Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 272

⎧
⎨

1
τ 0 + λ

m B 2

1
2

− , ,

one-dimensional case

1
τ 0 + λ

mn B 2

1
2

α mnk = − , two-dimensional case ,

⎩

1

τ 0 + λ mnk B 2

1
2

− , .

three-dimensional case

⎧
⎨

2 4

1

λ m A 2 − 4 α

m

, one-dimensional case ,

2 4

1

β mnk =

λ mn A 2 − 4 α

mn

, two-dimensional case ,

⎩

4

1
2

2

λ mnk A 2 − 4 α

mnk ,

three-dimensional case .

⎧
⎨

f ( x , τ ) , one-dimensional case ,

f τ = f ( M , τ )= f ( x , y , τ ) , two-dimensional case ,

⎩

f ( x , y , z , τ ) , three-dimensional case .

W ψ ( M , t ) is the solution at f = ϕ = 0, i.e.

⎧
⎨
⎩

)= m , n , k B mnk F mnk ( M ) e α mnk t sin β mnk t , u = W ψ ( M , t

(6.92)

1
M mnk β mnk

B mnk =

Ω ψ (

M ) F mnk ( M ) d Ω .

Here the integral

is the definite integral, the double integral and the triple inte-
gral depending on the dimensions of

Ω

Ω .Thed Ω stands for d x ,d x d y and d x d y d z ,

respectively.

Remarks

1. To apply the solution structure theorem and the Fourier method of expansion, the
boundary conditions must be linear, homogeneous, separable and with constant
coefficients. The ordinals m , n and k cannot be confused. Their starting value (0
or 1) depends on the boundary conditions. We should also pay attention to the
symbol change of h 1 and h 2 etc.
2. The μ n and μ k

μ m , μ n and μ k carry different meanings. We should use to distin-

guish if necessary. Note also that the μ m have different definitions in Rows 3, 6

and9inTable2.1.

m , n is a double summation in the two-dimensional case. Taking the starting values
of m and n equal to 1 as the example, coefficients B mn read

3.