑊𝑒^−((𝑝−𝑞)/𝑚)𝑥,1
5	𝑣𝑡=𝑒^(𝑝−𝑞)𝑥𝑚𝑣𝑥^𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝛼𝑣+𝛽+𝛾𝑓1^(2)(𝑥)	𝜂=𝑣𝑥𝑥+𝑝−𝑞𝑣𝑚−1^𝑥	𝑊𝑓1^(2)(𝑥),1
6	𝑣𝑡=𝑒^(𝑝−𝑞)𝑥𝑚−𝑝−𝑚𝑞𝑣𝑝−𝑞^𝑥𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝑝−𝑞−𝑝−𝑚𝑞𝑝−𝑚𝑞𝑣𝑝−𝑞^𝑥𝑚+1−𝑝−𝑞𝑝−𝑚𝑞𝛼𝑣+𝛽+𝛾𝑒^−((𝑝−𝑞)/(𝑚−1))𝑥𝑝,𝑞≠𝑝,𝑚	𝜂=𝑣𝑥𝑥+𝑝−𝑞𝑣𝑚−1^𝑥	𝑊𝑒^−((𝑝−𝑞)/(𝑚−1))𝑥,1
7	𝑣𝑡=𝑒^(𝑝−𝑞)𝑥𝐴(𝑣)𝑣𝑥^𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝛼𝑣+𝛽+𝛾𝑒^−(𝑝/𝑚)𝑥	𝜂=𝑣𝑥𝑥+𝑝𝑚𝑣𝑥	𝑊𝑒^−(𝑝/𝑚)𝑥,1
8	𝑣𝑡=𝑒^((𝑚+1)/𝑚)𝑝𝑥𝑚𝑣𝑥^𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝑣𝑥^𝑚+1+𝛼𝑣+𝛽+𝛾𝑒^−(𝑝/𝑚)𝑥	𝜂=𝑣𝑥𝑥+𝑝𝑚𝑣𝑥	𝑊𝑒^−(𝑝/𝑚)𝑥,1
9	𝑣𝑡=𝑒^((𝑚+1)/𝑚)𝑝𝑥𝑚𝑚𝑣𝑣𝑚+𝑘−1^𝑥𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝑚+𝑘−1𝑚𝑚𝑣𝑚+𝑘−1^𝑥𝑚+1+𝑚+𝑘−1𝑚𝛼𝑣+𝛽+𝛾𝑒^−(𝑝/𝑚)𝑥	𝜂=𝑣𝑥𝑥+𝑝𝑚𝑣𝑥	𝑊𝑒^−(𝑝/𝑚)𝑥,1
10	𝑣𝑡=𝑒^((𝑚+1)/𝑚)𝑝𝑥𝑚𝑣^2(−𝑣𝑥)^𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥−(−𝑣𝑥)^𝑚+1+𝛼𝑣+𝛽+𝛾𝑒^−(𝑝/𝑚)𝑥	𝜂=𝑣𝑥𝑥+𝑝𝑚𝑣𝑥	𝑊𝑒^−(𝑝/𝑚)𝑥,1
11	𝑣𝑡=𝑒^((𝑚+1)/𝑚)𝑝𝑥𝑚𝑣^2𝑚+1𝑣𝑚+𝑘−1^𝑥𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+(2𝑚+𝑘)(𝑚+𝑘−1)(𝑚+1)^2𝑣𝑚+1𝑣𝑚+𝑘−1^𝑥𝑚+1+𝑚+𝑘−1𝑚+1𝛼𝑣+𝛽+𝛾𝑒^−(𝑝/𝑚)𝑥	𝜂=𝑣𝑥𝑥+𝑝𝑚𝑣𝑥	𝑊𝑒^−(𝑝/𝑚)𝑥,1
12	𝑣𝑡=𝑒^((𝑚+1)/𝑚)𝑝𝑥𝑚𝑚𝑣𝑣𝑚+𝑘−1^𝑥𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝑚+𝑘−1𝑚𝑚𝑣𝑚+𝑘−1^𝑥𝑚+1+𝑚+𝑘−1𝑚(𝛼𝑣+𝛽+𝛾𝑓1^(3)(𝑥))	𝜂=𝑣𝑥𝑥+𝑝(𝑚+1)𝑒^−(𝑝𝑥/𝑚)−1𝑚𝑚𝑒^−(𝑝𝑥/𝑚)𝑣−1^𝑥	𝑊𝑓1^(3)(𝑥),1
13	𝑣𝑡=𝑒^((𝑚+1)/𝑚)𝑝𝑥𝑚𝑣𝑣𝑥^𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝛼𝑣+𝛽+𝛾𝑓1^(3)(𝑥)	𝜂=𝑣𝑥𝑥+𝑝(𝑚+1)𝑒^−(𝑝𝑥/𝑚)−1𝑚𝑚𝑒^−(𝑝𝑥/𝑚)𝑣−1^𝑥	𝑊𝑓1^(3)(𝑥),1
14	𝑣𝑡=𝑒^((𝑚+1)/𝑚)𝑝𝑥𝑚𝑣𝑥^𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝑣𝑥^𝑚+1+𝛼𝑣+𝛽+𝛾𝑓1^(3)(𝑥)	𝜂=𝑣𝑥𝑥+𝑝(𝑚+1)𝑒^−(𝑝𝑥/𝑚)−1𝑚𝑚𝑒^−(𝑝𝑥/𝑚)𝑣−1^𝑥	𝑊𝑓1^(3)(𝑥),1
15	𝑣𝑡=𝑒^((𝑚+1)/𝑚)𝑝𝑥𝑚𝑣𝑥^𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥+𝛼𝑣+𝛽+𝛾𝑓1^(3)(𝑥)	𝜂=𝑣𝑥𝑥+𝑝(𝑚+1)𝑒^−(𝑝𝑥/𝑚)−1𝑚𝑚𝑒^−(𝑝𝑥/𝑚)𝑣−1^𝑥	𝑊𝑓1^(3)(𝑥),1
16	𝑣𝑡=𝑒^((𝑚+1)/𝑚)𝑝𝑥𝑚𝑣^23𝑚+2𝑣𝑚+1^𝑥𝑚−1𝑣𝑥𝑥+𝑝𝑚𝑣𝑥−𝑚(𝑚+1)(3𝑚+2)^2𝑒^((𝑚+1)/𝑚)𝑝𝑥−()𝑞×𝑣𝑝−𝑚+1𝑣𝑝−𝑞^𝑥𝑚+1−𝑝−𝑞𝑝−(𝑚+1)𝑞𝛼𝑣+𝛽+𝛾𝑚𝑒^−(𝑝𝑥/𝑚)−1^(𝑚+1)/𝑚,𝑚≠−1	𝜂=𝑣𝑥𝑥+𝑝(𝑚+1)𝑒^−(𝑝𝑥/𝑚)−1𝑚𝑚𝑒^−(𝑝𝑥/𝑚)𝑣−1^𝑥	𝑊𝑚𝑒^−(𝑝𝑥/𝑚)−1^(𝑚+1)/𝑚,1