Page 403 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Let the solution of heat equation be
V , x t XT ...(xi)

where X is a function of x only and T that of t only.
Substituting this solution in (i) as we have done in Case I, we get the following three solutions:

(i) X Ax B T C

(ii) X Ae x Be x T Ce 2 2
C t
(iii) X A cos x B sin x T Ce 2 2
C t
Hence (ii) does not constitute the solution of (i), since in this case V x ,t XT increase indefinitely
with time, which is not the case. (iii) is also inadequate to give complete solution since in this
case temps tends to zero as t tends to infinity. Hence the complete solution must be a compilation
of (i) and (iii) Therefore
V , x t V x V t , x t ...(xii)
3
where V (x) denotes the temperature distribution after a long period of time when the rod has
s
reached a steady state of temperature distribution, V (x, t) denotes the transient effects which die
t
down with the passage of time. These two must be the solutions of the types (i) and (iii)
respectively.
It is obvious that when the end x = 0 is maintained at temperature V = 0 and the end x = L at
V = t ultimately there will be uniform gradation of temperature.
0
t
Therefore V x 0 . x
s
L
(xii) then becomes

t
V , x t 0 x V t , x t
L
with the help of (viii), (ix) and (x) the boundary and initial conditions for V x ,t are as follows:
t
V 0,t V t 0,t 0 ...(xiii)

V , L t t 0 V t , L t t 0

or V t , L t 0

t
and V x ,0 0 x V x ,0 t i ...(xiv)
t
L
t 0
or V x ,0 t i . x ...(xv)
t
L
Therefore let us take

2 2
V , x t A 'cos x B 'sin x e c t ...(xvi)
t

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