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.

587

1991 Mathematics Subject Classification. 73B30, 73C25.

1072-947X/1994/1100-0587$07.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1994 Plenum Publishing Corporation

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) = (v_{1}, v_{2}, v_{3}) = ∥v_{k}∥_{3×1} (onecolumn matrix), the temperature change v_{4}(x, t) and the ”chemical potential” of the medium v_{5}(x, t); C(x, t) = γ_{2} div v(x, t)+a_{12}v_{4}(x, t)+a_{2}v_{5}(x, t), where C(x, t) is the diffusing substance concentration; x = (x_{1}, x_{2}, x_{3}) is a point in the Euclidean space ℝ^{3}, t ≥ 0 is the time and X = (X_{1}, X_{2}, X_{3}), X_{4}, X_{5} are the given functions. We consider a system of partial differential equations of the generalized elastothermodiffusion theory written in the form

A(\frac{∂}{∂x})v − ∑_{k=1}^{2} γ_{k} grad v_{3+k} + X = ρ\frac{∂^{2}v}{∂t^{2}} +

+τ^{1} ∑_{k=1}^{2} γ_{k} \frac{∂}{∂t} grad v_{3+k},

δ_{1}Δv_{4} + X_{4} = a_{1}(1 + τ^{0} \frac{∂}{∂t})\frac{∂v_{4}}{∂t} + γ_{1} \frac{∂}{∂t} div v +

+a_{12}(1 + τ^{0} \frac{∂}{∂t})\frac{∂v_{5}}{∂t} ,

δ_{2}Δv_{5} + X_{5} = a_{2}(1 + τ^{0} \frac{∂}{∂t})\frac{∂v_{5}}{∂t} + γ_{2} \frac{∂}{∂t} div v +

+a_{12}(1 + τ^{0} \frac{∂}{∂t})\frac{∂v_{4}}{∂t} ,

(1)

where A(\frac{∂}{∂x}) ≡ ∥μδ_{jk}Δ + (λ + μ)\frac{∂^{2}}{∂x_{j}∂x_{k}} ∥_{3×3} is the statical operator of Lamé [8], δ_{jk} being the Kroneker symbol. The elastic, thermal, diffusion and relaxation constants satisfy the natural restrictions

μ > 0, 3λ + 2μ > 0, ρ > 0, a_{k} > 0, δ_{k} > 0, γ_{k} > 0, k = 1, 2, (2)
a_{1}a_{2} − a_{12}^{2} > 0, τ^{1} ≥ τ^{0} > 0.

In particular, for relaxation constants τ^{1} = τ^{0} = 0 we have the classical case.

Let D_{1} ⊂ ℝ^{3} be a finite domain bounded by the closed Liapunov surface S and D_{2} = ℝ^{3}\D̄_{1} be an infinite domain, n = (n_{1}, n_{2}, n_{3}) is the unit normal on S. Elastothermodiffusion constants of the domain D_{j} will be denoted by the left-hand subscripts _{j}λ, _{j}μ, _{j}ρ, _{j}τ^{0}, _{j}τ^{1}, …, j = 1, 2.

Problem A^{t}. Define in the infinite cylinder Z_{∞} = {(x, t) : x ∈ D_{1} ∪ D_{2}, t ∈]0, ∞[} the regular vetor V = (v, v_{4}, v_{5}) ∈ C^{1}(Z̄_{∞}) ∩ C^{2}(Z_{∞}) from the conditions

∀(x, t) ∈ Z_{∞} : _{j}μΔv(x, t) + (_{j}λ +_{j}μ) grad div v −

NON-STATIONARY PROBLEMS 589

− ∑_{k=1}^{2}_{j}γ_{k} grad v_{3+k} +_{j}X_{j} = ρ\frac{∂^{2}v}{∂t^{2}} +_{j}τ^{1} ∑_{k=1}^{2}_{j}γ_{k} \frac{∂}{∂t} grad v_{3+k},

jδ_{1}Δv_{4}(x, t) +_{j}X_{4} = _{j}a_{1}(1 +_{j}τ^{0} \frac{∂}{∂t})\frac{∂v_{4}}{∂t} +

+_{j}γ_{1} \frac{∂}{∂t} div v +_{j}a_{12}(1 +_{j}τ^{0} \frac{∂}{∂t})\frac{∂v_{5}}{∂t} ,

jδ_{2}Δv_{5}(x, t) +_{j}X_{5} = _{j}a_{2}(1 +_{j}τ^{0} \frac{∂}{∂t})\frac{∂v_{5}}{∂t} +

+_{j}γ_{2} \frac{∂}{∂t} div v +_{j}a_{12}(1 +_{j}τ^{0} \frac{∂}{∂t})\frac{∂v_{4}}{∂t} ,

x ∈ D_{j}, j = 1, 2,  (3)

∀x ∈ D_{j} : lim_{t→+0} V (x, t) = _{j}φ^{(0)}(x), j = 1, 2,

lim_{t→+0} \frac{∂V (x, t)}{∂t} = _{j}φ^{(1)}(x), j = 1, 2,

∀(y, t) ∈ S^{∞} ≡ {(y, t) : y ∈ S, t ∈ [0, ∞[} : [V]_{S}^{±} ≡ V^{+}(y, t) − V^{−}(y, t) = f(y, t),

[RV]_{S}^{±} ≡ [_{1}R(\frac{∂}{∂y} , n)V (y, t)]^{+} −

− [_{2}R(\frac{∂}{∂y} , n)V (y, t)]^{−} = F (y, t),

for large values of t and x ∈ D_{2}:

|D_{x,t}^{α}V (x, t)| ≤ \frac{const}{1 + |x|^{1+|α|}} e^{σ_{0}t}, |α| = \overline{0, 2}, σ_{0} ≥ 0,

D_{x,t}^{α} ≡ \frac{∂^{|α|}}{∂x_{1}^{α_{1}}∂x_{2}^{α_{2}}∂x_{3}^{α_{3}}∂t^{α_{4}}} , |α| = ∑_{k=1}^{4} α_{k},

where α = (α_{1}, α_{2}, α_{3}, α_{4}) is a multi-index; jφ^{(0)}(x) = (_{j}φ_{1}^{(0)},_{j}φ_{2}^{(0)},_{j}φ_{3}^{(0)},_{j}φ_{4}^{(0)},_{j}φ_{5}^{(0)}), jφ^{(1)}(x) = (_{j}φ_{1}^{(1)},_{j}φ_{2}^{(1)},_{j}φ_{3}^{(1)},_{j}φ_{4}^{(1)},_{j}φ_{5}^{(1)}), f(y, t) = (f_{1}, f_{2}, f_{3}, f_{4}, f_{5}), F (y, t) = (F_{1}, F_{2}, F_{3}, F_{4}, F_{5}), t ≥ 0, y ∈ S,

are the given real functions; _{j}R(\frac{∂}{∂x}, n) is a stress operator in the thermodiffusion theory for the medium D_{j} (5 × 5 matrix):

jR(\frac{∂}{∂x}, n) = ∥_{j}R_{kl}∥_{k,l=\overline{1,5}},

where

jR_{kl} = _{j}μδ_{lk} \frac{∂}{∂n(x)} +_{j}λn_{l}(x)\frac{∂}{∂x_{k}} +_{j}μn_{k}(x)\frac{∂}{∂x_{l}} , k, l = \overline{1, 3},

jR_{kl} = −_{j}γ_{l−3}n_{k}(1 +_{j}τ^{l−3} \frac{∂}{∂t}), k = \overline{1, 3}, l = 4, 5,

jR_{kl} = _{j}δ_{k−3}δ_{kl} \frac{∂}{∂n(x)}, k = 4, 5, l = \overline{1, 5},

here n(x) is C^{∞}-extention of n onto ℝ^{3};

V^{+}(y, t) = lim_{D_{1}∋x→y∈S} V (x, t), V^{−}(y, t) = lim_{D_{2}∋x→y∈S} V (y, t),

[_{1}R(\frac{∂}{∂y} , n(y))V (y, t)]^{+} = lim_{D_{1}∋x→y∈S} _{1}R(\frac{∂}{∂x}, n(y))V (x, t),

[_{2}R(\frac{∂}{∂y} , n(y))V (y, t)]^{−} = lim_{D_{2}∋x→y∈S} _{2}R(\frac{∂}{∂x}, n(y))V (x, t).

It is easy to verify that

R(\frac{∂}{∂x}, n)V = (T v − γ_{1}(1 + τ^{1} \frac{∂}{∂t})nv_{4} −

γ_{2}(1 + τ^{1} \frac{∂}{∂t})nv_{5}, δ_{1} \frac{∂v_{4}}{∂n} , δ_{2} \frac{∂v_{5}}{∂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),

1φ^{(1)}(y) −_{2}φ^{(1)}(y) = lim_{t→+0} \frac{∂f(y, t)}{∂t},

1R(\frac{∂}{∂y} , n)_{1}φ^{(0)}(y) −_{2}R(\frac{∂}{∂y} , n)_{2}φ^{(1)}(y) = F (y, 0),

1R(\frac{∂}{∂y} , n)_{1}φ^{(1)}(y) −_{2}R(\frac{∂}{∂y} , n)_{2}φ^{(1)}(y) = lim_{t→+0} \frac{∂F (y, t)}{∂t}.

The dynamic Problem A^{t} 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 ”higher order compatibility” conditions. The latter have the form

\begin{aligned}\frac{∂^{m}f(y, t)}{∂t^{m}} |_{t=0} & = _{1}φ^{(m)}(y) −_{2}φ^{(m)}(y), \\ \frac{∂^{m}F (y, t)}{∂t^{m}} |_{t=0} & = _{1}R_{1}φ^{(m)}(y) −_{2}R_{2}φ^{(m)}(y), \end{aligned} m = \overline{2, 7},

where

jφ^{(m)}(x) ≡ (_{j}φ_{1}^{(m)}(x),_{j}φ_{2}^{(m)}(x),_{j}φ_{3}^{(m)}(x)) =
= _{j}ρ^{−1}[_{j}μΔ(_{j}φ_{1}^{(m−2)},_{j}φ_{2}^{(m−2)},_{j}φ_{3}^{(m−2)}) +
+(_{j}λ +_{j}μ) grad div(_{j}φ_{1}^{(m−2)},_{j}φ_{2}^{(m−2)},_{j}φ_{3}^{(m−2)}) −
−_{j}γ_{1} grad_{j}φ_{4}^{(m−2)} −_{j}γ_{1}_{j}τ^{1} grad_{j}φ_{4}^{(m−1)} −

−_{j}γ_{2} grad_{j}φ_{5}^{(m−2)} −_{j}γ_{2}_{j}τ^{1} grad_{j}φ_{5}^{(m−1)} + \frac{∂^{m−2}_{j}X}{∂t^{m−2}} |_{t=0}],

ja_{1}_{j}τ^{0}_{j}φ_{4}^{(m)}(x) +_{j}a_{12}_{j}τ^{0}_{j}φ_{5}^{(m)}(x) = _{j}δ_{1}Δ_{j}φ_{4}^{(m−2)} −_{j}a_{1}_{j}φ_{4}^{(m−1)} −

−_{j}a_{12}_{j}φ_{5}^{(m−1)} −_{j}γ_{1} div(_{j}φ_{1}^{(m−1)},_{j}φ_{2}^{(m−1)},_{j}φ_{3}^{(m−1)}) + \frac{∂^{m−2}_{j}X_{4}}{∂t^{m−2}} |_{t=0},

ja_{12}_{j}τ^{0}_{j}φ_{4}^{(m)}(x) +_{j}a_{2}_{j}τ^{0}_{j}φ_{5}^{(m)}(x) = _{j}δ_{2}Δ_{j}φ_{5}^{(m−2)} −_{j}a_{2}_{j}φ_{5}^{(m−1)} −

−_{j}a_{12}_{j}φ_{4}^{(m−1)} −_{j}γ_{2} div(_{j}φ_{1}^{(m−1)},_{j}φ_{2}^{(m−1)},_{j}φ_{3}^{(m−1)}) + \frac{∂^{m−2}_{j}X_{5}}{∂t^{m−2}} |_{t=0}.

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 A^{t} satisfy the above-given ”higher order compatibility” conditions, then Problem A^{t} has the unique classical solution which is represented by the Laplace–Mellin integral

V (x, t) = \frac{1}{2πi} ∫_{σ−i∞}^{σ+i∞}e^{ζt}V̂(x, ζ)dζ,

where V̂(x, ζ) is the solution of the corresponding problem for elliptic system represented in the form

V̂(x, ζ) = ∑_{k=0}^{∞} a_{k}(ζ) Ω^{k} (x, ζ) + Ω(x, ζ).

The series converges uniformly; a_{k}(ζ), Ω^{k} (x, ζ), Ω(x, ζ) are the given vectorfunctions (constructed explicitly) and ζ = σ + iq, where σ ≥ σ_{0}^{∗} > σ_{0}; σ_{0}^{∗} is the defined constant.

Consider the Laplace transform

V̂(x, ζ) = ∫_{0}^{∞}e^{−ζt}V (x, t)dt,  (4)

where ζ = σ + iq is a complex parameter.

Using formally transform (4), the dynamic problem A^{t} is reduced to the corresponding problem with the complex parameter ζ (spectral problem) for V̂(x, ζ).

Problem A(ζ). Define for each ζ ∈ Π_{σ_{0}^{∗}} ≡ {ζ : Re ζ > σ_{0}^{∗} > σ_{0}} in D = D_{1} ∪ D_{2} the regular vector V̂ = (v̂, v̂_{4}, V̂_{5})) = V̂(·, ζ) ∈ C^{1}(\overline{D_{1} ∪ D_{2}}) ∩ C^{2}(D_{1} ∪ D_{2}) from the conditions

∀x ∈ D_{j}, j = 1, 2 :
jμΔv̂ + (_{j}λ +_{j}μ) grad div v −

− ∑_{k=1}^{2}_{j}γ_{k}(1 +_{j}τ^{1}ζ) grad v̂_{3+k} −_{j}ρζ^{2}v̂ = _{j}X̃,

jδ_{1}Δv̂_{4} −_{j}a_{1}(1 +_{j}τ^{0}ζ)v̂_{4} −_{j}a_{12}ζ(1 +_{j}τ^{0}ζ)v̂_{5} −

−_{j}γ_{1}ζ div v̂ = _{j}X̃_{4},

jδ_{2}Δv̂_{5} −_{j}a_{2}(1 +_{j}τ^{0}ζ)v̂_{5} −_{j}a_{12}ζ(1 +_{j}τ^{0}ζ)v̂_{4} −

−_{j}γ_{2}ζ div v̂ = _{j}X̃_{5},

|D_{x}^{β}V̂(x, ζ)| ≤ \frac{const}{1 + |x|^{1+|β|}} , |β| = \overline{0, 2},

(5)

where β = (β_{1}, β_{2}β_{3}) is a multi-index,

jX̃ = −_{j}X̂ −_{j}ρ(_{j}φ_{1}^{(1)},_{j}φ_{2}^{(1)}, _{j}φ_{3}^{(1)}) −

−_{j}ρζ(_{j}φ_{1}^{(0)},_{j}φ_{2}^{(0)}, _{j}φ_{3}^{(0)}) − ∑_{k=1}^{2}_{j}γ_{kj} grad_{j}φ_{3+k}^{(0)},

jX̃_{4} = −_{j}X̂_{4} −_{j}a_{1}_{j}φ_{4}^{(0)} −_{j}a_{1}_{j}τ^{0}(_{j}φ_{4}^{(1)} + ζ_{j}φ_{4}^{(0)}) −

−_{j}a_{12}_{j}φ_{5}^{(0)} −_{j}a_{12}_{j}τ^{0}(_{j}φ_{5}^{(1)} + ζ_{j}φ_{5}^{(0)}) −

− γ_{1} div(_{j}φ_{1}^{(0)},_{j}φ_{2}^{(0)},_{j}φ_{3}^{(0)}),

jX̃_{5} = −_{j}X̂_{5} −_{j}a_{2}_{j}φ_{5}^{(0)} −_{j}a_{2}_{j}τ^{0}(_{j}φ_{5}^{(1)} + ζ_{j}φ_{5}^{(0)}) −

−_{j}a_{12}_{j}φ_{4}^{(0)} −_{j}a_{12}_{j}τ^{0}(_{j}φ_{4}^{(1)} + ζ_{j}φ_{4}^{(0)}) −

−_{j}γ_{2} div(_{j}φ_{1}^{(0)},_{j}φ_{2}^{(0)},_{j}φ_{3}^{(0)});

∀y ∈ S : V̂^{+}(y, ζ) − V̂^{−}(y, ζ) = f̂(y, ζ),

[_{1}R(\frac{∂}{∂y} , n)V̂(y, ζ)]^{+} − [_{2}R(\frac{∂}{∂y} , n)V̂(y, ζ)]^{−} = F̃(y, ζ),

F̃(y, ζ) = F̂(y, ζ) −_{1}γ_{1} _{1}τ^{1}n(y)_{1}φ_{4}^{(0)} +_{2}γ_{1} _{2}τ^{1}n(y)_{2}φ_{4}^{(0)} −

−_{1}γ_{2} _{1}τ^{1}n(y)_{1}φ_{5}^{(0)} +_{2}γ_{2} _{2}τ^{1}n(y)_{2}φ_{5}^{(0)},

jR̂(\frac{∂}{∂y} , n)V̂ = (T v̂ − n(y) ∑_{k=1}^{2}_{j}γ_{k}(1 +_{j}τ^{1}ζ)v̂_{3+k},_{j}δ_{1} \frac{∂v̂_{4}}{∂n} ,_{j}δ_{2} \frac{∂v̂_{5}}{∂n} ).

Let L(\frac{∂}{∂x}, ζ) be a matrix differential operator of Problem A(ζ) and Φ(x, ζ)

= ∥Φ_{jk}(x, ζ)∥_{5×5} = ∥ Φ^{1}, Φ^{2}, …, Φ^{5} ∥_{5×5} be a matrix of fundamental solutions

of this operator L(\frac{∂}{∂x}, ζ), Φ^{k} (x, ζ)=(Φ_{1k}, Φ_{2k},…, Φ_{5k}), k = \overline{1, 5}, be column vectors. The matrix Φ(x, ζ) is constructed explicitly in terms of elementary functions [8]. Namely:

Φ(x, ζ) ≡ L̂(\frac{∂}{∂x}, ζ)φ(x, ζ) ≡

≡ L̂_{0}(\frac{∂}{∂x}, ζ)(Δ + λ_{4}^{2})φ(x, ζ) ≡ L̂_{0}(\frac{∂}{∂x}, ζ)φ̂(x, ζ),

φ̂(x, ζ) = ∑_{k=1}^{4} c_{k} \frac{exp(iλ_{k}|x|)}{|x|},

where λ_{k}, c_{k}, k = \overline{1, 4} are constants, L̂(\frac{∂}{∂x}, ζ) is a matrix connected with L(\frac{∂}{∂x}, ζ) : L̂L ≡ LL̂ ≡ I · det L, I is the unit 5 × 5 matrix.

In the above assumptions the sense of the notations_{j}L(\frac{∂}{∂x}, ζ) and_{j}Φ(x, ζ) becomes quite clear.

Thus we have to construct the solution of

Problem A(ζ).

V̂ = (v̂, v̂_{4}, v̂_{5}) ∈ C^{1}(D̄) ∩ C^{2}(D),
∀x ∈ D_{j} : _{j}L(\frac{∂}{∂x}, ζ)V̂(x, ζ) = _{j}χ(x), j = 1, 2,  (6)

\begin{matrix}∀y ∈ S : [V̂]_{S}^{±} ≡ V̂^{+}(y, ζ) − V̂^{−}(y, ζ) = f̂(y, ζ), \\ [R̂V̂]_{S}^{±} ≡ [_{1}R(\frac{∂}{∂y} , n)V̂(y, ζ)]^{+} − [_{2}R̂(\frac{∂}{∂y} , n)V̂(y, ζ)]^{−} = F̃(y, ζ), \end{matrix} (7)

|D_{x}^{β}V̂(x, ζ)| ≤ \frac{const}{1 + |x|^{1+|β|}} , |β| = \overline{0, 2},  (8)

where _{j}χ(x) = (_{j}X̃,_{j}X̃_{4},_{j}X̃_{5}) is the given vector.

Let V̂(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 ∈ D_{1} : V̂(x, ζ) = ∫_{S} _{1}Φ(x − z, ζ)(_{1}R̂V̂)^{+}d_{z}S −

− ∫_{S} (_{1} ˜R̂_{1}Φ^{∗})^{∗}V̂^{+}d_{z}S − ∫_{D_{1}} _{1}Φ_{1}χdz,  (9)

∀x ∈ D_{2} : 0 = ∫_{S} _{1}Φ(_{1}R̂V̂)^{+}d_{z}S − ∫_{S} (_{1} ˜R̂_{1}Φ^{∗})^{∗}V̂^{+}d_{z}S −

− ∫_{D_{1}} _{1}Φ_{1}χdz,  (10)

∀x ∈ D_{2} : V̂(x, ζ) = − ∫_{S} _{2}Φ(_{2}R̂V̂)^{+}dS + ∫_{S} (_{2} ˜R̂_{2}Φ^{∗})^{∗}V̂^{+}dS −

− ∫_{D_{2}} _{2}Φ_{2}χdz + ∫_{S} _{2}ΦF̃dS − ∫_{S} (_{2} ˜R̂_{2}Φ^{∗})^{∗}f̂dS,  (11)

∀x ∈ D_{1} : 0 = − ∫_{S} _{2}Φ(_{1}R̂V̂)^{+}dS + ∫_{S} (_{2} ˜R̂_{2}Φ^{∗})^{∗}V̂^{+}dS −

− ∫_{D_{2}} _{2}Φ_{2}χdz + ∫_{S} _{2}ΦF̃dS − ∫_{S} (_{2} ˜R̂_{2}Φ^{∗})^{∗}f̂dS,  (12)

where the superscripts ^{∗} and denote transposition and Lagrange’s conjugation, respectively.

It is clear that by substituting V̂^{+} and (_{1}R̂V̂)^{+} found from (10) and (12) 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 ∈ ℝ^{3},

1Ψ(x, z, ζ)=∥ (_{1}R̃_{1}Φ^{∗}(x−z, ζ))^{∗} _{5×5}, −_{1}Φ(x−z, ζ) _{5×5} ∥_{5×10}, (13)
2Ψ(x, z, ζ)=∥ (_{2}R̃_{2}Φ^{∗}(x−z, ζ))^{∗} _{5×5}, −_{2}Φ(x−z, ζ) _{5×5} ∥_{5×10}, (14)

ψ(x, ζ) = ∥ψ_{k}∥_{10×1} = (V̂^{+}, (_{1}R̂Ṽ)^{+}) is the sought for vector. Now relations (10) and (12) can be rewritten in the form

∀x ∈ D_{2} : ∫_{S} _{1}Ψ(x, z, ζ)ψ(z, ζ)d_{z}S = _{1}F (x),  (15)

∀x ∈ D_{1} : ∫_{S} _{2}Ψ(x, z, ζ)ψ(z, ζ)d_{z}S = _{2}F (x),  (16)

where

1F (x) = − ∫_{D_{1}} _{1}Φ_{1}χ dz,

2F (x) = ∫_{D_{2}} _{2}Φ_{2}χ dz − ∫_{S} _{2}ΦF̃ dS + ∫_{S} (_{2} ˜R̂_{2}Φ^{∗})^{∗}f̂dS

are the given vectors.

Let us construct auxiliary domains and surfaces in the following manner: D̃_{1} is a domain bounded by S̃_{1} located strictly in D_{1}, i.e, \overline{D̃}_{1} ⊂ D_{1}; D̃_{2} is an infinite domain bounded by S̃_{2} located strictly in D_{2}. It is clear that S̃_{1} ∩ S = ∅, S̃_{2} ∩ S = ∅.

Let {_{j}x^{k}}_{k=1}^{∞}, j = 1, 2, be a countable, dense everywhere, set of points on the auxiliary surface S̃_{j}, j = 1, 2. From (15) and (16) we have

∫_{S} _{1}Ψ(_{2}x^{k}, z, ζ)ψ(z, ζ)d_{z}S = _{1}F (_{2}x^{k}), k = \overline{1, ∞},  (17)

∫_{S} _{2}Ψ(_{1}x^{k}, z, ζ)ψ(z, ζ)d_{z}S = _{2}F (_{1}x^{k}), k = \overline{1, ∞}.  (18)

We denote the rows of the matrix_{j}Ψ considered as ten-component vectors by _{j}Ψ^{1},_{j}Ψ^{2},_{j}Ψ^{3},_{j}Ψ^{4},_{j}Ψ^{5} and consider the countably infinite set of vectors

{_{1}Ψ^{l}(_{2}x^{k}, z, ζ)}_{k=1, l=1}^{∞, 5} ⋃ {_{2}Ψ^{l}(_{1}x^{k}, z, ζ)}_{k=1, l=1}^{∞, 5}.  (19)

It is proved that (19) is linearly independent and complete in the space L_{2}(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}^{∞}.  (20)

We have, for example, performed enumeration like this:

ψ^{k}(z) ≡ _{a_{k}}Ψ^{l_{k}}(_{b_{k}}x^{q_{k}}, z, ζ), k = \overline{1, ∞},

where

a_{k} = k − 2[\frac{k − 1}{2}], b_{k} = 2[\frac{k + 1}{2}] − k + 1,

l_{k} = [\frac{k + 1}{2}] − 5 [\frac{[\frac{k+1}{2}] − 1}{5}] , q_{k} = [\frac{[\frac{k+1}{2}] + 4}{5}];

[k] is the integer part of the number k. It is clear that by virtue of (17) and (18) the scalar product

(ψ^{k}, ψ̄) = ∫_{S} ψ^{k}ψ dS = (ψ, ψ̄^{k})

is known for any k. Using our notations, we have

∫_{S} ψ^{k}ψ dS = _{a_{k}}F_{l_{k}}(_{b_{k}}x^{q_{k}}), k = \overline{1, ∞}.

Obviously, the complex conjugate system

{ψ̄^{k}(z)}_{k=1}^{∞}  (21)

is also complete.

Now we have to find coefficients α_{k}, k = \overline{1, N} assuming that the meansquare norm

∥ψ(z) − ∑_{k=1}^{N} α_{k}ψ̄^{k}(z)∥_{L_{2}(S)}

is minimal. As is well-known, for this it is necessary and sufficient that

(ψ(z) − ∑_{k=1}^{N} α_{k}ψ̄^{k}(z), ψ̄^{j}(z)) = 0, j = \overline{1, N}.

Hence we arrive at an algebraic system of equations

∑_{k=1}^{N} α_{k}(ψ̄^{k}, ψ̄^{j}) = (ψ, ψ̄^{j}), j = \overline{1, N},

with the known right-hand side and Gram’s determinant differing from zero, which defines coefficients α_{k}. Therefore, due to the property of the space L_{2}(S), we have

lim_{N→∞} ∥ψ(z) − ∑_{k=1}^{N} α_{k}ψ̄^{k}(z)∥_{L_{2}(S)} = 0.  (22)

NON-STATIONARY PROBLEMS 597

Let us introduce the notation

ψ^{N} (z) = ∑_{k=1}^{N} α_{k}ψ̄^{k}(z),

N V̂^{+} = (ψ^{N}_{1}, ψ^{N}_{2}, …, ψ^{N}_{5}) ≡ ∑_{k=1}^{N} α_{k}(ψ̄_{1}^{k}, ψ̄_{2}^{k}, …, ψ̄_{5}^{k}),

N(_{1}R_{τ} V̂)^{+} = (ψ^{N}_{6}, ψ^{N}_{7}, …, ψ^{N}_{10}) ≡ ∑_{k=1}^{N} α_{k}(ψ̄_{6}^{k}, ψ̄_{7}^{k}, …, ψ̄_{10}^{k}).

Then we have in the sense of the metric of L_{2}(S):

ψ(z) = lim_{N→∞} ψ^{N} (z), V̂^{+} = lim_{N→∞}_{N} V̂^{+}, (_{1}RV̂)^{+} = lim_{N→∞}_{N}(_{1}RV̂)^{+}.

Substituting the obtained approximate values in (9) and (11) and denoting the result of the substitution by _{N} V̂(x, ζ), we get

∀x ∈ D_{1} :_{N} V̂(x, ζ) = ∫_{S} _{1}Φ( ∑_{k=1}^{N} α_{k}(ψ̄_{6}^{k}, ψ̄_{7}^{k}, …, ψ̄_{10}^{k}))dS −

− ∫_{S} (_{1} ˜R̂_{1}Φ^{∗})^{∗}( ∑_{k=1}^{N} α_{k}(ψ̄_{1}^{k}, ψ̄_{2}^{k}, …, ψ̄_{5}^{k}))dS − ∫_{D_{1}} _{1}Φ_{1}χdz,

∀x ∈ D_{2} :_{N} V̂(x, ζ) = − ∫_{S} _{2}Φ( ∑_{k=1}^{N} α_{k}(ψ̄_{6}^{k}, ψ̄_{7}^{k}, …, ψ̄_{10}^{k}))dS +

+ ∫_{S} (_{2} ˜R̂_{2}Φ^{∗})^{∗}( ∑_{k=1}^{N} α_{k}(ψ̄_{1}^{k}, ψ̄_{2}^{k}, …, ψ̄_{5}^{k}))dS −

− ∫_{D_{2}} _{2}Φ_{2}χdz + ∫_{S} _{2}ΦF̃ dS − ∫_{S} (_{2} ˜R̂_{2}Φ^{∗})^{∗}f̂dS.

Now for any ε ≥ 0 we can give a positive number N(ε) such that for N > N(ε) we will have

|V̂(x, ζ) −_{N} V̂(x, ζ)| < ε,

x ∈ D̄′ ⊂ D; V̂(x, ζ) is the exact solution of the problem, i.e.,

V̂(x, ζ) = lim_{N→∞}_{N} V̂(x, ζ), x ∈ D̄′;

the convergence to the limit is uniform in D̄′.

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