In this paper the inverse spectral problem method is applied to integrate a nonlinear Hirota-type equation with a self-consistent series source in the class of periodic infinite-gap functions. An infinite system of differential equations is derived that describes the evolution of spectral data for the periodic Dirac operator. The solvability of the Cauchy problem for this system in the class of six-times continuously differentiable periodic infinite-gap functions is proved. In addition, an algorithm for finding infinite-gap solution to the Cauchy problem for a Hirota-type equation with a self-consistent series source is proposed, and an exact solution is found in the case of the single-gap case for the Dirac operator.

The existence and uniqueness of a classical solution of a mixed problem in a half-strip for the nonlinear Liouville equation is studied. The solution is presented in implicit form as a solution of an integral equation whose solvability is proved with the use of the Leray–Schauder theorem, and the corresponding a priori estimate is obtained by energy methods.

The paper focuses on a boundary value problem for a system of two homogeneous Helmholtz equations with mixed boundary conditions arising in the theory of electromagnetic wave scattering on structures covered with two-dimensional materials. The problem is reduced to a boundary integral equation on the entire real line involving a hypersingular integral operator. Introducing new variables and unknown function, one passes to an integral equation on an interval. To find an approximate solution of the resulting equation, a collocation method using Fourier–Chebyshev series to represent the solution is suggested; hypersingular integrals are calculated analytically. In the main part of the paper, we prove the Fredholm property of the hypersingular operator involved in the integral equation under consideration.

On a Riemannian manifold of dimension $$m\ge 3$$ , we consider a nonempty compact set $$A$$ that is invariant for some $$C^1 $$ -smooth flow. Sufficient conditions under which $$A $$ is a hyperbolic set of the flow are proposed.

A computer virus poses significant risks to individual computer systems. To mitigate these risks, various mathematical models have been developed. Several techniques, including the installation of antivirus software and the implementation of preventive measures based on epidemiological studies, can reduce the impact of virus attacks. This study focuses on the reaction-diffusion computer virus model. A qualitative analysis of the model is conducted, and the positivity of the model is established. Additionally, the boundedness of the model’s solutions is demonstrated. The stability region is determined by analyzing the variational matrix across different parametric spaces for both temporal and diffusive components of the model. It is observed that the stability region expands under the influence of diffusion phenomena. For the numerical investigation of the model, three computational schemes are employed: the forward Euler scheme, the backward Euler operator splitting scheme, and the non-standard finite difference (NSFD) scheme. The NSFD scheme is analyzed in terms of positivity, consistency, and convergence, with positivity being proven using M-matrix theory. A test problem is utilized to obtain numerical solutions, and various parameter values are explored to identify virus-free and virus-endemic equilibrium states. The NSFD scheme is shown to preserve all essential characteristics of the continuous model.

We consider the Cauchy problem for a system of two first-order integro-differential equations with memory in finite-dimensional Hilbert spaces, where the integral term contains a difference kernel. Such a mathematical model is typical for nonstationary electromagnetic processes taking into account the electric field dispersion effects. To obtain an approximate solution of the considered nonlocal problem, a transformation to a local Cauchy problem for a system of first-order equations is applied, based on approximating the difference kernel by a sum of exponentials. Two-level operator-difference schemes in Hilbert spaces are constructed and analyzed for stability.

The paper considers the stress-strain state of a layered geological medium under the influence of an external dynamic load. Each layer is described by an isotropic linear elastic model with specified mechanical parameters. For the numerical simulation of the wave propagation process in the two-dimensional statement of the problem, a grid-characteristic scheme of high approximation order is constructed. The approximation of boundary and contact conditions, as well as the accuracy deterioration problem for spatial splitting schemes, is addressed. Numerical solution results for test problems are presented.

A boundary value problem for quasi–one-dimensional hemodynamic equations for a straight circular vessel in which unsteady pressure and flow functions are set as a boundary condition is considered. Conditions for the existence of bounded and unbounded solutions as well as for the existence of a retrograde blood flow in the vessel are stated and proved.