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Question 4. Consider the equation и, — ихх — их —D 0, 0 < x < 1, t > 0 u(0, 1) %3D 0, и(1,г) %3 1, t > 0, и(х, 0) 3 0, 0

Question 4. Consider the equation и, — ихх — их —D 0, 0 < x < 1, t > 0 u(0, 1) %3D 0, и(1,г) %3 1, t > 0, и(х, 0) 3 0, 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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Question 4. Consider the equation И, — ихх — их — 0, 0 < x < 1, t > 0 u(0, г) %3D 0, и(1,г) —D 1, 1>0, и(х, 0) — 0, 0 <x<л. (a) Consider the change of coordinates u(x,t) = v(x,t)eax+bt¸ and choose a, b to reduce the equation to a standard heat equation. (b) Solve the corresponding equation for v(x,1) using the Fourier series. Remark. You may use the following statement without proof: the solution to the ODE y'(t) + ay(t) = f(t) is y(t) = e a' y(0) + e af | eas f(s)ds. -at
Transcribed Image Text:Question 4. Consider the equation И, — ихх — их — 0, 0 < x < 1, t > 0 u(0, г) %3D 0, и(1,г) —D 1, 1>0, и(х, 0) — 0, 0 <x<л. (a) Consider the change of coordinates u(x,t) = v(x,t)eax+bt¸ and choose a, b to reduce the equation to a standard heat equation. (b) Solve the corresponding equation for v(x,1) using the Fourier series. Remark. You may use the following statement without proof: the solution to the ODE y'(t) + ay(t) = f(t) is y(t) = e a' y(0) + e af | eas f(s)ds. -at
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