Given the initial condition u(x,0) = sin(x)+0.5sin(3x), xe[0,27], for a rod with ends fixed at zero temperature, write down the solution u(x,t) of the heat equation u, = 3u without doing any integration. xx
Given the initial condition u(x,0) = sin(x)+0.5sin(3x), xe[0,27], for a rod with ends fixed at zero temperature, write down the solution u(x,t) of the heat equation u, = 3u without doing any integration. xx
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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How would i do this question for PDE, I think it just involves knowing general soln. but I need to know how to do any these without doing integrals to practice for midterm. Thanks
![### Problem Statement
**b.** Given the initial condition \( u(x,0) = \sin(x) + 0.5 \sin(3x) \), \( x \in [0, 2\pi] \), for a rod with ends fixed at zero temperature, write down the solution \( u(x,t) \) of the heat equation \( u_t = 3u_{xx} \) without doing any integration.
### Explanation
This problem involves solving a heat equation with given initial conditions. The function \( u(x, t) \) represents the temperature distribution along a rod at position \( x \) and time \( t \).
- **Initial Condition:** The initial temperature distribution along the rod is defined by \( u(x, 0) = \sin(x) + 0.5 \sin(3x) \).
- **Domain:** \( x \) ranges from 0 to \( 2\pi \).
- **Boundary Conditions:** Both ends of the rod are kept at zero temperature.
- **Heat Equation:** The evolution of temperature is governed by the partial differential equation \( u_t = 3u_{xx} \). Here, \( u_t \) represents the time derivative, and \( u_{xx} \) represents the second spatial derivative.
### Approach
For such problems, the solution can often be written in terms of eigenfunctions that satisfy the boundary conditions. Since the initial condition is given as a linear combination of sine functions, the solution will typically take the form:
\[ u(x, t) = a_1 e^{-k_1 t} \sin(x) + a_2 e^{-k_2 t} \sin(3x) \]
**where:**
- \( k_1 \) and \( k_2 \) are constants determined by the heat equation.
- \( a_1 \) and \( a_2 \) are coefficients derived from the initial condition.
In this case, since the equation is \( u_t = 3u_{xx} \), each term of the form \( \sin(nx) \) evolves over time as \( e^{-3n^2t} \). Thus, the solution can be directly written as:
\[ u(x, t) = e^{-3t} \sin(x) + 0.5 e^{-27t} \sin(3x) \]
This solution satisfies both the initial and boundary conditions](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0548f28d-2867-4ce8-91e2-809a65372be3%2F469060e9-1604-4201-bd19-54176664e610%2Fubp7am9j_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**b.** Given the initial condition \( u(x,0) = \sin(x) + 0.5 \sin(3x) \), \( x \in [0, 2\pi] \), for a rod with ends fixed at zero temperature, write down the solution \( u(x,t) \) of the heat equation \( u_t = 3u_{xx} \) without doing any integration.
### Explanation
This problem involves solving a heat equation with given initial conditions. The function \( u(x, t) \) represents the temperature distribution along a rod at position \( x \) and time \( t \).
- **Initial Condition:** The initial temperature distribution along the rod is defined by \( u(x, 0) = \sin(x) + 0.5 \sin(3x) \).
- **Domain:** \( x \) ranges from 0 to \( 2\pi \).
- **Boundary Conditions:** Both ends of the rod are kept at zero temperature.
- **Heat Equation:** The evolution of temperature is governed by the partial differential equation \( u_t = 3u_{xx} \). Here, \( u_t \) represents the time derivative, and \( u_{xx} \) represents the second spatial derivative.
### Approach
For such problems, the solution can often be written in terms of eigenfunctions that satisfy the boundary conditions. Since the initial condition is given as a linear combination of sine functions, the solution will typically take the form:
\[ u(x, t) = a_1 e^{-k_1 t} \sin(x) + a_2 e^{-k_2 t} \sin(3x) \]
**where:**
- \( k_1 \) and \( k_2 \) are constants determined by the heat equation.
- \( a_1 \) and \( a_2 \) are coefficients derived from the initial condition.
In this case, since the equation is \( u_t = 3u_{xx} \), each term of the form \( \sin(nx) \) evolves over time as \( e^{-3n^2t} \). Thus, the solution can be directly written as:
\[ u(x, t) = e^{-3t} \sin(x) + 0.5 e^{-27t} \sin(3x) \]
This solution satisfies both the initial and boundary conditions
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