GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 1, 1997, 1-18
NONSTATIONARY INITIAL BOUNDARY VALUE
CONTACT PROBLEMS OF GENERALIZED
ELASTOTHERMODIFFUSION
T. BURCHULADZE
Abstract. The initial boundary value problems with mixed boundary conditions are considered for a system of partial differential equations of generalized electrothermodiffusion. Approximate solutions
are constructed and a mathematical substantiation of the method is
given.
It is well known that a wide range of problems of mathematical physics
includes boundary value and initial boundary value problems for differential equations. Such problems are rather difficult to solve because of an
enormous variety of geometrical forms of the investigated objects and the
complexity of boundary conditions. Hence an important and timely task
has arisen to develop sufficiently effective methods and tools for solving the
above-mentioned problems and obtaining their numerical solutions.
Until recently no methods were known for solving problems of elasticity
by means of conjugate fields. However, in the past few years there have
appeared and keep on appearing numerous published works dedicated to
this topic. The results of our studies in this direction are presented mainly
in [1, 2, 3].
In this paper, using the generalized Green–Lindsay theory of elastothermodiffusion as an example, the approaches to the solution of nonstationary
initial boundary value contact problems with mixed boundary conditions
are described for a system of partial differential equations of this theory in
a nonhomogeneous medium. These approaches are based on the method of
discrete singularities known as the Riesz–Fischer–Kupradze method. The
particular case is treated in [4].
1991 Mathematics Subject Classification. 73B30, 73C25.
Key words and phrases. Dynamic problems of elastothermodiffusion, Laplace transform, a singular fundamental solution, Riesz–Fischer–Kupradze method of approximate
solution.
1
c 1997 Plenum Publishing Corporation
1072-947X/97/0100-0001$12.50/0 
2
T. BURCHULADZE
Let us consider a three-dimensional isotropic elastic medium in which the
thermodiffusive process is taking place. The deformed state is described
by the displacement vector v(x, t) = v = (v1 , v2 , v3 )T = kvk k3×1 , temperature change v4 (x, t), and “chemical potential of the medium” v5 (x, t).
Here x = (x1 , x2 , x3 ) is a point of the Euclidean space R3 , t the time,
C(x, t) = γ2 div v(x, t) + a12 v4 (x, t) + a2 v5 (x, t), C(x, t) the diffusing substance concentration, and T denotes the operation of transposition.
The object of our investigation is a system of partial differential equations
of the generalized theory of elastothermodiffusion having the form [4]
L
where
 ∂ ∂‘
,
V (x, t) = H(x, t),
∂x ∂t
(1)
V = (v1 , v2 , . . . , v5 )T = (v, v4 , v5 )T = kvk k5×1
is an unknown vector V


LV = (LV )k 5×1 ,

∂
∂2v



µ∆vk + (λ + µ)
div v − ρ 2 +


∂xk
∂t


2

‘ ∂v

X

∂

3+l

, k = 1, 2, 3,
−
γl 1 + τ 1
∂t ∂xk
(LV )k =
l=1


‘ ∂v

∂

k
0 ∂

δ
∆v
−
a
τ
− γk−3 div v−
1
+

k−3
k
k−3


∂t
∂t
∂t



∂ ‘ ∂v9−k


−a12 1 + τ 0
, k = 4, 5;
∂t
∂t
€
T
H(x, t) = X(x, t), X4 (x, t), X5 (x, t)
= (X1 , . . . , X5 )T = kXk k5×1 is a
given vector-function; the elastic, thermal, diffusive, and relaxational constants τ 0 , τ 1 satisfy the natural restrictions [3]:
µ > 0,
3λ + 2µ > 0,
ρ > 0,
ak > 0, δk > 0, γk > 0, k = 1, 2,
a1 a2 − a212 > 0, τ 1 ≥ τ 0 > 0.
In particular, the classical case is realized for τ 1 = τ 0 = 0.
In this theory the differential operator of stresses which is a 5 × 5 matrix
is determined by the formula
‘
 ∂ ∂
, , n(x) V (x, t) =
∂x ∂t
“T
’ 
2
‘

X
∂
∂v4
∂v5
∂‘
= T
, n(x) v(x, t) − n(x)
v3+l , δ1
, δ2
,
γl 1 + τ 1
∂x
∂t
∂n
∂n
Rτ
l=1
NONSTATIONARY INITIAL BOUNDARY VALUE CONTACT PROBLEMS
3
∂
∂
∂
where T ( ∂x
, n(x)) ≡ kµδjk ∂n
+ λnj ∂x∂ k + µnk ∂x
k3×3 is the matrix difj
ferential operator of elastic stresses in the classical theory [1], n(x) =
(n1 (x), n2 (x), n3 (x)) the unit vector δjk the Kronecker symbol.
Clearly,
2
 ∂ ∂
‘
 ∂
‘
X
∂v3+k
Rτ
, , n(x) V ≡ Rτ
, 0, n(x) V −
,
rk (τ )
∂x ∂t
∂x
∂t
k=1
where rk (τ ) = γk τ 1 (n; 0, 0)T .
Assume that D is a finite domain with the boundary S which is a compact
closed surface of the Liapunov class Λ2 (α), α > 0 [1]; D11 ⊂ D and D12 ⊂ D
are the nonintersecting domains with the Liapunov boundaries S11 and S12 ,
respectively; D21 and D22 are the nonintersecting finite domains outside
S with the boundary Liapunov surfaces S21 and S22 , respectively; D1 is
a finite domain bounded by the surfaces S ∪ S11 ∪ S12 ; D2 is an infinite
domain bounded by the surfaces S ∪ S21 ∪ S22 . Consider the infinite domain
2
D1 ∪ S ∪ D2 ≡ R3 \ ∪ Dlk ; S is the contacting surface.
l,k=1
Let D1 and D2 be two different homogeneous isotropic physical media.
The elastothermodiffusive constants for Dj will be denoted by the lower
left indices j λ, j µ, j ρ, j γl , . . . , j τ 1 , j τ 0 , . . . while the differential operators by
∂
∂
∂
∂
j L( ∂x , ∂t ), j Rτ ( ∂x , ∂t , n(x)), . . . , where j = 1, 2.
Problem At . In the infinite cylinder
n
o
1,2
= (x, t) : x ∈ D1 ∪ D2 , t ∈ ]0, ∞[
Z∞
1,2
→ R5 of the class
determine a regular vector V = (v, v4 , v5 )T : Z∞
1,2
1
2
1,2
C (Z ∞ ) ∩ C (Z∞ ) using the boundary conditions
 ∂ ∂‘
1,2
: jL
∀(x, t) ∈ Z∞
,
V (x, t) = j H(x, t),
∂x ∂t
∀x ∈ Dj : lim V (x, t) = j ϕ(0) (x),
x ∈ Dj , j = 1, 2,
t→+0
∂V (x, t)
= j ϕ(1) (x), j = 1, 2,
∂t
t ≥ 0 : [V ]S ≡ V + (y, t) − V − (y, t) = f (y, t)
”
 ∂ ∂ ‘ •+
‚
ı
, ,n V
Rτ V S ≡ 1 Rτ
−
∂y ∂t
•−
”
∂ ∂ ‘
, , n V (y, t) = F (y, t),
− 2 Rτ
∂y ∂t
lim
∀y ∈ S,
t→+0
±
∀y ∈ Sj1 , t ≥ 0 : V + (y, t) = j f (y, t),
j = 1, 2,
(2)
4
T. BURCHULADZE
∀y ∈ Sj2 , t ≥ 0 :
”
j Rτ
for large t and x ∈ D2 :
 ∂ ∂ ‘ •+
, ,n V
= j F (y, t),
∂y ∂t
σ0 t
Œ
Œ α
ŒDx,t V (x, t)Œ ≤ const e
,
1+|α|
1 + |x|
α
Dx,t
≡
|α| = 0, 2,
∂ |α|
α2
α3 α4 ,
1
∂xα
1 ∂x2 ∂x3 ∂t
j = 1, 2;
σ0 ≥ 0,
|α| =
4
X
α4 ,
k=1
where α = (α1 , α2 , α3 , α4 ) is the multi-index.
Here
V + (y, t) =
V − (y, t) =
lim
V (x, t),
lim
V (x, t),
D1 3x→y∈S
D2 3x→y∈S
 ∂ ∂
‘
, , n(y) V (x, t),
D1 3x→y∈S
∂x ∂t
‘
 ∂ ∂
‚
ƒ−
, , n(y) V (x, t).
=
lim
2 Rτ V
2 Rτ
D2 3x→y∈S
∂x ∂t
‚
1 Rτ V
ƒ+
=
lim
1 Rτ
In a similar manner we determine the limit values on the surfaces Sjk ,
1,2
j, k = 1, 2; here j ϕ(0) , j ϕ(1) : Dj → R5 , f, F : S × [0, ∞[→ R5 , j H : Z∞
→
5
5
5
R , j f : Sj1 × [0, ∞[→ R , j F : Sj2 × [0, ∞[→ R are given real-valued
vector-functions of the definite classes [6].
For a classical (regular) solution to exist it is necessary that the condition
of “natural agreement” of the initial data be fulfilled. Here these conditions
have the form
∀y ∈ S : 1 ϕ(0) (y) − 2 ϕ(0) (y) = f (y, 0),
1ϕ
(1)
(1)
(y) − 2 ϕ
Œ
∂f (y, t) ŒŒ
(y) =
,
∂t Œt=0
•
2
‘
∂
X
(1)
(0)
k
, 0, n 1 ϕ (y) −
1 Rτ
1 r (τ )1 ϕ3+k (y) −
∂y
k=1
”
•
2
∂
‘
X
(1)
k
(0)
− 2 Rτ
, 0, n 2 ϕ (y) −
2 r (τ )2 ϕ3+k (y) = F (y, 0),
∂y
k=1
Œ
∂ j f (y, t) ŒŒ
∀y ∈ Sj1 : j ϕ(0) (y) = j f (y, 0), j ϕ(1) (y) =
Π, j = 1, 2,
∂t
t=0
‘
∂
(0)
, 0, n j ϕ (y) −
∀y ∈ Sj2 : j Rτ
∂y
”
(3)
NONSTATIONARY INITIAL BOUNDARY VALUE CONTACT PROBLEMS
−
2
X
jr
k
(1)
(τ )j ϕ3+k (y) = j F (y, 0),
5
j = 1, 2.
k=1
To investigate the dynamic problem At we shall use the Laplace transform
with respect to t. However, in our case these “natural conditions” are not
sufficient for substantiating the method. So we must additionally use the
“higher order” agreement conditions [2, 3] (with m = 1, 7):
Œ
∂ m f (y, t) ŒŒ
∀y ∈ S :
= 1 ϕ(m) (y) − 2 ϕ(m) (y),
∂tm Œt=0
Œ
”
‘
∂
∂ m F (y, t) ŒŒ
(m)
=
n
R
,
0,
(y) −
1 τ
1ϕ
∂tm Œt=0
∂y
•
2
X
(m+1)
k
−
1 r (τ )1 ϕ3+k (y) −
k=1
”
∂
‘
, 0, n 2 ϕ(m) (y) −
∂y
•
2
X
(m+1)
k
−
2 r (τ )2 ϕ3+k (y) ,
−
2 Rτ
(4)
k=1
Œ
∂ m j f (y, t) ŒŒ
∀y ∈ Sj1 :
= j ϕ(m) (y), j = 1, 2,
Œ
∂tm
t=0
Œ
∂
‘
∂ m j F (y, t) ŒŒ
(m)
=
R
,
0,
n
(y) −
∀y ∈ Sj2 :
j
τ
jϕ
Œ
∂tm
∂y
t=0
−
where (m ≥ 2)
= j ρ−1

−
=
k
(m+1)
(τ )j ϕ3+k (y),
j = 1, 2,
k=1
‘T
=
‘T

(m−2)
(m−2)
(m−2)
+
, j ϕ3
, j ϕ2
j µ∆ j ϕ1
”

(m−2)
(m−2)
(m−2)
, j ϕ3
, j ϕ2
j ϕ1
(m−2)
j γk grad j ϕ3+k
k=1
(m−2)
(x)
j δ1 ∆j ϕ4
jr
(m)
(m)
(m)
j ϕ1 (x), j ϕ2 (x), j ϕ3 (x)
+(j λ + j µ) grad div
2
X
2
X
j a1j τ
−
0
−
2
X
‘T
Œ •
∂ m−2 j X ŒŒ
,
j γkj τ grad j ϕ3+k −
∂tm−2 Œt=0
1
k=1
(m)
(m)
0
j ϕ4 (x) + j a12j τ j ϕ5 (x)
(m−1)
(x)
j a1j ϕ4
−
−
(m−1)
(x)
j a12j ϕ5
=
−
(5)
6
T. BURCHULADZE
−j γ1 div

(m−1)
(m−1)
(m−1)
, j ϕ2
, j ϕ3
j ϕ1
(m)
0
j a12j τ j ϕ4 (x)
(m−2)
= j δ2 ∆j ϕ5
(m)
‘T ∂ m−2 X (x, t) ŒŒ
j 4
Œ
−
Π,
∂tm−2
t=0
+ j a2j τ 0 j ϕ5 (x) =
(m−1)
(m−1)
(x) −
(x) − j a2j ϕ5
(x) − j a12j ϕ4

‘T ∂ m−2 X (x, t) ŒŒ
j 5
(m−1)
(m−1)
(m−1)
Œ
−j γ2 div j ϕ1
, j ϕ2
, j ϕ3
−
Π.
∂tm−2
t=0
Using the method of potential and the theory of multidimensional singular integral equations developed in [1, 2, 3], it can be proved with the aid of
the integral Laplace transform, that the structural conditions of agreement
(3), (4) are sufficient for the existence of a classical solution of the dynamic
problem At . The proof is similar to that for the case of a homogeneous
medium [3, 5]. Here our aim is to construct a solution agorithm for the
above-indicated composite nonhomogeneous medium bounded by several
closed surfaces and, using this example, we shall realize the method of an
approximate construction of Riesz–Fischer–Kupradze solutions (the method
of discrete singularities). Incidentally, in the limiting case this method can
serve as a tool for proving existence theorems.
Theorem 1. If the initial data of Problem At satisfy the agreement conditions (3), (4) and certain smoothness conditions (see [6]), then there exists
a unique classical solution of the dynamic problem At , and in the complex
half-plane Re ζ > σ0∗ this solution is represented by the Laplace–Mellin integral
σ+i∞
Z
1
V (x, t) =
eζt Vb (x, ζ)dζ,
2πi
σ−i∞
where Vb (x, ζ) is a solution of the corresponding elliptic problem, which is
represented as a series
Vb (x, ζ) =
∞
X
ak (ζ)k U (x, ζ) + U (x, ζ).
k=1
This series converges uniformly (with respect to the metric of the space
C) in the domain x ∈ D0 b D1 ∪ D2 ; ak (ζ), k U (x, ζ), U (x, ζ) are the
known functions and vector-functions (constructed explicitly), ζ = σ + iq,
σ > σ0∗ > σ0 and σ0∗ is the known constant.
By the formal application of the Laplace transform
Vb (x, ζ) =
Z∞
0
e−ζt V (x, t)dt
(6)
NONSTATIONARY INITIAL BOUNDARY VALUE CONTACT PROBLEMS
7
the dynamic problem At is reduced to the corresponding elliptic problem
b
A(ζ)
with a complex parameter ζ (the spectral problem) for the image
b
V (x, ζ):
b
Determine a regular vector
Problem A(ζ).
ˆ
‰
v , vb4 , vb5 )T , ∀ζ ∈ Πσ0∗ ≡ ζ : Re ζ > σ0∗ > σ0 ,
Vb (x, ζ) = (b
Vb ∈ C 1 (D1 ∪ D2 ) ∩ C 2 (D1 ∪ D2 ),
in D1 ∪ D2 using the conditions
 ∂ ‘
e
∀x ∈ Dj : j L
, ζ Vb (x, ζ) = j H(x,
ζ), j = 1, 2,
∂x
‚ ƒ±
∀y ∈ S : Vb S ≡ Vb + (y, ζ) − Vb − (y, ζ) = fb(y, ζ),
”
•+
∂
‘
ı
‚
, ζ, n Vb (y, ζ) −
Rτ Vb S ≡ 1 Rτ
∂y
”
•−
∂
‘
b
− 2 Rτ
, ζ, n V (y, ζ) = Fe(y, ζ),
∂y
∀y ∈ Sj1 : Vb + (y, ζ) = j fb(y, ζ), j = 1, 2,
”
•+
‘
∂
+
b
∀y ∈ Sj2 : j Rτ
, ζ, n V (y, ζ) = j Fe(y, ζ), j = 1, 2,
∂y
(7)
(8)
(9)
(10)
where
e =
H

e
2
X
γk grad j ϕ3+k ,
e = jX
b − jρ
jX
− jτ1
‘T
e
e
j X, j X4 , j X5

,
(1)
(1)
(1)
j ϕ1 , j ϕ2 , j ϕ3
(0)
‘T
− j ρζ

(0)
(0)
(0)
j ϕ1 , j ϕ2 , j ϕ3
k=1
€ (1)
(0) 
(0)
0
e
b
−
j X4 = j X4 − j a1 jϕ4 (x) − j a1j τ j ϕ4 + ζ j ϕ4
€ (1)
(0) 
(0)
0
− j a12j ϕ5 − j a12j τ j ϕ5 + ζ j ϕ5 −

‘T
(0)
(0)
(0)
− j γ1 div j ϕ1 , j ϕ2 , j ϕ3
,
€
(1)
(0) 
(0)
0
e
b
−
j X5 = j X5 − j a2 jϕ5 (x) − j a2j τ j ϕ5 + ζ j ϕ5

€
(0)
(1)
(0)
− j a12j ϕ4 − j a12j τ 0 j ϕ4 + ζ j ϕ4 −

‘T
(0)
(0)
(0)
− j γ2 div j ϕ1 , j ϕ2 , j ϕ3
,
(0)
(0)
Fe = Fb − 1 γ11 τ 1 n0 (y)1 ϕ4 + 2 γ12 τ 1 n0 (y)2 ϕ4 −
‘T
−
8
T. BURCHULADZE
(0)
(0)
− 1 γ21 τ 1 n0 (y)1 ϕ5 + 2 γ22 τ 1 n0 (y)2 ϕ5 ,
e = j Fb − j γ1j τ 1 n0 (y)j ϕ(0) − j γ2j τ 1 n0 (y)j ϕ(0) ,
4
5
jF
n0 (y) = (n(y), 0, 0)
Œ
Œ β
ŒDx Vb (x, ζ)Œ ≤
j = 1, 2,
T
const
, |β| = 0, 2,
1 + |x|1+|β|
β = (β1 , β2 , β3 ) is the multi-index.
Now we introduce the following Green matrix-functions.
Let j Gτ (x, y; ζ, j R3 ) be the Green tensor (5×5 matrix) for the differential
∂
, ζ), the infinite homogeneous domain j R3 = R3 \(Dj1 ∪ Dj2 )
operator j L( ∂x
and mixed boundary conditions (9), (10) on Sj1 ∪ Sj2 , respectively.
We have
3
j Gτ (x, y; ζ, j R )
= j Φτ (x − y, ζ) − j gτ (x, y; ζ, j R3 ),
where j Φτ (x − y, ζ) is a matrix of fundamental solutions of the operator
∂
j L( ∂x , ζ) which is constructed explicitly in terms of elementary functions
[4, 7], and j gτ (x, y; ζ, j R3 ) are regular matrices.
b
Let Vb (x, ζ) be a regular solution of Problem A(ζ).
Applying the Green
matrices j Gτ , the formula of general representation of a regular vector takes,
in view of the contact and boundary conditions, the form
2χD1 (x)Vb (x, ζ) =
−
Z
S
”
•+
∂
‘
b
, ζ, n V (z, ζ) dz S −
1 Gτ (x, z; ζ, 1 R ) 1 Rτ
∂z
3
Z h
3
T
e
1 Rτ 1 Gτ (x, z; ζ, 1 R )
S
Z
−
D1
−
S11
+
S12
2χD2 (x)Vb (x, ζ) = −
+
Z
S
−
iT
3
T
e
(x,
z;
ζ,
R
)
fb(x, ζ)dz S +
R
G
1 τ1 τ
1
1 Gτ (x, z; ζ, 1 R
Z h
S
Vb + (z, ζ)dz S −
3
e
1 Gτ (x, z; ζ, 1 R )1 H(z, ζ)dz
Z h
Z
iT
3
)1 Fe(z, ζ)dz S,
2 Gτ (x, z; ζ, 2 R
3
x ∈ 1 R3 \S;
(11)
”
•+
∂
‘
) 1 Rτ
, ζ, n V (z, ζ) dz S +
∂z
iT
3
T
e
2 Rτ 2 Gτ (x, z; ζ, 2 R )
Vb + (z, ζ)dz S −
NONSTATIONARY INITIAL BOUNDARY VALUE CONTACT PROBLEMS
−
Z
2 Gτ (x, z; ζ, 2 R
3
D2
+
Z
S
−
e
2 Rτ
S
−
Z h
S21
+
Z
e ζ)dz +
)2 H(z,
3 e
2 Gτ (x, z; ζ, 2 R )F (z, ζ)dz S
Z ”
9
−
•T
‘
∂
, ζ, n 2 GτT (x, z; ζ, 2 R3 ) fb(x, z)dz S −
∂z
T
3
e
2 Rτ 2 Gτ (x, z; ζ, 2 R )
2 Gτ (x, z; ζ, 2 R
S22
3
iT
fb(x, ζ)dz S +
)2 Fe(z, ζ)dz S, x ∈ 2 R3 \S,
(12)
where χDj (x) is the characteristic function of the domain Dj , equal to 1 for
x ∈ Dj and to 0 for x∈Dj , j = 1, 2.
It is easy to check that if the vectors Vb + and (1 Rτ Vb )+ are found by (11)
for x ∈ D2 and by (11) for x ∈ D1 , then the substitution of these values in
b
(11) for x ∈ D1 and in (12) for x ∈ D2 will give a solution of Problem A(ζ).
To this end we introduce the following matrices:
j Ψ(x, z, ζ) =


T
3 T
3
e
=
 (j Rτ j Gτ (x, z; ζ, j R )) 5×5 , −j Gτ (x, z; ζ, j R )
T
€
ψ(x, ζ) = kψk k10×1 = Vb + , (1 Rτ Vb )+ .




5×5
, (13)
5×10
It is easy to notice that by virtue of (11) and (12) we shall have
Z
∀x ∈ D2 :
1 Ψ(x, z, ζ)ψ(z, ζ)dz S = 1 Θ(x),
(14)
S
∀x ∈ D1 :
Z
2 Ψ(x, z, ζ)ψ(z, ζ)dz S
= 2 Θ(x),
S
where
1 Θ(x)
=−
Z
D1
−
Z h
S11
1 Gτ (x, z; ζ, 1 R
3
e ζ)dz −
)1 H(z,
3
T
e
1 Rτ 1 Gτ (x, z; ζ 1 , R )
iT
b
1 f (z, ζ)dz S
+
(15)
10
T. BURCHULADZE
+
Z
S12
2 Θ(x)
=
Z
D2
Z
−
S
+
3
e
1 Gτ (x, z; ζ, 1 R )1 F (z, ζ)dz S,
3
e
2 Gτ (x, z; ζ, 2 R )2 H(z, ζ)dz
e
2 G τ F dz S +
Z h
S21
T
e
2 Rτ 2 Gτ
iT
Z h
S
T
e
2 Rτ 2 Gτ
b
2 f dz S −
Z
iT
−
fbdz S +
2 Gτ (x, z; ζ, 2 R
S22
3
)2 Fedz S
are given vector-functions.
Now we shall construct auxiliary surfaces in the manner as follows: Se1
is the closed surface which lies strictly inside the domain D1 covering the
surface S11 ∪ S12 ; Se2 is the closed surface lying strictly inside the domain
D2 covering the surface S. Clearly,
Se1 ∩ S = ∅, Se2 ∩ S = ∅.
Let {j xk }∞
k=1 , j = 1, 2, be an everywhere dense countable set of points
on the auxiliary surface Sej , j = 1, 2.
By (14) and (15) we have
Z
k
k
(16)
1 Ψ(2 x , z, ζ)ψ(z, ζ)dz S = 1 Θ(2 x ), k = 1, ∞,
S
Z
2 Ψ(1 x
k
, z, ζ)ψ(z, ζ)dz S = 2 Θ(1 xk ), k = 1, ∞,
(17)
S
Denote the rows of the matrix j Ψ considered as ten-component vectors
(columns) by j Ψ1 , j Ψ2 , . . . , j Ψ5 (1 × 10 matrices) and investigate an infinite
countable set of vectors
o∞,5
o∞,5
n
n
k
l
.
(18)
∪ 2 Ψl (1 xk , z, ζ)
1 Ψ (2 x , z, ζ)
k=1, l=1
k=1, l=1
Theorem 2. Set (18) is linearly independent and complete in the vector
Hilbert space L2 (S), i.e., it forms a basis in this space.
For the proof of this theorem see [8].
Rewrite (18) as follows:
ˆ k ‰∞
ψ (z) k=1 ,
where
ψ k (z) ≡ ak Ψlk
€
qk
bk x , z, ζ

,
(19)
NONSTATIONARY INITIAL BOUNDARY VALUE CONTACT PROBLEMS 11
hk − 1i
hk + 1i
bk = 2
− k + 1,
2
"
# 2
#
k+1
k+1
hk + 1i
1
−
+
4
2
2
−5
, qk =
,
lk =
2
5
5
ak = k − 2
,
"
[a] is the integer part of the number a.
Clearly, by virtue of (16) and (17) the scalar product
Z
€ k 
€
k
ψ , ψ = [ψ k ]T ψ ds = ψ, ψ
S
is known for any k; namely,
Z

€
[ψ k ]T ψ ds = ak Θlk bk xqk , k = 1, ∞.
S
It is obvious that the complex-conjugate system
n k o∞
ψ (z)
(20)
k=1
is complete, too.
Now let us determine the coefficients αk , k = 1, N , assuming that the
norm reduces to minimum (in L2 (S) with respect to system (20):


N
X


k


min ψ(z) −
αk ψ (z)
αk
.
L2 (S)
k=1
As is well known, for this it is necessary and sufficient that
’
ψ(z) −
N
X
k
j
αk ψ (z), ψ (z)
k=1
“
= 0, j = 1, N .
Hence we come to an algebraic system of equations
N
X
k=1
€ k j €
j
αk ψ , ψ = ψ, ψ ,
j = 1, N ,
with the right-hand side known and Gram determinant differing from zero,
which determines the coefficients αk . Therefore by virtue of the property
of the space L2 (S) we have


N
X


k
ψ(z) −
lim 
αk ψ (z)

N →∞
k=1
L2 (S)
= 0.
(21)
12
T. BURCHULADZE
Let us introduce the notation
N
ψ(z) =
N
X
k
αk ψ (z),
k=1
b+
NV
N
 k k
N N
‘T
X
N ‘T
k
≡
= ψ1, ψ2, . . . , ψ5,
αk ψ 1 , ψ 2 , . . . , ψ 5 , ,
k=1
N
 k k
‘T X
N N
‘T
N

€
k
b + = ψ 6 , ψ 7 , . . . , ψ 10 ,
≡
αk ψ 6 , ψ 7 , . . . , ψ 10 , .
N 1 Rτ V
k=1
We have in the sense of the metric of the space L2 (S)
N
ψ(z) = lim ψ(z), Vb + = lim N Vb + ,
N →∞
N →∞
€
+
€
+
b
= lim N 1 Rτ Vb .
1 Rτ V
N →∞
By substituting the obtained approximate values in (11) for x ∈ D1 and in
(12) for x ∈ D2 and denoting the substitution result by N Vb (x, ζ) we have
∀x ∈ D1 : 2N Vb (x, ζ) =
−
Z
3
1 Gτ (x, z; ζ, 1 R )
S
×
k=1
S
Z h
”X
N
 k
‘ •
k T
αk ψ 6 , . . . , ψ 10
dz S −
iT
3
T
e
1 Rτ 1 Gτ (x, z; ζ, 1 R )
”X
N
k=1
×
 k
‘ •
k T
αk ψ 1 , . . . , ψ 5
dz S + 1 Θ(x),
(22)
”X
Z
N
 k
‘ •
k T
αk ψ 6 , . . . , ψ 10
dz S +
∀x ∈ D2 : 2N Vb (x, ζ) = − 2 Gτ (x, z; ζ, 2 R3 )
S
+
Z h
S
×
k=1
iT
3
T
e
2 Rτ 2 Gτ (x, z; ζ, 2 R )
”X
N
k=1
×
‘ •
 k
k T
dz S − 2 Θ(x).
αk ψ 1 , . . . , ψ 5
(23)
Thus determining the difference between (11)–(22) and (12)–(23) and applying the Cauchy–Buniakovsky inequality we find by virtue of (21) that
for any ε > 0 there is a positive number N (ε) such that for N > N (ε) we
shall have
Œ
Œ
Œ
Œb
ŒV (x, ζ) − N Vb (x, ζ)Œ < ε,
NONSTATIONARY INITIAL BOUNDARY VALUE CONTACT PROBLEMS 13
where x ∈ D 0 b D1 ∪D2 (a strictly internal subdomain), Vb (x, ζ) is an exact
solution of the problem, i.e.,
Vb (x, ζ) = lim
N →∞
b (x, ζ), x ∈ D 0
NV
tends to the limit uniformly (with respect to the metric C) in D 0 .
As for the convergence with respect to the metric L2 , one can easily verify
that the relation




=0
lim Vb (x, ζ) − N Vb (x, ζ)
N →∞
L2 (D1 ∪D2 )
is valid.
This method can be generalized to other more complicated problems.
The foregoing discussion clearly implies that the solution of Problem
At is constructed by means of the Green tensor j Gτ (x, y; ζ, j R3 ) whose
approximate values of can be constructed explicitly.
We shall give an algorithm of constructing Green tensors.
Clearly, to construct the tensor j Gτ it is sufficient to construct the matrix
j gτ . We have
 (1) (2)

(5) 

0
3
,
j gτ (x, x ; ζ, j R ) = j g τ , j g τ , . . . , j g τ 
5×5
(s)
where the column-vectors j g τ (x, x0 ; ζ, j R3 ), s = 1, 5, are a solution of the
following problem: In the infinite domain j R3 ≡ R3 \(Dj1 ∪ Dj2 ) determine
(s)
the vector j g τ by the conditions (j = 1, 2)
 ∂ ‘ (s)
∀x ∈ j R3 : j L
, ζ j g τ (x, x0 ; ζ, j R3 ) = 0 (x0 ∈ j R3 ),
∂x
(s)
(s)
∀y ∈ Sj1 : j g τ (y, x0 ; ζ, j R3 ) = j Φ τ (y − x0 , ζ),
”
•+
∂
‘ (s)
0
3
, ζ, n j g τ (y, x ; ζ, j R ) =
∀y ∈ Sj2 : j Rτ
∂y
‘ (s)
∂
=j Rτ
, ζ, n j Φ τ (y − x0 , ζ),
∂y
where s ∈ {1, . . . , 5} is a fixed number.
Let us construct auxiliary domains and surfaces in the manner as follows:
e jk , k = 1, 2, is the domain wholly lying in Djk , Sejk are the boundaries of
D
2
e jk , and j Se = ∪ Sejk .
D
k=1
We introduce a 5 × 5 matrix
j Mτ (y
 (1) (2)
(5) 


− x, ζ) = j M τ , j M τ , . . . , j M τ 
5×5
,
14
T. BURCHULADZE
where

j Φτ (y − x, ζ),
‘
∂
j Mτ (y − x, ζ) =
j Rτ
, ζ, n j Φτ (y − x, ζ),
∂y
y ∈ Sj1 ,
y ∈ Sj2 ,
x ∈ R3 .
ek }∞
Let {j x
k=1 be an everywhere dense countable set of points on an auxiliary
e
surface j S.
Theorem 3. A countable set of vectors
n
(l)
jMτ
€
o∞, 5
y − jx
ek , ζ
2
k=1, l=1
, y ∈ ∪ Sjm
(24)
m=1
2
is linearly independent and complete in the Hilbert space L2 ( ∪Sjm ).
m=1
For the proof see [3, pp. 57, 123].
Enumerate the elements of (24) as follows:
k
j ψ(y)
(pk )

y − jx
e[
hk − 1i
pk = k − 5
,
5
k
=jM
τ
k+4
5 ]
‘
, ζ , k = 1, ∞,
(25)
and assume {j ϕ(y)}∞
k=1 to be the system obtained from (25) by orthonor2
malization on ∪ Sjm by the Schmidt mehod, i.e.,
m=1
k
j ϕ(y)
=
k
X
s
j aksj ψ(y),
s=1
2
y ∈ ∪ Sjm , k = 1, ∞,
m=1
where j aks are the orthonormalization coefficients.
It is easy to verify that
 k
‘
(ps )
X

s+4


y ∈ Sj1 ,
e[ 5 ] , ζ ,

j aksj Φ y − j x

k
s=1
j ϕ(y) =
k
∂
‘ (ps )
‘
X

[ s+4

5 ], ζ ,

y
Φ
−
x
e
a
R
y ∈ Sj2 .
,
ζ,
n
j
j
ksj
τ
j

∂y
s=1
We introduce the notation
(s)
j
Ω
(1)
(s)
(z, x0 ) = j Φ τ (z − x0 , ζ),
z ∈ Sj1 ,
(26)
(27)
NONSTATIONARY INITIAL BOUNDARY VALUE CONTACT PROBLEMS 15
(2)
(s)
j
Ω
(z, x0 ) = j Rτ
Let
∂
‘ (s)
, ζ, n j Φ τ (z − x0 , ζ),
∂y
(k)
(s)
(s)
0
j Ω (z, x ) = j Ω
(z, x0 ),
z ∈ Sjk ,
z ∈ Sj2 .
k = 1, 2.
Clearly, by the property of the matrixt of fundamental solutions we have
(s)
j
x 0 ∈ j R3 .
Ω (·, x0 ) ∈ C ∞ (Sj1 ∪ Sj2 ),
(s)
By decomposing j Ω (z, x0 ) in a Fourier series with respect to the complete
orthonormal system of vectors (26) we obtain
(s)
j
where
0
Ω (z, x ) ≈
(s)
j
0
Ω k (x ) =
∞
X
(s)
j
2
k
Ω k (x0 )j ϕ(z),
z ∈ ∪ Sjl ,
l=1
k=1
Z
h
j
Sj1 ∪Sj2
(s)
Ω k (y, x0 )
iT
k
j ϕ(y)dy S.
2
Therefore by the property of the space L2 ( ∪ Sjl ) we have
l=1
 (s)

N (s)
X


0 k
0


lim j Ω (z, x ) −
j Ω k (x )j ϕ(z)
N →∞ 
= 0.
2
L2 ( ∪ Sjl )
k=1
l=1
We introduce the vectors
j
(N )
(s)
g τ (x, x0 )
=
N X
k
X
(s)
j
(pm )
Ω k (x0 )j akm j Φ
τ
k=1 m=1

‘
m+4
e[ 5 ] , ζ ,
x − jx
x, x0 ∈ j R3 , s = 1, 5.
Clearly, all the coefficients are uniquely defined. Now it is easy to show that
the desired approximate value of the Green tensor is written as
(N )
(s)
j Gτ
(s)
(s)
(N )
(x, x0 ; ζ, j R3 ) = j Φ τ (x − x0 , ζ) − j g τ
(s)
= j Φ τ (x − x0 , ζ) −
×j
N X
k
X
k=1 m=1

‘
m+4
Φ τ x − jx
e[ 5 ] , ζ ,
(pm )
(x, x0 ) =
(s)
j
Ω k (x0 )j akm ×
(28)
where j = 1, 2, s = 1, 5, x, x0 ∈ R3 , N > 0 is a natural number. For the
mathematical substantiation of this fact see [3, pp. 122–125].
16
T. BURCHULADZE
(N )
We would like to emphasize that for fixed N the Green tensor j Gτ
given explicitly by a finite number of quadratures.
Finally, on substituting (28) in (22) and (23), we respectively have
is
•
”X
Z
N
€ k
k T
e
)
3
(N
b
∀x ∈ D1 : 2N,Ne V (x, ζ) = 1 Gτ (x, z; ζ, 1 R )
dz S −
αk ψ 6 , . . . , ψ 10
k=1
S
−
Z h
”X
N
k=1
∀x ∈ D2 : 2N,Ne Vb (x, ζ) = −
×
+
Z
S
”X
N
k=1
Z h
”X
N
k=1
i
e (x, z; ζ, R3 ) T ×
1
(N )
•
€ k
k T
dz S + 1,Ne Θ(x),
αk ψ 1 , . . . , ψ 5
e)
(N
3
1 Gτ (x, z; ζ, 2 R )
(29)
×
•
€ k
k T
αk ψ 6 , . . . , ψ 10
dz S +
e
T
2 Rτ 2 Gτ
S
×
e
1 Rτ 1 Gτ
S
×
T
e (x, z; ζ, R3 )
2
(N )
k
k T
αk ψ 1 , . . . , ψ 5
€
•
iT
×
dz S − 2,Ne Θ(x).
(30)
Thus we obtain the following relation with respect to the metric of the
space C in D 0 b D1 ∪ D2 :
Vb (x, ζ) = lim N,Ne Vb (x, ζ).
N →∞
e→∞
N
(31)
As for an estimate in the closed domain D1 ∪ D2 , these estimates hold in
the space L2 (D1 ∪ D2 ).
Remark. Fundamental differential operators of the theory developed here
have the form
’ 
 ∂ ∂
‘
‘
∂
Pτ (k)
, , n(x) V (x, t) = T
, n(x) v(x, t) −
∂x ∂t
∂x
2

X
∂‘
v3+l ,
− n(x)
γl 1 + τ 1
∂t
l=1
− (δ1k + δ2k )v4 + (δ3k + δ0k )δ1
∂v4
,
∂n
NONSTATIONARY INITIAL BOUNDARY VALUE CONTACT PROBLEMS 17
“T
∂v5
− (δ1k + δ3k )v5 + (δ2k + δ0k )δ2
,
∂n
’
 ∂
‘
∂v4
, n(x) V (x, t) = v, (δ1k + δ2k )δ1
+ (δ3k + δ0k )v4 ,
Q(k)
∂x
∂n
“T
∂v5
(δ1k + δ3k )δ2
+ (δ2k + δ0k )v5 ,
∂n
where k = 0, 3, Rτ ≡ Pτ (0) , δjk is the Kronecker symbol.
Let 1 Dk , k = 0, 7, be the nonintersecting domain within S with the
Liapunov boundaries 1 Sk (accordingly, 1 Dk ∩ 1 Dm = ∅, k 6= m); 2 Dk
k = 0, 7, be the nonintersecting finite domains outside S with the Liapunov
boundaries 2 Sk (accordingly, 2 Dk ∩ 2 Dm = ∅, k 6= m); 1 D be a finite
7
domain bounded by the surfaces S ∪ ∪ 1 Sk , 2 D be an infinite domain
k=0
7
bounded by the surfaces S ∪ ∪ 2 Sk . Consider a nonhomogeneous infinite
k=0
2
7
j=1
k=0
domain 1 D ∪ S ∪ 2 D ≡ R3 \ ∪ ∪ ∪ j Dk , where S is the contacting surface.
The method described can be generalized as well to the case where contact conditions on S and boundary conditions on j Sk have the form
 ∂
‘
, n(y) V (x, t) −
∂x
1 D3x→y∈S
 ∂
‘
−
, n(y) V (x, t) = f (y, t),
lim
2 Q(k0 )
∂x
2 D3x→y∈S
 ∂ ∂
‘
lim
, , n(y) V (x, t) −
1 Pτ (k0 )
∂x ∂t
1 D3x→y∈S
 ∂ ∂
‘
lim
P
,
,
n(y)
−
V (x, t) = F (y, t),
2 τ (k0 )
∂x ∂t
2 D3x→y∈S
∀y ∈ j Sk , t ≥ 0 :
lim
j Q(k) V (x, t) = j f(k) (y, t),
∀y ∈ S,
t≥0:
lim
1 Q(k0 )
j D3x→y∈j Sk
∀y ∈ j Sk ,
t≥0:
lim
j D3x→y∈j Sk
k = 0, 3, j = 1, 2,
=
P
V
(x,
t)
j τ (k)
j F(k) (y, t),
k = 4, 7, j = 1, 2,
where k0 ∈ {0, 1, . . . , 7} is a fixed number.
References
1. V. D. Kupradze, T.G. Gegelia, M.O. Basheleishvili, and T.V. Burchuladze. Three-dimensional problems of the mathematical theory of elasticity
18
T. BURCHULADZE
and thermoelasticity (Translated from Russian). North-Holland Publishing
Company, Amsterdam–New York–Oxford, 1979; Russian original: Nauka,
Moscow, 1976.
2. V.D. Kupradze and T.V. Burchuladze. Dynamical problems of the
theory of elasticity and thermoelasticity. (Russian) Current Problems in
Mathematics (Russian), v. 7, 163–294. Itogi Nauki i Tekhniki Akad. Nauk
SSSR, Vsesoyuzn. Inst. Nauchn. i Tekhnich. Inform., Moscow, 1975;
English translation: J. Soviet Math. 7(1977), No. 3.
3. T.V. Burchuladze and T.G. Gegelia. The development of the method
of potential in the elasticity theory. (Russian) Metsniereba, Tbilisi, 1985.
4. T. V. Burchuladze. Nonstationary problems of generalized elastothermodiffusion for nonhomogeneous media. (Russian) Georgian Math.
J. 1(1994), No. 6, 575–586.
5. T.V. Burchuladze. Green formulas in the generalized theory of elastothermodiffusion and their applications. (Russian) Trudy Tbilis. Mat.
Inst. Razmadze 90(1988), 25–33.
6. T.V. Burchuladze. Mathematical theory of boundary value problems
of thermodiffusion in deformed solid elastic bodies. (Russian) Trudy Tbilis.
Mat. Inst. Razmadze 65(1980), 5–23.
7. T. V. Burchuladze. To the theory of dynamic mathematical problems
of generalized elastothermodiffusion. (Russian) Trudy Tbilis. Mat. Inst.
Razmadze 100(1992), 10–38.
8. T. V. Burchuladze. Approximate solutions of boundary value problems of thermodiffusion of solid deformable elastic bodies. (Russian) Trudy
Tbilis. Mat. Inst. 71(1982), 19–30.
(Received 28.11.1994)
Author’s address:
A. Razmadze Mathematical Institute
Georgian Academy of Sciences
1, M. Alexidze St., Tbilisi 380093
Republic of Georgia