This paper is devoted to the study of the problem of unique solvability of a nonlocal problem associated with fractional derivatives in the context of boundary conditions for an equation of mixed elliptic-hyperbolic type. An important feature of the equation under consideration is that its order degenerates along the line of type change. On the elliptic part of the boundary of the domain, we establish the Dirichlet condition, while on the characteristic part of the boundary, the domain specifies a condition that connects the Riemann–Liouville fractional derivatives pointwise with the values of the solution on the characteristics. The order of the derivative depends on the order of degeneration of the equation, as well as on the values of the solution and its derivative on the degeneration line located inside the domain. To prove the uniqueness of the solution to this problem, we apply the principles of extremum. The question of the existence of a solution is reduced to the study of the solvability of a singular integral equation with a Cauchy kernel. The paper also presents a condition that guarantees the existence of a regularizer that allows transforming a singular equation into a Fredholm equation of the second kind. Given the possibility of reducing the problem to an equivalent Fredholm integral equation of the second kind, as well as the proven uniqueness of the desired solution, we can conclude that there is a solution to the problem in the required class of functions.

This paper investigates the solvability of natural initial-boundary value problems for linear second-order hyperbolic equations with involutive argument deviation in higher-order terms. We establish sufficient conditions for the existence and uniqueness of regular solutions - - solutions possessing all generalized derivatives in the sense of S.L. Sobolev that appear in the equations.

A redefined system of integral equations with singular kernels consisting of Volterra type integral equation with a special line and singular integral equation of Vekua type on the lower base of a cylinder is considered, for which an explicit solution is found, conditions of jointness are determined, properties of solutions are studied and Cauchy type problems are posed and solved.

We study the initial-boundary value problem for a third-order mixed-type differential equation. The issues of correctness of the problem statement are considered, and the existence and uniqueness of solutions are analyzed. Theorems on the conditional stability of the solution depending on the correctness set are proved. To obtain approximate solutions, we use a regularization method based on the calculation of a priori estimates. The study uses spectral analysis, which allows us to obtain numerical solutions and evaluate their stability. The results can be useful for further research in the field of mathematical physics and computational mathematics.

The weight w(e) of an edge e in a 3-polytope is the degree-sum of its endvertices. An edge e = uv is an (i, j)-edge if d(u) ≤ i and d(v) ≤ j. In 1940 Lebesgue proved that every 3-polytope has a (3, 11)-edge, or (4, 7)-edge, or (5, 6)-edge, where 7 and 6 are best possible. In 1955, Kotzig proved that every 3-polytope has an edge e with w(e) ≤ 13, which bound is sharp. Borodin (1987), answering Erd˝os’ question of 1976, proved that every plane graph without vertices of degree less than 3 has such an edge. Moreover, Borodin (1991) refined this by proving that there is either a (3, 10)-edge, or (4, 7)-edge, or (5, 6)-edge.

Given a 3-polytope, the minimum weight of all its edges is denoted by w, of those incident with just one 3-face and called semi-weak is w∗, and those incident with two 3-faces and called weak, is w∗∗. Borodin (1996) proved that if w∗∗ = ∞, that is there are no weak edges, then either w∗ ≤ 9 or w ≤ 8, where both bounds are sharp.

Recently, we refined this fact by proving that w∗∗ = ∞ implies either a semi-weak (3, 6)-edge, or semi-weak (4, 4)-edge, or else a strong (3, 5)-edge, which description is tight. (Note that if (3, 5)-edges are allowed, then there may be no 3-faces, and hence semi-weak edges, at all.)

The purpose of our note is to further refine these results by proving that in fact w∗∗ = ∞ implies either a semi-weak (3, 6)-edge, or semi-weak (4, 4)-edge, or a strong (3, 5)-edge incident with a 4-face, or else a strong (3, 3)-edge incident with a 5-face, where no parameter can be improved.

In this short note we provided new sharp results concerning the action of so called integral Toeplits operators in holomorphic Lizorkin-Triebel function spaces in the unit polydisk. Holomorphic function spaces of Lizorkin-Triebel type in the unit polydisk extend well known Bergman and Hardy spaces simultuanously. Because of this it is interesting to study these new spaces in higher dimension. Our results may have various interesting applications in complex function theory. Of one and several complex variables. Our proofs in particular are based on several standard estimates of complex function theory of several complex variables. We also mention that Toeplits operators were mainly studied previously only in one dimension namely in the unit disk case. Our results may have also interesting applications in operator theory.

The model of the formation of linear short capillary waves on the liquid-gas surface under the action of cavitation created by ultrasonic vibrations was proposed. The equations of propagation of capillary waves were constructed in the formulation of classical and generalized functions (of slow growth) that take into account: the viscosity of the liquid phase; attenuation of wave vibrations over time due to the viscosity of the liquid phase, which implies a limited amplitude of the waves (despite the fact that in the absence of attenuation, the wave can oscillate indefinitely over time). It was proved, that for equations in generalized functions for the case of collapse of a set of bubbles in a limited volume of liquid, the displacement profile (as generalized function, with is integral in the sense of the principal Cauchy value) of the interfacial surface is a regular generalized function of slow growth. The estimated dependences of the average increase in the interfacial surface on the parameters of ultrasonic action and the viscosity of the liquid are constructed. The dependences showed an increase in the interfacial surface up to 1.6 times or more for a liquid with a viscosity close to water. The obtained value is similar to the experimental data. The existence of a limiting viscosity has been established, starting from which the effect ceases to be noticeable. This indicates the need for research at different ambient temperatures. Since, on the one hand, with increasing temperature, the viscosity of the liquid phase decreases, and on the other hand, the degree of cavitation development decreases. Apparently, there may be an optimal temperature in this regard.

This work is devoted to the construction of the Green’s function of the classical Dirichlet, Neumann and Robin problems for the Poisson equation in a multidimensional ball, a constructive method for constructing the Green’s function of the Dirichlet problem for a polyharmonic (biharmonic) equation in a multidimensional ball and a description of correct boundary value problems for polyharmonic operators.

An integro-differential equation, which sets the density of one random process applied in biological problems, is considered. The Laplace transform is used to find a fundamental solution to this equation. The formula of the fundamental solution is found explicitly for the case when the kernel of the integral term is a sum of densities of exponential distributions with different parameters. The resulting formula allows to find solutions for different initial data.

For one system of n hyperbolic equations with weakly singular characteristics on the plane, a solution to the Goursat problem is constructed using the Riemann method. The existence and uniqueness of the obtained solution is proved. The Riemann matrix is obtained in the form of a hypergeometric function of matrix arguments. The properties of this function are investigated.

A class of nonlinear identification problems for differential equations in Banach spaces resolved with respect to the highest Dzhrbashyan — Nersesyan fractional derivative is investigated. The existence and the uniqueness of a local solution of the identification problem are proved by the method of contraction maps.

A method for constructing explicit solutions to Darboux problems for Bianchi equations is considered, which is a generalization of the Riemann-Hadamard method for a secondorder hyperbolic equation. The definition of the Riemann — Hadamard function for the Bianchi equation in the general case is given. The existence and uniqueness of the solution of the Darboux problems for the Bianchi equations is proved by reduction to the Volterra integral equation with partial integrals.

For an evolution equation with a Caputo–Fabrizio derivative in a Banach space, the solvability of linear inverse coefficient problems is studied. For a linear inverse problem with unknown time-dependent coefficients, conditions for unique solvability are obtained. The wellposedness of this inverse problem is also investigated.

The question of well-posedness in Sobolev spaces of parabolic inverse problems on recovering the source is considered. The unknowns are a solution to the equation and its the right-hand side which is representable in the form of a sum of the products of functions depending on time and Dirac delta-functions.

In the present article we study integro–differential equations with integral conditions on the lateral boundary and prove existence theorems for regular solutions. The method of regularization and the method of continuation in a parameter are employed to establish solvability.

Singular integral operators of Gevrey problems for high-order parabolic equations with weighted conjugation (gluing) conditions are considered. In contrast to the classical case, the singular Cauchy operator together with the noncompact integral operators of a special form whose kernels are approximately homogeneous of degree −1 are among these operators. The Fredholm property criterion is established for these operators as well as formulas for their indexes. Examples of singular integral equations arising in the study of boundary value problems for forward- backward equations are displayed.

In study the question of regular solvability in Sobolev spaces of parabolic inverse coefficient problems in layered media with transmissions conditions of imperfect contact type. The solution has all generalized derivatives included in the equation summable with some degree. The proof is based on a priori estimates and the fixed point theorem.

The question of regular solvability in Sobolev spaces of parabolic inverse problems is considered. The unknowns are coefficient of the equations and the right-hand side which are representable in the form of a finite segments of the series in some basis whose coefficients depending on time are to be determined.

Using the simplest one-dimensional example, we analyze the transition from the kinetic model of cold plasma (Landau, Iordansky) to the hydrodynamic model. During the transition, the class of initial Cauchy data corresponding to a globally smooth solution is narrowed. We construct moment chains that satisfy similar first-order equations and offer an explanation of why cutting them off at each next step expands the class of initial data corresponding to a smooth solution.

The paper considers such nonlinear phenomena in condensed matter physics as discrete breezers and delocalized nonlinear vibrational modes. The properties of delocalized nonlinear vibrational modes and discrete breezers are described, the relationship between them and their influence on the macroscopic properties of model nonlinear lattices and crystals are analyzed using computer modeling methods. A phenomenon such as the manifestation of modulation instability of delocalized nonlinear oscillatory modes, leading to the excitation of chaotic discrete breezers, has also been studied.

The evolutionary linear equations resolved with respect to the m-th order derivative on R, without initial conditions, are investigated. A class of bisectorial operators has been introduced. The existence of a unique solution is proved for an equation with a bisectorial operator.

The paper solves the problem of constructing a regression model in the form of a power-law two-factor form and a correlation model of staffing performance in the form of transfer functions. The input indicators for the models are the number of employees and the ratio of the average monthly salary in PJSC NK Rosneft to the average salary in Russia, and the output parameter of work efficiency is the volume of oil and gas condensate production.

An inverse initial-boundary value problem on a bounded interval is considered for a system of odd-order quasilinear evolution equations. Right-hand sides of equations of a special type are chosen as controls and integral conditions as overdetermination. Results on existence and uniqueness of solutions under small input data are established.

The search is carried out for necessary and sufficient conditions on a linear closed operator for the existence of a strongly continuous resolving family to an equation resolved with respect to the Riemann — Liouville fractional derivative, with the corresponding operator at the unknown function.