5. Let L>1/2. Consider the diffusion equation Uų = Kurr, -L 0, u(-L,t) = u(L, t) = 0, u(x, 0) = 4 3 By using the maximum principle to show 1 (a) Ju(x, t)| < for -L
5. Let L>1/2. Consider the diffusion equation Uų = Kurr, -L 0, u(-L,t) = u(L, t) = 0, u(x, 0) = 4 3 By using the maximum principle to show 1 (a) Ju(x, t)| < for -L
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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I need help on 5
![1. Use the coordinate method to solve \( 2u_x - u_y + (x + 2y)u = 0 \) with \( u(x, 0) = e^{-2x^2/5} \).
2. Show that the steady state of the diffusion equation
\[
\begin{cases}
u_t = Ku_{xx}, & 0 < x < L, \, t > 0, \\
u_x(0, t) = u_x(L, t) = 0, \\
u(x, 0) = \phi(x)
\end{cases}
\]
is given by
\[
u_s(x) = \frac{1}{L} \int_0^L \phi(x) \, dx.
\]
3. Classify equation \( u_{xx} + u_{xt} - 20u_{tt} = 0 \), and solve the equation with initial conditions: \( u(x, 0) = \phi(x) \) and \( u_t(x, 0) = \psi(x) \).
4. Find the general solution of \( w_{tt} = w_{xx} + \sin(x + t) \) by using the coordinate method.
5. Let \( L > 1/2 \). Consider the diffusion equation
\[
\begin{cases}
u_t = Ku_{xx}, & -L < x < L, \, t > 0, \\
u(-L, t) = u(L, t) = 0, \\
u(x, 0) = \frac{x}{4} - \frac{x^3}{3}
\end{cases}
\]
By using the maximum principle, show:
(a) \( |u(x,t)| \le \frac{1}{12} \) for \( -L \le x \le L \) and \( t \ge 0 \);
(b) \( u(-x,t) = -u(x,t) \) for \( -L \le x \le L \) and \( t \ge 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe81da00a-4cea-4000-a024-f288b328857b%2Fa7e96ade-98a6-4250-a0a7-e9f60af71492%2Fj0p7bim_processed.png&w=3840&q=75)
Transcribed Image Text:1. Use the coordinate method to solve \( 2u_x - u_y + (x + 2y)u = 0 \) with \( u(x, 0) = e^{-2x^2/5} \).
2. Show that the steady state of the diffusion equation
\[
\begin{cases}
u_t = Ku_{xx}, & 0 < x < L, \, t > 0, \\
u_x(0, t) = u_x(L, t) = 0, \\
u(x, 0) = \phi(x)
\end{cases}
\]
is given by
\[
u_s(x) = \frac{1}{L} \int_0^L \phi(x) \, dx.
\]
3. Classify equation \( u_{xx} + u_{xt} - 20u_{tt} = 0 \), and solve the equation with initial conditions: \( u(x, 0) = \phi(x) \) and \( u_t(x, 0) = \psi(x) \).
4. Find the general solution of \( w_{tt} = w_{xx} + \sin(x + t) \) by using the coordinate method.
5. Let \( L > 1/2 \). Consider the diffusion equation
\[
\begin{cases}
u_t = Ku_{xx}, & -L < x < L, \, t > 0, \\
u(-L, t) = u(L, t) = 0, \\
u(x, 0) = \frac{x}{4} - \frac{x^3}{3}
\end{cases}
\]
By using the maximum principle, show:
(a) \( |u(x,t)| \le \frac{1}{12} \) for \( -L \le x \le L \) and \( t \ge 0 \);
(b) \( u(-x,t) = -u(x,t) \) for \( -L \le x \le L \) and \( t \ge 0 \).
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