Cauchy Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 216

By the solution structure theorem, we can first seek the solution of
⎧
⎨

u t
τ 0 +

A 2

u tt = Δ u , 0 < r < + ∞ , 0 < t ,

(5.89)

⎩

u ( r , 0 )= 0 , u t ( r , 0 )= ψ ( r ) .

To eliminate the u t -term, consider a function transformation

t
2τ 0

e − u ( r , t )= v ( r , t ) ,

PDS (5.89) is thus transformed to
v tt

A 2 c 2 v

1
2

= Δ v + , c =

τ 0 ,

0 < r < + ∞ , 0 < t ,

(5.90)

( , )= , ( , )= ψ ( ) .

v r 0 0 v t r 0 r

Applying a spherical Bessel transformation to (5.90) and using F j 0 [ Δ v ]=

2 v

− ω ( ω , t ) yields an initial-value problem of the ordinary differential equation

v tt

A 2

c 2 v

2

( ω , )+ − ( ω , )= ,

t ω t 0

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

Its solution can be readily obtained as

sin √ A 2 c 2 t ψ ( ω ) ¯ ω

2

−

√ A 2

v ( ω , t )= . (5.91)

2

c 2

ω −

Its inverse spherical Bessel transformation will lead to the solution of PDS (5.90),
and consequently, the solution of PDS (5.89)

2
π

+ ∞

2 v

u ( r , t )= W ψ ( r , t )= ω ( ω , t ) j 0 ( ω r ) d ω

0

sin √ A 2 c 2 t

2
π

+ ∞

ψ ( ω )

¯

2

−

ω

ω sin ω r

= √ A 2 d ω . (5.92)

r

ω

2

− c 2

0

Finally, the solution of PDS (5.88) is, by the solution structure theorem,

1 W ϕ (

t

τ 0 + ∂

u = r , t )+ W ψ ( r , t )+ W f τ ( r , t − τ ) d τ ,

∂ t

0

where f τ = f ( r , τ ) .