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u(x, t) = X(x)T(t) = C3eute1+i)z. The above solution is complex, but temperature is a real number. Find the real part of u(x, t) and verify that it also satisfies the heat equation.

u(x, t) = X(x)T(t) = C3eute1+i)z. The above solution is complex, but temperature is a real number. Find the real part of u(x, t) and verify that it also satisfies the heat equation.

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

PDE: u_t = k*u_xx, 0 < x < ∞, t > 0

BC: u(0, t) = u_0(t), t > 0

boundedness: |u(x, t)| < ∞, 0 < x < ∞, t > 0

Separtation of variable: kX''(x)/X(x)=T'(t)/T(t)=λ

u(x, t) = X(x)T(t) = C3eiuste (1+i)x The above solution is complex, but temperature is a real number. Find the real part of u(x, t) and verify that it also satisfies the heat equation.
Transcribed Image Text:u(x, t) = X(x)T(t) = C3eiuste (1+i)x The above solution is complex, but temperature is a real number. Find the real part of u(x, t) and verify that it also satisfies the heat equation.
You are looking for a bounded solution that oscillates in time, so set the above constant to be pure imaginary. Show that if A = iw, where w > 0, then T(t) Coetut and X(x) Cje^L+i)» + Cze¯ -(1+i)x = for some real number y > 0. What is y? Hint: you will need to calculate Viw/k. Refer to Section 17.2 of the textbook if you have forgotten how to find roots of complex numbers.
Transcribed Image Text:You are looking for a bounded solution that oscillates in time, so set the above constant to be pure imaginary. Show that if A = iw, where w > 0, then T(t) Coetut and X(x) Cje^L+i)» + Cze¯ -(1+i)x = for some real number y > 0. What is y? Hint: you will need to calculate Viw/k. Refer to Section 17.2 of the textbook if you have forgotten how to find roots of complex numbers.
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