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

Table 1

Solution for 𝑛>−1 with 𝑚=𝑛≠−1.

𝑓(𝑥)	Parameter	Solution
𝑥^𝑎	𝑎 arbitrary	𝜃=𝑐1𝑥^(1−𝑎)/2𝐽(1−𝑎)/(𝑎−2)𝑖2𝑀√𝑛+1𝑥^1−(𝑎/2)𝑎−2+𝑐2𝑥^(1−𝑎)/2𝑌(1−𝑎)/(𝑎−2)𝑖2𝑀√𝑛+1𝑥^1−(𝑎/2)𝑎−2^1/(𝑛+1)
	𝑎≠2
𝑥^𝑎	𝑎=0	𝜃=𝑐1sin𝑖𝑀√𝑛+1𝑥+𝑐2cos𝑖𝑀√𝑛+1𝑥^1/(𝑛+1)
𝑥^𝑎	1𝑎=2	𝜃=𝑐1𝑥^1/4𝐽1/3𝑖4𝑀√𝑛+1𝑥^3/43+𝑐2𝑥^1/4𝑌1/3𝑖4𝑀√𝑛+1𝑥^3/43^1/(𝑛+1)
𝑥^𝑎	𝑎=1	𝜃=𝑐1𝐽0𝑖2𝑀(𝑛+1)𝑥+𝑐2𝑌0𝑖2𝑀(𝑛+1)𝑥^1/(𝑛+1)
𝑥^𝑎	𝑎=2	𝜃=𝑐1𝑥^−((1/2)+(1/2))1+4𝑀2(𝑛+1)+𝑐2𝑥^−((1/2)−(1/2))1+4𝑀2(𝑛+1)1/(𝑛+1)
𝑥^𝑎	𝑎=3	𝑐𝜃=^1𝐽2𝑖2𝑀(𝑛+1)/𝑥+𝑐2𝑌2𝑖2𝑀(𝑛+1)/𝑥𝑥^1/(𝑛+1)
𝑒^𝑎𝑥	𝑎≠0	𝜃=𝑐1𝑒^−(𝑎𝑥/2)𝐽1𝑖2𝑀√𝑛+1𝑒^−(𝑎𝑥/2)𝑎+𝑐2𝑒^−(𝑎𝑥/2)𝑌1𝑖2𝑀√𝑛+1𝑒^−(𝑎𝑥/2)𝑎^1/(𝑛+1)
sin𝑥		𝜃^𝑛+1=𝑐1HeunG2,−(𝑛+1)𝑀^21,0,1,2+1,1,sin𝑥+1√()cos𝑥1−sin𝑥^3/4×𝑐21+√1−cos^2𝑥^1/4HeunG2,−(𝑛+1)𝑀^2+54,12,32,32,1,sin𝑥+1
cos𝑥		𝜃^𝑛+1=𝑐1HeunG2,−(𝑛+1)𝑀^21,0,1,2+1,1,cos𝑥+1√()sin𝑥1−cos𝑥^3/4×𝑐21+1−sin^2𝑥^1/4HeunG2,−(𝑛+1)𝑀^2+54,12,32,32,1,sin𝑥+1