Cauchy Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 199

Finally, the solution of PDS (5.62) is

u = W ψ ( x , y , t )

ch A

2 2 2

( At ) − ( x − ξ ) − ( y − η )

1

t

e −

=

2 τ 0

ψ ( ξ , η ) d ξ d η . (5.67)

2 π A 2 2 2

( At ) − ( x − ξ ) − ( y − η )

D At

5.4.3 Solution of PDS (5.61)

By the solution structure theorem, we obtain the solution of PDS (5.61)

1 W ϕ (

t

τ 0 + ∂

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

∂ t

0

where f τ = f ( x , y , τ ) . Here the solution of

⎧
⎨

u t
τ 0 +

A 2 R 2 u tt = Δ u , × ( 0 , + ∞ ) ,

(5.68)

⎩

u ( x , y , 0 )= ϕ ( x , y ) , u t ( x , y , 0 )= 0

is

1 W ϕ (

τ 0 + ∂

u = x , y , t )

∂ t

1
2

1

τ 0 + ∂

t
2τ 0

e −

=

2 π A ∂ t

ch A

2 2 2

( At ) − ( x − ξ ) − ( y − η )

· ϕ ( ξ , η ) d ξ d η . (5.69)

2 2 2

( At ) − ( x − ξ ) − ( y − η )

D At

The solution of

⎧
⎨

u t
τ 0 +

A 2 R 2 u tt = Δ u + f ( x , y , t ) , × ( 0 , + ∞ ) ,

⎩

u ( x , y , 0 )= 0 , u t ( x , y , 0 )= 0