ди = c t > 0, 0

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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The excess pore-water pressure u(x,t) in an infinite layer of clay satisfies the diffusion equation
ди
t > 0, 0<x< 4
= C
where t is the time in minutes, x is the vertical height in meters from the base clay layer and the
positive constant c is the coefficient of consolidation. There is complete drainage at the top and
bottom of the clay layer which is of thickness 4 m. The distribution of excess pore-water pressure
u(x,t) is initially Ax). The boundary value problem governing the excess pore-water pressure u(x,t) is
given by
ди
0<x< 4 and t > 0
C-
with
u(0,t)=0 and u(4,1)=0, DO
and
u(x,0)=f(x),
0<r<4
Here, f(x) is a non-trivial function.
a) Using the method of separation of variables, show that
en²n?
NIT X
00
u(x,t)= C, sin-
-t
16
n=1
4
Note: You must solve all three cases for 2.
b) Hence, find the values of the constant C, that satisfy the initial condition given that
[, 5, 0<x<3
f(x)=
|0, 3<x<4
Transcribed Image Text:The excess pore-water pressure u(x,t) in an infinite layer of clay satisfies the diffusion equation ди t > 0, 0<x< 4 = C where t is the time in minutes, x is the vertical height in meters from the base clay layer and the positive constant c is the coefficient of consolidation. There is complete drainage at the top and bottom of the clay layer which is of thickness 4 m. The distribution of excess pore-water pressure u(x,t) is initially Ax). The boundary value problem governing the excess pore-water pressure u(x,t) is given by ди 0<x< 4 and t > 0 C- with u(0,t)=0 and u(4,1)=0, DO and u(x,0)=f(x), 0<r<4 Here, f(x) is a non-trivial function. a) Using the method of separation of variables, show that en²n? NIT X 00 u(x,t)= C, sin- -t 16 n=1 4 Note: You must solve all three cases for 2. b) Hence, find the values of the constant C, that satisfy the initial condition given that [, 5, 0<x<3 f(x)= |0, 3<x<4
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