Symposium

on Differential Equations and

Mathematical Physics

Dedicated to the 100th Birthday Anniversary

of Academician V. Kupradze and

90th Birthday Anniversary of Academician N. Vekua

A B S T R A C T S

Tbilisi, Georgia, December 24 – 25, 2003

Tbilisi 2003

This is a collection of abstracts of the section talks submitted to the Program Committee of the Symposium on Differential Equations and Mathematical Physics (Tbilisi, Georgia, December 24 – 25, 2003).

All abstracts are assorted to the following sections:

• Differential Equations;

• Mathematical Physics. Abstracts are reproduced as presented by the authors.

Program Committee: I. Kiguradze (Chairman) R. Bantsuri V. Kokilashvili

Academic Secretary: N. Partsvania

http://www.rmi.acnet.ge/DEMPh

\begin{matrix}c \\ ◯ \end{matrix} A. Razmadze Mathematical Institute

S E C T I O N I

DIFFERENTIAL EQUATIONS

L. Alkhazishvili

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

OPTIMAL PROBLEMS WITH INCOMMENSURABLE DELAYS AND WITH CONTINUOUS AND DISCONTINUOUS INITIAL CONDITIONS

Optimal problems for a non-linear controlled differential equation with variable delays in phase coordinates and controls, non-fixed initial moment and discontinuous and continuous initial conditions are considered. Necessary conditions of optimality of initial function and control in the form of linearized integral maximum principle and necessary conditions of optimality of initial moment in the form of equality and inequality are obtained.

M. Ashordia and N. Kekelia

Sukhumi Branch of I. Javakhishvili Tbilisi State University Tbilisi, Georgia

ON THE ASYMPTOTIC STABILITY OF A CLASS OF SYSTEMS OF GENERALIZED LINEAR ORDINARY DIFFERENTIAL EQUATIONS

We study the stability in the Liapunov sense of the system of generalized linear ordinary differential equations

dx(t) = dA(t) · x(t) + df(t),  (1)

where A : R_{+} → R^{n×n}, f : R_{+} → R^{n} are matrix- and vector-functions with locally bounded variation components.

To a considerable extent, the interest to the theory of generalized ordinary differential equations has been stimulated also by the fact that this theory enables one to investigate systems of ordinary differential, difference and impulse equations.

By a solution of system (1) we understand a vector-function x whose components have bounded total variations on every closed segment from

R_{+} and x(t) = x(s) + ∫_{s}^{t} dA(τ) · x(τ) (0 ≤ s ≤ t < +∞).

Let d_{1}A(t) = A(t) − A(t−), d_{2}A(t) = A(t+) − A(t), S_{0}(A)(t) = A(t) −

∑_{0<τ≤t}d_{1}A(τ) − ∑_{0≤τ<t}d_{2}A(τ) and det(I_{n}+(−1)^{j}d_{j}A(t))≠0 for t∈R_{+} (j =1, 2).

For the concrete matrix-function A we give some effective conditions for its stability.

Theorem. Let α_{ik} ∈ R (i, k = 1, …, n), and let μ_{i} : R_{+} → R_{+} (1, …, n) be nondecreasing functions such that s_{0}(μ_{i}) ∈ C̃_{loc}(R_{+}; R_{+}) (i = 1, …, n),

lim_{t→+∞} a_{0}(t) = +∞ and σ_{i} = lim inf_{t→+∞} (α_{ii}d_{2}μ_{i}(t)) > −1 (i = 1, …, n),

where a_{0}(t) ≡ ∫_{0}^{t} η_{0}(s)ds + ∑_{0<s≤t} ln(1 − η_{1}(s)) − ∑_{0≤s<t} ln(1 + η_{2}(s)), η_{0}(t) ≡

min{|α_{ii}|(s_{0}(μ_{i})(t))′ : i = 1, …, n}, η_{j}(t) ≡ max {α_{ii}d_{j}μ_{i}(t) : i = 1, …, n} (j = 1, 2). Then the condition

α_{ii} < 0 (i = 1, …, n), r(H) < 1,  (2)

where H = (\frac{1−δ_{ik}}{1+σ_{i}} \frac{|α_{ik}|}{|α_{ii}|})_{i,k=1}^{n}, is sufficient for the matrix-function A(t) = (α_{ik}μ_{i}(t))_{i,k=1}^{n} to be asymptotically stable; and if in addition α_{ik} ≥ 0 (i ≠ k;

i, k = 1, …, n) and ∑_{k=1,i≠k}^{n}α_{ik}d_{1}μ_{i}(t) < min {1 − α_{ii}d_{1}μ_{i}(t), |1 + α_{ii}d_{1}μ_{i}(t)|}

for t ∈ R_{+} (i = 1, …, n), then condition (2) is necessary as well.

Supported by GRDF (Grant No. 3318).

Z. Gegechkori, J. Rogava, and M. Tsiklauri

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

THE THIRD AND FOURTH ORDER ACCURACY DECOMPOSITION FORMULAS FOR A SEMIGROUP

The first decomposition formula for an exponential matrix function was constructed by Lie in 1875. Namely, he showed that

lim_{n→∞} (exp(A_{1}/n) exp(A_{2}/n))^{n} = exp(A_{1} + A_{2}),

where A_{1} and A_{2} are finite dimensional matrices. In 1959, Trotter generalized this formula for exponential operator function – semigroup exp(t(A_{1} + A_{2})) (A_{1} and A_{2} are self-adjoined positively defined operators). Resolvent analog to this formula for the first time was constructed by Chernoff in 1968.

Note that Lie’s formula is of the first order accuracy decomposition formula, that is: (exp(A_{1}/n) exp(A_{2}/n))^{n} − exp(A_{1} + A_{2}) = Op(1/n), where the norm of the matrix Op(1/n) is of the O(1/n) order. There are also known the second order accuracy decomposition formulas. The third and fourth order accuracy decomposition formulas for two-dimensional splitted operator was constructed by B. O. Dia and M. Schatzman in 1996. Note that the formulas constructed by these authors are not automatically stable decomposition formulas. Decomposition formula is called automatically stable if the sum of absolute values of the split coefficients is equal to one. Q. Sheng has proved that in the real number field there does not exist such automatically stable splitting of the exponential operator function-semigroup whose accuracy order is higher than two.

We introduce the notation: T_{1}(λ) = exp(−τλA_{1}) ⋯ exp(−τλA_{m}) and T_{2}(λ) = exp(−τλA_{m}) ⋯ exp(−τλA_{1}), where λ is a scalar, and A_{1}, …, A_{m} (m ≥ 2) are linear (generally unbounded) operators; A = A_{1} + ⋯ + A_{m}, τ = t/n (t ≥ 0, n – is a natural number). The following formulas are true:

exp(−tA) − (\frac{1}{2}(T_{1}(α)T_{2}(\overline{α}) + T_{2}(α)T_{1}(\overline{α})))^{n} = Op(τ^{3}),
exp(−tA) − [\frac{1}{2}(T_{1}(\frac{α}{2})T_{2}(\frac{\overline{α}}{2})T_{1}(\frac{\overline{α}}{2})T_{2}(\frac{α}{2})+

+ T_{2}(\frac{α}{2})T_{1}(\frac{\overline{α}}{2})T_{2}(\frac{\overline{α}}{2})T_{1}(\frac{α}{2}))]^{n} = Op(τ^{4}),

where α = \frac{1}{2} ± i\frac{1}{2\sqrt{3}}; operators (−αA_{j}), (−\overline{α}A_{j}) (j = 1, …, m) and (−A) generate strongly continuous semigroups; Op(τ^{k}) is an operator whose norm is of the fourth order with respect to τ (more precisely, in case of unbounded operators ∥Op(τ^{k})φ∥ = O(τ^{k}) for any φ from the domain of definition of the operator Op(τ^{k})).

Note that the decomposition formulas constructed by us are automatically stable.

G. Giorgadze

Institute of Cybernetics, Georgian Academy of Sciences, Tbilisi, Georgia

FUCHSIAN SYSTEMS ON RIEMANN SURFACES

Dedicated to the memory of Andrey Bolibruch

The fundamental work of A.Bolibruch gave a new impetus to the study of Fuchsian systems [1]. In particular, A.Bolibruch invented some powerful tools for investigation of the famous Riemann-Hilbert problem and global properties of Fuchsian system in terms of associated holomorphic vector bundles with meromorphic connections. This technique later was successfully applied for solving the Riemann-Hilbert problem for compact Riemann surfaces of higher genus (see [2] and references therein). The results presented in the talk belong to the same direction.

Denote by Λ_{X}^{1}(log S) the sheaf of 1-forms holomorphic over X \ S, where X is a compact Riemann surface and S is a finite subset. We consider admissible pairs over (X, S) consisting of a holomorphic vector bundle E → X and holomorphic connection ∇ : Λ^{0}(E) → Λ^{0} ⊗ Λ_{X}^{1}(log S). For such pairs one can define the monodromy representation and splitting type [1]. A logarithmic connection (E, ▽) is called quasi-Fuchsian if there exists a splitting of E such that −2g(j − 1) ≤ k_{j−1} − k_{j} ≤ 2g, i = 1, …, n − 1. If k_{1} = 0, then logarithmic connection (E, ∇) is Fuchsian. The following theorem explicates some previously known results of Fuchsian systems with prescribed monodromy.

Theorem. 1) If the monodromy matrix ρ(γ_{j}) is diagonalisable for some j, then for representation ρ there exists a Fuchsian system whose monodromy representation coincides with ρ.

2) For any two dimensional representation ρ there exists a rank-two Fuchsian system whose monodromy representation coincides with ρ.

3) Let (E, ∇) be a stable pair. Then there exists a semi-stable pair (E′, ∇′) such that deg(E′) = 0, ∇′ has the same singular points as ∇, and the monodromy representations induced by the ∇ and ∇′ coincide.

We also present a similar result for holomorphic principal G-bundles with holomorphic connection and G-system on Riemann surfaces where G is a compact Lie group (see [3]). Such generalization of Fuchsian systems opens wide perspectives for studying certain differential equations of modern mathematical physics, for example, the two-dimensional Yang–Mills equations.

References

1. A. A. Bolibruch, The Riemann–Hibert problem. (Russian) Russian Math. Surveys 45(1990), No. 2, 1–47.

2. A. A. Bolibruch, Stable vector bundles with logarithmic connections and Riemann–Hilbert problem. (Russian) Dokl. Akad. Nauk 381(2001), No. 1, 10–13.

3. G. Giorgadze, Regular systems on Riemann surfaces. J. Math. Sci. 118 (2003), No. 5, 5347–5399.

J. Gvazava

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON THE NONLINEAR GOURSAT PROBLEM WITH A FREE CHARACTERISTIC

An alternative nonlinear version of the Goursat characteristic problem is considered for the quasilinear mixed type hyperbolic-parabolic equation

x{u_{t}u_{xx} − (u_{x} − u_{t})u_{xt} − u_{x}u_{tt}} + u_{x}(u_{x} + u_{t}) = 0  (1)

with order degeneracy on x = 0.

This problem consists in finding a solution of equation (1) by the characteristic condition

u|_{t=x} = φ(x), 0 ≤ x ≤ a,  (2)

simultaneously with the domain of its definition provided that this solution satisfies the condition

αu_{x} + βu_{t}|_{γ} = ν(x), a ≤ x ≤ b,  (3)

on the unknown characteristic curve γ.

Under the conditions φ ∈ C^{2}[0, a], ν ∈ C^{1}[a, b] an explicit representation of the desired characteristic curve γ by the relation

t = x + (α − β) ∫_{a}^{x} \{\frac{zν(z)}{αaφ′(a)} − 1\}^{−1}dz  (4)

is constructed.

Using representation (4), a nonlinear version of the Asgeirssonian mean value property and general representation of solutions for equation (1), we prove the unique solvability of problem (1)–(3).

T. Jangveladze and Z. Kiguradze

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

ON A SYSTEM OF NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

In the cylinder Q = (0, 1)×(0, ∞), we consider the initial boundary value problem

\frac{∂U}{∂t} = \frac{∂}{∂x} [a(S) \frac{∂U}{∂x} ], \frac{∂V}{∂t} = \frac{∂}{∂x} [a(S) \frac{∂V}{∂x} ], (x, t) ∈ Q,  (1)
U(0, t) = U(1, t) = V (0, t) = V (1, t) = 0, t ≥ 0,  (2)
U(x, 0) = U_{0}(x), V (x, 0) = V_{0}(x), x ∈ [0, 1],  (3)

where

S(t) = ∫_{0}^{t} ∫_{0}^{1} [(\frac{∂U}{∂x} )^{2} + (\frac{∂V}{∂x} )^{2}] dx dτ,  (4)

or

S(x, t) = ∫_{0}^{t} [(\frac{∂U}{∂x} )^{2} + (\frac{∂V}{∂x} )^{2}] dτ.  (5)

Theorem 1. If a(s) ≥ a_{0} = const > 0, U_{0}, V_{0} ∈ W^{∘} _{2}^{1}(0, 1), then for problem (1)–(4) the following estimate is true

∥U∥_{W_{2}^{1}} + ∥V ∥_{W_{2}^{1}} ≤ Ce^{− \frac{a_{0}}{2} t}.

Consider the boundary conditions

U(0, t) = V (0, t) = 0, U(1, t) = ψ_{1}, V (1, t) = ψ_{2}, t ≥ 0.  (6)

Then the following statement is true.

Theorem 2. If a(S) = (1 + S)^{p}, p ∈ (−1/2, 0) or p ∈ (0, 1], U_{0}(0) = V_{0}(0) = 0, U_{0}(1) = ψ_{1}, V_{0}(1) = ψ_{2}, ψ_{1}^{2} + ψ_{2}^{2} ≠ 0, U_{0}, V_{0} ∈ W_{2}^{2}(0, 1), then for problem (1)–(4) and (1), (3), (5), (6) the following estimates are true:

\frac{∂U(x, t)}{∂x} = ψ_{1} + O(t^{−1−p}), \frac{∂V (x, t)}{∂x} = ψ_{2} + O(t^{−1−p}),

\frac{∂U(x, t)}{∂t} = O(t^{−1}), \frac{∂V (x, t)}{∂t} = O(t^{−1})

as t → ∞, uniformly in x on [0, 1].

The numerical solutions of problems (1)–(4) and (1), (3), (5), (6) are also obtained. The numerical experiments are agreed with the theoretical conclusions.

G. Kharatishvili

Institute of Cybernetics, Georgian Academy of Sciences Tbilisi, Georgia

GENERAL SOLUTIONS FOR SOME CLASSES OF LAME GENERALIZED AND CLASSICAL EQUATIONS

Theorem 1. Let us consider the Lame generalized equation

α(t)ẍ + (αt^{2} + βt + γ)ẋ + (δt + η)x = f(t), α, β, γ, δ, η – ∀ const, (1)

where particular class is defined by

α = −δ, β = \frac{γδ}{η} − η.  (2)

General solution of equation (1) with condition (2) has the form

\begin{matrix}x = (t − \frac{γ}{η} )(c_{1} − ∫ (t − \frac{γ}{η} )^{−2}e^{− ∫ \frac{αt^{2}+βt+γ}{α(t)} dt}× \\ × (c_{2} − ∫ e^{∫ \frac{αt^{2}+βt+γ}{α(t)} dt}(t − \frac{γ}{η} )\frac{f(t)}{α(t)} dt)dt). \end{matrix}  (3)

If the following condition is satisfied

α(t) = (t − τ)(t − θ)(t − ξ), τ, θ, ξ – ∀ const,  (4)

then equation (1) pass into the Lame nonhomogeneous equation (algebraic form).

From Theorem 1 follows

Theorem 2. If condition (2) is satisfied, then the general solution of the Lame nonhomogeneous equation

(t − τ)(t − θ)(t − ξ)ẍ + (αt^{2} + βt + γ)ẋ + (δt + η)x = f(t)  (5)

has form (3), where α(t) has form (4).

Since

\frac{αt^{2} + βt + γ}{(t − τ)(t − θ)(t − ξ)} = \frac{k}{t − τ} + \frac{l}{t − θ} + \frac{m}{t − ξ} ,

where

k = \frac{ατ^{2} + βτ + γ}{(τ − θ)(τ − ξ)}, l = \frac{αθ^{2} + βθ + γ}{(θ − τ)(θ − ξ)}, m = \frac{αξ^{2} + βξ + γ}{(ξ − τ)(ξ − θ)},

formulae (3) has the form

\begin{matrix}x = (t − \frac{γ}{η} )(c_{1} − ∫ (t − τ)^{−k}(t − θ)^{−l}(t − ξ)^{−m}(t − \frac{γ}{η} )^{−2}× \\ × (c_{2} − ∫ (t − τ)^{k−1}(t − θ)^{l−1}(t − ξ)^{m−1}(t − \frac{γ}{η} )f(t) dt) dt). \end{matrix} (6)

S. Kharibegashvili

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON SOME NONLOCAL PROBLEMS FOR A WAVE EQUATION

Some nonlocal problems are formulated and investigated for a wave equation in the case of two or three spatial variables.

G. Khimshiashvili

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ELLIPTIC BOUNDARY PROBLEMS FOR CAUCHY-RIEMANN SYSTEMS

We deal with boundary value problems for elliptic first order systems of partial differential equations with constant coefficients. The concrete problem which will be addressed, is concerned with the description of those systems which permit elliptic local boundary conditions of Riemann–Hilbert type.

In this context we consider the so-called generalized Cauchy–Riemann systems introduced by E. Stein and G. Weiss [3]. They are characterized by the property that each component of every differentiable solution is a harmonic function. For a long time it was known that not all of such systems possess elliptic Riemann-Hilbert problems, but there did not exist neither a reasonable explanation of this phenomenon nor an effective algorithm to verify if a given system has this property. So it was considered as an important problem to give a complete description of those cases when elliptic Riemann–Hilbert problems do exist (see, e.g., [4]).

We describe a partial solution to this problem based on Clifford analysis [2] and topological invariants introduced by P. Baum, R. Douglas and M. Taylor in the framework of operator K-theory [1].

In particular, we will present a comprehensive list of generalized Cauchy– Riemann systems possessing elliptic Riemann–Hilbert problems. This list is complete for systems in the space of any dimension different from 5 and 6.

Some further results related with the index formulae and homotopy classification for elliptic Riemann–Hilbert problems will be also presented.

References

1. P. Baum, R. G. Douglas, and M. E. Taylor, Cycles and relative cycles in analytic K-homology. J. Differential Geom. 30(1989), No. 3, 761–804.

2. F. Bracks, R. Delange, and F. Sommen, Clifford analysis. Pitman Res. Notes Math. 76(1982).

3. E. M. Stein and G. Weiss, Generalization of the Cauchy–Riemann equations and representations of the rotation group. Amer. J. Math. 90(1968), 163–196.

4. I. Stern, On the existence of Fredholm boundary value problems for generalized Cauchy–Riemann systems in the space. Complex Variables Theory Appl. 21(1993), No. 1–2, 19–38.

E. Khmaladze

Victorian University of Wellington, New Zealand

STOCHASTIC DIFFERENTIAL EQUATION FOR BROWNIAN MOTION UNDER BOUNDARY CONDITIONS

In ℝ^{d}, consider an arbitrary star-shaped set C, containing O and Wiener random measure b on the class of Borel subsets B(C) of C. The values of this measure, b(A), A ∈ B(C), are Gaussian random variables with expectation O and covariance

Eb(A)b(A′) = μ_{(d)}(A ∩ A′),  (1)

where μ_{(d)} is Lebesgue measure in ℝ^{d}. Given the boundary values of b this random measure is not Wiener random measure any more and has more complicated covariance structure than (1).

Using 1-dimensional innovation argument on each ray it is shown that certain filtration with 1-dimensional time parameter allows construction of unique innovation random measure w on B(C), which is again Wiener random measure and is in one-to-one correspondence with b, given values of b on the boundary.

I. Kiguradze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON A MULTI-POINT BOUNDARY VALUE PROBLEM FOR HIGHER ORDER SINGULAR DIFFERENTIAL EQUATIONS

Optimal sufficient conditions are found for the solvability and unique solvability of the boundary value problem

u^{(n)} = f(t, u, ⋯, u^{(n−1)}),  (1)
u^{(i−1)}(t_{0})=0 (i=1, …, n − 1), ∑_{k=1}^{n}α_{k}u^{(k−1)}(a_{k})+∑_{k=1}^{n}β_{k}u^{(k−1)}(b_{k})=0. (2)

Here n ≥ 2, a ≤ a_{k} < t_{0} < b_{k} ≤ b (k = 1, …, n), (−1)^{n−k}α_{k} ≥ 0,

β_{k} ≥ 0 (k = 1, …, n), ∑_{k=1}^{n}(|α_{k}| + β_{k}) > 0, and f : I_{t_{0}} × R^{n} → R is a

measurable in the first argument and continuous in the last n arguments function, I_{t_{0}} = [a, b] \ {t_{0}}. Moreover, the function f, generally speaking, is nonintegrable in the first argument on [a, b] as it has the singularity at the point t_{0}.

In particular, the following theorem is proved.

Theorem. Let there exist measurable functions g_{k}, \overline{g}_{k} : I_{t_{0}} → [0, +∞[ and an integrable function g : [a, b] → [0, +∞[ such that

∫_{a}^{b} |t − t_{0}|^{n−k}\overline{g}_{k}(t) dt < +∞ (k = 1, …, n),

\begin{matrix}∑_{k=1}^{n} \frac{1}{(n − k)!} ∫_{a}^{t_{0}}(t_{0} − t)^{n−k}g_{k}(t) dt ≤ 1, \\ ∑_{k=1}^{n} \frac{1}{(n − k)!} ∫_{t_{0}}^{b}(t − t_{0})^{n−k}g_{k}(t) dt ≤ 1, \end{matrix}  (3)

and on the set I_{t_{0}} × R^{n} the following inequalities are satisfied:

f(t, x_{1}, ⋯, x_{n}) sgn[(t − t_{0})^{n−1}x_{1}] ≥ − ∑_{k=1}^{n} g_{k}(t)|x_{k}| − g(t),

|f(t, x_{1}, ⋯, x_{n})| ≤ ∑_{k=1}^{n} \overline{g}_{k}(t)|x_{k}| + g(t).

Then problem (1), (2) has at least one solution.

Note that in the right-hand sides of inequalities (3) we cannot write down 1 + ε instead of 1 no matter how small ε > 0 would be.

The above results are obtained jointly with Prof. R. P. Agarwal.

Supported by GRDF (Grant No. 3318).

T. Kiguradze

I. Javakhishvili Tbilisi State University, Tbilisi, Georgia

ON INITIAL-BOUNDARY VALUE PROBLEMS IN AN INFINITE STRIP FOR A NONLINEAR HYPERBOLIC EQUATION OF THIRD ORDER

Let I ⊂ ℝ be a compact interval containing zero. For the nonlinear equation

u^{(2,1)} = f(x, t, u, u^{(1,0)}, u^{(2,0)}, u^{(0,1)})  (4)

consider the initial–boundary value problems u(x, 0) = φ(x) for x ∈ [0, +∞),

u(0, t) = ψ(t), sup{|u(x, t)| : x ∈ [0, +∞)} < +∞ for t ∈ I; (2_{1})

u(x, 0) = φ(x) for x ∈ ℝ, sup{|u(x, t)| : x ∈ ℝ} < +∞ for t ∈ I. (2_{2})

Here u^{(j,k)}(x, y) = \frac{∂^{j+k}u(x,y)}{∂x^{j}∂y^{k}}, f : ℝ × I × ℝ^{4} → ℝ is a continuous function, φ : ℝ → ℝ is a twice continuously differentiable function, and ψ : I → ℝ is a continuously differentiable function such that φ(0) = ψ(0).

The linear case of equation (1) arises in study of nonsteady simple shearing flow of second order fluids (c.f. [1]) and also in the theory of seepage of homogeneous fluids through fissured rocks ([2]).

We assume that f satisfies the following three conditions: (i) there exists a continuous function δ : I → ℝ such that (f(x, t, u, v, w, z_{1}) − f(x, t, u, v, w, z_{2})) sgn(z_{1} − z_{2}) ≥ δ(t)|z_{1} − z_{2}|; (ii) f is locally Lipschitz continuous with respect to w; (iii) there exist continuous functions l and g : I → ℝ_{+} such that |f(x, t, u, v, w, z)| ≤ l(t)(|u| + |v| + |w| + |z|) + g(t).

Theorem. Let δ(t) > 0 for t ∈ I. Then problem (1), (2_{k}) (k = 1, 2) is solvable. Moreover an arbitrary solution u_{k} of problem (1), (2_{k}) (k = 1, 2) admits the estimate

∥u_{1}∥_{C^{(2,1)}(ℝ_{+}×I)} ≤ M(∥φ∥_{C^{2}(ℝ_{+})} + ∥ψ∥_{C^{1}(I)} + ∥g∥_{C(I)}) for k = 1,
∥u_{2}∥_{C^{(2,1)}(ℝ×I)} ≤ M(∥φ∥_{C^{2}(ℝ)} + ∥g∥_{C(I)}) for k = 2,

where M > 0 is a constant independent of φ, ψ and g. Furthermore if f is locally Lipschitz continuous with respect to u and v, then problem (1), (2_{k}) (k = 1, 2) is uniquely solvable.

The above results are obtained jointly with Prof. V. Lakshmikantham.

Supported by GRDF (Grant No. 3318).

References

1. B. D. Coleman, R. J. Duffin, and V. J. Mizel, Instability, uniqueness, and nonexistence theorems for the equation u_{t} = u_{xx} − u_{xtx} on a strip. Arch. Rat. Mech. Anal. 19(1965), 100–116.

2. G. Barenblat, I. Zheltov, and I. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24(1960), 1286–1303.

M. Klebanskaya

Georgian Technical University, Tbilisi, Georgia

A NONLINEAR VERSION OF ASGEIRSSONIAN MEAN

VALUE PROPERTY AND ITS APPLICATION TO CHARACTERISTIC PROBLEMS

A nonlinear version of known Asgeirsson mean value property is considered for the special class of second order quasilinear hyperbolic equations with parabolic degeneracy, whose general solutions are represented by means of superposition of two arbitrary functions. In the simplest case of the equation

2y(u_{x} − 2y)u_{xx} + (u_{y} − 2yu_{x} − 2y)u_{xy} − u_{x}u_{yy} = 2u_{x}(u_{x} − 1), (1)

the general solution is

u = y^{2} + f[y + g(x − y^{2})].  (2)

Characteristic families of (1) are represented by relations: x − y^{2} = c, u − y^{2} = c.

The analogue of Asgeirsson principle for nonlinear equations is as follows: the sums of ordinates on the opposite vertexes of arbitrary characteristic quadrangle are equal. This principle can be used for investigation of different nonlinear hyperbolic problems for equation (1), including Goursat characteristic problem, which in this case means:

to define the regular solution u(x, y) of equation (1) simultaneously with its domain of definition if it satisfies the condition

u|_{x=y^{2}} = φ(y), 0 ≤ y ≤ a, φ ∈ C^{2}[0, a]  (3)

and the arc of the curve

x = ψ(y), 0 ≤ y ≤ b  (4)

is characteristic along this solution.

Using the mean value property, under the condition ψ′(y) − 2y > 0, characteristic of u − y^{2} = c and corresponding solution are constructed in the explicit form:

x=ψ{y−y_{0}+Ψ(x_{0}−y_{0}^{2})}−2y{y_{0}+Ψ(x_{0}−y_{0}^{2})}+{y_{0}+Ψ(x_{0}−y_{0}^{2})}^{2}, (5)
u = y^{2} + φ{y − Ψ(x − y^{2})} − {y − Ψ(x − y^{2})}^{2},  (6)

where (x_{0}, y_{0}) are coordinates of an arbitrary point between characteristic curves x = y^{2}, x = ψ(y), x − y^{2} = ψ(b) − b^{2} and x = ψ(y − a) + 2ay − a^{2}. Characteristic quadrangle bounded by these characteristic arcs in compliance with data (3), (4) is the definition domain of solution (6).

Theorem. If the relation ψ(y) − y^{2} = z has only one solution as the functional equation with respect to y, then problem (1), (3), (4) is uniquely solvable.

V. Kokilashvili

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON THE SOLVABILITY OF DIVERGENCE EQUATION IN THE THEORY OF INCOMPRESSIBLE FLUIDS

Let Ω ⊂ R^{n}, n ≥ 2, be a bounded domain with Lipschitz boundary and let x_{0} ∈ Ω.

Let p : Ω → R^{1} be a measurable function satisfying the following conditions:

i) 1 < \underline{p} ≤ p(x) ≤ \overline{p} < ∞; ii) |p(x) − p(y)| ≤ \frac{C}{− ln |x−y|}, |x − y| < \frac{1}{2}.

Put p′(x) = \frac{p(x)}{p(x)−1}.

By L_{ρ}^{p(·)}(Ω) we denote a Banach function space L_{ρ}^{p(·)}, i.e., a space of all measurable functions for which

∥fρ∥_{L^{p(·)}} < ∞,

where ρ(x) = |x − x_{0}|^{α}.

For the definition of the norm in L^{p(·)} see, for example, [1]. Define

L^{∘}_{ρ}^{p(·)} = {f ∈ L_{ρ}^{p(·)}(Ω) : ∫_{Ω} f(x) dx = 0}.

By W_{ρ}^{1,p(·)} we denote the weighted Sobolev space. Theorem. Let p satisfy the conditions (i) and (ii). Assume that \frac{1}{p(x_{0})} <

α < \frac{1}{p′(x_{0})}.

Then for each f ∈ L^{∘}_{ρ}^{p(·)} the divergence equation

div u = f

is solvable in vectorial W_{ρ}^{1,p(·)} space and the estimate

∥∇u∥_{L_{ρ}^{p(·)}} ≤ c∥f∥_{L_{ρ}^{p(·)}}

holds.

References

1. L. Diening and M. Ružićka, Calderon–Zygmund operators in generalized Lebesgue spaces L^{p(·)} and problems related to fluid dynamics. Abert-Ludwig Universität Freiburg, Preprint Nr. 21/2002–04.07.2002.

G. Kvinikadze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH PROPERTY A

The differential equation

u^{(n)}(t) + ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)}u(s) d_{s}r_{i}(s, t) = 0  (1)

is considered, where n ≥ 2, σ_{i}, τ_{i} ∈ C(R_{+}; (0, ∞)), τ_{i}(t) ≤ σ_{i}(t), while the functions r_{i} : R_{+} × R_{+} → R are nondecreasing in the first argument and Lebesgue integrable in the second argument on any finite subsegment of [0, +∞).

We say that equation (1) has Property A if any of its solutions is oscillatory when n is even and either is oscillatory or satisfies

|u^{(i)}(t)| ↓ 0 for t ↑ +∞ (i = 0, …, n − 1)

when n is odd. Sufficient conditions are established for equation (1) to have Property A. The obtained results are new even for a special case of equation (1) – the differential equation with deviated arguments.

L. Malaguti^{∗}, C. Marcelli^{∗∗}, and N. Partsvania^{∗∗∗}

∗University of Modena and Reggio Emilia, Reggio Emilia, Italy ∗∗University of Ancona, Ancona, Italy

∗∗∗A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON TRANSITIONAL SOLUTIONS OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS

We consider the differential equation

u′′ = f(t, u, u′),  (1)

where f : R × [0, 1] × R → R is a continuous function such that f(t, 0, 0) = f(t, 1, 0) = 0 for t ∈ R, and for this equation we study the problem on the existence of at least one so-called transitional solution, i. e. a solution satisfying the conditions

lim_{t→−∞} u(t) = 0, lim_{t→+∞} u(t) = 1, 0 ≤ u(t) ≤ 1 for t ∈ R.  (2)

On the basis of I. Kiguradze’s theorem from [1] new and in a certain sense optimal conditions are obtained for the solvability of problem (1), (2). Some of these results generalize the previous ones from [2] and [3].

References

1. I. T. Kiguradze and B. L. Shekhter, Singular boundary value problems for second-order ordinary differential equations. (Russian) Itogi Nauki i Tekhniki, Sovrem. Probl. Mat., Noveĭshie Dostizh. 30(1987), 105–201; translated in J. Soviet Math. 43(1988), No. 2, 2340–2417.

2. Yu. A. Klokov, Boundary value problems with a condition at infinity for equations of mathematical physics. (Russian) Riga, 1963.

3. L. Malaguti and C. Marcelli, Heteroclinic orbits in plane dynamical systems. Arch. Math. (Brno) 38(2002), No. 3, 183–200.

M. Mania^{∗} and R. Tevzadze^{∗∗}

∗A. Razmadze Mathematical Institute, Georgian Academy of Sciences

Tbilisi, Georgia ∗∗Institute of Cybernetics, Georgian Academy of Sciences Tbilisi, Georgia

AN EXPONENTIAL MARTINGALE EQUATION

Let (Ω, F, P) be a probability space with continuous filtration F = (F_{t}, t ∈ [0, T]). Denote by M a stable subspace of the space of square integrable martingales and let M^{⊥} be its strongly orthogonal complement.

We consider the following exponential equation

E_{T} (m)E_{T}^{α}(m^{⊥}) = c exp{η},  (1)

where η is a given F_{T} -measurable random variable and α is a given real number. A solution of equation (1) is a triple (c, m, m^{⊥}), where c is strictly positive constant, m ∈ M and m^{⊥} ∈ M^{⊥}. Here E(X) is the Doleans–Dade exponential of X.

Equations of such type arose in mathematical finance and they are used to characterize optimal martingale measures.

Our aim is to prove the existence of a unique solution of equation (1) for arbitrary α ≠ 0 and η of a general structure, assuming that it satisfies the following boundedness condition:

A) η is an F_{T} -measurable random variable of the form η = ξ + γA_{T} , where ξ ∈ L^{∞}, γ is a constant and A = (A_{t}, t ∈ [0, T]) is a continuous F - adapted increasing process such that E(A_{T} − A_{τ} /F_{τ}) ≤ C for all stopping times τ for a constant C > 0.

Note that if α ≠ 0 (if α = 0, solution of (1) does not exist in general), equation (1) is equivalent to the following semimartingale backward equation with the square generator

Y_{t} = Y_{0} − \frac{γ}{2} A_{t} − ⟨L⟩_{t} − \frac{1}{α} ⟨L^{⊥}⟩_{t} + L_{t} + L_{t}^{⊥}, Y_{T} = \frac{1}{2} ξ.  (2)

Theorem 1. Let α ≠ 0 and condition A) be satisfied. Then there is a constant γ_{0} > 0 such that for any |γ| ≤ γ_{0} there exists a unique triple (c, m, m^{⊥}), where c ∈ R_{+}, m ∈ BMO ∩ M, m^{⊥} ∈ BMO ∩ M^{⊥}, that satisfies equation (1).

If the filtration generated by 2-dimensional Wiener processs (w, w^{⊥}) and

ξ = 2g(w_{T} , w_{T}^{⊥}), A_{t} = ∫_{0}^{t} f(s, w_{s}, w_{s}^{⊥})ds for some continuous, bounded func-

tions g(x, y), f(s, x, y), equation (2) is equivalent to the PDE

V_{t}+ \frac{1}{2} V_{xx}+ \frac{1}{2} V_{yy} +|V_{x}|^{2}+ \frac{1}{α} |V_{y}|^{2}+ \frac{γ}{2} f(t, x, y)=0, V (T, x, y)=g(x, y) (3)

and solutions of (2) and (3) are related as Y_{t} = V (t, w_{t}, w_{t}^{⊥}).

T. Tadumadze^{∗} and I. Ramishvili^{∗∗}

∗I. Vekua Institute of Applied Mathematics, Tbilisi, Georgia ∗∗Georgian Technical University, Tbilisi, Georgia

FORMULAS OF VARIATION OF SOLUTION FOR QUASI-LINEAR CONTROLLED NEUTRAL DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS AND CONTINUOUS

INITIAL CONDITIONS

Formulas of variation of solution play an important role in proving necessary conditions of optimality for optimal problem [1].

In the present work, formulas of variation of solution for the neutral controlled differential equation

ẋ(t) = ∑_{j=1}^{k} A_{j}(t)ẋ(η_{j}(t)) + f(t, x(τ_{1}(t)), …, x(τ_{s}(t)), u(θ_{1}(t)), …, u(θ_{ν}(t)))

with the discontinuous (continuous) initial condition

x(t) = φ(t), t < t_{0}, x(t_{0}) = x_{0} ( x(t) = φ(t), t ≤ t_{0} )

are obtained.

Finally we note that the formulas of variation of solution for various classes of neutral differential equations are given in [2]–[6].

References

1. R. Gamkrelidze, Principles of optimal control theory. Mathematical Concepts and Methods in Science and Engineering, Vol. 7. Plenum Press, New York-London, 1978.

2. N. Gorgodze and T. Tadumadze, Differentiability of solutions of neutral type quasi-linear differential equations with respect to the initial data and righthand side. Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics 13(1998), No. 3, 17–20.

3. G. Kharatishvili, T. Tadumadze, and N. Gorgodze, Continuous dependence and differentiability of solution with respect to initial data and right-hand side for differential equations with deviating argument. Mem. Differential Equations Math. Phys. 19(2000), 3–105.

4. N. Gorgodze, Local Representations for the variations of solutions of some class neutral differential equations. Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics 15(2000), No. 3, 58–61.

5. I. Ramishvili, Formulas of variation of solution for quasi-linear controlled neutral differential equations. Mem. Differential Equations Math. Phys. 29(2003), 156–159.

6. T. Tadumadze and I. Ramishvili, Formulas of variation of solution for quasilinear controlled neutral differential equations with continuous initial condition. Georgian Math. J. 11(2004) (to appear).

Z. Tsintsadze

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

NECESSARY CONDITIONS OF OPTIMALITY TO LINEAR CONTROL SYSTEMS IN THE PRESENCE OF MIXED AND

INTERMEDIATE RESTRICTIONS

We consider the problem

I = ∫_{t_{0}}^{t_{1}} f_{0}(x(t), u(t)) dt → inf  (1)

under the restrictions

ẋ(t) = f(x(t), u(t)),  (2)
g(x(t), u(t)) ≤ 0,  (3)
q(x(t_{(1)}), …, x(t_{(N)})) = 0,  (4)

where [t_{0}, t_{1}] is e fixed segment from R^{1}, x = x(t) ∈ W_{1,1}^{n}[t_{0}, t_{1}], u = u(t) ∈ L_{1}^{r}[t_{0}, t_{1}], f = (f_{1}, ..., f_{n})′, g = (g_{1}, ..., g_{m})′, q = (q_{1}, ..., q_{s})′, t_{(p)} ∈ [t_{0}, t_{1}], p = \overline{1, N},t_{(1)} = t_{0}, t_{(N)} = t_{1}. The functions f, g, q are linear with respect to all arguments. Moreover, the restrictions (2), (3) are fulfilled almost everywhere on [t_{0}, t_{1}] and the restrictions (3) satisfy the conditions of generality, i.e., for any (x, u) satisfying (3) the system of vectors grad_{u}g^{j}(x, u), j ∈ J(x, u) is linearly independent. Here by J(x, u) we denote the set of such indices j (j = \overline{1, m}) for which g^{j}(x, u) = 0.

Since the restrictions (3) are equivalent to equality type restrictions y^{2} + g(x, u) = 0,where y = y(t) ∈ L_{2}^{m}[t_{0}, t_{1}], y^{2} = ((y^{1})^{2}, …, (y^{m})^{2})′, therefore it is easy to see that problem (1)–(4) is a particular case of the general extremal problem considered in [1]. Using Theorem 1 from [1] we obtain the necessary conditions of optimality for problem (1)–(4).

References

1. Z. Tsintsadze, The Lagrange principle of taking restrictions off and its application to linear optimal control problems in the presence of mixed restrictions and delays. Mem. Differential Equations Math. Phys. 11(1997), 105-128.

S E C T I O N II MATHEMATICAL PHYSICS

R. Bantsuri and N. Shavlakadze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

THE CONTACT PROBLEM OF INTERACTION OF ELASTIC BEAM OF VARIABLE RIGIDITY WITH ELASTIC BASIS

A contact problem for anisotropic plate in the shape of angle (−Q < arg z < 0) is investigated when a semi-infinite beam of variable bending rigidity dx^{α} (d > 0, α ≥ 0) leans on one of the edges and the other edge of the angle is free. A distribution of strains in the plate and flexure in the beam must be determined in case when a normal load of intensity P_{0}(x) effects on the beam.

Applied problem is reduced to the Karleman boundary-value problem for a strip, an exact solution for any value α is obtained.

It is proved that an unknown contact normal strain in the vertex of angle for any Q and in case of α ≥ 3 admits the estimate: P (x)−P_{0}(x) = O(x^{α−3}), x → 0 and for large x : P (x) − P_{0}(x) = O(x^{−(1+τ_{0}^{+})}), τ_{0}^{+} > 0.

In case of 2 < α < 3, in the vertex of the angle we get: P (x) − P_{0}(x) = O(x^{−(1+μ_{0}^{−})}), −1 < μ_{0}^{−} < 0.

For α < 2 a normal contact strain can be bounded or unbounded near the point x = 0 and for large x : P (x) − P_{0}(x) = O(x^{−(3−α)}).

Concrete results for different values of Q and also the results in case of an orthotropic plate are obtained.

G. Berikelashvili

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON A NONLOCAL GENERALIZATION OF THE DIRICHLET PROBLEM

Let Ω = {(x_{1}, x_{2}) : 0 < x_{k} < 1, k = 1, 2} be a unit square with a boundary Γ, and let Γ_{1} = {(0, x_{2}) : 0 < x_{2} < 1}, Γ_{∗} = Γ\Γ_{1}, ξ ∈ (0; 1], ε ∈ (0; 1).

Consider the nonlocal boundary value problem

Lu = f(x), x ∈ Ω, u(x) = 0, x ∈ Γ_{∗}, l(u) = 0, 0 < x_{2} < 1,  (1)

where

Lu := ∑_{i,j=1}^{2} \frac{∂}{∂x_{i}} (a_{ij} \frac{∂u}{∂x_{j}} ) − a_{0}u, l(u) := ∫_{0}^{ξ} β(x)u(x) dx_{1}, β(x) := εx_{1}^{ε−1}/ξ^{ε}

and the constant coefficients satisfying the following conditions

∑_{i,j=1}^{2} a_{ij}t_{i}t_{j} ≥ ν(t_{1}^{2} + t_{2}^{2}), ν > 0, a_{0} ≥ 0.

By L_{2}(Ω, ρ) we denote the weighted Lebesgue space; ρ(x) := (x_{1}/ξ)^{ε} for x_{1} < ξ, ρ(x) := 1 for x_{1} ≥ ξ.

We denote by W_{2}^{1}(Ω, ρ) the weighted Sobolev space with the norm

∥u∥_{W_{2}^{1}(Ω,ρ)} = (∥u∥_{L_{2}(Ω,ρ)}^{2} + ∥\frac{∂u}{∂x_{1}} ∥_{L_{2}(Ω,ρ)}^{2} + ∥\frac{∂u}{∂x_{2}} ∥_{L_{2}(Ω,ρ)}^{2})^{1/2}.

Define the subspace of the space W_{2}^{1}(Ω, ρ) which can be obtained by closing the set

C^{∗}^{∞}(\overline{Ω}) = \{u ∈ C^{∞}(\overline{Ω}) : supp u ∩ Γ_{∗} = ∅, l(u) = 0, 0 < x_{2} < 1\}

with the norm ∥ · ∥_{W_{2}^{1}(Ω,ρ)}. Denote it by W^{∗} _{2}^{1}(Ω, ρ).

Let the right-hand side f(x) in equation (1) be a linear continuous func-

tional on W^{∗} _{2}^{1}(Ω, ρ) which can be represented as

f = f_{0} + ∂f_{1}/∂x_{1} + ∂f_{2}/∂x_{2}, f_{k}(x) ∈ L_{2}(Ω, ρ), k = 0, 1, 2.

Theorem 1. Problem (1) has a unique solution from W^{∗} _{2}^{1}(Ω, ρ).

By passing to the limit ξ → 0 the nonlocal condition l(u) = 0 transforms to u(0, x_{2}) = 0, while Theorem 1 to the well-known theorem on the existence and uniqueness of a solution of the Dirichlet problem. In this sense, the nonlocal problem (1) can be regarded as a generalization of the Dirichlet boundary value problem.

G. Chavchanidze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

NON-NOETHER SYMMETRIES IN HAMILTONIAN SYSTEMS

We discuss some geometric aspects of the concept of non-Noether symmetry. It is shown that in regular Hamiltonian systems such a symmetry canonically leads to a Lax pair on the algebra of linear operators on cotangent bundle over the phase space. Correspondence between the non-Noether symmetries and other wide spread geometric methods of generating conservation laws such as bi-Hamiltonian formalism, bidifferential calculi and Frölicher–Nijenhuis geometry is considered. It is proved that the integrals of motion associated with the continuous non-Noether symmetry are in involution whenever the generator of the symmetry satisfies a certain Yang–Baxter type equation. Non-Noether symmetries of the Toda model, nonlinear Schödinger equation and Korteweg-de Vries equations (KdV and mKdV) are discussed. It appears that these symmetries yield the complete sets of conservation laws in involution and lead to the bi-Hamiltonian realizations of the above mentioned models.

O. Chkadua

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ASYMPTOTICS OF SOLUTIONS OF MIXED BOUNDARY VALUE PROBLEMS OF ELECTROELASTICITY NEAR THE

CUSPIDAL EDGES

The asymptotics of solutions of spatial mixed boundary value problems of statics of the electroelasticity theory is established for anisotropic homogeneous media in domains having boundaries with closed cuspidal edges (dihedral angle is equal to 2π).

In this case we take into account the piezoelectric effect.

The complete asymptotics of solutions are considered near the cuspidal edge.

The properties of exponent of the first term of asymptotic expansion of solutions are established in the special class of the transversally-isotropic piezoelectric bodies.

A. Dzhishkariani

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

PROJECTIVE METHODS FOR THE ELLIPTIC BOUNDARY VALUE PROBLEMS

Projective methods originate in Ritz’s (1908) and Galerkin’s (1915) works. These methods have been developed in the Friedrichs, Kantorovich, Michlin, Vainikho, Marchuk, Agoshkov, Siarle, Strang, Fix and others works.

In 1963, we obtained the general estimate for the equation Au = f, u ∈ D(A), f ∈ H,

∥u − u_{n}∥_{L_{2}(Ω)} ≤ Cλ_{n+1}^{−1}∥Au_{n} − f∥,

A is a self-adjoint positive definite operator in the Hilbert space H ≡ L_{2}, 0 < λ_{1} ≤ ⋯ ≤ λ_{n} → ∞, n → ∞, Bφ_{k} = λ_{k}φ_{k}, k = 1, 2, …, D(B) = D(A), ∥Au_{n} − f∥ → 0, n → ∞, φ_{k} are the basic functions.

The basic estimate of F.E.M. (Strang, Fix, Siarle, etc.) is

∥u − u_{h}∥_{H_{A}} = O(h^{k−m}),

where h is a mesh size, 2m is the order of differential operator A, u ∈ W_{2}^{k}(Ω), k − 1 is the order of finite polynomials, H_{A} is the energetic space. By us was obtained the estimate (DAN USSR, 1980) O(h^{k−s}), m < s ≤ q, when u_{h} ∈ W_{2}^{q}(Ω).

In 1993, Porter and Stirling proposed projective-iterative scheme (cyclic) for the second kind equation u + T u = f, u, f ∈ H.

This scheme for the equation Au + Ku = f, u ∈ D(A), f ∈ H, is generalized by us, when the form of inverse operator A is known. The estimates

∥u − u_{h,l}∥_{L_{2}(Ω)} = O(h^{k+m(l+1)}),
∥u − u_{h,l}∥_{C(Ω)} = O(h^{k+m(l+1)− \frac{p}{2}} ), u_{m} > p, k ≥ 2m,

where u ∈ W_{2}^{k}(Ω), l is a number of cycle, p is a dimension of the domain Ω, are obtained.

G. Jorjadze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

POISSON STRUCTURE OF WZNW THEORY

We investigate local and causal character of non-equal time Poisson brackets of SL(2, R) WZNW model and its coset theories. The quantum realisation is discussed.

D. Kapanadze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ASYMPTOTICS OF POTENTIALS ON MANIFOLDS WITH EDGE SINGULARITIES

Boundary value problems on manifolds with conical singularities or edges contain potential operators as well as trace and Green operators which play a similar role as the corresponding operators in (pseudo-differential) boundary value problems on a smooth manifold. There is then a specific asymptotic behaviour of these operators close to the singularities. We characterise potential operators in terms of actions of cone or edge pseudo-differential operators (in the neighbouring space) on densities supported by submanifolds which also have conical or edge singularities. As a byproduct we show the continuity of such potentials as continuous operators between cone or edge Sobolev spaces and subspaces with asymptotics.

G. Kapanadze

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

ON THE PROBLEM OF SEARCH FOR SMOOTH STIFF CONTOUR IN CASE OF PLATE BENDING

The article studies the problem of search for smooth stiff contour inside the square, when the plate is bent by principal normal bending moments of the same value, fixed at the square sides. The searched contour is free of external loading and it must be determined on condition that the tangential normal moment at it gets the constant V, o value and the cutting force, normal and twisting moments equate with zero. The methods of conformal reflections and linear conjugation boundary problem theory are applied to study the task. On their basis it is proved that in this case under examination, one of the complex potentials is a linear function and the other complex potential and the equation of searched contour is formulated effectively (in analytical terms). A formula is worked out for the parameter V, o and the limits of its variation and the type of contour (to some extent of approximation) for various values of the mentioned parameter are determined.

G. Khatiashvili

N. Muskhelishvili Institute of Computational Mathematics, Georgian Academy of Sciences Tbilisi, Georgia

DIRICHLET’S AND NEUMAN’S PROBLEMS FOR LAYERED ELLIPTIC MEDIUMS

In the first part, there are considered the Dirichlet’s and Neuman’s plane problems for isotropic and anisotropic mediums, with exterior elliptic boundary and interfaces, occupying following domains: a) two-layered elliptic ring, b) two-layered elliptic ring strenghtened by isotropic or anisotropic solid elliptic kernel, c) isotropic or anisotropic infinite medium (environment) strenghtened by indicated layered elliptic ring.

In the second part, the obtained results are generalized for the multilayered elliptic ring and infinite medium (environment). It is assumed that each layer is made from materials with different physical characteristics and all ellipses are confocal.

All problems are solved using Faber’s polynomials.

J. Peradze

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

THE SOLVABILITY OF A NONLINEAR INITIAL BOUNDARY VALUE PROBLEM FOR A BEAM AND THE ALGORITHM OF CONSTRUCTION OF ITS APPROXIMATE

SOLUTION

The problem

\begin{matrix}\frac{∂^{2}w}{∂t^{2}} = (cd − a + b ∫_{0}^{1} (\frac{∂w}{∂x} )^{2} dx) \frac{∂^{2}w}{∂x^{2}} − cd \frac{∂ψ}{∂x} , \\ \frac{∂^{2}ψ}{∂t^{2}} = c \frac{∂^{2}ψ}{∂x^{2}} − c^{2}d(ψ − \frac{∂w}{∂x} ), \\ 0 < x < 1, 0 < t ≤ T, \end{matrix}  (1)

\frac{∂^{s}w}{∂t^{s}} (x, 0) = w^{s}(x), \frac{∂^{s}ψ}{∂t^{s}} (x, 0) = ψ^{s}(x), s = 0, 1,  (2)
w(0, t) = w(1, t) = 0, \frac{∂ψ}{∂x} (0, t) = \frac{∂ψ}{∂x} (1, t) = 0,  (3)

describing the vibration of a beam with hinged ends, is considered in the Timoshenko model. Here a, b, c, d, T are the given positive constants, cd − a > 0, and w^{s}(x), ψ^{s}(x) are the given functions, s = 0, 1.

Assuming that w^{s}(x) and ψ^{s}(x) are analytic functions of the forms

w^{s}(x) = ∑_{i=1}^{∞} a_{i}^{s} sin iπx, ψ^{s}(x) = \frac{b_{0}^{s}}{\sqrt{2}} + ∑_{j=1}^{∞} b_{j}^{s} cos jπx, s = 0, 1,

and applying S. Bernstein’s approach, it is proved that there exist functions w(x, t) and ψ(x, t) which are a solution of problem (1)–(3) and which are analytic functions with respect to x for all 0 < t ≤ T.

Further, a numerical algorithm of finding a solution of problem (1)–(3) is proposed. This algorithm consists in representing the solution as

w_{n}(x, t) = ∑_{i=1}^{n} w_{ni}(t) sin iπx, ψ_{n}(x, t) = \frac{ψ_{n0}(t)}{\sqrt{2}} + ∑_{j=1}^{n} ψ_{nj}(t) cos jπx

and using Galerkin’s method. As a result, we obtain the Cauchy problem for a system of ordinary differential equations with respect to functions w_{ni}(t) and ψ_{nj}(t), i = 1, 2, …, n, j = 0, 1, …, n, which is solved by means of the Crank–Nicholson implicit difference scheme. Therefore on the time grid layers we have to solve a system of nonlinear algebraic equations. For this we use the Picard iteration method. The error of each of three constituent parts of the algorithm is estimated.

L. Shapakidze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

THE EFFECT OF A TRANSVERSE PRESSURE GRADIENT

ON THE STABILITY OF FLOW BETWEEN TWO PERMEABLE CYLINDERS

Secondary regimes arising after the loss of stability of flow between rotating permeable cylinders in the presence of transversal pressure gradient are studied.

It is stated that after the loss of stability of the main flow, depending both on the pressure gradient and on radial Reynold’s number, there may take place either rotating-symmetrical stationary flows, or three-dimensional autooscillations with different period in azimuthal direction.

J. Sharikadze

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

EQUATIONS OF DYNAMIC, TEMPERATURE AND MAGNETIC BOUNDARY LAYERS AND THEIR INTEGRATION

Space in which usually happens the action of magnetic field on the motion of conducting fluid for the large meaning of Reynolds magnetic number is called magnetic boundary layer.

We have two kinds of magnetic boundary layer. Vector tensity of magnetic field in the first kind of magnetic boundary layer is in the plane of vector velocity. And the angle between the magnetic field and the velocity is small. Magnetic field in the second kind of magnetic boundary layer is normal for the both components of vector velocity.

We can write equations of dynamic, temperature and magnetic boundary layers for the both cases:

\frac{∂u}{∂t} + u\frac{∂u}{∂x} + v \frac{∂u}{∂y} = \frac{∂u_{∞}}{∂t} + u_{∞} \frac{∂u_{∞}}{∂x} + ν \frac{∂^{2}u}{∂y^{2}} + \frac{α}{ρμ_{0}} (B_{x} \frac{∂B_{x}}{∂x} + B_{y} \frac{∂B_{x}}{∂y} ),

ρc_{τ} (\frac{∂T}{∂t} + u\frac{∂T}{∂x} + v \frac{∂T}{∂y} ) = λ\frac{∂^{2}T}{∂y^{2}} + η(\frac{∂u}{∂y})^{2} + \frac{1}{σμ_{0}^{2}} (\frac{∂B_{x}}{∂y} )^{2},

\frac{∂B_{x}}{∂t} + u\frac{∂B_{x}}{∂x} + v \frac{∂B_{x}}{∂y} = α(B_{x} \frac{∂u}{∂x} + B_{y} \frac{∂u}{∂y}) + ν_{m} \frac{∂^{2}B_{x}}{∂y^{2}} ,

\frac{∂u}{∂x} + \frac{∂v}{∂y} = 0, α(\frac{∂B_{x}}{∂x} + \frac{∂B_{y}}{∂y} ) = 0,

and conditions:

u = v = 0, T = T_{w}(x, t), B_{x} = B_{w}(x, t), y = 0,
u = u_{∞}(x, t), T = T_{∞} = const, B_{x} = B_{y} = 0, y → ∞,

u = v = T = B_{x} = B_{y} = 0, t = 0; B_{w}(x, 0) = u_{∞}(x, 0) = T_{w}(x, 0) = 0.

In the case, where α = 1, we have the equations of the first kind of magnetic boundary layer, and in the case, where α = 0 and B_{x} = B_{z}, we have the equations of the second kind of magnetic boundary layer.

There are two methods of solving a system of equations in the report. They are based on the asymptotical approximation of boundary layer and on the method of finiteness thickness of boundary layer.

In the first case the solution of the systems reduce to the solution of an integro-differential equation, in which the method of successive approximations is used. In the second case method of successive approximations helps us to determine thickness of boundary layer: δ_{u}, δ_{T}, δ_{B}. For them we may write the system of partial equations of the first order.

All physical properties of boundary layer are found.

M. Svanadze

I. Vekua Institute of Applied Mathematics Tbilisi, Georgia

POTENTIAL METHOD IN THE THEORY OF THERMOELASTICITY WITH MICROTEMPERATURES

The linear theory for elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures was constructed by D. Iesan and R. Quintanilla.

In this paper, the fundamental solution of the system of the equations of steady oscillations is constructed in terms of elementary functions. The Sommerfeld-Kupradze type radiation conditions are established. The existence of eigenfrequencies of the interior homogeneous boundary-value problems of steady oscillations is studied. The basic properties of wave numbers are treaded. The connection between plane waves and eigenfrequencies is established. Theorems on the uniqueness and existence of a solution of boundary-value problems by means of the potential method and the theory of multidimensional singular integral equations are proved.

A. Tsitskishvili and R. Tsitskishvili

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON THE CONSTRUCTION OF SOLUTIONS OF CERTAIN SPATIAL AXISYMMETRIC MIXED PROBLEMS OF

FILTRATION WITH PARTIALLY UNKNOWN BOUNDARIES

The axis of symmetry is assumed to be the x-axis directed downwards, the distance to the x-axis is denoted by y, and the velocity vector is expressed as follows: V⃗ (u, v) = grad φ. The conditions for incompressibility and potentiality of a moving liquid are of the form div (yV⃗) = 0 (1), \frac{∂u}{∂y} − \frac{∂v}{∂x} = 0 (2). The streamline equation vdx − udy = 0 multiplied by y becomes the exact differential of the stream function ψ(x, y), where u = \frac{∂φ}{∂x} = y^{−1} \frac{∂ψ}{∂y} (3), v = \frac{∂φ}{∂y} = −y^{−1} \frac{∂ψ}{∂x} (4). The functions φ(x, y) and ψ(x, y) with respect to y are even functions (surface of rotation). Therefore y^{−1} \frac{∂φ}{∂y} and y^{−1} \frac{∂ψ}{∂y} tend to finite limits, as y → ∞. The domain S(z) with the boundary ℓ(z), occupied by a moving liquid we combine with the plane z = x + iy. The boundary ℓ(z) consists of an unknown curve and known segments, lines and of their portions. We seek for the functions ω(z) = φ(x, y)+iψ(x, y), w(z) = u(x, y)+iv(x, y), where φ(x, y) and ψ(x, y) must satisfy equations (3),(4) and also the following boundary conditions: ψ(x, y) = const along nonpermeable boundaries; φ(x, y) = const along water boundaries; φ(x, y)− kx = const, ψ(x, y) = Q, k = const, Q = const along an unknown curve; φ(x, y)− kx = const along the leaking interval; ψ(x, y) = 0, (x, y) ∈ ℓ(z), along the axis of symmetry. To the above-mentioned conditions are added to the corresponding equations of curves.

Let us consider the right half of the plane axisymmetric domain S_{0}(z) with the boundary ℓ_{0}(z) which coincides with the domain S(z) with the boundary ℓ(z) and respectively with the boundary conditions. The equation of depression curve in the plane problem must be the function y^{2}. Assume that to the domain S_{0}(z) there correspond the domains S(ω_{0}) and S(w_{0}) of a complex potential ω_{0}(z) = φ_{0}(x, y) + iψ_{0}(x, y) and of a complex velocity w_{0}(z) = \frac{dω_{0}(z)}{dz}. Geometrical characteristics respectively of the domains S(z) = S(z_{0}), S(ω) = S(ω_{0}), S(w) = S(w_{0}) and boundaries ℓ(z) = ℓ(z_{0}), ℓ(ω) = ℓ(ω_{0}), ℓ(w) = ℓ(w_{0}) coincide. The boundary conditions are likewise coincide. Here we emphasize that the functions ω_{0}(z) and w_{0}(z) are holomorphic, while the functions ω(z) and w(z) are analytic ones.

The half-plane Im (ζ) > 0 of the plane ζ = ξ + iη is conformally mapped onto the domains S(z_{0}), S(ω_{0}) and S(w_{0}). Conformally mapping functions we denote by z_{0}(ζ), ω_{0}(ζ) and w_{0}(ζ). Generalized functions φ(ξ, η) and ψ(ξ, η) must satisfy the equations \frac{∂φ}{∂ξ} = y^{−1} \frac{∂ψ}{∂η} (5), \frac{∂φ}{∂η} = y^{−1} \frac{∂ψ}{∂ξ} (6), or Δφ(ξ, η) + y^{−1}( \frac{∂y}{∂ξ} \frac{∂φ}{∂ξ} + \frac{∂y}{∂η} \frac{∂φ}{∂η}) = 0 (7), Δψ(ξ, η) − y^{−1}( \frac{∂y}{∂ξ} \frac{∂ψ}{∂ξ} + \frac{∂y}{∂η} \frac{∂ψ}{∂η}) = 0 (8). We consider it possible to construct holomorphic functions z_{0}(ζ), ω_{0}(ζ), w_{0}(ζ) efficiently. A solution of

(7) (analogously, of (8)) is sought in the form φ(ξ, η) = ∑_{k=0}^{∞} φ_{k}(ξ, η) (9), where

φ_{0}(ξ, η) (analogously, ψ_{0}(ξ, η)) is known, φ_{k}(ξ, η) is defined by the Poisson formula φ_{k}(ξ, η) = ∫∫_{Im(ζ)≥0} G(ξ, η; ξ_{1}, η_{1})φ_{k−1}(ξ,_{1} , η_{1})dξ_{1}dη_{1}, k = \overline{1, ∞} (10), where G is Green’s function for the half-plane with a coefficient. It remains to prove the convergence of series (9). Equation (8) is treated analogously.

T. Vashakmadze

I. Vekua Institute of Applied Mathematics Tbilisi, Georgia

TO SOME THERMO–DYNAMICAL NONLINEAR SPATIAL PROBLEMS OF PIEZO–PORO–ELASTICITY FOR BINARY

MIXTURE

The main objective of this report is construction and justification of the new mathematical models for anisotropic nonhomogeneous visco-poro-elastic media and their application in case of thin-walled structures with variable thickness in thermodynamic and stationary nonlinear problems of determination of stress-strain state. Particularly it will be studied the case, when thin-walled structure is poro-piezo-electric and electrically conductive elastic binary mixture. For thermo-dymanical and stationary cases by using and developing discrete projective methods will be constructed computational models and on their basis may be created effective program systems. For dynamic problems will be considered the finite and semi-finite timedepending cases. When the interval is finite, the third and fourth order accuracy new schemes of Rogava, Gegechkori and Tsiklauri for thermoporo-elastic structures will be used. In the semi-finite case using Laguerre polynomial system we consrtuct a new stable implicit and explicit schemes, which are convergent unlike the classical method for relatively non -smooth classes of functions. Spatial mathematical models considered here generalize the well-known theories, e.g. contemporary theories of Biot, Frenkel, Novojilov, Naghdi, Antman. On the basis of the above-mentioned spatial mathematical models there are constructed a system of nonlinear equations of von Karman-Mindlin-Reissner type for anisotropic nonhomogeneous piezoporo-elastic thin-walled structures with variable thickness and justified their ”Physical Soundness”(in Truesdell-Ciarlet sense). This question is yet an unsolved problem even in the case of von Karman equations’ system for thin isotropic homogeneous plate with a constant thickness. Finally there will be reported results respect V-elliptic of some above considering linear stationary systems of differential equations, in particular, for systems of equations corresponding to spatial anisotropic theory of elasticity.

D. Zeragia

N. Muskhelishvili Institute of Computational Mathematics, Georgian Academy of Sciences Tbilisi, Georgia

CONSTRUCTION OF SMALL SOLUTIONS OF THE BOUNDARY VALUE PROBLEM FOR A SECOND ORDER NONLINEAR DIFFERENTIAL EQUATION CONNECTED WITH TRANSVERSE-LONGITUDINAL ROD BENDING

We consider the problem on bifurcation of equilibrium forms of the axis of a rod whose one end is clamped and the other end is free, loaded with constant compressing force P and constant distributed force of intensity γ.

In the given case the bent rod axis is described by the nonlinear differential equation with two real parameters P and γ and zero boundary conditions (see [1], [2]).

The case, in which an auxiliary boundary value problem is Fredholmian one with a number of zeros n = 1, is considered. The Liapunov–Schmidt branching equation, taking in this case the form of equation with one unknown and two real parameters, is investigated. Solutions of equation of branching allowing one to construct approximate solutions of the initial boundary value problem are given in the first approximation.

The problems on the existence of points of bifurcation and continuous branches-solutions of that boundary value problem, when the force of axial compression is proportional to the transverse load (P = mγ, m = const > 0), are studied. For finding quantities of critical forces the functionals are constructed. Asymptotical representations of small solutions, expressing an approximate type of supercritical deformation of the rod, are given.

References

1. V. V. Eliseev, D. P. Zeragia, // Proc. Georgian Polytech. Inst., 1989, No. 6(348), 148–150.

2. V. V. Eliseev, Mechanics of elastic rods. St.-Petersburg State Techn. Univ., St.-Petersburg, 1994.

Author Index

Alkhazishvili, L., 3 Ashordia, M., 4 Bantsuri, R., 25 Berikelashvili, G., 26 Chavchanidze, G., 27 Chkadua, O., 28 Dzhishkariani, A., 29 Gegechkori, Z., 5 Giorgadze, G., 6 Gvazava, J., 7 Jangveladze, T., 8 Jorjadze, G., 30 Kapanadze, D., 31 Kapanadze, G., 32 Kekelia, N., 4 Kharatishvili, G., 9 Kharibegashvili, S., 10 Khatiashvili, G., 33 Khimshiashvili, G., 11 Khmaladze, E., 12 Kiguradze, I., 13 Kiguradze, T., 14 Kiguradze, Z., 8 Klebanskaya, M., 15 Kokilashvili, V., 16 Kvinikadze, G., 17 Malaguti, L., 18 Mania, M., 19 Marcelli, C., 18 Partsvania, N., 18 Peradze, J., 34 Ramishvili, I., 20 Rogava, J., 5 Shapakidze, L., 35 Sharikadze, J., 36 Shavlakadze, N., 25 Svanadze, M., 37 Tadumadze, T., 20 Tevzadze, R., 19 Tsiklauri, M., 5 Tsintsadze, Z., 21 Tsitskishvili, A., 38 Tsitskishvili, R., 38 Vashakmadze, T., 39 Zeragia, D., 40