Page 395 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 2
d 2 0. ...(ix)
d 2

2
d T 2 2
c T 0. ...(x)
dt 2
Equation (ix) has the solution of the form

Ae i ...(xi)
Substituting new variable s = r in equation (vii), we have

2
d R 1 dR 2
1 R 0
ds 2 s ds s 2
which is Bessel s equation whose general solution is
R C J s C Y s
2
1
or R C J r C Y r
2
1
But since the deflection of the membrane is always finite while Y becomes infinite as r 0
hence we cannot use Y and must choose C = 0.
2
Now using boundary condition (iii)

u a , ,t R a T t

R a 0

Otherwise if 0 or T t 0, u 0

R a GJ a 0

or J a 0 ...(xii)

Let 1 , 2 be the positive root of (xii),
The corresponding solution of (viii)

T A n cose nt B n sinC nt
Thus we get the general solution as

u r , ,t A n cosC n t B n sin n t e i J n r ...(xiii)
1 n 1
which satisfies the boundary condition (iii).
Considering the solution of the wave equation (ii) which are radially symmetric i.e. when the
solution is independent of , we get the general solution as

u , r t A n cosC n t B n sinC n t J 0 n r ...(xiv)
n 1

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