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

Table 13

Original variables for 𝑚≠𝑛,  𝑛≠−1 and various 𝑓(𝑥).

Fin profile (parameter 𝑎)	Parameter 𝑛	Solution
	𝑓(𝑥)=𝑥^𝑎
Rectangular
𝑎=0	𝑛=−3𝑚−4	𝜃=1+(𝑚+1)𝑀^2(𝑥^2−1)^1/(2(𝑚+1))
Convex parabolic
1𝑎=2	𝑛=−4𝑚−5	𝜃=𝑥^1/21𝑐1cosh±5𝑀𝑐12(𝑚+1)311−√𝑥^1/(5(𝑚+1))
Triangular
𝑎=1	𝑚=−1	𝜃=2cosh^2tanh^−1±(1/2𝑀)2/(𝑛+1)−±𝑀(𝑛+1)/2ln2(𝑥+1)cosh^2tanh^−1±(1/2𝑀)2/(𝑛+1)−±𝑀(𝑛+1)/2ln(𝑥+1)^1/(𝑛+1)
Concave parabolic
𝑎=2	𝑛=−𝑚−2	1𝑢=𝑡±∫𝑐+(𝑚+1)𝑀^2ln𝑡dt
	𝑓(𝑥)=𝑒^𝑎𝑥
Exponential
𝑎 arbitrary	𝑛=−2𝑚−3	𝜃=−9(𝑚+1)𝑀^2(𝑎𝑒^𝑎𝑥+𝑐)^2𝑒^3𝑎𝑥2𝑎^41/(3(𝑚+1))