Let us consider the following heat equation problem | ди --k=f in (0, L)× (0,T), ôx? a'u at (1) u(0,t) = a(t),u(1,t)= B(t),0
Let us consider the following heat equation problem | ди --k=f in (0, L)× (0,T), ôx? a'u at (1) u(0,t) = a(t),u(1,t)= B(t),0
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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![Let us consider the following heat equation problem
| ди
--k=f in (0, L)× (0,T),
ôx?
a'u
at
(1)
u(0,t) = a(t),u(1,t)= B(t),0<t<T,
u(x,0) = u,(x),0< x< L,
where in (1): (i) k is a positive constant. (ii)fand uo are given functions of (x,f) and x, respectively.
(iii) The functions a and ßare given, differentiable on (0, T) and continuous at t= 0. Let define
the functions uapand w by
(2)
U(x,f) = B(t) +[a(t) – B(1)],*,v(x,t) =[0,L]×[0,T'),
(t)].
(3)
w-и — Иар,
respectively.
da
dв
1) Find the variant of system (1) verified by function w (denote
and
- by å and ß ,
dt
dt
respectively).
2) Take advantage of question 1) to solve problem (1) using a sine expansion, if: L = 1, k = 1,
Ax, t) = xt, uo(x) = sinx, a(t) = t and A(t) = e".](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd3662bb-e706-4e7e-b70a-a9d3a5ea138f%2Fffa67c24-abec-41ca-90d8-9074edaaca95%2F3mmv96.png&w=3840&q=75)
Transcribed Image Text:Let us consider the following heat equation problem
| ди
--k=f in (0, L)× (0,T),
ôx?
a'u
at
(1)
u(0,t) = a(t),u(1,t)= B(t),0<t<T,
u(x,0) = u,(x),0< x< L,
where in (1): (i) k is a positive constant. (ii)fand uo are given functions of (x,f) and x, respectively.
(iii) The functions a and ßare given, differentiable on (0, T) and continuous at t= 0. Let define
the functions uapand w by
(2)
U(x,f) = B(t) +[a(t) – B(1)],*,v(x,t) =[0,L]×[0,T'),
(t)].
(3)
w-и — Иар,
respectively.
da
dв
1) Find the variant of system (1) verified by function w (denote
and
- by å and ß ,
dt
dt
respectively).
2) Take advantage of question 1) to solve problem (1) using a sine expansion, if: L = 1, k = 1,
Ax, t) = xt, uo(x) = sinx, a(t) = t and A(t) = e".
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