Page 401 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 2
V 2 V
c ...(i)
t x 2
Equation (i) is known as one dimensional heat equation.
Now we shall find out the solution of equation (i) under different initial and boundary conditions.
Case I: Let L is length of the rod whose ends are kept at zero temperature and whose initial
temperature is f(x).
The boundary conditions are

V 0,t 0 ...(ii)

V , L t 0 for all t ...(iii)
The initial conditions are

V x ,0 f x 0 x L ...(iv)
Let the solution of equation (i) is of the form

V , x t X x T t

V XT say ...(v)

where X is a function of x only and T is that of t only.
Substituting this solution in equation (i), we get
2
1 d X 1 dT
2
X dx 2 c T dt
since L.H.S. is a function of x and R.H.S. is a function of t, hence both sides will be equal only
when both reduces to same constant. Therefore

2
1 d X 1 dT 0 or or 2
2
2
X dx 2 c t dt
and hence in these three cases, we have
2
d X dT
(a) 2 0 and 0 ,
dx dt
2
d X 2 dT 2 2
(b) 2 X 0 and c t 0,
dx dt
2
d X 2 dT 2 2
(c) 2 X 0 and c t 0
dx dt
The general solution in these three cases are

(i) X Ax B T c

2
(ii) X Ae x Be x T c e x 2 c t
(iii) X A cos x B sin x , T Ce 2 2
c t

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