Georgian Mathematical Journal
1(1994), No. 6, 587-598
NON-STATIONARY PROBLEMS OF GENERALIZED
ELASTOTHERMODIFFUSION FOR INHOMOGENEOUS
MEDIA
T. BURCHULADZE
Abstract. The method of investigation of non-stationary boundary
value problems of the theory of thermodiffusion using the Laplace
integral transform is described. In the classical theory of elasticity
this method was first used by V. Kupradze and the author.
The interconnection of deformation, thermal conductivity and diffusion
processes in an elastic isotropic solid body is described by a system of five
scalar partial differential equations of general type. In the classical case this
system is hyperbolic with respect to some part of components of an unknown
vector function and parabolic with respect to the rest components. A system
of equations of the conjugate (connected) theory of thermoelasticity is a
particular case [1–4].
In the classical theory of elastothermodiffusion it is assumed that propagation velocity of heat and of diffusing substance is infinitely large.
In particular, however, it is often necessary to take into account the fact
that heat propagates not with an infinitely large but with a finite velocity.
The heat flux does not occur in the body instantly but is characterized by
the finite relaxation time.
The consideration of these physical factors makes the main system of
differential equations very complicated. There exist various generalizations
of this theory. Three-dimensional non-stationary problems of non-classical
(generalized) thermodiffusion are treated in [5–8].
In this paper the Green–Lindsay theory is generalized to problems of
elastothermodiffusion. Initial boundary value problems are investigated
for the considered physical system of differential equations in piecewisehomogeneous media with boundary and contact conditions; a substantiation
of the Riesz–Fischer–Kupradze method is given and approximate solutions
are considered.
1991 Mathematics Subject Classification. 73B30, 73C25.
587
c 1994 Plenum Publishing Corporation
1072-947X/1994/1100-0587$07.00/0 
588
T. BURCHULADZE
Let us consider a three-dimensional homogeneous isotropic elastic medium in which a thermodiffusion process takes place. The deformed state is
described by the displacement vector v(x, t) = (v1 , v2 , v3 ) = kvk k3×1 (onecolumn matrix), the temperature change v4 (x, t) and the ”chemical potential” of the medium v5 (x, t); C(x, t) = γ2 div v(x, t)+a12 v4 (x, t)+a2 v5 (x, t),
where C(x, t) is the diffusing substance concentration; x = (x1 , x2 , x3 ) is a
point in the Euclidean space R3 , t ≥ 0 is the time and X = (X1 , X2 , X3 ),
X4 , X5 are the given functions. We consider a system of partial differential equations of the generalized elastothermodiffusion theory written in the
form
A
2
X
€ ∂ 
∂2v
v−
γk grad v3+k + X = ρ 2 +
∂x
∂t
k=1
+τ 1
2
X
k=1
γk
∂
grad v3+k ,
∂t
€
∂
∂  ∂v4
+ γ1 div v +
δ1 ∆v4 + X4 = a1 1 + τ 0
∂t ∂t
∂t
€

0 ∂ ∂v5
+a12 1 + τ
,
∂t ∂t
€
∂
∂  ∂v5
δ2 ∆v5 + X5 = a2 1 + τ 0
+ γ2 div v +
∂t ∂t
∂t

€
0 ∂ ∂v4
,
+a12 1 + τ
∂t ∂t
(1)
2
∂
) ≡ kµδjk ∆ + (λ + µ) ∂x∂j ∂xk k3×3 is the statical operator of
where A( ∂x
Lamé [8], δjk being the Kroneker symbol. The elastic, thermal, diffusion
and relaxation constants satisfy the natural restrictions
µ > 0, 3λ + 2µ > 0, ρ > 0, ak > 0, δk > 0, γk > 0, k = 1, 2,
a1 a2 −
a212
> 0,
1
(2)
0
τ ≥ τ > 0.
In particular, for relaxation constants τ 1 = τ 0 = 0 we have the classical
case.
Let D1 ⊂ R3 be a finite domain bounded by the closed Liapunov surface
S and D2 = R3 \D̄1 be an infinite domain, n = (n1 , n2 , n3 ) is the unit
normal on S. Elastothermodiffusion constants of the domain Dj will be
denoted by the left-hand subscripts j λ, j µ, j ρ, j τ 0 , j τ 1 , . . . , j = 1, 2.
Problem At . Define in the infinite cylinder Z∞ = {(x, t) : x ∈ D1 ∪
D2 , t ∈]0, ∞[} the regular vetor V = (v, v4 , v5 ) ∈ C 1 (Z̄∞ ) ∩ C 2 (Z∞ ) from
the conditions
∀(x, t) ∈ Z∞ : j µ∆v(x, t) + (j λ + j µ) grad div v −
NON-STATIONARY PROBLEMS
−
2
X
k=1
589
2
X
∂
∂2v
+ jτ1
grad v3+k ,
j γk
j γk grad v3+k + j Xj = ρ
2
∂t
∂t
k=1
€

0 ∂ ∂v4
+
j δ1 ∆v4 (x, t) + j X4 = j a1 1 + j τ
∂t ∂t
€
∂  ∂v5
∂
,
+ j γ1 div v + j a12 1 + j τ 0
∂t
∂t ∂t
€

0 ∂ ∂v5
+
j δ2 ∆v5 (x, t) + j X5 = j a2 1 + j τ
∂t ∂t
€
∂
∂  ∂v4
+ j γ2 div v + j a12 1 + j τ 0
,
∂t
∂t ∂t
x ∈ Dj , j = 1, 2,
(0)
∀x ∈ Dj : lim V (x, t) = j ϕ
t→+0
(3)
(x), j = 1, 2,
∂V (x, t)
= j ϕ(1) (x), j = 1, 2,
t→+0
∂t
∀(y, t) ∈ S ∞ ≡ {(y, t) : y ∈ S, t ∈ [0, ∞[} :
lim
+
−
[V ]±
S ≡ V (y, t) − V (y, t) = f (y, t),
∂
, n)V (y, t)]+ −
∂y
∂
− [2 R( , n)V (y, t)]− = F (y, t),
∂y
[RV ]±
S ≡ [1 R(
for large values of t and x ∈ D2 :
α
|Dx,t
V (x, t)| ≤
α
≡
Dx,t
const
eσ 0 t ,
1 + |x|1+|α|
|α| = 0, 2, σ0 ≥ 0,
4
X
∂ |α|
αk ,
α1
α2
α3 α4 , |α| =
∂x1 ∂x2 ∂x3 ∂t
k=1
where α = (α1 , α2 , α3 , α4 ) is a multi-index;
(0)
(x)
jϕ
jϕ
(0)
(0)
(0)
(0)
(0)
(1)
(1)
(1)
(1)
(1)
= (j ϕ1 , j ϕ2 , j ϕ3 , j ϕ4 , j ϕ5 ),
(1)
(x) = (j ϕ1 , j ϕ2 , j ϕ3 , j ϕ4 , j ϕ5 ),
f (y, t) = (f1 , f2 , f3 , f4 , f5 ),
F (y, t) = (F1 , F2 , F3 , F4 , F5 ),
t ≥ 0,
y ∈ S,
∂
, n) is a stress operator in the thermodare the given real functions; j R( ∂x
iffusion theory for the medium Dj (5 × 5 matrix):
jR

€ ∂
, n = kj Rkl kk,l=1,5 ,
∂x
590
T. BURCHULADZE
where
j Rkl
∂
∂
∂
+ j λnl (x)
+ j µnk (x)
, k, l = 1, 3,
∂n(x)
∂xk
∂xl
l−3 ∂
), k = 1, 3, l = 4, 5,
j Rkl = −j γl−3 nk (1 + j τ
∂t
∂
, k = 4, 5, l = 1, 5,
j Rkl = j δk−3 δkl
∂n(x)
= j µδlk
here n(x) is C ∞ -extention of n onto R3 ;
V + (y, t) =
lim
D1 3x→y∈S
V (x, t), V − (y, t) =
lim
D2 3x→y∈S
V (y, t),
€ ∂
€∂

ƒ+

lim
, n(y) V (y, t) =
, n(y) V (x, t),
1R
D1 3x→y∈S
∂y
∂x
€ ∂

ƒ−

‚ €∂
lim
R
, n(y) V (y, t) =
, n(y) V (x, t).
2R
2
D2 3x→y∈S
∂y
∂x
‚
1
R
It is easy to verify that
R

€ ∂

€
∂
, n V = T v − γ1 1 + τ 1
nv4 −
∂x
∂t
€
∂v4
∂
∂v5 ‘
γ2 1 + τ 1
, δ2
,
nv5 , δ1
∂t
∂n
∂n
where T is the ”classical” stress operator.
For a classical (regular) solution to exist, it is necessary that the conditions of ”natural compatibility” of initial data be fulfilled. These conditions
have the form
∀y ∈ S : 1 ϕ(0) (y) − 2 ϕ(0) (y) = f (y, 0),
(1)
(y)
1ϕ
− 2 ϕ(1) (y) = lim
t→+0
∂f (y, t)
,
∂t
∂
∂
, n)1 ϕ(0) (y) − 2 R( , n)2 ϕ(1) (y) = F (y, 0),
∂y
∂y
∂F (y, t)
∂
∂
, n)1 ϕ(1) (y) − 2 R( , n)2 ϕ(1) (y) = lim
.
1 R(
t→+0
∂y
∂y
∂t
1 R(
The dynamic Problem At is investigated by the Laplace transform method. However, the ”natural compatibility” conditions of this method are
not sufficient for our purpose and should therefore be complemented with
NON-STATIONARY PROBLEMS
591
”higher order compatibility” conditions. The latter have the form
where
∂ m f (y, t) ŒŒ
= 1 ϕ(m) (y) − 2 ϕ(m) (y),
Œ
∂tm
t=0
m = 2, 7,
∂ m F (y, t) ŒŒ
(m)
(m)
= 1 R1 ϕ (y) − 2 R2 ϕ (y),
Œ
∂tm
t=0
(m)
(m)
(m)
(m)
(x) ≡ (j ϕ1 (x), j ϕ2 (x), j ϕ3 (x)) =
h
(m−2)
(m−2)
(m−2)
= j ρ−1 j µ∆(j ϕ1
, j ϕ3
)+
, j ϕ2
jϕ
(m−2)
+(j λ + j µ) grad div(j ϕ1
(m−2)
, j ϕ2
(m−2)
, j ϕ3
(m−2)
−j γ2 grad j ϕ5
(m)
0
j a1j τ j ϕ4 (x)
(m−1)
−
)−
(m−1)
−
− j γ1j τ 1 grad j ϕ4
Πi
m−2
∂
jX Œ
(m−1)
1
+
,
Œ
j γ2j τ grad j ϕ5
m−2
∂t
t=0
(m−2)
−j γ1 grad j ϕ4
(m)
(m−2)
+ j a12j τ 0 j ϕ5 (x) = j δ1 ∆j ϕ4
(m−1)
(m−1)
(m−1)
(m−1)
− j a1j ϕ4
∂
Œ
j X4 Œ
m−2 Œ
m−2
−
,
∂t
t=0
(m−1)
(m)
(m−2)
(m)
0
0
−
− j a2j ϕ5
j a12j τ j ϕ4 (x) + j a2j τ j ϕ5 (x) = j δ2 ∆j ϕ5
Œ
m−2
∂
j X5 Œ
(m−1)
(m−1)
(m−1)
(m−1)
−j a12j ϕ4
− j γ2 div(j ϕ1
, j ϕ2
)+
, j ϕ3
Π.
∂tm−2 t=0
−j a12j ϕ5
− j γ1 div(j ϕ1
, j ϕ2
, j ϕ3
)+
These conditions of ”quantitative nature” are sufficient for the existence
of the classical solution. We will not dwell on this here but proceed to
the construction of approximate solutions by the Riesz–Fischer–Kupradze
method.
Theorem. If the initial data of Problem At satisfy the above-given
”higher order compatibility” conditions, then Problem At has the unique
classical solution which is represented by the Laplace–Mellin integral
1
V (x, t) =
2πi
σ+i∞
Z
σ−i∞
eζt Vb (x, ζ)dζ,
where Vb (x, ζ) is the solution of the corresponding problem for elliptic system
represented in the form
Vb (x, ζ) =
∞
X
k=0
k
ak (ζ) Ω (x, ζ) + Ω(x, ζ).
592
T. BURCHULADZE
k
The series converges uniformly; ak (ζ), Ω (x, ζ), Ω(x, ζ) are the given vectorfunctions (constructed explicitly) and ζ = σ + iq, where σ ≥ σ0∗ > σ0 ; σ0∗ is
the defined constant.
Consider the Laplace transform
Vb (x, ζ) =
Z∞
e−ζt V (x, t)dt,
(4)
0
where ζ = σ + iq is a complex parameter.
Using formally transform (4), the dynamic problem At is reduced to the
corresponding problem with the complex parameter ζ (spectral problem)
for Vb (x, ζ).
Problem A(ζ). Define for each ζ ∈ Πσ0∗ ≡ {ζ : Re ζ > σ0∗ > σ0 } in

v , vb4 , Vb5 ) = Vb (·, ζ) ∈ C 1 (D1 ∪ D2 ) ∩
D = D1 ∪ D2 the regular vector Vb = (b
C 2 (D1 ∪ D2 ) from the conditions
∀x ∈ Dj ,
j = 1, 2 :
v + (j λ + j µ) grad div v −
j µ∆b
−
v4
j δ1 ∆b
v5
j δ2 ∆b
2
X
j γk (1
k=1
e
+ j τ 1 ζ) grad vb3+k − j ρζ 2 vb = j X,
− j a1 (1 + j τ 0 ζ)b
v4 − j a12 ζ(1 + j τ 0 ζ)b
v5 −
e4 ,
− j γ1 ζ div vb = j X
v5 − j a12 ζ(1 + j τ 0 ζ)b
− j a2 (1 + j τ 0 ζ)b
v4 −
e5 ,
− j γ2 ζ div vb = j X
const
,
|Dxβ Vb (x, ζ)| ≤
1 + |x|1+|β|
|β| = 0, 2,
where β = (β1 , β2 β3 ) is a multi-index,
e = −j X
b − j ρ(j ϕ(1) , j ϕ(1) , j ϕ(1) ) −
2
1
3
jX
(0)
(0)
(0)
− j ρζ(j ϕ1 , j ϕ2 , j ϕ3 ) −
2
X
j γkj
(0)
grad j ϕ3+k ,
k=1
(0)
(1)
(0)
0
e
b
j X4 = −j X4 − j a1j ϕ4 − j a1j τ (j ϕ4 + ζ j ϕ4 ) −
(1)
(0)
(0)
− j a12j ϕ5 − j a12j τ 0 (j ϕ5 + ζ j ϕ5 ) −
(0)
(0)
(0)
− γ1 div(j ϕ1 , j ϕ2 , j ϕ3 ),
b5 − j a2j ϕ(0) − j a2j τ 0 (j ϕ(1) + ζ j ϕ(0) ) −
e = −j X
5
5
5
j X5
(5)
NON-STATIONARY PROBLEMS
(1)
(0)
593
(0)
− j a12j ϕ4 − j a12j τ 0 (j ϕ4 + ζ j ϕ4 ) −
(0)
(0)
(0)
− j γ2 div(j ϕ1 , j ϕ2 , j ϕ3 );
∀y ∈ S : Vb + (y, ζ) − Vb − (y, ζ) = fb(y, ζ),
‚ €∂ 
ƒ+ ‚ € ∂ 
ƒ−
, n Vb (y, ζ) − 2 R
, n Vb (y, ζ) = Fe(y, ζ),
1R
∂y
∂y
(0)
(0)
1
e
b
F (y, ζ) = F (y, ζ) − 1 γ1 1 τ n(y)1 ϕ + 2 γ1 2 τ 1 n(y)2 ϕ −
4
4
− 1 γ2 1 τ
1
(0)
n(y)1 ϕ5

€

b ∂ , n Vb = T vb − n(y)
∂y
jR
2
X
+ 2 γ2 2 τ
j γk (1
k=1
1
(0)
n(y)2 ϕ5 ,
v3+k , j δ1
+ j τ 1 ζ)b
v4
∂b
∂b
v5 ‘
, j δ2
.
∂n
∂n
∂
Let L( ∂x
, ζ) be a matrix differential operator of Problem A(ζ) and Φ(x, ζ)
1
2
5
= kΦjk (x, ζ)k5×5 = k Φ, Φ, . . . , Φ k5×5 be a matrix of fundamental solutions
k
∂
of this operator L( ∂x
, ζ), Φ (x, ζ) = (Φ1k , Φ2k ,. . ., Φ5k ), k = 1, 5, be column
vectors. The matrix Φ(x, ζ) is constructed explicitly in terms of elementary
functions [8]. Namely:

€
b ∂ , ζ ϕ(x, ζ) ≡
Φ(x, ζ) ≡ L
∂x
€
€ ∂ €


2
b 0 ∂ , ζ ϕ(x,
b0
≡L
b ζ),
, ζ ∆ + λ4 ϕ(x, ζ) ≡ L
∂x
∂x
4
X
exp(iλk |x|)
ϕ(x,
b ζ) =
ck
,
|x|
k=1
b ∂ , ζ) is a matrix connected with
where λk , ck , k = 1, 4 are constants, L(
∂x
∂
b ≡ I · det L, I is the unit 5 × 5 matrix.
b ≡ LL
L( ∂x , ζ) : LL
∂
In the above assumptions the sense of the notations j L( ∂x
, ζ) and j Φ(x, ζ)
becomes quite clear.
Thus we have to construct the solution of
Problem A(ζ).
Vb = (b
v , vb4 , vb5 ) ∈ C 1 (D̄) ∩ C 2 (D),
€ ∂
(6)
, ζ)Vb (x, ζ) = j χ(x), j = 1, 2,
∀x ∈ Dj : j L
∂x
b
b+
b−
∀y ∈ S : [Vb ]±
S ≡ V (y, ζ) − V (y, ζ) = f (y, ζ),
(7)
‚ €∂ 
ƒ
ƒ−

‚ €
b
bVb ]± ≡ 1 R ∂ , n Vb (y, ζ) + − 2 R
[R
, n Vb (y, ζ) = Fe (y, ζ),
S
∂y
∂y
594
T. BURCHULADZE
|Dxβ Vb (x, ζ)| ≤
const
, |β| = 0, 2,
1 + |x|1+|β|
(8)
e jX
e4 , j X
e5 ) is the given vector.
where j χ(x) = (j X,
b
Let V (x, ζ) be a regular solution of Problem A(ζ). Taking into account
the contact conditions, by virtue of the formulas for general representation
of the solution [8] we have
∀x ∈ D1 : Vb (x, ζ) =
−
Z
S
∀x ∈ D2 : 0 =
−
Z
Z
1 Φ(x
S
eb ∗ ∗ b +
(1 R
1 Φ ) V dz S −
Z
S
∀x ∈ D1 : 0 = −
−
D2
Z
S
2 Φ2 χdz
D2
Z
1 Φ1 χdz,
Z
S
Z
S
bb +
2 Φ(2 RV ) dS +
+
Z
S
e
2 ΦF dS
bb +
2 Φ(1 RV ) dS +
2 Φ2 χdz
+
Z
S
(9)
eb ∗ ∗ b +
(1 R
1 Φ ) V dz S −
1 Φ1 χdz,
∀x ∈ D2 : Vb (x, ζ) = −
−
Z
D1
b b )+ dz S −
1 Φ(1 RV
D1
Z
bVb )+ dz S −
− z, ζ)(1 R
−
Z
S
Z
S
Z
S
(10)
eb ∗ ∗ b +
(2 R
2 Φ ) V dS −
eb ∗ ∗ b
(2 R
2 Φ ) f dS,
eb ∗ ∗ b +
(2 R
2 Φ ) V dS −
e
2 ΦF dS −
Z
S
eb ∗ ∗ b
(2 R
2 Φ ) f dS,
(11)
(12)
where the superscripts ∗ and e denote transposition and Lagrange’s conjugation, respectively.
bVb )+ found from (10) and (12)
It is clear that by substituting Vb + and (1 R
in (9) and (11) we will solve Problem A(ζ). It appears that (10) and (12)
can be used for constructing approximate values of the unknown vectors.
We introduce the following notations: z ∈ S, x ∈ R3 ,


∗

e ∗
1 Ψ(x, z, ζ) =  (1 R1 Φ (x−z, ζ))


∗

e ∗
2 Ψ(x, z, ζ) =  (2 R2 Φ (x−z, ζ))



, (13)
, −1 Φ(x−z, ζ)
5×5
5×5 5×10



, (14)
, −2 Φ(x−z, ζ)

5×5
5×5
5×10
NON-STATIONARY PROBLEMS
595
bVe )+ ) is the sought for vector. Now relations
ψ(x, ζ) = kψk k10×1 = (Vb + , (1 R
(10) and (12) can be rewritten in the form
Z
(15)
∀x ∈ D2 :
1 Ψ(x, z, ζ)ψ(z, ζ)dz S = 1 F (x),
∀x ∈ D1 :
S
Z
2 Ψ(x, z, ζ)ψ(z, ζ)dz S
= 2 F (x),
(16)
S
where
1 F (x) = −
2 F (x)
=
Z
D2
2 Φ2 χ dz
−
Z
S
Z
1 Φ1 χ dz,
D1
e dS +
2 ΦF
Z
S
eb ∗ ∗ b
(2 R
2 Φ ) f dS
are the given vectors.
Let us construct auxiliary domains and surfaces in the following manner:
e 2 is
e 1 ⊂ D1 ; D
e
D1 is a domain bounded by Se1 located strictly in D1 , i.e, D
e
an infinite domain bounded by S2 located strictly in D2 . It is clear that
Se1 ∩ S = ∅, Se2 ∩ S = ∅.
Let {j xk }∞
k=1 , j = 1, 2, be a countable, dense everywhere, set of points
on the auxiliary surface Sej , j = 1, 2. From (15) and (16) we have
Z
k
k
(17)
1 Ψ(2 x , z, ζ)ψ(z, ζ)dz S = 1 F (2 x ), k = 1, ∞,
S
Z
2 Ψ(1 x
k
, z, ζ)ψ(z, ζ)dz S = 2 F (1 xk ),
k = 1, ∞.
(18)
S
We denote the rows of the matrix j Ψ considered as ten-component vectors
by j Ψ1 , j Ψ2 , j Ψ3 , j Ψ4 , j Ψ5 and consider the countably infinite set of vectors
[ˆ
ˆ l k
‰∞, 5
‰∞, 5
l
k
(19)
2 Ψ (1 x , z, ζ) k=1, l=1 .
1 Ψ (2 x , z, ζ) k=1, l=1
It is proved that (19) is linearly independent and complete in the space
L2 (S); i.e., forms the basis in this space.
Let us enumerate set (19) arbitrarily and denote the resulting countable
set by
{ψ k (z)}∞
k=1 .
We have, for example, performed enumeration like this:
ψ k (z) ≡ ak Ψlk (bk xqk , z, ζ), k = 1, ∞,
(20)
596
T. BURCHULADZE
where
hk − 1i
hk + 1i
, bk = 2
− k + 1,
2
2 "
"
#
#
k+1
hk + 1i
[ k+1
1
[
]
−
]
+
4
2
2
−5
, qk =
;
lk =
2
5
5
ak = k − 2
[k] is the integer part of the number k. It is clear that by virtue of (17) and
(18) the scalar product
Z
(ψ k , ψ̄) = ψ k ψ dS = (ψ, ψ̄ k )
S
is known for any k. Using our notations, we have
Z
ψ k ψ dS = ak Flk (bk xqk ), k = 1, ∞.
S
Obviously, the complex conjugate system
∞
{ψ̄ k (z)}k=1
(21)
is also complete.
Now we have to find coefficients αk , k = 1, N assuming that the meansquare norm
N
X


ψ(z) −
αk ψ̄ k (z)L2 (S)
k=1
is minimal. As is well-known, for this it is necessary and sufficient that
N
‘

X
ψ(z) −
αk ψ̄ k (z), ψ̄ j (z) = 0,
j = 1, N .
k=1
Hence we arrive at an algebraic system of equations
N
X
αk (ψ̄ k , ψ̄ j ) = (ψ, ψ̄ j ), j = 1, N ,
k=1
with the known right-hand side and Gram’s determinant differing from zero,
which defines coefficients αk . Therefore, due to the property of the space
L2 (S), we have
N
X


αk ψ̄ k (z)L (S) = 0.
lim ψ(z) −
2
N →∞
k=1
(22)
NON-STATIONARY PROBLEMS
597
Let us introduce the notation
N
ψ (z) =
N
X
αk ψ̄ k (z),
k=1
N
N
N
X
N
b + = (ψ 1 , ψ 2 , . . . , ψ 5 ) ≡
NV
N
N
k=1
N
b )+ = (ψ 6 , ψ 7 , . . . , ψ 10 ) ≡
N (1 Rτ V
αk (ψ̄1k , ψ̄2k , . . . , ψ̄5k ),
N
X
k
).
αk (ψ̄6k , ψ̄7k , . . . , ψ̄10
k=1
Then we have in the sense of the metric of L2 (S):
N
ψ(z) = lim ψ (z),
N →∞
Vb + = lim
N →∞
b +,
(1 RVb )+ = lim
NV
N →∞
b )+ .
N (1 R V
Substituting the obtained approximate values in (9) and (11) and denoting the result of the substitution by N Vb (x, ζ), we get
∀x ∈ D1 : N Vb (x, ζ) =
Z
−
S
Z
eb ∗ ∗
(1 R
1Φ )
+
S
−
Z
D2
N
X
k=1
S
∀x ∈ D2 : N Vb (x, ζ) = −
Z
1Φ
N
X
k=1
Z
‘
k
) dS −
αk (ψ̄6k , ψ̄7k , . . . , ψ̄10
Z
‘
αk (ψ̄1k , ψ̄2k , . . . , ψ̄5k ) dS − 1 Φ1 χdz,
D1
N
X
2Φ
k=1
S
‘
k
αk (ψ̄6k , ψ̄7k , . . . , ψ̄10
) dS +
‘
 N
eb ∗ ∗ X
αk (ψ̄1k , ψ̄2k , . . . , ψ̄5k ) dS −
(2 R
2Φ )
k=1
2 Φ2 χdz +
Z
S
e
2 ΦF dS −
Z
S
eb ∗ ∗ b
(2 R
2 Φ ) f dS.
Now for any ε ≥ 0 we can give a positive number N (ε) such that for
N > N (ε) we will have
Œ
Œ
ŒVb (x, ζ) − N Vb (x, ζ)Œ < ε,
x ∈ D̄0 ⊂ D; Vb (x, ζ) is the exact solution of the problem, i.e.,
Vb (x, ζ) = lim
N →∞
b (x, ζ),
NV
the convergence to the limit is uniform in D̄0 .
x ∈ D̄0 ;
598
T. BURCHULADZE
The method presented here can also be generalized for other more complicated problems.
References
1. V.D. Kupradze, T.G. Gegelia, M.O. Basheleishvili and T.V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity
and thermoelasticity. (Translated from Russian) North-Holland Publishing
Company, Amsterdam–New-York–Oxford, 1979; Russian origonal: 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, v.7, (Russian) 163-294, Itogi Nauki i Tekhniki Akad. Nauk
SSSR, Vsesoyuzn. Inst. Nauchn. i Tekhnich. Inform., Moscow, 1975.
3. T.V. Buchukuri, Some oscillation boundary value problems of elasticity theory with allowance of thermodiffusion. (Russian) Dokl. Akad. Nauk
SSSR 235(1977), No. 2, 310-312.
4. ”—–”, Boundary value problems of oscillation of moment thermodiffusion. (Russian) Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 7(1980),
5-56.
5. T.V. Burchuladze, To the theory of dynamic problems of non-classical
thermoelasticity. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze, 23(1983),
12-24.
6. ”—–”, Three-dimensional problems of non-classical thermoelasticity.
(Russian) Trudy Tbiliss. Mat. Inst. Razmadze, 25(1984), 4-21.
7. ”—–”, Dynamic problems of the non-classical theory of thermoelasticity. (Russian) Z. Anal. Anwendungen, 3(1984), No. 5, 441-455.
8. T.V. Burchuladze and T.G. Gegelia, The development of the method
of potential in the elsticity theory. (Russian) Metsniereba, Tbilisi, 1985.
9. T.V. Burchuladze, To the theory of dynamic problems of thermodiffusion of deformed elastic solid bodies. (Russian) Bull. Acad. Sci. Georgian
SSR 89(1978), No.2, 329-332.
(Received 15.01.1993)
Author’s address:
A. Razmadze Mathematical Institute
Georgian Academy of Sciences
1, Z. Rukhadze St., Tbilisi, 380093
Republic of Georgia