Some Exact Solutions of Nonlinear Fin Problem for Steady Heat Transfer in Longitudinal Fin with Different Profiles

Table 10

Lie bracket of the admitted symmetry algebra for 𝑚≠𝑛,𝑛≠−1 and various 𝑓(𝑥).

𝑓(𝑥)=𝑥^𝑎	𝑎=0	𝑛 arbitrary	𝑓(𝑥)=𝑥^𝑎	𝑎=0	𝑛=−3𝑚−4
[𝑋𝑖,𝑋𝑗]	𝑋1	𝑋2	[𝑋𝑖,𝑋𝑗]	𝑋1	𝑋2	𝑋3
𝑋1	0	𝑋1	𝑋1	0	𝑋1	𝑋3
𝑋2	𝑋1	0	𝑋2	−𝑋1	0	2𝑋2
			𝑋3	−𝑋3	−2𝑋2	0
𝑓(𝑥)=𝑥^𝑎	1𝑎=2	𝑛=−4𝑚−5	𝑓(𝑥)=𝑥^𝑎	𝑎=1	𝑚=−1
[𝑋𝑖,𝑋𝑗]	𝑋1	𝑋2	[𝑋𝑖,𝑋𝑗]	𝑋1	𝑋2
𝑋1	0	𝑋1	𝑋1	0	𝑋1
𝑋2	−𝑋1	0	𝑋2	−𝑋1	0
𝑓(𝑥)=𝑥^𝑎	𝑎=2	𝑛=−𝑚−2	𝑓(𝑥)=𝑒^𝑎𝑥	any 𝑎	𝑛 arbitrary
[𝑋𝑖,𝑋𝑗]	𝑋1	𝑋2	[𝑋𝑖,𝑋𝑗]	𝑋1	𝑋2
𝑋1	0	𝑋1	𝑋1	0	𝑋1
𝑋2	−𝑋1	0	𝑋2	𝑋1	0