If u(x,t) satisfies ∂u/∂t + ∂³u/∂x³ = 0, with u(x,0) = f(x), and u, ∂u/∂x, ∂²u/∂x² -> 0 as |x| -> ∞, use Fourier transform methods to show that u(x,t) = (3t)^(-1/3) ∫f(y) Ai[(x-y)/((3t)^(-1/3))] dy (integral from -∞ to ∞), where Ai(x) is the Airy function, for which Ai(x) = 1/π ∫cos(ω³/3 + ωx) dω (integral from 0 to ∞).
Attempt:
all my integrals are from now on are -∞ to ∞
F{u(x,t)} = U(ω,t) = 1/(2π) ∫u(x,t) e^[iωx] dx
F{u_t} = ∂U/∂t(ω,t)
F{u_xxx} = (-iω)³U(ω,t)
=> ∂U/∂t + iω³U = 0
∂U/U = -iω³∂t
(i'm going to use w for ω for more convience)
U(w,t) = c(w)*e^[-iw³t] where c(w) arb fn of w.
=> u(x,t) = ∫c(w) e^[-iw³t] e^[-iwx] dw
The IC gives
u(x,0) = f(x) = ∫c(w) e^[-iwx] dw
=> c(w) = F{f(x)} = 1/(2π) ∫f(x)e^[iwx] dx
u(x,t) = ∫1/(2π) ∫f(y) e^[iwy] dy e^[-iw³t] e^[-iwx] dw
interchanging order of integration
u(x,t) = 1/(2π)∫f(y)[∫e^[-iw(x-y)] e^[-iw³t] dw] dy
and i'm stuck...
did i make a mistake somewhere? if not how do i get it to the form they want?
ie u(x,t) = (3t)^(-1/3) ∫f(y) Ai[(x-y)/((3t)^(-1/3))] dy
Thank you for your help