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2. Exercise §2.5 #4. Here is a direct relationship between the wave and diffusion equations. Let u(z,t) solve the wave equation on the whole line with bounded second derivatives. Let (a) Show that v(x, t) (b) Show that limt (TT XXX. C v(x, t) = √kt__e-s²2²/4kªu(x,s)ds -8 solves the diffusion equation! ov(x, t) = u(x, 0). ∞ II/ iffusion

2. Exercise §2.5 #4. Here is a direct relationship between the wave and diffusion equations. Let u(z,t) solve the wave equation on the whole line with bounded second derivatives. Let (a) Show that v(x, t) (b) Show that limt (TT XXX. C v(x, t) = √kt__e-s²2²/4kªu(x,s)ds -8 solves the diffusion equation! ov(x, t) = u(x, 0). ∞ II/ iffusion

Advanced Engineering Mathematics
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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A 2. Exercise §2.5 #4. Here is a direct relationship between the wave and diffusion equations. Let u(z,t) solve the wave equation on the whole line with bounded second derivatives. Let e-s²c²/4ktu(x, s)ds C v(r, t) = Arkt 00 e (a) Show that v(x, t) solves the diffusion equation! (b) Show that limt ov(x, t) = u(x, 0). (Hint: (a) Write the formula as v(x, t) equation with constant k/c² for t > 0. H(s, t)u(x, s)ds, where H(r, t) solves the diffusion (b) Use the fact that H(s, t) is essentially the source function of the diffusion equation with the spatial variable s.)
Transcribed Image Text:A 2. Exercise §2.5 #4. Here is a direct relationship between the wave and diffusion equations. Let u(z,t) solve the wave equation on the whole line with bounded second derivatives. Let e-s²c²/4ktu(x, s)ds C v(r, t) = Arkt 00 e (a) Show that v(x, t) solves the diffusion equation! (b) Show that limt ov(x, t) = u(x, 0). (Hint: (a) Write the formula as v(x, t) equation with constant k/c² for t > 0. H(s, t)u(x, s)ds, where H(r, t) solves the diffusion (b) Use the fact that H(s, t) is essentially the source function of the diffusion equation with the spatial variable s.)
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