Page 417 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 4. Find the temperature u(x, t) in a slab whose ends x = 0 and x = L are kept at temperature
zero and whose initial temperature F(x) is given by

L
x
f ( ) A when 0 x
2
1
0 when L x . L
2
Answers: Self Assessment

1. l 2 m 2 n 2 k 2

n ct n x
t
x
2. u ( , ) D n sin L sin L
n 1
where
L
2 n x
x
D g ( )sin dx
n
n c c
0
m x n y rt
x
y
t
4. V ( , , ) A mn sin sin e
l l
where
rl 2 2 m 2 n 2 and
4 l l m x n y
y
A mn 2 f ( , )sin sin dxdy
x
l x 0 y 0  
23.6 Further Readings

Books H.T.H. Piaggio, Differential Equation
L.N. Sneddon, Elements of Partial Differential Equations

Louis A. Pipes, and L. R. Harnvill, Applied Mathematics for Engineers and
Physicists

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