Heat conduction in a thin circular ring is described by the following PDE U₁=Uzz for t≥ 0. with periodic boundary conditions in space (x € [0, 2π]): u(0, t) = u(2ñ, t), u₂(0, t) = u₂(2ñ, t), etc. Solve using separation of variables, (i.e. u(x, t) = X(x)T(t)). Find a general solution (i.e. for arbitraty initial condition u(x,0) = f(x)) and describe the asymptotic behavior of all solutions as t→ ∞. Provide a physical interpretation of this behavior. u(0, t) = u(2n, t) ux(0,t)=ux (2n, t) U₂ = Uxx

Heat conduction in a thin circular ring is described by the following PDE U₁=Uzz for t≥ 0. with periodic boundary conditions in space (x € [0, 2π]): u(0, t) = u(2ñ, t), u₂(0, t) = u₂(2ñ, t), etc. Solve using separation of variables, (i.e. u(x, t) = X(x)T(t)). Find a general solution (i.e. for arbitraty initial condition u(x,0) = f(x)) and describe the asymptotic behavior of all solutions as t→ ∞. Provide a physical interpretation of this behavior. u(0, t) = u(2n, t) ux(0,t)=ux (2n, t) U₂ = Uxx

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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Question

Heat conduction in a thin circular ring is described by the following PDE
Ut=Uzz for t≥ 0.
with periodic boundary conditions in space (r = [0, 2π]):
u(0, t) = u(2n, t), ux(0, t) = uz (2π, t), etc.
Solve using separation of variables, (i.e. u(x, t) = X(x)T(t)). Find a general solution (i.e.
for arbitraty initial condition u(x,0) = f(x)) and describe the asymptotic behavior of all
solutions as t→ ∞o. Provide a physical interpretation of this behavior.
u(0, t) = u(2n, t)
ux (0,t) = ux (2n, t)
Ut = Uxx

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