The determinants of the block band matrices based on the n-dimensional Fourier equation. Part 2

THE DETERMINANTS OF THE BLOCK BAND MATRICES BASED ON THE n-DIMENSIONAL FOURIER EQUATION

PART 2

Grzegorz Biernat, Sylwia Lara-Dziembek, Edyta Pawlak Institute of Mathematics, Czestochowa University of Technology

Czestochowa, Poland

grzegorz.biernat@im.pcz.pl, sylwia.lara@im.pcz.pl, edyta.pawlak@im.pcz.pl

Abstract. This work is a continuation of the considerations concerning the determinants of the band block matrices on the example of the n-dimensional Fourier equation (work Part 1). The discussion will concern the special case called the three-dimensional Fourier equation.

Keywords: block matrices, band matrices, determinant, Fourier equation

Introduction

The 3D Fourier equation can describe some of the transport phenomena which are typical of the irreversible processes occurring in nature. These phenomena depend on the transport energy, matter, momentum or electric charge on a macro- scopic scale.

The process of transport at the time could be presented by a partial differential equation

2 2 2

2 2 2

1 2 3

K

x x x t

∂ Φ ∂ Φ ∂ Φ ∂ Φ

+ + =

∂ ∂ ∂ ∂ (1)

where K is a constant characterizing the process.

This equation describes the propagation of a certain scalar quantity

(

x1,x2,x3,t

)

Φ .

Therefore, the Fourier equation appears in both the heat transfer and, for example, hydrogeology.

In hydrogeology the three-dimensional Fourier equation

2 2 2

2 2 2

H H H S H

x y z T t

∂ ∂ ∂ ∂

+ + =

∂ ∂ ∂ ∂

2 2 2

H H H S H

x y z T t

∂ ∂ ∂ ∂

+ + =

∂ ∂ ∂ ∂ , where H - height of hydraulic, S - water capacity, T - conductivity and t - time, describes the transient filtering in a homogeneous and isotropic domain without internal sources.

In this paper we consider the transient diffusion of heat in the 3D domain.

1. Solution of the problem

The Fourier equation describing the heat flow is in the form [1]

( ) ( ) ( )

( )

2 2 2

1 2 3 1 2 3 1 2 3

2 2 2

1 2 3

1 2 3

, , , , , , , , ,

, , ,

T x x x t T x x x t T x x x t

x x x

T x x x t c

t λ

ρ

∂ ∂ ∂ 

 + + =

 ∂ ∂ ∂ 

 

= ∂

∂

(2)

where λ is a thermal conductivity, c is a specific heat, ρ is a mass density and T, x₁, x₂, x₃, t denotes the temperature, geometrical co-ordinates and time.

Let us consider the following inner element

(

∆x1,∆x2,∆x3

)

and the internal nodes of the spatial grid (Fig. 1).

Fig. 1. An example of the inner element and the internal nodes of the spatial grid x2

∆

x1

x2

x3

I1

I3

I2

x1

∆

x3

∆

The second differential spatial derivatives are as follows:

( )

( )

( )

1 2 3 1 2 3 1 2 3

1

1 2 3 1 2 3 1 2 3

2

1 2 3 1 2 3 1 2 3

2 1, , , , , , 1, , ,

1 1

2 2

1 2

, 1, , , , , , 1, ,

2 2

2 2

2

2 , , 1, , , , , , 1,

3 3

2 2

3 3

2

, 1 1

2

, 1 1

2

, 1 1

i i i l i i i l i i i l

i i i l i i i l i i i l

i i i l i i i l i i i l

T T T

T

i m

x x

T T T

T i m

x x

T T T

T i m

x x

− +

− +

− +

− +

∂ = ≤ ≤ −

∂ ∆

− +

∂ = ≤ ≤ −

∂ ∆

− +

∂ = ≤ ≤ −

∂ ∆

(3)

and the first differential derivative of the time has a form

1,2,3, 1,2,3, 1 i i i l i i i l , 1

T T

T l q

t t

− −

∆ = ≤ ≤

∆ ∆ (4)

Therefore, the Finite Difference Method leads to the internal system of equations

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 3 1 2 3 1 2 3

1 2 3 1 2 3 1 2 3

1 2 3 1 2 3 1 2 3

1 2 3 1 2 3

1, , , , , , 1, , ,

2 2 2

1 1 1

, 1, , , , , , 1, ,

2 2 2

2 2 2

, , 1, , , , , , 1,

2 2 2

3 3 3

, , , , ,

2

2

2

i i i l i i i l i i i l

i i i l i i i l i i i l

i i i l i i i l i i i l

i i i l i i i

T T T

x x x

T T T

x x x

T T T

x x x

c c

T T

t t

λ λ λ

λ λ λ

λ λ λ

ρ ρ

− +

− +

− +

− + +

∆ ∆ ∆

+ − + +

∆ ∆ ∆

+ − + =

∆ ∆ ∆

= −

∆ ∆ ^{,l −1}

(5)

in each time step l [2].

Then, for example 1≤i₁≤3, 1≤i₂≤3, 1≤i₃≤ we obtain 3

( ) ( )

( )

( ) ( ^{( )} )

2 2 2 2

3 1 1 1 1 1 2 1

2 2

2 2

1 2 1 1 1 2 1 1

det det det 2 det 2

det 2 det 2

A A A d I A d I

A d d I A d d I

= ⋅ − ⋅ − ⋅

⋅ − + ⋅ − −

(6)

Using the characteristic polynomials of the matrix A₁² we get the following form

( ) ( ) ( ( ) ) ( ^{( )} )

2 2 2 2

1 1 1 1

3

2 2

2 2

1 1 2 2 1 2 1

det

det 2 2 2 2

A A A A

A

A W d W d W d d W d d

=

= ⋅ ⋅ ⋅ + ⋅ −

(7)

If f is any monic polynomial of one variable, A is any square matrix of degree n,

1, 2, ... ,

λ λ λ are the eigenvalues of the matrix A the matrix n f A has eigenvalues

( )

( ) ( )

1 , 2 , ...,

( )

f λ f λ f λn [3].

So, the characteristic polynomial of matrix f A has the following form

( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )

( ) 1 2

1 2

1 1 2 2 1 2

1 2

1 ...

1 , ,..., , ,..., ...

( 1) , ,...,

n

f A n

n n n n

n n

n

n n

W λ λ f λ λ f λ λ f λ

λ τ λ λ λ λ τ λ λ λ λ

τ λ λ λ

− −

     

= −  −  ⋅ − ⋅ ⋅ − =

= −  − + + +

+ − 

(8)

where 1

(

1, 2, ... ,

)

, 2

(

1, 2, ... ,

)

, ...,

(

1, 2, ... ,

)

n n n n

τ λ λ λ τ λ λ λ τ λ λ λ are the fun-

damental symmetric polynomials [4].

Using the fact above we determine the particular characteristic polynomials

( ) ( ) ( ( ) ) ( ^{( )} )

2 2 2 2

1 1 1 1

2 2

2 2

1 2 2 1 2 1

2 , 2 , 2 , 2

A A A A

W d W d W d +d W d −d (9)

In our case n = 3 and

( )

( )

( )

1 1 2 3 1 2 3 1 1

2 2

2 1 2 3 1 2 1 3 2 3 1 1

3 2

3 1 2 3 1 2 3 1 1 1 1

, , tr 3

, , 3 2

, , det 2

A a

a b

A a a b

τ λ λ λ λ λ λ

τ λ λ λ λ λ λ λ λ λ

τ λ λ λ λ λ λ

= + + = =

= + + = −

= = = −

(10)

Then

( )

( ) ( )

2 1

3 2 2 2 2 2 2 2 2 2 2 2 2 2

1 2 3 1 2 1 2 2 3 1 2 3

WA p

p λ λ λ p λ λ λ λ λ λ p λ λ λ

=

 

= − − + + + + + − 

(11)

Designating the following order

( ) ( )

( ) ( )

( ) ( )

2 2 2 2 2 2

1 2 3 1 2 3 1 2 1 3 2 3 1 1

2 2 2 2 2 2 2

1 2 1 2 2 3 1 2 1 3 2 3 1 2 3 1 2 3

4 4

1 1

2 2

2 2 2 3 2

1 2 3 1 2 3 1 1 1

2 3 4

2

3 4

2

a b

a b

a a b

λ λ λ λ λ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ

λ λ λ λ λ λ

+ + = + + − + + = +

+ + = + + − + + =

= +

= = −

(12)

and putting in place p adequate values from (9) and (12) we obtain

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )( )

( )( ) ( )

( )

12

2 1

2 1

2 1

2 6 2 2 4 4 4 2 3 2 2

1 1 1 1 1 1 1 1 1 1 1

2 6 2 2 4 4 4 2 3 2 2

2 2 1 1 2 1 1 2 1 1 1

2 6 2 2 4

2 1 2 1 1 1 2 1

2 2

4 4 3 2

1 1 2 1 1 1 1

2

2 1

2 8 4 3 4 2 3 4 2

2 8 4 3 4 2 3 4 2

2 8 4 3 4

2 3 4 2

2

A

A

A

A

W d d a b d a b d a a b

W d d a b d a b d a a b

W d d d d a b d d

a b d d a a b

W d d

= − + + − + + −

= − + + − + + −

 + = − + + + + +

 

 

− + + + −

 − = −

 

 

( ) ( )( )

( )( ) ( )

6 2 2 4

2 1 1 1 2 1

2 2

4 4 3 2

1 1 2 1 1 1 1

8 4 3 4

2 3 4 2

d d a b d d

a b d d a a b

− + + − +

− + − + −

(13)

So, the determinant (7) is in the form

3 2

3 1 1 1

2 3 2 2 2 2 2 4 4 3 2 2

1 1 1 1 1 1 1 1 1 1

2 3 2 2 2 2 2 4 4 3 2 2

2 1 1 2 2 1 1 1 1 1

2 3 2 2 2 2

1 2 1 1 1 2

4 4 2

1 1 1 2

det ( 2 )

[(2 ) (3 4 )(2 ) 2 (3 4 ) ( 2 ) ]

[(2 ) (3 4 )(2 ) 2 (3 4 ) ( 2 ) ]

{[2( ) ] (3 4 )[2( ) ]

2(3 4 )( )

A a a b

d a b d d a b a a b

d a b d d a b a a b

d d a b d d

a b d d

= − ⋅

⋅ − + + + − − ⋅

⋅ − + + + − − ⋅

⋅ + − + + +

+ + + − ₁³ _{1 1}^{2 2}

2 3 2 2 2 2

1 2 1 1 1 2

4 4 2 3 2 2

1 1 1 2 1 1 1

( 2 ) }

{[2( ) ] (3 4 )[2( ) ]

2(3 4 )( ) ( 2 ) }

a a b

d d a b d d

a b d d a a b

− ⋅

⋅ − − + − +

+ + − − −

(14)

Finally, after transformation we have the following form of this determinant

2 2 2 2 2 2

3 1 1 2 1 1 1 1 2 1 1 2

2 2 2 2 2 2

1 1 2 1 1 1 2 2 1

2 2 2 2 2 2

1 1 1 1 1 1 2 1 1 1

2 2 2 2 2 2

1 1 1 1 1 2 1 1 2

det [ 2( ) ] [ 2( ) ][ 2( ) ]

[ 2( ) ][ 2( ) ]( 2 )

[ 2( ) ][ 2( ) ][ 2( ) ]

( 2 )[ 2( ) ][ 2( ) ](

A a d d a a b d d a b d

a b d a b d d d a

a b d a b d d a b d

b a a b d d a d d a

= − + − + − − + ⋅

⋅ − − − + + − + ⋅

⋅ − + − − − − − ⋅

⋅ − + − − + − − ²₁−2d₁²)

(15)

Conclusion

The procedure given in this article constitutes a special case (n = 3) of the general procedure for calculating the determinants of the block band matrix appear- ing in the n-dimensional Fourier equation when the FDM is used.

References

[1] Mochnacki B., Suchy J.S., Numerical methods in computations of foundry processes, Polish Foundrymen’s Technical Association, Kraków 1995.

[2] Biernat G., Lara-Dziembek S., Pawlak E., The determinants of the block matrices in the 3D Fourier equation, Scientific Research of the Institute of Mathematics and Computer Science 2012, 4(11), 5-10.

[3] Lang S., Algebra, Springer Science Business Media Inc., 2002.

[4] Mostowski A., Stark M., Elementy algebry wyższej, PWN, Warszawa 1975.