In Dunkl theory on \BbbR n which generalizes classical Fourier analysis, we study the solution of the Klein–Gordon-equation defined by: Მ²_t u - Δ_ku = - m²u, u(x, 0) = g(x), Მ_tu(x, 0) = f(x), где m > 0, а Მ²_t u denoting the second derivative of the solution u with respect to t, and Δ_ku u the Dunkl  Laplacian with respect to x where f and g being two functions  in Ѕ(Rⁿ), defining  the initial conditions. An integral representation for its solution is obtained, which makes it possible to study certain properties. As a specific result, the energies associated with the Dunkl–Klein–Gordon equation are studied.

For a time-fractional wave equation with an integral term of the convolution type, we study the direct Cauchy problem and the inverse problem of finding a multidimensional kernel of the integral, depending not only on the time variable, but also on the first n — 1 components of the spatial variable x = (x1,x₂,... ,x_n) e Rⁿ. In this case, the known problems are the Cauchy data specified at time t = 0 and the redefinition condition on the hyperplane x_n = 0. Problems are equivalently reduced to problems that are convenient for further study. Using the fundamental solution of the time-fractional wave operator, which contains the generalized hypergeometric Fox function, the solution to the direct problem is written in the form of an integral equation of Volterra type and its properties are studied. Using the results of the direct problem, the solution to the inverse problem is also represented as a nonlinear integral equation. By applying the contraction mapping principle to this equation, the local solvability of the problem is established.

We study an inverse problem for a time-fractional diffusion equation with initial-boundary and overdetermination conditions.This inverse problem aims to determine a time varying coefficient and source in the equation with overdetermination integral conditions. First, we establish the unique existence of the classical solution using the Fourier method, Gronwall inequality for direct problem. Second, by using the fixed point theorem in Banach space, the local existence and uniqueness of this inverse problem are obtained. To verify the theoretical results, a numerical solution to the problem was constructed using the finite difference method. Finally, a numerical example is presented to show the effectiveness of the proposed method.

Let D be a bounded domain in Cⁿ (n > 1) with a real analytic connected boundary dD = Г. The Bochner-Martinelli integral (integral operator) M(f) is considered for real analytic functions f о n Г. It is shown th at the integral M (f) is real analytic up to Г. Iterations of the Bochner-Martinelli integral M^k(f) are considered. It is proved that they converge to a function holomorphic in D at k ^ to. The Bochner-Маrtinelli transform M(T)(z) is defined for analytical functionals T. It is proved that the iterations of M^k(T)(z) converge weakly to a CR-functional at k ^ to.

In this paper, a boundary value problem for a mixed integro-differential equation is considered. The unique solvability of the posed problem is proved. The proof is based on the spectral method.

The set of solvable linear differential equations is expanded, cases when solutions can be constructed are highlighted.

The behavior of trajectories of solutions to piecewise linear second order differential equations is being studied. These equations are widely used in mechanics, electrical engineering and automatic control theory. Of particular interest are the conditions for the emergence of limit cycles in the vicinity of the rest region of a second order piecewise linear differential equation with a discontinuous switching line. It has been established that if a region of rest (consisting of rest points) exists, then it remains inside the limit cycle. One of the primary tasks is to determine the region of rest that appears on the line of stitching solutions. In the course of the work, new relations were obtained that provide limited solutions to piecewise linear equations. Using these new conditions, phase portraits are constructed that take into account the coefficients of the equations. Conditions have also been found under which there is no rest region. To solve these problems, the method of stitching solutions from two half-planes was used.