Cauchy Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 224

t

t 2 A 2 c 2

r 2

2 2 2

2

c 2 =( − A ω t ) +( i ct ) = ω − or r = ( A ω ) − ,

d At

e i r cos ϕ cos θ = e − i A ω t cos θ = e − iωβ ,

sin θ d θ = − ,

At 2

1

sin θ = − cos 2 θ = 1 − , ( sin θ ≥ 0 , θ ∈ [ 0 , π ])

Substituting these and J 0 ( i x )= I 0 ( x ) into Eq. (5.110) yields a very important for-

mula for Fourier transformation
sin t

c 2

⎛

2 ⎞
⎠ e − iωβ d

A

At

(

ω )

2

−

A

1
2 A

⎝ c

(

t 2

= I 0 − β . (5.111)

A ω )

2

− c 2

− At

5.8.2 Fourier Transformation for Three-Dimensional Problems

Consider the PDS

⎧
⎨

u t
τ 0 +

A 2 R 3

u tt = Δ u , × ( 0 , + ∞ ) ,

(5.112)

⎩

u ( M , 0 )= 0 , u t ( M , 0 )= ψ ( M ) ,

R 3 . It is transformed to, by the function transfor- where M stands for point ( x , y , z ) ∈

t
2τ 0 ,

e − mation u ( M , t )= v ( M , t )

v tt =

A 2 c 2 v

1

R 3

Δ v + , c =

τ 0 ,

× ( 0 , + ∞ ) ,

2

(5.113)

v ( M , 0 )= 0 , v t ( M , 0 )= ψ ( M ) .

Applying a triple Fourier transformation to PDS (5.113) yields an initial-value prob-
lem of the ordinary differential equation
v tt ( ω ,

A 2

2

c 2

t )+( ω − ) v ( ω , t )= 0 ,

v ( ω , 0 )= 0 , v t ( ω , 0 )= ψ ( ω ) , ¯

2 1 2 3 . Its solution is

where ¯ ψ ( ω )= F [ ψ ( M )] and ω = ω + ω + ω

sin t c 2

2

( A ω ) −

v ( ω , t )= ψ ( ω ) . ¯

( A ω )

2

− c 2