Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 115

and

u 2 | t = 0 = 0 .

Hence, u 2 also satisfies the boundary and initial conditions of (3.2), so that u 2 is
indeed the solution of (3.2).
2. Since PDS (3.3) is linear, the principle of superposition is valid. Applying this
principle to (3.3) shows that u 3 is the solution of (3.3).
Remark 1. All Remarks made in Section 2.1 are also valid here.

Remark 2. The solution structure theorem reduces the development of solutions
for mixed problems to solving PDS (3.1). Similar to the procedure in Chapter 2,
PDS (3.1) can be solved by the Fourier method or the method of separation of
variables. A comparison of PDS (3.1) with PDS (2.2) in Section 2.1 reveals the
difference between the two: PDS (3.1) here only involves the first derivative of u
with respect to t and requires only one initial condition. This only varies the T

-
equation in the solution of separable variables and reduces the second-order ODE
for PDS (2.2) in Section 2.1 to a first-order ODE. For PDS (2.2) in Section 2.1,

( t )

T ( a 2 T t )+ λ ( t )= 0

with the general solution

C 1 cos √ C 2 sin √ T ( t )= λ at + λ at .

Here,

T ( a 2 T t )+ λ ( t )= 0

with a general solution

Ce − λ a 2 t

T ( t )= .

Therefore, we can readily write out solutions of mixed problems of heat-conduc-
tion equations based on the solutions of wave equations in Chapter 2, simply by
changing the trigonometric functions of t to an exponential functions of t .
Remark 3. The solution structure theorem is also valid for Cauchy problems. How-
ever, for Cauchy problems the structure of W ϕ (

M , t ) differs from that for the mixed

problems.