Solve the heat equation ku əx² u(x, t) = = u(0, t) = 0, u(L, t) = 0, t > 0 u(x, 0) = x(L-X), 0 0 (see (1) in section 12.3) subject to the given conditions. Assume a rod of length L. at' n = 1
Solve the heat equation ku əx² u(x, t) = = u(0, t) = 0, u(L, t) = 0, t > 0 u(x, 0) = x(L-X), 0 0 (see (1) in section 12.3) subject to the given conditions. Assume a rod of length L. at' n = 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Here is the problem and the conditions from 12.3 (1)
Please write out the explanation on paper NOT typed because if it's typed once it comes out it is VERY confusing!
Thank you so much!!
![The image shows the one-dimensional heat equation, which is given by:
\[ k \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}, \quad 0 < x < L, \quad t > 0 \quad (1) \]
**Explanation:**
- \( u \) is the temperature at position \( x \) and time \( t \).
- \( k \) is the thermal diffusivity constant of the material.
- The equation states that the change in temperature over time (\(\frac{\partial u}{\partial t}\)) is proportional to the second spatial derivative of temperature (\(\frac{\partial^2 u}{\partial x^2}\)).
- The expression \( 0 < x < L \) indicates the spatial domain, while \( t > 0 \) denotes time after the initial condition.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76aaab3c-5a4a-4ff8-814d-d2e8a1f30217%2F08b392e3-e48b-445f-b921-7902978c2def%2F1w42tqk_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows the one-dimensional heat equation, which is given by:
\[ k \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}, \quad 0 < x < L, \quad t > 0 \quad (1) \]
**Explanation:**
- \( u \) is the temperature at position \( x \) and time \( t \).
- \( k \) is the thermal diffusivity constant of the material.
- The equation states that the change in temperature over time (\(\frac{\partial u}{\partial t}\)) is proportional to the second spatial derivative of temperature (\(\frac{\partial^2 u}{\partial x^2}\)).
- The expression \( 0 < x < L \) indicates the spatial domain, while \( t > 0 \) denotes time after the initial condition.
![**Heat Equation Problem:**
**Problem Statement:**
Solve the heat equation:
\[ k \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}, \quad 0 < x < L, \quad t > 0 \]
(Refer to equation (1) in section 12.3) subject to the given conditions. Assume a rod of length \( L \).
**Boundary Conditions:**
- \( u(0, t) = 0, \quad u(L, t) = 0, \quad t > 0 \)
- \( u(x, 0) = x(L - x), \quad 0 < x < L \)
**Solution Form:**
\[ u(x, t) = \, ( \text{solution expression} ) \, + \sum_{n=1}^{\infty} \left( \, ( \text{series terms} ) \, \right) \]
**Explanation:**
This problem involves solving the heat equation, which describes how heat diffuses through a given medium over time. The boundary conditions specify that the temperature at both ends of the rod is maintained at zero, and the initial temperature distribution along the rod is given by \( x(L - x) \).
The solution will involve finding the expression for \( u(x, t) \) using separation of variables or a similar method and will most likely involve a series expansion (as indicated by the summation notation in the solution form).
This problem is a part of mathematical physics and is typically covered in sections dealing with partial differential equations and heat conduction problems in physics or engineering courses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76aaab3c-5a4a-4ff8-814d-d2e8a1f30217%2F08b392e3-e48b-445f-b921-7902978c2def%2F4sav96f_processed.png&w=3840&q=75)
Transcribed Image Text:**Heat Equation Problem:**
**Problem Statement:**
Solve the heat equation:
\[ k \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}, \quad 0 < x < L, \quad t > 0 \]
(Refer to equation (1) in section 12.3) subject to the given conditions. Assume a rod of length \( L \).
**Boundary Conditions:**
- \( u(0, t) = 0, \quad u(L, t) = 0, \quad t > 0 \)
- \( u(x, 0) = x(L - x), \quad 0 < x < L \)
**Solution Form:**
\[ u(x, t) = \, ( \text{solution expression} ) \, + \sum_{n=1}^{\infty} \left( \, ( \text{series terms} ) \, \right) \]
**Explanation:**
This problem involves solving the heat equation, which describes how heat diffuses through a given medium over time. The boundary conditions specify that the temperature at both ends of the rod is maintained at zero, and the initial temperature distribution along the rod is given by \( x(L - x) \).
The solution will involve finding the expression for \( u(x, t) \) using separation of variables or a similar method and will most likely involve a series expansion (as indicated by the summation notation in the solution form).
This problem is a part of mathematical physics and is typically covered in sections dealing with partial differential equations and heat conduction problems in physics or engineering courses.
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