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

Table 2

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