A hypercomplex method for solving the piecewise-continuous Dirichlet problem for the Lamé system in domains with corner points has been developed, which is based on the representation of solutions using the components of a monogenic function in a commutative biharmonic algebra. Using a biharmonic integral of the Cauchy type, the Dirichlet problem has been reduced to a system of integral equations.
We study invariant regions and route dynamics of a four-dimensional discrete Lotka-Volterra operator acting on the simplex S^3 . Using the sign structure of the interaction vector, the simplex is decomposed into invariant signature regions describing different dynamical regimes. It is shown that the relation between the parameters af and cd determines two different geometric configurations of the phase space. In the degenerate case, the simplex decomposes into four invariant regions, while in the nondegenerate case, a more complicated nine-region structure arises. Adjacency relations between neighboring regions are described, and trajectory transitions are determined by coordinate iterations and sign changes. The spectral structure of the interaction matrix is also investigated and connected with oscillatory trajectory behaviour. A biological interpretation of the obtained dynamics is proposed within a discrete SEIRS-type epidemic framework.
The convective Cahn–Hilliard–Oono equation with homogeneous Dirichlet boundary conditions is considered. Local bifurcations near the zero equilibrium state are studied in the case where the stability spectrum of this state is close to a spectrum with a simple zero eigenvalue or a double zero eigenvalue. Sufficient conditions for the existence of spatially inhomogeneous equilibrium states are obtained, and asymptotic formulas for these states are derived. Their stability is examined. The analysis uses methods of the theory of dynamical systems with an infinite-dimensional phase space.
In this work, a boundary value problem with nonlocal conditions for a fourth-order mixed-type equation is considered. A peculiarity of the equation is that its type changes depending on the sign of the time variable, which leads to additional difficulties in studying the correctness of the problem. For the analysis, a method of separation of variables is used, allowing the initial problem to be reduced to a spectral problem for a non-self-adjoint operator. The properties of the eigenfunctions and associated functions that forms a biorthogonal system are studied, and their completeness in the corresponding functional space is determined. Based on the obtained expansions, the solution to the problem is constructed and the uniqueness theorem is proven. The obtained results can be applied to a wide class of nonlocal boundary value problems for equations of mixed type of higher order.
We study the rate of Maxwellization for processes described by the discrete kinetic Broadwell model. When analyzing the conditions for the maximization of such processes, we face difficulties in the stabilization of the Fourier coefficients for indices (k, l) such that (k^2-l^2)l=0 . These difficulties are shown to be caused by an inconsistency between the right- and left-hand sides of the one-dimensional model and by the related “cross” problem in the two-dimensional model. However, this inconsistency can be removed: if, for different groups of particles with equal group velocities, the components of their equilibrium positions satisfy certain relations, then it is possible to obtain uniform estimates for the Fourier coefficients of the solution of the one-dimensional model, which allows one to pass to the weak limit in the definition of a weak solution.
