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

Table 7

Symmetries for 𝑚=𝑛=−1,𝑓(𝑥)=𝑥^𝑎.

𝑓(𝑥)	Symmetries
1	𝑋1=𝜕𝜕𝜃, 𝑋2=2𝑥^2𝜕𝜕𝑥+(2ln𝜃+𝑥^2𝑀^2𝜕)𝜃𝜕𝜃	𝑋3𝜕=𝑥𝜕𝑥+𝑥^2𝑀^2𝜃𝜕𝜕𝜃
	𝑋4=2(𝑥^2𝑀^2𝜕−2ln𝜃)𝑥𝜕𝑥+(𝑥^4𝑀^4−4ln^2𝜕𝜃)𝜃𝜕𝜃	𝑋5=(2ln𝜃−𝑥^2𝑀^2𝜕)𝜃𝜕𝜃
	𝑋6=(3𝑥^2𝑀^2𝜕−2ln𝜃)𝜕𝑥+2𝑥^3𝑀^4𝜃𝜕𝜕𝜃	𝑋7𝜕=−𝑥𝜃𝜕𝜃, 𝑋8𝜕=−𝜃𝜕𝜃
√𝑥	𝑋1𝜕=−𝜃𝜕𝜃, 𝑋2=−4𝑥√𝑥𝜕8𝜕𝑥−(3𝑥^2𝑀^2=2√𝜕𝑥ln𝜃)𝜃𝜕𝜃	𝑋3𝜕=𝑥√𝜕𝑥+𝑥𝑥𝑀^2𝜃𝜕𝜕𝜃
	𝑋4=√2𝑥ln𝜃−3𝑥^2𝑀^2𝜕+2𝜕𝑥𝑥ln𝜃−3𝑥^2√𝑥𝑀^2𝑀^2𝜃𝜕𝜕𝜃	𝑋5=(3√𝑥ln𝜃−2𝑥^2𝑀^2)3√𝑥𝜃𝜕𝜕𝜃
	𝑋6=2√𝜕𝑥𝜃𝜕𝜃,	𝑋8=23𝑥√𝑥𝑀^2𝑥𝜕−ln𝜃𝜕𝑥
	𝑋71=−√𝑥𝜕𝜕𝑥−𝑥𝑀^2𝜃𝜕−1𝜕𝜃2ln^24𝜃−9𝑥^3𝑀^4+13𝑥√𝑥𝑀^2𝜃𝜕ln𝜃𝜕𝜃
𝑥	𝑋1=−𝑥^2𝜕ln𝑥+𝜕𝑥𝑥𝑀^2−𝑥ln𝑥𝑀^2𝜕−ln𝜃𝜃ln𝑥𝜕𝜃	𝑋2𝜕=𝑥ln𝑥𝜕𝑥+𝑥ln𝑥𝑀^2𝜃𝜕𝜕𝜃
	𝑋3=ln𝜃−𝑥𝑀^2𝑥𝜕+𝜕𝑥ln𝜃−𝑀^2𝑥𝑥𝜃𝑀^2𝜕𝜕𝜃	𝑋4𝜕=−𝑥𝜕𝑥−𝑥𝑀^2𝜃𝜕𝜕𝜃
	𝑋5𝜕=−𝜃𝜕𝜃, 𝑋6𝜕=𝜃ln𝑥𝜕𝜃,	𝑋8=−ln𝜃−𝑥𝑀^2𝜕𝑥ln𝑥𝜕𝑥
	𝑋7=ln𝜃−𝑥𝑀^2𝜃𝜕𝜕𝜃−𝜃ln^2𝜃+ln𝜃ln𝑥𝑀^2𝑥−2𝑥ln𝜃𝑀^2−𝑥^2𝑀^4ln𝑥+𝑥^2𝑀^4𝜕𝜕𝜃
𝑥^2	𝑋1𝜕=−𝑥𝜕𝑥, 𝑋2𝜕=−+𝜕𝑥ln𝜃−ln𝑥𝑀^2−𝑀^2𝑥𝜃𝜕𝜕𝜃	𝑋3=−𝑥^2𝜕𝜕𝑥−𝑥𝜃𝑀^2𝜕𝜕𝜃
	𝑋4=𝑥^2(ln𝑥𝑀^2𝜕−ln𝜃)𝜕𝑥+𝑥𝜃(ln𝑥𝑀^2−ln𝜃)𝑀^2𝜕𝜕𝑥	𝑋5=(ln𝜃−𝑀^2𝜕ln𝑥)𝜃𝜕𝜃
	𝑋6=𝑥^2𝑀^2𝜕−ln𝑥−ln𝜃𝜕𝑥+𝜃ln𝜃−𝑀^2ln𝑥^2𝜕𝜕𝜃	𝑋7𝜃=−𝑥𝜕𝜕𝜃, 𝑋8𝜕=−𝜃𝜕𝜃
𝑒^𝑎𝑥	𝑋1𝜕=−𝜃𝜕𝜃, 𝑋2𝜃=−𝑎𝑒^𝑎𝑥𝜕𝜕𝜃, 𝑋3𝑒=−^𝑎𝑥𝑎𝜕−𝜕𝑥𝑥𝜃𝑀^2𝑎𝜕𝜕𝜃	𝑋4=𝜕𝜕𝑥+(𝑎𝑥+1)𝑀^2𝜕𝜕𝜃
	𝑋5=1𝑎^2𝑒^𝑎𝑥𝜕𝜕𝑥+(𝑎^2𝑒^𝑎𝑥ln𝑥+𝑀^2𝜕)𝜃𝜕𝜃	𝑋6=𝑎^2𝑒^𝑎𝑥ln𝜃+(𝑎𝑥+1)𝑀^2𝑎^2𝑒^𝑎𝑥𝜃𝜕𝜕𝜃
	𝑋7=𝑎𝑒^𝑎𝑥ln𝜃+𝑥𝑀^2𝑎^2𝜃𝜕𝜕𝑥+𝑥𝜃𝑀^2𝑎^2𝑒^𝑎𝑥ln𝜃+𝑥𝑀^2𝑎^2𝑒^𝑎𝑥𝜃𝜕+𝑎𝜕𝜃3𝑒^2𝑎𝑥ln^2𝜃+𝑎^2𝑥𝑒^𝑎𝑥𝑀^2ln𝜃+𝑎𝑒^𝑎𝑥𝑀^2ln𝜃+𝑥𝑀^4𝑎𝑒^𝑎𝑥𝜃𝜕𝜕𝜃	𝑋8=𝑎𝑒^𝑎𝑥ln𝜃+𝑥𝑀^2𝑎𝑒^𝑎𝑥𝜃𝜕𝜕𝑥