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

Table 5

Solution for 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)]