Vol 53, No 7 (2017)

We suggest a simple method for reducing problems with an integral condition for evolution equations to a Volterra integral equation of the first kind. For Volterra equations of the convolution type, we indicate necessary and sufficient solvability conditions for the case in which the right-hand side lies in some classes of functions of finite smoothness. We use these conditions to construct examples of nonexistence of a local solution for the heat equation with an integral condition.

We consider the Cauchy problem for the Bessel–Struve equation in a Banach space. A sufficient condition for the solvability of this problem is proved, and the solution operator is written in explicit form via the Bessel and Struve operator functions. A number of properties is established for the solutions.

We derive boundary conditions on multilayer films bounding a ball and consisting of infinitely thin strongly and weakly permeable layers and obtain formulas expressing the solutions of boundary value problems for the Laplace equation in a ball bounded with two-layer films by single quadratures via the solutions of the classical Dirichlet and Neumann problems for the Laplace equation in the ball (without the films).

We consider a spectral problem that is nonlinear in the spectral parameter for a self-adjoint vector differential equation of order 2n. The boundary conditions depend on the spectral parameter and are self-adjoint as well. Under some conditions of monotonicity of the input data with respect to the spectral parameter, we present a method for counting the eigenvalues of the problem in a given interval. If the boundary conditions are independent of the spectral parameter, then we define the notion of number of an eigenvalue and give a method for computing this number as well as the set of numbers of all eigenvalues in a given interval. For an equation considered on an unbounded interval, under some additional assumptions, we present a method for approximating the original singular problem by a problem on a finite interval.

For a hyperbolic equation, we consider an inverse coefficient problem in which the unknown coefficient occurs in both the equation and the initial condition. The solution values on a given curve serve as additional information for determining the unknown coefficient. We suggest an iterative method for solving the inverse problem based on reduction to a nonlinear operator equation for the unknown coefficient and prove the uniform convergence of the iterations to a function that is a solution of the inverse problem.

We develop a mathematical model of the boundary value problem describing magnetic field shielding by a cylindrical thin-walled shell (screen) made of materials whose permeability depends nonlinearly on the magnetic field intensity. Integral boundary conditions on the shell surface are used. A numerical method is suggested for solving a nonlinear boundary value problem of magnetostatics with integral boundary conditions. The shielding efficiency coefficient characterizing the external magnetic field attenuation when passing into the interior of the cylindrical screen is studied numerically.

For a linearized finite-difference scheme approximating the Dirichlet problem for a multidimensional quasilinear parabolic equation with unbounded nonlinearity, we establish pointwise two-sided solution estimates consistent with similar estimates for the differential problem. These estimates are used to prove the convergence of finite-difference schemes in the grid L₂ norm.