Let u be the solution of the initial boundary value problem for the Heat Equation, дли(t, х) %3D 2 д; и(t, х), 1€ (0, о), х€ (0,5); with non-homogenous Dirichlet boundary conditions u(t, 0) = 5 and u(t, 5) = 3, and with initial condition (0). e 0, 5, u(0, x) = f(x) = 3, The solution u of the problem above, with the conventions given in class, has the form 00 u(t, x) = uE(x) +E en Un(t) wn(x), n=1 where ug is the equilibrium solution with the boundary conditions above and the functions vn, w, satisfy the the normalization conditions v„(0) = 1 and = 1. Find the functions uE, Un, Wn, and the constants Cn. ug(x) = 2/5pi Σ vn(t) = e^-((2n^2pi^2)/25t) Σ w„(x) = sin((pixn)/5) Σ Cn = (2(-2cos((pin)/2)+5-3(-1)^n))/(pin) Σ
Let u be the solution of the initial boundary value problem for the Heat Equation, дли(t, х) %3D 2 д; и(t, х), 1€ (0, о), х€ (0,5); with non-homogenous Dirichlet boundary conditions u(t, 0) = 5 and u(t, 5) = 3, and with initial condition (0). e 0, 5, u(0, x) = f(x) = 3, The solution u of the problem above, with the conventions given in class, has the form 00 u(t, x) = uE(x) +E en Un(t) wn(x), n=1 where ug is the equilibrium solution with the boundary conditions above and the functions vn, w, satisfy the the normalization conditions v„(0) = 1 and = 1. Find the functions uE, Un, Wn, and the constants Cn. ug(x) = 2/5pi Σ vn(t) = e^-((2n^2pi^2)/25t) Σ w„(x) = sin((pixn)/5) Σ Cn = (2(-2cos((pin)/2)+5-3(-1)^n))/(pin) Σ
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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Transcribed Image Text:Let u be the solution of the initial boundary value problem for the Heat Equation,
дли(t, х) %3D 2 д; и(t, х), 1€ (0, о), х€ (0,5);
with non-homogenous Dirichlet boundary conditions u(t, 0) = 5 and u(t, 5) = 3, and with initial condition
(0).
e 0,
5,
u(0, x) = f(x) =
3,
The solution u of the problem above, with the conventions given in class, has the form
00
u(t, x) = uE(x) +E en Un(t) wn(x),
n=1
where ug is the equilibrium solution with the boundary conditions above and the functions vn, w, satisfy the the normalization conditions
v„(0) = 1 and
= 1.
Find the functions uE, Un, Wn, and the constants Cn.
ug(x) = 2/5pi
Σ
vn(t) = e^-((2n^2pi^2)/25t)
Σ
w„(x) = sin((pixn)/5)
Σ
Cn = (2(-2cos((pin)/2)+5-3(-1)^n))/(pin)
Σ
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