Volume 59, Nº 1 (2023)

We study the uniqueness of the solution of a time-regular problem for the operator-differential equation  with the Tricomi operator. The order of the differential expression  is considered to be an arbitrary positive integer, and the regular boundary conditions are given with respect to the time variable. The operator  is generated by the Tricomi equation. The boundary conditions for the Tricomi operator are given by the Dirichlet condition on the elliptic part and by the fractional derivative traces of the solution along the characteristics. It is indicated that this operator is a self-adjoint operator in the space. The self-adjointness of the operator guarantees the existence of a complete system of eigenfunctions orthonormal in if is a domain bounded by a Lyapunov curve and by characteristics of the wave equation.

A criterion for the existence of solutions of the second boundary value problem for the -Laplacian on Riemannian parabolic manifolds with a model end is obtained.

We obtain conditions for the existence of a global solution and the blow-up of the solution of the Cauchy problem on a finite time interval for a nonlinear third-order partial differential equation generalizing the equation of torsion vibrations of a cylindrical rod with allowance for internal and external damping modeling the propagation of longitudinal stress waves along a one-dimensional viscoelastic rod whose the material obeys Voigt–Kelvin medium deformation law.

We consider an optimal control problem for a linear system with constant coefficients with an integral convex performance index containing a small parameter multiplying the integral term in the class of piecewise continuous controls with smooth geometric constraints. Such problems are called cheap control problems. It is shown that the limit problem will be a problem with a terminal performance index. It is established that if the terminal term of the performance index is a convex (strictly convex) and continuously differentiable function, then the performance functional in the limit problem has similar properties. It is proved that, in the general case, convergence with respect to the performance functional is valid, and under the condition of strict convexity of the terminal term of the performance index in the original problem, convergence to the minimum point of the terminal summand of the performance index in the limit problem is valid. The limit of the defining vector in the original problem is found as the small parameter tends to zero. In particular, it is shown that the first component of the defining vector in the original problem converges to the defining vector in the limit problem. The problems of controlling a point of low mass in a medium with and without resistance with a terminal part depending on both slow and fast variables are considered in detail, and complete asymptotic expansions of the defining vectors in these problems are constructed.

For an isotropic stratified elastic strip we consider the Poincaré–Steklov operator that maps normal stresses into normal displacements on part of the boundary. A new approach is proposed for constructing the transform of the kernel of the integral representation of this operator. A variational formulation of the boundary value problem for the transforms of displacements is obtained. A definition is given and the existence and uniqueness are proved for a generalized solution of the problem. An iteration method for solving variational equations is constructed, and conditions for its convergence are obtained based on the contraction mapping principle. The variational equations are approximated by the finite element method. As a result, at each step of the iteration method, it is required to solve two independent systems of linear algebraic equations, which are solved using the tridiagonal matrix algorithm. A heuristic algorithm is proposed for choosing the sequence of parameters of the iteration method that ensures its convergence. Verification of the developed computational algorithm is carried out, and recommendations on the use of adaptive finite element grids are given

We consider a nonlinear second-order ordinary differential equation of a special form whose particular case arises when constructing exact solutions of the nonlinear heat equation with a power-law coefficient. Conditions are obtained for the parameters under which the equation admits a single integration. A number of examples of constructing exact solutions expressed in terms of elementary functions or in terms of the Lambert function are given.