Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 83

Finally, the solution of PDS (2.54) follows from the solution structure theorem,

t

= ∂
∂

u

t W Φ +

W Ψ ( r , θ , ϕ , t )+ W F τ ( r , θ , ϕ , t − τ ) d τ ,

0

where F τ = F ( r , θ , ϕ , τ ) .

Remark 1. We can apply the Fourier method of expansion to easily solve 27 mixed
problems in a cylindrical domain by using the eigenfunctions in Table 2.1 and in
Sect. 2.5.2.

Remark 2. Mixed problems of wave equations can also be solved using the method
of Laplace transformation, which is discussed in Appendix B.

2.7 Methods of Solving One-Dimensional Cauchy Problems

Clearly, the solution structure theorem is also valid for Cauchy problems. In this
section, we consider the PDS
u tt

a 2 u xx

= + f ( x , t ) , − ∞ < x < + ∞ , 0 < t ,

(2.60)

u ( x , 0 )= ϕ ( x ) , u t ( x , 0 )= ψ ( x ) .

In particular, we discuss methods of solving PDS (2.60) and examine properties of
the solution.

2.7.1 Method of Fourier Transformation

The Fourier transformation and the Laplace transformation are two important inte-
gral transformations and find applications in a variety of fields. Readers are referred
to Appendix B for a discussion of these two transformations. Here we apply the
Fourier transformation to solve PDS. This approach is called the method of Fourier
transformation and different from the Fourier method of expansion discussed be-
fore.
The solution structure theorem has reduced the problem of finding a solution
of PDS (2.60) to seeking W ψ ( x , t ) , the solution for the case ϕ = f = 0. Applying

a Fourier transformation with respect to x to u ( x , t ) in PDS (2.60) yields

F a 2 u xx

2 u 2 u

a 2 = ( i ω ) ( ω , t )= − ( ω a ) ( ω , t ) ,

i.e.

2 u

u tt ( ω , t )+( ω a ) ( ω , t )= 0 ,