Find a formal solution to the following initial-boundary value problem. du = 4– t>0, 00, u(x,0) = f(x), 0
Find a formal solution to the following initial-boundary value problem. du = 4– t>0, 00, u(x,0) = f(x), 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Initial-Boundary Value Problem
Find a formal solution to the following initial-boundary value problem.
\[
\frac{\partial u}{\partial t} = 4 \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < \pi, \quad t > 0,
\]
\[
\frac{\partial u}{\partial x}(0,t) = 0, \quad u(\pi,t) = 0 \quad t > 0,
\]
\[
u(x,0) = f(x), \quad 0 < x < \pi
\]
### Boundary Value Problem for \( X(x) \)
With \( u(x,t) = X(x)T(t) \), what is the associated boundary value problem for \( X(x) \)?
- A. \[ X''(x) - \lambda X(x) = 0, \quad X'(0) = X(0) = 0 \]
- B. \[ X''(x) - \lambda X(x) = 0, \quad X'(0) = X(\pi) = 0 \]
- C. \[ X''(x) + \lambda X(x) = 0, \quad X'(0) = X(0) = 0 \]
- D. \[ X''(x) + \lambda X(x) = 0, \quad X'(0) = X(\pi) = 0 \]
The correct answer is:
\[ \boxed{\text{C.} \quad X''(x) + \lambda X(x) = 0, \quad X'(0) = X(0) = 0} \]
### Formal Solution
Find a formal solution.
\[ u(x,t) = \sum_{n=0}^{\infty} a_n \]
where \( f(x) = \sum_{n=0}^{\infty} a_n \).
This problem involves solving the heat equation with given initial and boundary conditions by using separation of variables. The correct form of the boundary value problem for \( X(x) \) includes solving the differential equation \( X''(x) + \lambda X(x) = 0 \) with the boundary conditions \( X'(0) = 0 \) and \( X(0) = 0 \). Further steps would include finding the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62cf0025-2a72-406a-bc17-39c2a0537bca%2Fdccbfd5a-624d-41d9-91fa-c563ae8777a2%2F030sxil_processed.png&w=3840&q=75)
Transcribed Image Text:### Initial-Boundary Value Problem
Find a formal solution to the following initial-boundary value problem.
\[
\frac{\partial u}{\partial t} = 4 \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < \pi, \quad t > 0,
\]
\[
\frac{\partial u}{\partial x}(0,t) = 0, \quad u(\pi,t) = 0 \quad t > 0,
\]
\[
u(x,0) = f(x), \quad 0 < x < \pi
\]
### Boundary Value Problem for \( X(x) \)
With \( u(x,t) = X(x)T(t) \), what is the associated boundary value problem for \( X(x) \)?
- A. \[ X''(x) - \lambda X(x) = 0, \quad X'(0) = X(0) = 0 \]
- B. \[ X''(x) - \lambda X(x) = 0, \quad X'(0) = X(\pi) = 0 \]
- C. \[ X''(x) + \lambda X(x) = 0, \quad X'(0) = X(0) = 0 \]
- D. \[ X''(x) + \lambda X(x) = 0, \quad X'(0) = X(\pi) = 0 \]
The correct answer is:
\[ \boxed{\text{C.} \quad X''(x) + \lambda X(x) = 0, \quad X'(0) = X(0) = 0} \]
### Formal Solution
Find a formal solution.
\[ u(x,t) = \sum_{n=0}^{\infty} a_n \]
where \( f(x) = \sum_{n=0}^{\infty} a_n \).
This problem involves solving the heat equation with given initial and boundary conditions by using separation of variables. The correct form of the boundary value problem for \( X(x) \) includes solving the differential equation \( X''(x) + \lambda X(x) = 0 \) with the boundary conditions \( X'(0) = 0 \) and \( X(0) = 0 \). Further steps would include finding the
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 5 images
