Cauchy Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 209

5.6 Comparison of Fundamental Solutions of Classical
and Hyperbolic Heat-Conduction Equations

In this section we analyze and compare fundamental solutions of classical and hy-
perbolic heat-conduction equations by using the two-dimensional case as the ex-
ample. Such an analysis and comparison is useful for revealing what fundamental
properties of the classical equation are preserved in its hyperbolic version, what the
improvements from the classical equation to the hyperbolic one are and what are the
undesirable features of hyperbolic heat-conduction equations.

5.6.1 Fundamental Solutions of Two Kinds
of Heat-Conduction Equations

Classical Heat-Conduction Equation

By its definition, the fundamental solution of classical heat-conduction equations
satisfies (Eq. (3.54) in Section 3.5),
u t =

a 2 R 2

Δ u + δ ( x − x 0 , y − y 0 , t − t 0 ) , × ( 0 , + ∞ ) ,

u ( x , y , 0 )= 0 .

Its solution reads

t

u 1 ( x , y , t )= d τ V ( x , ξ , t − τ ) V ( y , η , t − τ ) δ ( ξ − x 0 , η − y 0 , τ − t 0 ) d ξ d η

0

R 2

e − ( x − x 0 ) 2 +( y − y 0 ) 2

1

4 a 2

= V ( x , x 0 , t − t 0 ) V ( y , y 0 , t − t 0 )=

( t − t 0 )

. (5.77)

4 a 2

π ( t − t 0 )

Hyperbolic Heat-Conduction Equation

For hyperbolic heat-conduction equations, the fundamental solution is the solution
of

u t

a 2 R 2

+ τ = + δ ( − , − , − ) , × ( , + ∞ ) ,

0 u tt Δ u x x 0 y y 0 t t 0 0

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

or

⎧
⎨

u t
τ

+ δ (

x − x 0 , y − y 0 , t − t 0 )

A 2 R 2

0 +

u tt = Δ u , × ( 0 , + ∞ ) ,

τ

0

⎩

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