Now we have an existence and uniqueness result for the heat equation (pending some housekeeping with unproven results). 24.5 Theorem. Let f = C(R) be bounded in the sense that there exists M> 0 such that |f(x) ≤ M for all x E R. Then the only solution to the heat IVP Su₁ = Uxx, -∞0 < x < ∞, t>0 | u(x, 0) = f(x), −xx<∞ (24.2) is u(x,t) = √ √ H(x − y, t)f(y) dy, x = R, t>0 f(x), x = R, t = 0, e-82/4t H(s, t) = So what else is this solution doing? First, no matter what the initial temperature distri- bution is, eventually everything "cools all the way down." 24.6 Problem. Let f = C(R) be bounded and let |f| be integrable. Let u solve (24.2). Prove that lim u(x,t) = 0 847 for each R. Go further and explain how this limit is "uniform" in x by finding a bound |u(x, t) M(t) valid for all x R and t>0 with lim₁→∞ M(t) = 0.
Now we have an existence and uniqueness result for the heat equation (pending some housekeeping with unproven results). 24.5 Theorem. Let f = C(R) be bounded in the sense that there exists M> 0 such that |f(x) ≤ M for all x E R. Then the only solution to the heat IVP Su₁ = Uxx, -∞0 < x < ∞, t>0 | u(x, 0) = f(x), −xx<∞ (24.2) is u(x,t) = √ √ H(x − y, t)f(y) dy, x = R, t>0 f(x), x = R, t = 0, e-82/4t H(s, t) = So what else is this solution doing? First, no matter what the initial temperature distri- bution is, eventually everything "cools all the way down." 24.6 Problem. Let f = C(R) be bounded and let |f| be integrable. Let u solve (24.2). Prove that lim u(x,t) = 0 847 for each R. Go further and explain how this limit is "uniform" in x by finding a bound |u(x, t) M(t) valid for all x R and t>0 with lim₁→∞ M(t) = 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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Transcribed Image Text:Now we have an existence and uniqueness result for the heat equation (pending some
housekeeping with unproven results).
24.5 Theorem. Let f = C(R) be bounded in the sense that there exists M> 0 such that
|f(x) ≤ M for all x E R. Then the only solution to the heat IVP
Su₁ = Uxx, -∞0 < x < ∞, t>0
| u(x, 0) = f(x), −xx<∞
(24.2)
is
u(x,t)
=
√ √ H(x − y, t)f(y) dy, x = R, t>0
f(x), x = R, t = 0,
e-82/4t
H(s, t)
=
So what else is this solution doing? First, no matter what the initial temperature distri-
bution is, eventually everything "cools all the way down."
24.6 Problem. Let f = C(R) be bounded and let |f| be integrable. Let u solve (24.2).
Prove that
lim u(x,t) = 0
847
for each R. Go further and explain how this limit is "uniform" in x by finding a bound
|u(x, t) M(t) valid for all x R and t>0 with lim₁→∞ M(t) = 0.
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