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2. Show that the steady state of the diffusion equation Uų = Kuge, 0 0, Uz (0, t) = u(L, t) = 0, u(x, 0) = $(x) %3D is given by 1 u,(2) = (2)dz. L

2. Show that the steady state of the diffusion equation Uų = Kuge, 0 0, Uz (0, t) = u(L, t) = 0, u(x, 0) = $(x) %3D is given by 1 u,(2) = (2)dz. L

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 12CR
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I need help with number 2 and 4

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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