Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 44

Therefore, u 3 also satisfies the boundary and initial conditions of (2.3), so that it
is indeed the solution of (2.3).
3. Since PDS (2.4) is linear, the principle of superposition is valid. Applying this
principle to (2.4) shows that u u 3 is the solution of (2.4).

Remark 1. The solution structure theorem reduces the task of finding the solution
of (2.4) to that of finding W ψ (

= u 1 + u 2 +

, the solution of (2.2). Problem (2.4) covers a
large number of problems by noting that:

M , t )

1. the spatial dimensions can be one, two, three or even higher;
2. one or two of the three functions f

( , )

M t (the source term), ϕ ( M ) and ψ ( M ) (the

initial values) can be vanished; and
3. the boundary conditions may have variety of forms.

Remark 2 . Mathematics is a discipline of high abstraction and wide application.
It enjoys the beauty of conciseness. Here, W ( M , t ) can be viewed as a kind of

function structure, similar to sin and ln .Once W ψ ( M , t ) is available, W ϕ ( M , t )

and W f τ ( should be regarded as known functions without the necessity of

writing them explicitly. After u

M , t − τ )

, the solution of (2.2) is available, we
may apply the solution structure theorem to readily write the solutions of (2.1),
(2.3) and (2.4) by using W . Therefore, it is of crucial importance to solve (2.2).
Remark 3. In developing the solution structure theorem, we assume that:

= W ψ ( M , t )

1. for all PDS there exist classical solutions of separable-variable type;
2. all derivatives are right-continuous so that all equations are also valid at t = 0;

and
3. all the high-order mixed derivatives are continuous. When we apply the solution
structure theorem, however, the solution sometimes only refers to the nominal
solution.

This remark is valid for all solution structure theorems in the topic unless otherwise
stated.

Remark 4. Without loss of generality, PDS (2.1)-(2.4) are all with homogeneous
boundary conditions. To apply the solution structure theorem for problems with
nonhomogeneous boundary conditions, the homogenization of boundary conditions
should first be made by some appropriate function transformations.

Remark 5. The solution structure can have other forms. Here, we emphasized the
importance of solving (2.2). In fact, after any one of three PDS (2.1)-(2.3) is solved,
the solutions of the other two and PDS (2.4) are readily available.