22.5 Problem. Repeat the work above for the transport IVP U₁ + Ux = 0, −∞
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![22.5 Problem. Repeat the work above for the transport IVP
U₁ + Ux = 0, −∞ <x, t< ∞
Ut
u(x, 0) = f(x), −x<x<∞
and recover the expected, beloved formula u(x,t) = f(x − t).
transform to u in x and get an ODE-type IVP for u. Solve it.
[Hint: apply the Fourier
Then recover u from its
Fourier transform via Untheorem 22.2. Do some algebra in the integrand and recognize the
integral as the Fourier transform of f.]
22.2 Untheorem. Let f: RC be "nice." Then
f(x) = √1/= Lº° ƒ(k)eik² dk.
2πT](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F690bc708-737a-4036-8bde-cd8ee17ec8dd%2F64141760-b2af-4967-9c38-222ef44a6c77%2Fca1gtro_processed.jpeg&w=3840&q=75)
Transcribed Image Text:22.5 Problem. Repeat the work above for the transport IVP
U₁ + Ux = 0, −∞ <x, t< ∞
Ut
u(x, 0) = f(x), −x<x<∞
and recover the expected, beloved formula u(x,t) = f(x − t).
transform to u in x and get an ODE-type IVP for u. Solve it.
[Hint: apply the Fourier
Then recover u from its
Fourier transform via Untheorem 22.2. Do some algebra in the integrand and recognize the
integral as the Fourier transform of f.]
22.2 Untheorem. Let f: RC be "nice." Then
f(x) = √1/= Lº° ƒ(k)eik² dk.
2πT
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