Mixed Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 143

Chapter 4
Mixed Problems of Hyperbolic
Heat-Conduction Equations

We first develop the solution structure theorem for solving mixed problems of hy-
perbolic heat-conduction equations. We then apply the Fourier method of expansion
to solve one-, two- and three-dimensional mixed problems.

4.1 Solution Structure Theorem

The hyperbolic heat-conduction equation reads

a 2 u t + τ 0 u tt = Δ u + F ( M , t ) , (4.1)

or

u t
τ

A 2

0 +

u tt = Δ u + f ( M , t ) , (4.2)

where A 2 a 2 )

in one-, two- and three-dimensional space, respectively. The wave equation and the
heat-conduction equation are two special cases of the hyperbolic heat-conduction
equation at

= / τ 0 , f ( M , t )= F ( M , t ) / τ 0 and M stands for point x , ( x , y ) and ( x , y , z

→ ∞ =

τ and τ 0, respectively. A unit analysis shows that τ 0 and A 0 0

have the dimensions of time and velocity, respectively. For most materials, τ 0 has a

small positive value so u has some weak properties of waves.
Con si der m ix ed problems of hyperbolic heat-conduction equations in a closed
region . Three kinds

of linear homogeneous boundary conditions may, therefore, be written as

Ω .Let Ω = Ω ∪ ∂Ω ,where ∂Ω is the boundary surface of Ω

L ( u , u n ) | ∂Ω = 0 ,

where u n stands for the normal derivative of u .