Solve the following problems:
1. Solve the governing equation dealing with the heat transfer phenomena in a heated rod using the FINITE DIFFERENCE Method.
??2?? /????2 = -h′(??∞ - ??) using the boundary conditions: ??(0) = ???? and ??(??) = ????. Use the following parameters: L = 10 m, h′ = 0.05 ??-2, ??∞ = 200 ??, ?????? ??h?? ???????????????? ????????????????????: ??(0) = 300 ?? ?????? ??(10) = 400 ??.
2. The heat transfer problems dealing with the radiation phenomena face nonlinear boundary conditions. Use the SHOOTING method to solve the governing equation: ??2??/????2 = -h′ (??∞ - ??) - ??′ (??∞4 - ??4), where ??′ = a bulk heat transfer parameter reflecting the relative impacts of radiation and conduction = 2.7 × 10-9??-3??-2 . This equation can serve to illustrate how the Finite - Difference Method is used to solve a two-point boundary value problem. The remaining problem conditions are as follows: L = 10 m, h′ = 0.05 ??-2 , ??∞ = 200 ??, T(0) = 300 ?? and T(10) = 400 K.
3. The governing differential equation for a hot rod under steady - state can be expressed: ??2??/????2 = 0.15 ?? . Obtain a solution for a 10 m rod, T(0) = 240 ?? and T(10) = 150 K using the following methods.
(a) Analytically,
(b) The shooting Method,
(c) The Finite - Difference Method with ??? = 1.