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

Table 9

Modified symmetries for 𝑓(𝒳) where 𝒳=𝑥+1 and 𝑚≠𝑛,𝑛≠−1.

Fin profile (parameter 𝑎)	Parameter 𝑛	Symmetries
	𝑓(𝒳)=𝒳^𝑎
Arbitrary	𝑛 arbitrary	𝑋1𝜕=𝒳+𝜕𝑥𝑎−2𝑦𝜕𝑛−𝑚𝜕𝑦
Rectangular	𝑛 arbitrary	𝑋1=𝜕𝜕𝑥, 𝑋2𝜕=𝒳−2𝜕𝑥𝜃𝜕𝑛−𝑚𝜕𝑦
𝑎=0	𝑛=−3𝑚−4	𝑋1=𝜕𝜕𝑥, 𝑋2=𝜕−𝑦𝜕𝑥2𝜕𝜕𝑦,
		𝑋3=1()𝒳𝜕𝑚+1−2𝑚+1𝜕𝜕𝑥−𝜃𝜕𝑦
Convex parabolic	𝑛 arbitrary	𝑋1=𝑚𝜕3(𝑚+1)2𝒳𝜕𝜕𝑥−3𝜃𝜕𝑦
1𝑎=2	𝑛=−4𝑚−5	𝑋1=4𝒳√𝒳𝜕√𝜕𝑥+2𝜕𝒳𝑦𝜕𝑦,
		𝑋2𝜕=−2𝒳−3𝜕𝑥5𝑦𝜕𝜕𝑦
Triangular	𝑛 arbitrary	𝑋1=𝑚)𝑋𝜕𝑚+1(𝑛−𝑚𝜕𝜕𝑥−𝜃𝜕𝑦
𝑎=1	𝑚=−1	𝑋1𝜕=−𝒳+𝜕𝜕𝑥𝜕𝑦,
		𝑋2𝜕=(2−ln𝒳)𝒳𝜕𝜕𝑥+ln𝒳𝜕𝑦
Concave parabolic	𝑛 arbitrary	𝑋1𝜕=−2𝒳𝜕𝑥
𝑎=2	𝑛=−𝑚−2	𝑋1=𝜕−𝑦𝜕𝑥𝒳𝜕𝜕𝑦, 𝑋2𝜕=𝒳𝜕𝑥
Cubic	𝑛 arbitrary	𝑋1𝜕=−2𝒳𝜕𝑥
𝑎=3	𝑛=−3𝑚−52	𝑋11=−𝜕2𝒳+𝑦𝜕𝑥𝒳^2𝜕𝜕𝑦, 𝑋2=𝒳2𝜕−𝑦𝜕𝑥5𝜕𝜕𝑦
	𝑓(𝒳)=𝑒^𝑎𝒳
Exponential	𝑛 arbitrary	𝑋1=𝑚)𝜕𝑎(𝑚+1)(𝑚−𝑛𝜕𝜕𝑥−𝑎𝑦𝜕𝑦
𝑎 arbitrary	𝑛=−2𝑚−3	𝑋1=𝑒^𝑎𝒳𝑎^2𝜕−𝑒𝜕𝑥𝑎^𝑎𝒳𝑦𝜕𝜕𝑦,
		𝑋2=1𝑎𝜕−𝑦𝜕𝑥3𝜕𝜕𝑦