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ôu ôu ôt ôx? u (0, 1) — и(1,1) - 0,1> 0, : 0 in (0,1) × (0,+∞), u(х,0) — и, (х),0

ôu ôu ôt ôx? u (0, 1) — и(1,1) - 0,1> 0, : 0 in (0,1) × (0,+∞), u(х,0) — и, (х),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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Use the *method of separation of variables* to solve \[ \begin{cases} \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0 \text{ in } (0,1) \times (0,+\infty), \\ u(0,t) = u(1,t) = 0, \, t > 0, \\ u(x,0) = u_0(x), \, 0 < x < 1, \end{cases} \] if: 1) \( u_0(x) = \sin 3\pi x \). 2) \( u_0(x) = x \).
Transcribed Image Text:Use the *method of separation of variables* to solve \[ \begin{cases} \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0 \text{ in } (0,1) \times (0,+\infty), \\ u(0,t) = u(1,t) = 0, \, t > 0, \\ u(x,0) = u_0(x), \, 0 < x < 1, \end{cases} \] if: 1) \( u_0(x) = \sin 3\pi x \). 2) \( u_0(x) = x \).
Expert Solution
Step 1

Given that

the heat equation

∂u∂t-∂2u∂t2=0withu(0,t)=u(1,t)=0with ICu(x,0)=u0(x)

 

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