b) By using $(x), solve the given heat Problem. Use: √1 x (2²²-1) Dun (7x) dx = Só Din (nitx) dx = Ḥ 00 Анбиль ^b)" Ulrit) = 2 x (x²-1) + Σ Bne pin(nitze) ทะ B₁ = 2 f¹ f(x) sin (mix) dx 2010 (1-(-1)) Tim c) By Considering that U (xit) = ) = $(x) + nal 6(-1)^ 77³03 1-(-1)^ TTO. b(-1)^ πT³m³ | 42₂ Gues | ≤ 1/2 2²³ Uz 4 (4 (uit) Σ Un (x it) Write down what Un(xit) is from your solution in (b). Then assume lo = b/TT² and show that (>1/772 Ul(xit) 2 $(x) d) In (c) you showed that the second term was small compared to the first, so (without Prof) write down the first term -17²4 + B₁ e sun (1x) approxisition
b) By using $(x), solve the given heat Problem. Use: √1 x (2²²-1) Dun (7x) dx = Só Din (nitx) dx = Ḥ 00 Анбиль ^b)" Ulrit) = 2 x (x²-1) + Σ Bne pin(nitze) ทะ B₁ = 2 f¹ f(x) sin (mix) dx 2010 (1-(-1)) Tim c) By Considering that U (xit) = ) = $(x) + nal 6(-1)^ 77³03 1-(-1)^ TTO. b(-1)^ πT³m³ | 42₂ Gues | ≤ 1/2 2²³ Uz 4 (4 (uit) Σ Un (x it) Write down what Un(xit) is from your solution in (b). Then assume lo = b/TT² and show that (>1/772 Ul(xit) 2 $(x) d) In (c) you showed that the second term was small compared to the first, so (without Prof) write down the first term -17²4 + B₁ e sun (1x) approxisition
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
![**Transcription for Educational Website:**
---
### Problem Statement
**b) By using \( \phi(x) \), solve the given heat problem.**
**Use:**
\[
\int_0^1 x(x^2-1) \sin(n \pi x) \, dx = \frac{6(-1)^n}{\pi^3 n^3}
\]
\[
\int_0^1 \sin(n \pi x) \, dx = \frac{1-(-1)^n}{n \pi}
\]
**Answer:**
\[
U(x,t) = \frac{b}{6} x(x^2-1) + \sum_{n=1}^{\infty} B_n e^{-\frac{n^2 \pi^2 t}{b}} \sin(n \pi x)
\]
\[
B_n = 2b \int_0^1 f(x) \sin(n \pi x) \, dx
= \frac{2b \left( 1-(-1)^n \right)}{n \pi} - \frac{b(-1)^n}{\pi^3 n^3}
\]
---
### Consideration and Solution Requirements
**c) By considering that**
\[
U(x,t) = \phi(x) + \sum_{n=1}^{\infty} U_n(x,t)
\]
**Write down what \( U_n(x,t) \) is from your solution in (b). Then assume \( U_0 = b \pi t^2 \) and show that:**
\[
\left| \frac{U_2 \sin t}{U_0(x,t)} \right| \leq \frac{1}{12} e^{-3}, \quad t \geq 1/\pi^2
\]
---
### Approximation
**d) In (c) you showed that the second term was small compared to the first, so (without proof) write down the first term approximation:**
\[
U(x,t) \approx \phi(x) + B_1 e^{-\pi^2 t} \sin(\pi x)
\]
---
This transcription includes mathematical expressions and steps to solve a heat equation problem using a series solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2F37d87784-febf-42b2-88b0-58578530f401%2Fwjzhs6k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
---
### Problem Statement
**b) By using \( \phi(x) \), solve the given heat problem.**
**Use:**
\[
\int_0^1 x(x^2-1) \sin(n \pi x) \, dx = \frac{6(-1)^n}{\pi^3 n^3}
\]
\[
\int_0^1 \sin(n \pi x) \, dx = \frac{1-(-1)^n}{n \pi}
\]
**Answer:**
\[
U(x,t) = \frac{b}{6} x(x^2-1) + \sum_{n=1}^{\infty} B_n e^{-\frac{n^2 \pi^2 t}{b}} \sin(n \pi x)
\]
\[
B_n = 2b \int_0^1 f(x) \sin(n \pi x) \, dx
= \frac{2b \left( 1-(-1)^n \right)}{n \pi} - \frac{b(-1)^n}{\pi^3 n^3}
\]
---
### Consideration and Solution Requirements
**c) By considering that**
\[
U(x,t) = \phi(x) + \sum_{n=1}^{\infty} U_n(x,t)
\]
**Write down what \( U_n(x,t) \) is from your solution in (b). Then assume \( U_0 = b \pi t^2 \) and show that:**
\[
\left| \frac{U_2 \sin t}{U_0(x,t)} \right| \leq \frac{1}{12} e^{-3}, \quad t \geq 1/\pi^2
\]
---
### Approximation
**d) In (c) you showed that the second term was small compared to the first, so (without proof) write down the first term approximation:**
\[
U(x,t) \approx \phi(x) + B_1 e^{-\pi^2 t} \sin(\pi x)
\]
---
This transcription includes mathematical expressions and steps to solve a heat equation problem using a series solution.
![**Problem 24**
Consider the following heat problem in dimensionless variables:
\[ U_t = U_{xx} - bx \quad 0 < x < 1, \, t > 0 \]
\[ U(0, t) = 0, \quad U(1, t) = 0 \, , \, t > 0 \]
\[ U(x, 0) = U_0 \quad , \quad 0 < x < 1 \]
where \( b > 0 \) and \( U_0 > 0 \) are constants.
- This is the heat equation with a negative source.
(a) Derive the steady-state solution \( \phi(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2F37d87784-febf-42b2-88b0-58578530f401%2Fc7apg2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 24**
Consider the following heat problem in dimensionless variables:
\[ U_t = U_{xx} - bx \quad 0 < x < 1, \, t > 0 \]
\[ U(0, t) = 0, \quad U(1, t) = 0 \, , \, t > 0 \]
\[ U(x, 0) = U_0 \quad , \quad 0 < x < 1 \]
where \( b > 0 \) and \( U_0 > 0 \) are constants.
- This is the heat equation with a negative source.
(a) Derive the steady-state solution \( \phi(x) \).
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