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Recall that the general solution to the following boundary value problem for the heat equation Ut = Uxx u(L, t) = TL u(0, t) = To, u(x,0) = (x) is of the form ∞ u(x, t) = v(x, t) + Σbne(¹7)²t sin(nπ- n=1 for v(x, t) = To + (TL - To), and for bn's being the Fourier sine series of p(x) − v(x, 0). Let L = T. a. Solve the boundary value problem when p(x) = x. To = Tπ = 2. b. Solve the boundary value problem with p(x) = sin(2x) — 2 sin x and To = T₁ = 0.

Recall that the general solution to the following boundary value problem for the heat equation Ut = Uxx u(L, t) = TL u(0, t) = To, u(x,0) = (x) is of the form ∞ u(x, t) = v(x, t) + Σbne(¹7)²t sin(nπ- n=1 for v(x, t) = To + (TL - To), and for bn's being the Fourier sine series of p(x) − v(x, 0). Let L = T. a. Solve the boundary value problem when p(x) = x. To = Tπ = 2. b. Solve the boundary value problem with p(x) = sin(2x) — 2 sin x and To = T₁ = 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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Recall that the general solution to the following boundary value problem for the heat equation Ut = Uxx u(L, t) = TL u(0, t) = To, u(x,0) = (x) is of the form ∞ u(x, t) = v(x, t) + bne ()²t sin (nn) n=1 for v(x, t) = To + (TL − To), and for bn's being the Fourier sine series of p(x) — v(x,0). Let L = T. a. Solve the boundary value problem when p(x) = x. To = T✩ = 2. b. Solve the boundary value problem with p(x) = sin(2x) — 2 sin x and To = T₁ = 0.
Transcribed Image Text:Recall that the general solution to the following boundary value problem for the heat equation Ut = Uxx u(L, t) = TL u(0, t) = To, u(x,0) = (x) is of the form ∞ u(x, t) = v(x, t) + bne ()²t sin (nn) n=1 for v(x, t) = To + (TL − To), and for bn's being the Fourier sine series of p(x) — v(x,0). Let L = T. a. Solve the boundary value problem when p(x) = x. To = T✩ = 2. b. Solve the boundary value problem with p(x) = sin(2x) — 2 sin x and To = T₁ = 0.
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