Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 113

Chapter 3
Heat-Conduction Equations

We first develop the solution structure theorem for mixed problems of heat-conduc-
tion equations, followed by methods of solving one-, two- and three-dimensional
mixed problems. For conciseness, we directly borrow the results in Chapter 2 for
developing the solutions of heat-conduction equations. Emphasis is also placed on
the difference between wave equations and heat-conduction equations. Finally, we
discuss methods of solving one-, two- and three-dimensional Cauchy problems.

3.1 The Solution Structure Theorem For Mixed Problems

Consider mixed problems of three-dimensional heat-conduction equations in a clos-
ed region

¯

.Let ¯

. Three kinds
of linear homogeneous boundary conditions can, therefore, be written as
L u

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

∂Ω =

, ∂

u

0 ,

∂ n

which refers to the boundary conditions at two ends for one-dimensional cases.
The solution structure theorem describes the relation among solutions of the fol-
lowing three PDS,

⎧
⎨

a 2 u t = Δ u , Ω × ( 0 , + ∞ ) ,

L u

∂Ω =

, ∂

u

(3.1)

0 ,

⎩

∂ n

( , )= ϕ ( ) ,

u M 0 M