Belan E. P.
Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 21-43
We consider a scalar parabolic equation in the circle of radius r. This problem is a gasless combustion phenomenological model in the surface of a cylinder of $r$ radius. We consider the problems of the existence, asymptotic form and stability of traveling waves and the nature of gaining, losing their stability.
Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 315–328
We investigate central manifolds of quasilinear parabolic equations of arbitrary order in an unbounded domain. We suggest an algorithm for the construction of an approximate central manifold in the form of asymptotically convergent power series. We describe the application of the results obtained in the theory of stability.
Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 11–24
We prove the existence of an m-parameter family of global solutions of a system of difference-differential equations. For difference-differential equations on a torus, we introduce the notion of rotation number. We also consider the problem of perturbation of an invariant torus of a system of difference-differential equations and study the problem of the existence of periodic and quasiperiodic solutions of second-order difference-differential equations.
Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1021-1036
Under certain assumptions, we prove the existence of an m-parameter family of solutions that form the central invariant manifold of a nonlinear parabolic equation. For this purpose, we use an abstract scheme that corresponds to energy methods for strongly parabolic equations of arbitrary order.
Integral manifolds and exponential splitting of linear parabolic equations with rapidly varying coefficients
Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1593–1608
We study linear parabolic equations with rapidly varying coefficients. It is assumed that the averaged equation corresponding to the source equation admits exponential splitting. We establish conditions under which the source equation also admits exponential splitting. It is shown that integral manifolds play an important role in constructing transformations that split the equations under consideration. To prove the existence of integral manifolds, we apply Zhikov's results on the justification of the averaging method for linear parabolic equations.
Ukr. Mat. Zh. - 1982. - 34, № 6. - pp. 738—741
Ukr. Mat. Zh. - 1976. - 28, № 4. - pp. 524–526
Ukr. Mat. Zh. - 1968. - 20, № 5. - pp. 654–660
Ukr. Mat. Zh. - 1968. - 20, № 4. - pp. 449–459
Construction of solutions of almost diagonal systems of linear differential equations using the accelerated convergence method
Ukr. Mat. Zh. - 1968. - 20, № 2. - pp. 166–175
Ukr. Mat. Zh. - 1967. - 19, № 3. - pp. 85–89