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Main
research directions:
Ordinary
differential equations (8 Dr's.)
In
the field of ordinary differential equations the investigations are
concentrated around main problems in the theory of nonlinear boundary
value problems, such as the existence of solutions, uniqueness and
multiplicity of solutions, properties of solutions. A specific feature
of the investigations is that a number of the results are sharp, for
instance, the definition of upper and lower functions for the second
order equations, estimates of nonlinearities in superlinear boundary
problems, estimates of the number of solutions in the second order
problems having solutions of oscillatory type (Yu.Klokov, A.Lepin,
F.Sadyrbaev).
A special class of exact solutions for the nonlinear wave equations
has been constructed, and geometrical properties of these solutions
are described (V.Gudkov).
Mathematical modeling of physical processes (3 Dr., 1 eng.)
In
the field of mathematical modeling main efforts are devoted to elaboration
of mathematical models, difference schemes and programs for describing
electro-kinetic processes, flows of gas and liquid matter in non-homogeneous
media with rapidly varying coefficients, and phase transitions.Transport
kinetic models are based on nonlinear Fokker-Planck,diffusion-absortion,
advection-diffusion, Schrodinger and master equations. Within the
framework of the Ginzburg-Landau model the asymptotic long-distance
behaviour of the two-point Green's function below the critical point,
as well as approaching the critical point from low temperatures are
investigated (J.Kaupuzs, J.Rimshans). Mathematical simulation of electrodiffusion
in electrolytes has been done (B.Martuzans, Yu.Skryl).
Theory
of optimal control for partial differential equations (2 Dr., 2 eng.)
In
the field of theory of optimal control problems for partial differential
equations investigations mainly concern various extensions (relaxation,
convexication, G-closure) of original problems. Effective descriptions
for successful extensions for optimal material layout problems in
the case of a single equation are obtained (U.Raitums).
Significant publications
1.V.V.Gudkov.
Geometrical properties of matrix solutions of the nonlinear Klein-Gordon
equation.- J.Phys.A: Math.Gen., 1999, v.32, p.L281-L284.
2.V.V.Gudkov. Matrix solutions of wave equations and Clifford algebras.
- J.Phys.A.: Math.Gen., 2000, v.33, No.39, p.6975-6979.
3.V.V.Gudkov. Torus as a geometrical image of matrix solutions of
wave equations. – Nonlinear Analysis, 2001, v.47, No.9, p.5945-5953.
4.J.Kaupuzs. Critical exponents predicted by grouping of Feynman diagrams
in phi^4 model. - Ann. Phys. (Leipzig), 2001, v.10, No.4, p.299-331.
5.J.Kaupuzs, J.Rimshans. Non steady state 2D numerical simulation
of gas-liquid phase transition in metal vapour. - Computer Assisted
Mechanics and Engineering Sciences, 2000, v.7, No.3, p.413-420.
6.Yu.A.Klokov. On Bernstein-Nagumo conditions for Neumann boundary
value problems for ordinary differential equations.- Diferencial’nye
Uravnenija (Differential Equations), 1998, v.34, No.2, p.184-188 (in
Russian).
7.Yu.A.Klokov, A.P.Mikhailov, M.M.Adjutov. Nonlinear mathematical
models and non-classical boundary value problem for ordinary differential
equations. - Fundamentals of Mathematical Modelling. Series Cybernetics,
Moscow, “Nauka”, 1998, p.98-197 (in Russian).
8.Yu.Klokov, F.Sadyrbaev. Rapid oscillations in sublinear problems.
- Funkcialaj Ekvacioj, 1999, v.42, No.3, p.339-353.
9.Yu.Klokov, F.Sadyrbaev. Sharp conditions for rapid nonlinear oscillations.
- Nonlinear Analysis: TMA, 2000, v.39, p.519 -533.
10.A.Ya.Lepin and A.D.Myshkis. General nonlocal nonlinear boundary
value problem for differential equation of 3rd order. - Nonlinear
Analysis: TMA, 1997, v.28, No.9, p.1533-1543.
11.A.Ya.Lepin, L.A.Lepin, A.D.Myshkis. Two-point boundary value problem
for nonlinear differential equation of nth order. - Nonlinear Analysis:TMA,
2000, v.40, p.397-406.
12.A.Ya.Lepin and A.D.Myshkis. Extension of the Bernstein Condition
to Systems of Ordinary Differential Equations of General Form. - Zeitschrift
für Analysis und ihre Anwendungen (Journal for Analysis and its
Application), 2000, v.19, No.1, p.279-284.
13.A.Lepin, V.Ponomarev. On a singular boundary value problem for
a second order ordinary differential equation. - Nonlinear Analysis:
TMA, 2000, v.42, p.949-960.
14.A.Ya.Lepin, F.Sadyrbaev. The Upper and Lower Functions Method for
Second Order Systems. - Zeitschrift für Analysis und ihre Anwendungen
(Journal for Analysis and its Applications), 2001, v.20, No.3, p.739-753.
15.B.Martuzans, Yu.Skryl. Concentration separation in the binary electrolytes.
- J.Chem.Soc., Faraday Trans., 1998, v.94, No.16, p.2411-2416.
16.B.Martuzans, Yu.Skryl. Hyperbolic diffusion in the liquid junctions
in the moderate dilution approximation. - Latvian Journal of Physics
and Technical Sciences, 2001, v.1, p.28-35.
17.U.Raitums. On the weak closure of sets of feasible states for linear
elliptic equations in the scalar case. - SIAM J. of Control and Optimization,
1999, v.37, No.4, p.1033-1047.
18.U.Raitums. On the local representation of G-closure. – Arch.Rational
Mech. Anal., 2001, v.158, p.213-234.
19.F.Sadyrbaev. Nonlinear boundary value problems of the calculies
of variations. – Discrete and Continuous Dynamical Systems, Supplement
Volume: Dynamical Systems and Differential Equations, Amer.Inst. of
Math. Sciences, 2003, p.760-771.
20.Yu. Skryl. The effect of the hyberbolic diffusion in liquid junction.
– Phys.Chem. Chem.Phys., 2000, No.2, p.2969-2976.
21.O.Zaytsev. On strong closure of the graphs associated with families
of elliptic operators. - Numerical Functional Analysis and Optimization,
1999, v.20, No.3-4, p.395-404.
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