Analytical Solutions for Steady Heat Transfer in Longitudinal Fins with Temperature-Dependent Properties

Table 1

Fundamental operations of the differential transform method.

Original function	Transformed function
𝑦(𝑡)=𝑥(𝑡)±𝑧(𝑡)	𝑌(𝑡)=𝑋(𝑡)±𝑍(𝑡)
𝑦(𝑡)=𝛼𝑥(𝑡)	𝑌(𝑡)=𝛼𝑋(𝑡)
𝑦(𝑡)=𝑑𝑦(𝑡)𝑑𝑡	𝑌(𝑡)=(𝜅+1)𝑌(𝜅+1)
𝑑𝑦(𝑡)=^2𝑦(𝑡)𝑑𝑡^2	𝑌(𝑡)=(𝜅+1)(𝜅+2)𝑌(𝜅+2)
𝑑𝑦(𝑡)=^𝑠𝑦(𝑡)𝑑𝑡^𝑠	𝑌(𝑡)=(𝜅+1)(𝜅+2)⋯(𝜅+𝑠)𝑌(𝜅+𝑠)
𝑦(𝑡)=𝑥(𝑡)𝑧(𝑡)	𝑌(𝑡)=∑^𝜅𝑖=0𝑋(𝑖)𝑍(𝜅−𝑖)
𝑦(𝑡)=1	𝑌(𝑡)=𝛿(𝜅)
𝑦(𝑡)=𝑡	𝑌(𝑡)=𝛿(𝜅−1)
𝑦(𝑡)=𝑡^𝑠	𝑌(𝑡)=𝛿(𝜅−𝑠)
𝑦(𝑡)=exp(𝜆𝑡)	𝜆𝑌(𝑡)=^𝜅𝜅!
𝑦(𝑡)=(1+𝑡)^𝑠	𝑌(𝑡)=𝑠(𝑠−1)⋯(𝑠−𝜅+1)𝜅!
𝑦(𝑡)=sin(𝜔𝑡+𝛼)	𝜔𝑌(𝑡)=^𝜅𝜅!sin𝜋𝜅2!+𝛼
𝑦(𝑡)=cos(𝜔𝑡+𝛼)	𝜔𝑌(𝑡)=^𝜅𝜅!cos𝜋𝜅2!+𝛼