Heat Conduction and One-Dimensional Wave Equations ∝ !! = ! vs. α !! = !! Heat Conduction: ∝ ! !! = ! Boundary conditions: (0, ) = 0, (, ) = 0 Case: Bar with both ends kept at 0 degree General Solution: , = ! ! !!! !∝ ! ! ! ! ! !/! ! !"# ! Steady State Solution: () = 0 Other info: ! = ! = 2 ! ! Heat Conduction: ∝ ! !! = ! Boundary conditions: ! (0, ) = 0, ! (, ) = 0 Case: Bar with both ends perfectly insulated General Solution: , = ! + ! ! !!! !∝ ! ! ! ! ! !/! ! !"# ! Steady State Solution: () = ! Other info: ! = ! ! ! ! = ! = 2 ! !
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Heat Conduction and One-Dimensional Wave Equations ∝𝟐 𝑢!! = 𝑢! vs. α𝟐𝑢!! = 𝑢!!
Heat Conduction: ∝! 𝑢!! = 𝑢!
Boundary conditions: 𝑢(0, 𝑡) = 0,𝑢(𝐿, 𝑡) = 0
Case: Bar with both ends kept at 0 degree General Solution: 𝑢 𝑥, 𝑡 = 𝐶!!
!!! 𝑒!∝!!!!!!/!!𝑠𝑖𝑛 !"#
!
Steady State Solution: 𝑣(𝑥) = 0 Other info:
𝐶! = 𝑏! =2𝐿
𝑓 𝑥 𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥
!
!
Heat Conduction: ∝! 𝑢!! = 𝑢!
Boundary conditions: 𝑢!(0, 𝑡) = 0,𝑢!(𝐿, 𝑡) = 0
Case: Bar with both ends perfectly insulated General Solution: 𝑢 𝑥, 𝑡 = 𝐶! + 𝐶!!
!!! 𝑒!∝!!!!!!/!!𝑐𝑜𝑠 !"#
!
Steady State Solution: 𝑣(𝑥) = 𝐶! Other info: 𝐶! =
!!!
𝐶! = 𝑎! =2𝐿
𝑓 𝑥 𝑐𝑜𝑠𝑛𝜋𝑥𝐿𝑑𝑥
!
!
Heat Conduction: ∝! 𝑢!! = 𝑢!
Boundary conditions: 𝑢 0, 𝑡 = 𝑇!,𝑢 𝐿, 𝑡 = 𝑇!
Case: Bar with 𝑇! degrees at the left end, and 𝑇!degrees at the right end General Solution: 𝑢 𝑥, 𝑡 = !!!!!
!𝑥 + 𝑇! + 𝐶!!
!!! 𝑒!∝!!!!!!/!!𝑠𝑖𝑛 !"#
!
Steady State Solution: 𝑣 𝑥 = !!!!!
!𝑥 + 𝑇!
Other info: 𝑣(𝑥) = 𝐴𝑥 + 𝐵 , and 𝑤 𝑥, 0 = 𝑓 𝑥 − 𝑣(𝑥)