1. Consider the heat equation du = k for - T 0 | with the periodic boundary conditions и(-п,t) — и(п, t) and du (-п,t) - du (7T, t), for t > 0, -IT and the initial value condition u(x, 0) = f(x) for æ E [-7, 7]. Solve for the following initial value conditions: a) f(x) = So re (-7,0] x € x € (0, 7) b) f(x) =
1. Consider the heat equation du = k for - T 0 | with the periodic boundary conditions и(-п,t) — и(п, t) and du (-п,t) - du (7T, t), for t > 0, -IT and the initial value condition u(x, 0) = f(x) for æ E [-7, 7]. Solve for the following initial value conditions: a) f(x) = So re (-7,0] x € x € (0, 7) b) f(x) =
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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![### Heat Equation Problem
1. **Consider the heat equation:**
\[
\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}
\]
for \(-\pi < x < \pi\), and \(t > 0\),
with the **periodic boundary conditions:**
\[
u(-\pi, t) = u(\pi, t) \quad \text{and} \quad \frac{\partial u}{\partial x}(-\pi, t) = \frac{\partial u}{\partial x}(\pi, t), \quad \text{for } t > 0,
\]
and the **initial value condition:**
\[
u(x, 0) = f(x) \quad \text{for } x \in [-\pi, \pi].
\]
**Solve for the following initial value conditions:**
a) \( f(x) = x \)
b)
\[
f(x) =
\begin{cases}
0 & x \in (-\pi, 0] \\
1 & x \in (0, \pi)
\end{cases}
\]
This exercise involves solving a partial differential equation with specified boundary and initial conditions. The solution techniques include separation of variables and Fourier series due to the periodic boundary conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25098126-90d0-49b3-878f-610b12d49610%2Fd381aa45-921d-4ff4-8352-048ac61cfacc%2F2qdfoe4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Heat Equation Problem
1. **Consider the heat equation:**
\[
\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}
\]
for \(-\pi < x < \pi\), and \(t > 0\),
with the **periodic boundary conditions:**
\[
u(-\pi, t) = u(\pi, t) \quad \text{and} \quad \frac{\partial u}{\partial x}(-\pi, t) = \frac{\partial u}{\partial x}(\pi, t), \quad \text{for } t > 0,
\]
and the **initial value condition:**
\[
u(x, 0) = f(x) \quad \text{for } x \in [-\pi, \pi].
\]
**Solve for the following initial value conditions:**
a) \( f(x) = x \)
b)
\[
f(x) =
\begin{cases}
0 & x \in (-\pi, 0] \\
1 & x \in (0, \pi)
\end{cases}
\]
This exercise involves solving a partial differential equation with specified boundary and initial conditions. The solution techniques include separation of variables and Fourier series due to the periodic boundary conditions.
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