Cauchy Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 200

is

t

u = W f τ ( x , y , t − τ ) d τ

0

t

1

t − τ

2τ 0 d

e −

= τ

2 π A 0

ch A A 2

( − τ )

2

− ( − ξ )

2

− ( − η )

2

t x y

A 2

· f ( ξ , η , τ ) d ξ d η . (5.70)

( t − τ )

2

− ( x − ξ )

2

− ( y − η )

2

D A ( t − τ )

A superposition of Eqs. (5.67) and (5.69) forms the counterpart of the Poisson for-
mula of two-dimensional wave equations. Similarly, Eq. (5.70) is the counterpart of
the Kirchhoff formula.
Remark. The unit of u is

Θ in Eqs. (5.67), (5.69) and (5.70). In Eq. (5.69), for

example

T − 1 LT − 1 [ c ]= , [ A ]= ,

c
A

2

2

2

( At ) − ( x − ξ ) − ( y − η ) = 1

and

⎡ ⎤

1
2

1
A τ 0 + ∂

ch

⎣ ⎦

[ u ]=

2 ϕ ( ξ , η )

d ξ d η

∂ t

2 2

( At ) − ( x − ξ ) − ( y − η )

TL − 1 T − 1 L − 1 L 2 = · · · Θ · = Θ .

5.4.4 Verification of CDS

We demonstrate here that the u in Eqs. (5.67), (5.69) and (5.70) satisfies the initial
conditions.

1. The u in Eq. (5.67) By Eq. (5.67), it is clear that u ( x , y , 0 )= 0 . For conciseness,

denote the integrand of double integral in Eq. (5.67) by K ( ψ ) . After using the

generalized mean value theorem for integral, we have

ch
D At

d ξ d η

K ( ψ ) d σ = ψ

2 . 2 2

( ) − ( − ξ ) − ( − η )

At x y

D At