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)

𝐶! = 𝑏! =2𝐿

(𝑓 𝑥 − 𝑣 𝑥 )𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥

!

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One-Dimensional Wave Equations: α!𝑢!! = 𝑢!!

Boundary conditions: 𝑢 0, 𝑡 = 0,𝑢 𝐿, 𝑡 =  0

Initial conditions: 𝑢(𝑥, 0) = 𝑓(𝑥),𝑢!(𝑥, 0) = 𝑔(𝑥)

Case: Undamped One-dimensional Wave Equation General Solution: 𝑢 𝑥, 𝑡 = (𝐴!!

!!! 𝑐𝑜𝑠 !"#$!+ 𝐵!𝑠𝑖𝑛

!"#$!)𝑠𝑖𝑛 !"#

!

Other info: ** See Special Cases Below **  

𝐴! = 𝑏! =2𝐿

𝑓 𝑥 𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥

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!

𝐵! =𝐿𝑎𝑛𝜋

𝑏! =2𝑎𝑛𝜋

𝑔 𝑥 𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥

!

!

One-Dimensional Wave Equations: α!𝑢!! = 𝑢!!

Boundary conditions: 𝑢 0, 𝑡 = 0,𝑢 𝐿, 𝑡 =  0

Initial conditions: 𝑢 𝑥, 0 = 0  ,𝑢!(𝑥, 0) = 𝑔(𝑥)

Case: Special Case of Undamped One-dimensional Wave Equation 𝑖𝑛𝑖𝑡𝑖𝑎𝑙  𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 = 0 General Solution: 𝑢 𝑥, 𝑡 = (𝐵!𝑠𝑖𝑛

!"#$!)𝑠𝑖𝑛 !"#

!

Other info: 𝐴! = 0

𝐵! =𝐿𝑎𝑛𝜋

𝑏! =2𝑎𝑛𝜋

𝑔 𝑥 𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥

!

!

One-Dimensional Wave Equations: α!𝑢!! = 𝑢!!

Boundary conditions: 𝑢 0, 𝑡 = 0,𝑢 𝐿, 𝑡 =  0

Initial conditions: 𝑢(𝑥, 0) = 𝑓(𝑥),𝑢!(𝑥, 0) = 0

Case: Special Case of Undamped One-dimensional Wave Equation 𝑖𝑛𝑖𝑡𝑖𝑎𝑙  𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 0 General Solution: 𝑢 𝑥, 𝑡 = (𝐴!!

!!! 𝑐𝑜𝑠 !"#$!)𝑠𝑖𝑛 !"#

!

Other info: 𝐵! = 0  

𝐴! = 𝑏! =2𝐿

𝑓 𝑥 𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥

!

!

𝐵! = 0