Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 124

Remark 2. The nominal solution contains an exponentially decaying factor in each
term. This relaxes the requirement for the smoothness of the initial distribution ϕ ( x )

in the classical solution. The constraint on ϕ ( x ) is dependent on the existence of the

integral of the product of ϕ ( x ) with the sine function over the domain [ 0 , l ] ,i.e.the

condition for using the Fourier series to expand.

3.3.2 Uniqueness

The proof of uniqueness and stability is based on the following important physi-
cal phenomenon. Consider one-dimensional heat conduction in a rod without any
internal heat source. Let M be the maximum temperature of two ends and initial
distribution. The temperature is then less than or equal to M at any point in the rod
and at any time instant. This can be physically understood and can also be proven
mathematically.

Theorem. Assume that u ( x , t ) is continuous in the domain D :0 ≤ x ≤ l , 0 ≤ t ≤ T

a 2 u xx inside D .The u ( T is any positive constant; Fig. 3.1) and satisfies u t = ( x , t )

can then take its maximum or minimum value only at the boundary Γ : x = 0 , x = l ,

t = 0. This is called the extremum principle of heat conduction .

Proof. Without loss of generality, we consider the case of maximum value only. We
must then start from the given condition to arrive at

max
D

u ( x , t )= max

Γ

u ( x , t ) (3.21)

, M

M . There would exist one point

Let M = max

D

u ( x , t ) = max

Γ

u ( x , t ) . Suppose M >

( x 0 , t 0 ) inside D such that

M

max
D

u ( x , t )= u ( x 0 , t 0 )= M > .

Define a new function v ( x , t ) by

Fig. 3.1 Domain D