Find the solution in the form of Fourier integrals: Ut- 3uxx = 0, u(0, t) = 0, |u(x, t) bounded as x → ∞o, sin a u(x, 0) -{$i = x € [0, π], x € (π, ∞0). x > 0, t > 0, t> 0, t> 0,
Find the solution in the form of Fourier integrals: Ut- 3uxx = 0, u(0, t) = 0, |u(x, t) bounded as x → ∞o, sin a u(x, 0) -{$i = x € [0, π], x € (π, ∞0). x > 0, t > 0, t> 0, t> 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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#17
![### Problem Statement on Fourier Integrals
#### Problem 17
**Objective:** Find the solution in the form of Fourier integrals for the following partial differential equation:
\[u_t - 3u_{xx} = 0, \quad x > 0, \ t > 0,\]
**Boundary and Initial Conditions:**
1. Boundary Condition at \( x = 0 \):
\[u(0, t) = 0, \quad t > 0,\]
2. Condition at infinity:
\[|u(x, t)| \text{ bounded as } x \to \infty, \quad t > 0,\]
3. Initial Condition:
\[ u(x, 0) = \begin{cases}
\sin x & \text{if } x \in [0, \pi], \\
0 & \text{if } x \in (\pi, \infty).
\end{cases}
\]
This problem involves solving the above partial differential equation (PDE) subject to the given initial and boundary conditions using the method of Fourier integrals. The solution will involve representing the initial condition as a Fourier integral and then proceeding to find the solution to the PDE.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2F0d501d82-0472-4b79-a9b5-e5abebb4be9a%2F1nkdl3i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement on Fourier Integrals
#### Problem 17
**Objective:** Find the solution in the form of Fourier integrals for the following partial differential equation:
\[u_t - 3u_{xx} = 0, \quad x > 0, \ t > 0,\]
**Boundary and Initial Conditions:**
1. Boundary Condition at \( x = 0 \):
\[u(0, t) = 0, \quad t > 0,\]
2. Condition at infinity:
\[|u(x, t)| \text{ bounded as } x \to \infty, \quad t > 0,\]
3. Initial Condition:
\[ u(x, 0) = \begin{cases}
\sin x & \text{if } x \in [0, \pi], \\
0 & \text{if } x \in (\pi, \infty).
\end{cases}
\]
This problem involves solving the above partial differential equation (PDE) subject to the given initial and boundary conditions using the method of Fourier integrals. The solution will involve representing the initial condition as a Fourier integral and then proceeding to find the solution to the PDE.
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