Cauchy Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 223

where

π
0 ( −

1
2

e i r cos ϕ cos θ r cos sin 2 I 1 = 1 ) J 1 ( r sin ϕ sin θ ) ϕ θ d θ ,

π
0 ( −

1
2

e i r cos ϕ cos θ r cos

I 2 = i ) J 0 ( r sin ϕ sin θ ) θ sin θ sin ϕ d θ .

Since d ( xJ 1 ( x )) = xJ 0 ( x ) d x ,

we obtain

1
r sin

[(

θ )

(

θ )]

d r sin ϕ sin J 1 r sin ϕ sin

ϕ

= r sin ϕ sin θ J 0 ( r sin ϕ sin θ ) cos θ d θ .

Thus

π

i
2 r sin

e i r cos ϕ cos θ d

I 2 = − [( r sin ϕ sin θ ) J 1 ( r sin ϕ sin θ )]

ϕ

0

e i r cos ϕ cos θ (

i
2 r sin

π

= − r sin ϕ sin θ ) J 1 ( r sin ϕ sin θ )

ϕ 0

π
0 (

e i r cos ϕ cos θ ( − − r sin ϕ sin θ ) J 1 ( r sin ϕ sin θ ) i r cos ϕ sin θ ) d θ

π

1
2

e i r cos ϕ cos θ r cos sin 2 = J 1 ( r sin ϕ sin θ ) ϕ θ d θ = − I 1 .

0

d I
d ϕ =

Hence 0, so that I is independent of ϕ .Also,

π

1
2

sin r
r

e i r cos θ sin I ( r , ϕ )= I ( r , 0 )= θ d θ = .

0

Finally, we obtain the integral formula of Bessel function of order zero

π

1
2

sin r
r

e i r cos ϕ cos θ sin

J 0 ( r sin ϕ sin θ ) θ d θ = . (5.110)

0

Consider now the varia ble transformation r cos ϕ = − A ω t , r sin ϕ = i ct ,

= √ −

At cos θ = β ,wherei 1.

Thus