5. Let L>1/2. Consider the diffusion equation Uų = Kurr, -L 0, u(-L,t) = u(L, t) = 0, u(x, 0) = 4 3 By using the maximum principle to show 1 (a) Ju(x, t)| < for -L

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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1. Use the coordinate method to solve 2u, – Uy + (x + 2y)u = 0 with u(x,0) = e-2"/5,
2. Show that the steady state of the diffusion equation
Uz = Kugr, 0 <x < L, t> 0,
Uz (0, t) = uz(L, t) = 0,
u(x, 0) = $(x)
%3D
is given by
u,(2) =
1
$(x)dx.
3. Classify equation ugz +Uzt – 20ut = 0, and solve the equation with initial conditions:
u(x,0) = 6(x) and u;(x,0) = (x).
4. Find the general solution of wt = Wra + sin(x +t) by using the coordinate method.
5. Let L>1/2. Consider the diffusion equation
Uų = Kura, -L <x < L, t >0,
u(-L, t) — и(L, t) %3D0,
u(x,0)
4
%3D
By using the maximum principle to show
1
(a) Ju(x, t)| <
for -L <x <Land t 2 0;
12
(b) u(-x, t) =-u(x, t) for -L <a<L and t > 0.
Transcribed Image Text:1. Use the coordinate method to solve 2u, – Uy + (x + 2y)u = 0 with u(x,0) = e-2"/5, 2. Show that the steady state of the diffusion equation Uz = Kugr, 0 <x < L, t> 0, Uz (0, t) = uz(L, t) = 0, u(x, 0) = $(x) %3D is given by u,(2) = 1 $(x)dx. 3. Classify equation ugz +Uzt – 20ut = 0, and solve the equation with initial conditions: u(x,0) = 6(x) and u;(x,0) = (x). 4. Find the general solution of wt = Wra + sin(x +t) by using the coordinate method. 5. Let L>1/2. Consider the diffusion equation Uų = Kura, -L <x < L, t >0, u(-L, t) — и(L, t) %3D0, u(x,0) 4 %3D By using the maximum principle to show 1 (a) Ju(x, t)| < for -L <x <Land t 2 0; 12 (b) u(-x, t) =-u(x, t) for -L <a<L and t > 0.
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