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1. Let C be a circle of radius 1 and let u(0, t) be the temperature at a point 0, at time t. To make this well-defined, we need that u(0 + 2π) = u(0). Say u(0, t) satisfies the heat equation ut = U00. Let πT E(t):= = 12 u²(0, t)do. -π a) Show that E' (t) ≤ 0. b) If the initial temperature u(0, 0) = 0, show that u(0, t) = 0 for all t ≥ 0.

1. Let C be a circle of radius 1 and let u(0, t) be the temperature at a point 0, at time t. To make this well-defined, we need that u(0 + 2π) = u(0). Say u(0, t) satisfies the heat equation ut = U00. Let πT E(t):= = 12 u²(0, t)do. -π a) Show that E' (t) ≤ 0. b) If the initial temperature u(0, 0) = 0, show that u(0, t) = 0 for all t ≥ 0.

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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1. Let C be a circle of radius 1 and let u(0, t) be the temperature at a point 0, at time t. To make this well-defined, we need that u(0 + 2π) = u(0). Say u(0, t) satisfies the heat equation ut = uoo. Let 1 E(t) := ½ [*_ u²(0, t)do. 2 π a) Show that E'(t) ≤ 0. b) If the initial temperature u(0, 0) = 0, show that u(0, t) = 0 for all t≥ 0.
Transcribed Image Text:1. Let C be a circle of radius 1 and let u(0, t) be the temperature at a point 0, at time t. To make this well-defined, we need that u(0 + 2π) = u(0). Say u(0, t) satisfies the heat equation ut = uoo. Let 1 E(t) := ½ [*_ u²(0, t)do. 2 π a) Show that E'(t) ≤ 0. b) If the initial temperature u(0, 0) = 0, show that u(0, t) = 0 for all t≥ 0.
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