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

Table 6

Modified general solution for 𝑓(𝒳) where 𝒳=𝑥+1 and 𝑚=𝑛=−1.

𝑓(𝒳)	Parameter	Solution
𝒳^𝑎	𝑎 arbitrary	𝜃=𝑐1𝑒^−(𝒳1−𝑎[(𝑎−2)𝑐2+(𝑎−1)𝑀2𝒳])/((𝑎−1)(𝑎−2))
	𝑎≠2
𝒳^𝑎	𝑎=0	𝜃=𝑐1𝑒^1/2𝒳(2𝑐2+𝑀2𝒳)
𝒳^𝑎	1𝑎=2	𝜃=𝑐1𝑒^√2/3𝑋(3𝑐2+𝑀2𝒳)
𝒳^𝑎	𝑎=1	𝜃=𝑐1𝑒^𝑀2𝒳𝒳^𝑐2
𝒳^𝑎	𝑎=2	𝜃=𝑐1𝒳^𝑀2𝑒^−(𝑐2/𝒳)
𝒳^𝑎	𝑎=3	𝜃=𝑐1𝑒^−1/2((𝑐2+2𝑀2𝒳)/𝒳2)
𝑒^𝑎𝒳	𝑎≠0	𝜃=𝑐1𝑒^−(𝑒−𝑎𝒳[𝑎𝑐2+(𝑎+1)(1+𝐼𝑒𝐼𝒳)2𝒳−𝑀𝒳])/𝑎2
sin𝒳		𝜃=𝑐11−𝐼𝑒^𝐼𝒳1+𝐼𝑒^𝐼𝒳𝑀2𝒳𝑒^𝐼𝑀2polylog2(−𝑒𝐼𝒳)−𝐼𝑀2polylog2(𝑒𝐼𝒳)−2𝑐2argtanh(𝑒𝐼𝑋)
cos𝒳		𝜃=𝑐11−𝐼𝑒^𝐼𝒳1+𝐼𝑒^𝐼𝒳𝑀2𝒳𝑒^𝐼[𝑀2dilog(1+𝐼𝑒𝐼𝒳)−𝑀2dilog(1+𝐼𝑒𝐼𝒳)−2𝑐2arctan(𝑒𝐼𝒳)]
ln𝒳		𝜃=𝑐1𝑒^−[𝑀2𝐸𝑖1(−2ln𝒳)+𝑐2𝐸𝑖1(−ln𝒳)]