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

Table 11

The type of second-order equations admitting 𝐿₂ for 𝑚≠𝑛,  𝑛≠−1 and various 𝑓(𝑥).

Fin profile (parameter 𝑎)	Parameter 𝑛	Canonical form of the equation
	𝑓(𝑥)=𝑥^𝑎
Rectangular	𝑛 arbitrary	𝑢^′′=−(𝑛−𝑚)2𝑢^′𝑡2(𝑛+𝑚)+4(𝑛−𝑚)^2−𝑀^2𝑢^′2
𝑎=0	𝑛=−3𝑚−4	𝑢^′′1=−2𝑢^′−1𝑡1+4(𝑚+1)𝑀^2(𝑢^′−1)^2
Convex parabolic
1𝑎=2	𝑛=−4𝑚−5	𝑢^′′𝑢=−^′−1𝑡35+40(𝑚+1)𝑀^2(𝑢^′−1)^2
Triangular
𝑎=1	𝑚=−1	𝑢^′′𝑢=−^′−1𝑡1−(𝑛+1)𝑀^2(𝑢^′−1)^22
Concave parabolic
𝑎=2	𝑛=−𝑚−2	𝑢^′′=−(𝑢^′−1)^3𝑡(𝑚+1)𝑀^2
Cubic
𝑎=3	𝑛=−3𝑚−52	𝑢^′′=−5(𝑛+1)𝑀^2𝑢16^′−1𝑡1625(𝑛+1)𝑀^2−(𝑢^′−1)^2
	𝑓(𝑥)=𝑒^𝑎𝑥
Exponential
𝑎 arbitrary	𝑛=−2𝑚−3	𝑢^′′𝑢=−^′−1𝑡13+3(𝑚+1)𝑀^2(𝑢^′−1)^22𝑎^4