Let us consider the following problem for the heat equation ôu ô’u = 0 in (0,1) × (0,+00), ốt u(x,0) = u,(x),Vx e (0,1), ди (0,t) = %3D (1) ди -(1,t) = 0, Vt > 0. Use the method of separation of variables to compute the solution of problem (1) if 1) u, (x) = 1, Vx e(0,1).
Let us consider the following problem for the heat equation ôu ô’u = 0 in (0,1) × (0,+00), ốt u(x,0) = u,(x),Vx e (0,1), ди (0,t) = %3D (1) ди -(1,t) = 0, Vt > 0. Use the method of separation of variables to compute the solution of problem (1) if 1) u, (x) = 1, Vx e(0,1).
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
Consider the following problem for the heat equation:
\[
\begin{cases}
\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0 & \text{in } (0,1) \times (0,+\infty), \\
u(x,0) = u_0(x), & \forall x \in (0,1), \\
\frac{\partial u}{\partial x}(0,t) = \frac{\partial u}{\partial x}(1,t) = 0, & \forall t > 0.
\end{cases}
\]
### Method of Separation of Variables
Use the method of separation of variables to compute the solution of problem (1) if:
1. \( u_0(x) = 1, \forall x \in (0,1). \)
2. \( u_0(x) = \cos 7\pi x, \forall x \in (0,1). \)
3. \( u_0(x) = 3 + 2 \cos 7\pi x, \forall x \in (0,1). \)
4. \( u_0(x) = \sin \pi x, \forall x \in (0,1). \)
### Explanation
The problem described involves solving a one-dimensional heat equation, which is a partial differential equation (PDE) representing heat distribution over time. The method of separation of variables is employed to find solutions considering different initial conditions \( u_0(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3b63b44-9302-458a-a5f0-5e786e8527ac%2Fd1330456-06d4-4e2f-a924-12e529673eec%2Fhqwec4r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Heat Equation Problem
Consider the following problem for the heat equation:
\[
\begin{cases}
\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0 & \text{in } (0,1) \times (0,+\infty), \\
u(x,0) = u_0(x), & \forall x \in (0,1), \\
\frac{\partial u}{\partial x}(0,t) = \frac{\partial u}{\partial x}(1,t) = 0, & \forall t > 0.
\end{cases}
\]
### Method of Separation of Variables
Use the method of separation of variables to compute the solution of problem (1) if:
1. \( u_0(x) = 1, \forall x \in (0,1). \)
2. \( u_0(x) = \cos 7\pi x, \forall x \in (0,1). \)
3. \( u_0(x) = 3 + 2 \cos 7\pi x, \forall x \in (0,1). \)
4. \( u_0(x) = \sin \pi x, \forall x \in (0,1). \)
### Explanation
The problem described involves solving a one-dimensional heat equation, which is a partial differential equation (PDE) representing heat distribution over time. The method of separation of variables is employed to find solutions considering different initial conditions \( u_0(x) \).
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