Cauchy Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 229

⎡

br I 1 b r 2

⎤

At

( )

2

−

At

1
2 A

⎣

r r ) d r +

⎦

¯

2 At ¯

= ( ψ ( ψ ( At )

At )

2

− r 2

− At

⎛
⎝

⎞
⎠

bI 1 b ( r 2

At

At )

2

−

1

M ) d r = ψ ( d S

8 π A

r 2

( At )

2

−

− At

S M
r

1

M ) + ψ ( d S

4 π A 2 t

S r

⎡ ⎛ ⎞

bI 1 b ( r 2

At

At )

2

−

1 1

2

⎣ ⎝ ⎠

M ) d r

= ψ ( d S

4 π A

2

r 2 ( At ) −

− At

S r

⎤

1
At

⎦ .

M ) + ψ ( d S

S r

Thus the solution of

⎧
⎨

u t
τ 0 +

A 2 R 3

u tt = Δ u , × ( 0 , + ∞ ) ,

(5.119)

⎩

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

is

⎡

1 1

At

t
2τ 0

⎣

A e −

M ) u ( M , t )= W ψ ( M , t )= ψ ( d S

4 π

S At

(5.120)

⎛
⎝

⎞
⎠

⎤
⎦ .

bI 1 b ( r 2

At

At )

2

−

1
2

M ) d r + ψ ( d S

r 2

( At )

2

−

− At

S M
r

Finally, the solution of PDS (5.115) comes from the solution structure theorem,

1 W ϕ (

t

τ 0 + ∂

u ( M , t )= M , t )+ W ψ ( M , t )+ W f τ ( M , t − τ ) d τ , (5.121)

∂ t 0

where f τ = f ( M , τ ) .