Description
Haoran Sun (haoransun@link.cuhk.edu.cn) Problem 1 (P352 Q1). Let
u˜(ξ,t) = ˆ dx e−iξxu(x,t)
Hence
u˜t(ξ,t) = −κξ2u˜(ξ,t) + iµξu˜(ξ,t) ⇒ u˜(ξ,t) = e−κξ2t+iµξtC(ξ)
which means
u dξ e−κξ2t+iµξteiξxC(ξ)
u dξ eiξxC(ξ) = ϕ(x)
⇒ C(ξ) = ˆ dx ϕ(x)e−iξx
then we solved the equation. Problem 2 (P352 Q2). Let
u˜(µ,y) = ˆ dx e−iµxu(x,y)
Hence
⇒ u˜(µ,y) = A(µ)sinhµy + B(µ)coshµy
which means
u dµ eiµx[A(µ)sinhµy + B(µ)coshµy] uy dµ eiµx[µA(µ)coshµy + µB(µ)sinhµy] uy dµ eiµxµA(µ) = h(x)
⇒ A ˆ dx e−iµxh(x)
µ
then we solved the equation.
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