Partial derivaives
𝑘 =±𝑖𝑤
∴ 𝐹(𝑥) = 𝑒 𝑖𝑤𝑥 𝑎𝑛𝑑 𝐹(𝑥) = 𝑒 −𝑖𝑤𝑥
∴ 𝐹(𝑥) = 𝐴𝑒 𝑖𝑤𝑥 +𝐵𝑒 −𝑖𝑤𝑥
Now we rewrite 𝐹(𝑥) in terms of trigonometric functions, since:
𝑒 𝑖𝑤𝑥 = cos(𝑤𝑥) + 𝑖 sin(𝑤𝑥) ,
𝑒 −𝑖𝑤𝑥 = cos(𝑤𝑥) − 𝑖 sin(𝑤𝑥)
This gives:
𝐹(𝑥) = 𝐴(cos(𝑤𝑥) + 𝑖 sin(𝑤𝑥)) + 𝐵(cos(𝑤𝑥) − 𝑖 sin(𝑤𝑥))
𝐹(𝑥) = 𝐴cos(𝑤𝑥)+𝐵sin(𝑤𝑥) 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑛𝑒𝑤 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠 𝐴,𝐵
The boundary conditions then require:
𝐴cos(0) + 𝐵sin(0) = 0 ,
𝐴 cos(𝑤𝐿) + 𝐵 sin(𝑤𝐿) = 0
Since sin(0) = 0, cos(0) = 1 then:
𝐴 =0,
∴ 𝐵 sin(𝑤𝐿) = 0
∴𝐵=0 𝑜𝑟 sin(𝑤𝐿)=0
We are looking for non-trivial solutions thus:
𝑛𝜋 𝐿
𝑤𝐿=𝑛𝜋 ⟹𝑤=
,
𝑛 ∈ℤ
𝑛𝜋 𝐿
∴ 𝐹(𝑥) = 𝐵 sin (
𝑥)
Likewise, 𝐺(𝑡) satisfies the ODE given by:
𝐺 ′ (𝑡) = −𝜆𝐾𝐺(𝑡)
𝑑𝐺 𝐺(𝑡)
⇒
= −𝜆𝐾𝑑𝑡
1 𝐺(𝑡)
⟹∫
𝑑𝐺 = ∫ −𝜆𝐾 𝑑𝑡
⟹ ln|𝐺(𝑡)| = −𝜆𝐾𝑡 + 𝐶
⟹ 𝐺(𝑡) = 𝐷𝑒 −𝜆𝐾𝑡 ,
𝑤ℎ𝑒𝑟𝑒 𝐷 = 𝑒 𝐶
𝑛 2 𝜋 2 𝐾 𝐿 2
(−
𝑡)
⟹ 𝐺(𝑡) = 𝐷𝑒
Since: 𝑇(𝑥, 𝑡) = 𝐹(𝑥)𝐺(𝑡) , we conclude that the nontrivial separable solutions of the heat equation that satisfy the boundary conditions are given by:
𝑛 2 𝜋 2 𝐾 𝐿 2
𝑛𝜋 𝐿
(−
𝑡)
𝑇(𝑥, 𝑡) = 𝑃 sin (
𝑥)𝑒
,
𝑤ℎ𝑒𝑟𝑒 𝑃 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
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