Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 136

3.5 Multiple Fourier Transformations for Two-
and Three-Dimensional Cauchy Problems

3.5.1 Two-Dimensional Case

Find the solution of

u t =

a 2 R 2 Δ u , × ( 0 , + ∞ ) ,

(3.50)

u ( x , y , 0 )= ϕ ( x , y ) ,

where R 2

× ( 0 , + ∞ ) stands for − ∞ < x , y < + ∞ . t ∈ ( 0 , + ∞ ) .

Solution. Apply a double Fourier transformationwith respect to x and y to PDS (3.50)
to obtain

2 u

u t +( ω a ) = 0 , u ( ω , 0 )= ϕ ( ω ) , ¯ (3.51)

2 1 2 (see Appendix B). The solution of

where u ( ω , t )= F [ u ( x , y , t )] , ω = ω + ω

Eq. (3.51) is

2 t 2 t

e − ( ω a ) e − ( ω a )

¯

u = A ( ω ) = ϕ ( ω ) , (3.52)

where ¯

ϕ ( ω )= F [ ϕ ( x , y )] . It follows from Section 3.4.1 that

F − 1 e − ( ω 1 a )

2 t

F − 1 e − ( ω 2 a )

2 t

1
2 a √ π

x 2
4 a 2 t

1
2 a √ π

y 2
4 a 2 t

t e − t e −

= , = ,

F − 1 e − ( ω a )

2 t

F − 1 e − ( ω

a 2 t

x 2 y 2

4 a 2 t

1
4 a 2

+

2
1 + ω

2
2 )

t e −

= = .

π

Therefore, the solution of Eq. (3.50) is, by a double inverse Fourier transformation
to Eq. (3.52) and the convolution theorem,

u = W ϕ ( x , y , t )= ϕ ( ξ , η ) V ( x , ξ , t ) V ( y , η , t ) d ξ d η . (3.53)

R 2

By the solution structure theorem,

t t

u = W f τ ( x , y , t − τ ) d τ = d τ f ( ξ , η , τ ) V ( x , ξ , t − τ ) V ( y , η , t − τ ) d ξ d η

0 0

R 2

(3.54)