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 $.

Given that the heat equation ∂u∂t-∂2u∂t2=0withu(0,t)=u(1,t)=0with ICu(x,0)=u0(x)

Answered: ôu ôu ôt ôx? u (0, 1) — и(1,1) - 0,1>…

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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

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

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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 $.

Expert Solution

Step 1

Given that

the heat equation

$\frac{\partial u}{\partial t} - \frac{\partial^{2} u}{\partial t^{2}} = 0 w i t h u (0, t) = u (1, t) = 0 w i t h I C u (x, 0) = u_{0} (x)$

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