Consider the partial differential equation du 10²u Ət 40x²¹ together with the boundary conditions u(0, t) = 0 and u(, t) = 0 for t 0 and the initial condition u(x,0) = x( ½ − x) for 0 < x < 1/1. (a) If n is a positive integer, show that the function 2 Un(x, t) = e¯n²t sin(2nx), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is u(x, t) = Σ B₂e-n²t sin (2nx). n=1 Write down (but do not evaluate) an integral that can be used to determine the constants Bn. =
Consider the partial differential equation du 10²u Ət 40x²¹ together with the boundary conditions u(0, t) = 0 and u(, t) = 0 for t 0 and the initial condition u(x,0) = x( ½ − x) for 0 < x < 1/1. (a) If n is a positive integer, show that the function 2 Un(x, t) = e¯n²t sin(2nx), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is u(x, t) = Σ B₂e-n²t sin (2nx). n=1 Write down (but do not evaluate) an integral that can be used to determine the constants Bn. =
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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![### Partial Differential Equation Problem
Consider the partial differential equation
\[
\frac{\partial u}{\partial t} = \frac{1}{4} \frac{\partial^2 u}{\partial x^2},
\]
together with the boundary conditions \( u(0, t) = 0 \) and \( u\left(\frac{\pi}{2}, t\right) = 0 \) for \( t \geq 0 \) and the initial condition \( u(x, 0) = x\left(\frac{\pi}{2} - x\right) \) for \( 0 < x < \frac{\pi}{2} \).
#### (a)
If \( n \) is a positive integer, show that the function
\[
u_n(x, t) = e^{-n^2 t} \sin(2nx),
\]
satisfies the given partial differential equation and boundary conditions.
#### (b)
The general solution of the partial differential equation that satisfies the boundary conditions is
\[
u(x, t) = \sum_{n=1}^{\infty} B_n e^{-n^2 t} \sin(2nx).
\]
Write down (but do not evaluate) an integral that can be used to determine the constants \( B_n \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25020e6a-8b25-4818-b603-e6c17a0cb59a%2F2efbd5e5-80c3-4a67-8d9e-a5af201124cd%2Fz8hz07_processed.png&w=3840&q=75)
Transcribed Image Text:### Partial Differential Equation Problem
Consider the partial differential equation
\[
\frac{\partial u}{\partial t} = \frac{1}{4} \frac{\partial^2 u}{\partial x^2},
\]
together with the boundary conditions \( u(0, t) = 0 \) and \( u\left(\frac{\pi}{2}, t\right) = 0 \) for \( t \geq 0 \) and the initial condition \( u(x, 0) = x\left(\frac{\pi}{2} - x\right) \) for \( 0 < x < \frac{\pi}{2} \).
#### (a)
If \( n \) is a positive integer, show that the function
\[
u_n(x, t) = e^{-n^2 t} \sin(2nx),
\]
satisfies the given partial differential equation and boundary conditions.
#### (b)
The general solution of the partial differential equation that satisfies the boundary conditions is
\[
u(x, t) = \sum_{n=1}^{\infty} B_n e^{-n^2 t} \sin(2nx).
\]
Write down (but do not evaluate) an integral that can be used to determine the constants \( B_n \).
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