dru – Kôrau = 0, u(t, 0) %3 и(t, 1) %3 0, (u(0, x) = 4.x(1 – x), (t, 2) € (0, оо) х (0, 1), te (0, 0), x E [0, 1]. %3| |

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Problem 3. Consider the classical solution to the initial boundary-value problem for the heat equation:
(t, x) E (0, 0) x (0,1),
te (0, 0),
л€ [0, 1].
dru – Kôrau = 0,
-
u(t,0) = u(t, 1) = 0,
(1.6)
u(0, x) = 4x(1 – x),
(1) Show that 0 < u(t, x) < 1 for all (t, x) E (0, ∞) × [0, 1].
(2) Show that u(t, x) = u(t, 1 – x) for all (t, x) E [0, c) × [0, 1].
(3) Use the energy method to show that
E(t) = | lu(t, x)|²dæ
(1.7)
is a decreasing function of time.
1
Transcribed Image Text:Problem 3. Consider the classical solution to the initial boundary-value problem for the heat equation: (t, x) E (0, 0) x (0,1), te (0, 0), л€ [0, 1]. dru – Kôrau = 0, - u(t,0) = u(t, 1) = 0, (1.6) u(0, x) = 4x(1 – x), (1) Show that 0 < u(t, x) < 1 for all (t, x) E (0, ∞) × [0, 1]. (2) Show that u(t, x) = u(t, 1 – x) for all (t, x) E [0, c) × [0, 1]. (3) Use the energy method to show that E(t) = | lu(t, x)|²dæ (1.7) is a decreasing function of time. 1
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