Considering an analog of the Boussinesq equation, we examine spatial non local boundary value problems with the Samarski˘ıcondition and prove the existence and uniqueness of regular solutions.

In a cylindrical domain of the space Rn+1 we study Vragov’s boundary value problem for a mixed type equation of the second order with a spectral parameter. Under certain conditions on the coefficients, we establish the a priori estimates that allow us to prove a unique solvability of this boundary value problem in the energy space. Some sufficient conditions are obtained for the Fredholm solvability of the boundary value problem in this space.

This is a continuation of the authors’ article [1] which is devoted to solvabil ity of conjugate problems (generalized diffraction problems) for some nonclassical higher order differential equations of composite type. We prove existence and uniqueness the orems of regular solutions to these problems.

We consider the two-phase inverse Stefan problem of reconstructing the right-hand side of the heat equation as a function of time given its spatial distribu tion. We propose a new method for accounting for the heat of the phase passage by introducing a heat source distributed in a neighborhood of the phase transition bound ary. An algorithm is constructed for computations based on transforming the original problem to a boundary value problem for the loaded heat equation and present examples of simulations.

We examine forward-backward parabolic equations of the second order with gluing conditions containing functions of variables t ∈ [0,T] with the use of the theory of singular integral equations. Solvability is established of boundary value problems in H¨older spaces. We also demonstrate that the H¨older classes of solutions depend on a noninteger H¨older exponent and the signs of coefficients occurring in the gluing conditions at the ends of the interval [0,T] provided that some necessary and sufficient conditions on the input data of the problem are fulfilled.

We consider the equilibrium problem for a two-dimensional viscoelastic body with a thin rigid inclusion. The differential statement of the problem involves an integral condition accounting for the action of external forces on the rigid-hand part. We give an equivalent statement with variational inequality and use it to establish the unique solvability of the original problem. The additional properties of the solutions enable us to simplify the interpretation of the integral condition.

We show that the strongly generalized nonclassical differential equations of higher order of the form 2p k=0 αk(t)D2p−k t u(x,t) − Au(x,t)=f(x,t) become well posed on releasing part of the boundary of the domain from boundary value conditions.

We consider two boundary value problems for an equation of the third or der with multiple characteristics and a nonlocal condition in time. In order to prove uniqueness, we use the method of energy integrals. By the method of potentials, the Green’s function is constructed and employed to prove the unique solvability of the prob lems in question. The influence is studied of the boundary conditions on smoothness of solutions.

We study a general boundary value problem for a linear elliptic equation. Existence and uniqueness theorems are proven under the corresponding boundary con ditions and the conjugate conditions on the interface between two media.

We study the distributions of deformation velocities and stress in neighbor hoods of the ends of a shell (the boundary effect) using Rabotnov’s two-layer shell model. We solve the system of ordinary differential equations using an iterative procedure and compare solutions for a shell of finite length and a semi-infinite shell.

We compare the currents and voltages induced in an aerial transmission line in the event of a nearby lightning strike, calculating them with the use of two different mathematical models. The first model describes the electrostatic component of the in duced currents, and the second, the electromagnetic component. We show that the peak values of these components are comparable and that in permafrost conditions the peak values of the electrostatic components of induced currents and voltage can be greater by orders of magnitude, and so more dangerous than in the areas without permafrost.

We present a mathematical model of jigging using the statical approach for describing the process and the theory of Brownian motion. The Fokker–Planck equation is obtained for fractions in a jigging machine. The distributions of the grainy rocks under study are calculated in various cases.