8. u(0,1)= 0, u(1, 1) = 0, t>0 u(x, 0) = 4 sin 3x + 8 sin 6mx, 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
Solve number 8.
![**Heat Equation Problems and Boundary Conditions**
In Problems 7 and 8, solve the heat equation (1) subject to the given conditions. [*Hint: A solution of the boundary-value problem (1)–(3) need not be an infinite series.*]
**Problem 7:**
- Boundary Conditions:
- \( u(0, t) = 0 \)
- \( u(\pi, t) = 0 \), \( t > 0 \)
- Initial Condition:
- \( u(x, 0) = 10 \sin 5x \), \( 0 < x < \pi \)
**Problem 8:**
- Boundary Conditions:
- \( u(0, t) = 0 \)
- \( u(1, t) = 0 \), \( t > 0 \)
- Initial Condition:
- \( u(x, 0) = 4 \sin 3\pi x + 8 \sin 6\pi x \), \( 0 < x < 1 \)
**Diagram Explanation:**
The diagram on the right shows a (a) thin rod of length \(2L\). It suggests that the heat distribution is modeled along the rod with boundary conditions applied at either end of the rod. The circular diagram illustrates the position \(x\) on the rod, centered at \(x=0\), indicating the spatial domain over which the heat equation applies.
[*Hint: Think full Fourier series.*]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F850c5fe0-78a8-4fb7-bf63-56ee551aa9a1%2F1432fce2-4074-447e-b7bc-8f7d019f0b70%2Fcgge70s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Heat Equation Problems and Boundary Conditions**
In Problems 7 and 8, solve the heat equation (1) subject to the given conditions. [*Hint: A solution of the boundary-value problem (1)–(3) need not be an infinite series.*]
**Problem 7:**
- Boundary Conditions:
- \( u(0, t) = 0 \)
- \( u(\pi, t) = 0 \), \( t > 0 \)
- Initial Condition:
- \( u(x, 0) = 10 \sin 5x \), \( 0 < x < \pi \)
**Problem 8:**
- Boundary Conditions:
- \( u(0, t) = 0 \)
- \( u(1, t) = 0 \), \( t > 0 \)
- Initial Condition:
- \( u(x, 0) = 4 \sin 3\pi x + 8 \sin 6\pi x \), \( 0 < x < 1 \)
**Diagram Explanation:**
The diagram on the right shows a (a) thin rod of length \(2L\). It suggests that the heat distribution is modeled along the rod with boundary conditions applied at either end of the rod. The circular diagram illustrates the position \(x\) on the rod, centered at \(x=0\), indicating the spatial domain over which the heat equation applies.
[*Hint: Think full Fourier series.*]
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