卷 57, 编号 12 (2017)

For matrices whose elements are scalar linear difference operators, algorithms for checking invertibility (unimodularity) and constructing an inverse matrix (if it exists) are proposed. Their complexity is lower than that of other available algorithms. The differences of these algorithms from their differential analogues are discussed.

Four problems are solved in which a high-frequency source in the one-dimensional heat equation with homogeneous initial–boundary conditions is recovered from partial asymptotics of its solution. It is shown that the source can be completely recovered from an incomplete (two-term) asymptotic representation of the solution. The formulation of each source recovery problem is preceded by constructing and substantiating asymptotics of the solution to the original initial–boundary value problem.

In a half-plane, a homogeneous Dirichlet boundary value problem for an inhomogeneous singularly perturbed convection–diffusion equation with constant coefficients and convection directed orthogonally away from the boundary of the half-plane is considered. Assuming that the right-hand side of the equation belongs to the space C^λ, 0 < λ < 1, and the solution is bounded at infinity, an unimprovable estimate of the solution is obtained in a corresponding Hölder norm (anisotropic with respect to a small parameter).

New second-order accurate monotone difference schemes on nonuniform spatial grids for two-dimensional stationary and nonstationary convection–diffusion equations are proposed. The monotonicity and stability of the solutions of the computational methods with respect to the boundary conditions, the initial condition, and the right-hand side are proved. Two-sided and corresponding a priori estimates are obtained in the grid norm of C. The convergence of the proposed algorithms to the solution of the original differential problem with the second order is proved.

The coalescence and fragmentation of particles in a dispersion system are investigated by applying kinetic theory methods, namely, by generalizing the Boltzmann kinetic equation to coalescence and fragmentation processes. Dynamic equations for the particle concentrations in the system are derived using the kinetic equations of motion. For particle coalescence and fragmentation, equations for the particle momentum, coordinate, and mass distribution functions are obtained and the coalescence and fragmentation coefficients are calculated. The equilibrium mass and velocity distribution functions of the particles in the dispersion system are found in the approximation of an active terminal group (Becker–Döring-type equation). The transition to a continuum description is performed.

In this paper, a variable-order fractional derivative nonlinear cable equation is considered. It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of class of fractional partial differential equation with variable coefficient of fractional differential equation in various continues functions of spatial and time orders. Our main aim is to generalize the Chebyshev cardinal operational matrix to the fractional calculus. Finally, illustrative examples are included to demonstrate the validity and applicability of the presented technique.