Том 57, № 6 (2017)

This paper deals with the function F(α; z) of complex variable z defined by the expansion \(F\left( {\alpha ;z} \right) = \sum\nolimits_{k = 0}^\infty {\frac{{{z^k}}}{{{{\left( {k!} \right)}^\alpha }}}} \) which is a natural generalization of the exponential function (hence the name). Primary attention is given to finding relations concerning the locations of its zeros for α ∈ (0,1). Note that the function F(α; z) arises in a number of modern problems in quantum mechanics and optics. For α = 1/2, 1/3,..., approximations of F(α; z) are constructed using combinations of degenerate hypergeometric functions ₁F₁(a; c; z) and their asymptotic expansions as z → ∞. These approximations to F(α; z) are used to approximate the countable set of complex zeros of this function in explicit form, and the resulting approximations are improved by applying Newton’s high-order accurate iterative method. A detailed numerical study reveals that the trajectories of the zeros under a varying parameter α ∈ (0,1] have a complex structure. For α = 1/2 and 1/3, the first 30 complex zeros of the function are calculated to high accuracy.

The method of solution continuation with respect to a parameter is used to solve an initial value problem for a system of ordinary differential equations with several limiting singular points. The solution is continued using an argument (called the best) measured along the integral curve of the problem. Additionally, a modified argument is introduced that is locally equivalent to the best one in the considered domain. Theoretical results are obtained concerning the conditioning of the Cauchy problem parametrized by the modified argument in a neighborhood of each point of its integral curve.

The solvability of inverse problems of finding the coefficients of a parabolic equation together with solving this equation is studied. In these problems, certain additional conditions on the boundary are used as overdetermination conditions. Existence and uniqueness theorems for regular solutions of such problems are proven.

A two-dimensional linearized model of coastal sediment transport due to the action of waves is studied. Up till now, one-dimensional sediment transport models have been used. The model under study makes allowance for complicated bottom relief, the porosity of the bottom sediment, the size and density of sediment particles, gravity, wave-generated shear stress, and other factors. For the corresponding initial–boundary value problem the uniqueness of a solution is proved, and an a priori estimate for the solution norm is obtained depending on integral estimates of the right-hand side, boundary conditions, and the norm of the initial condition. A conservative difference scheme with weights is constructed that approximates the continuous initial–boundary value problem. Sufficient conditions for the stability of the scheme, which impose constraints on its time step, are given. Numerical experiments for test problems of bottom sediment transport and bottom relief transformation are performed. The numerical results agree with actual physical experiments.

Solutions of a nonlinear hyperbolic system of equations describing weakly nonlinear quasitransverse waves in a weakly anisotropic elastic medium are studied. The influence of small-scale processes of dissipation and dispersion is investigated. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with a stationary structure. Among the discontinuities with a stationary structure, there are special ones that, in addition to relations following from conservation laws, satisfy additional relations required for the existence of their structure. In the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Special discontinuities lead to nonunique self-similar solutions of the Riemann problem. Asymptotics of non-self-similar problems for equations with dissipation and dispersion are found numerically. These asymptotics correspond to self-similar solutions of the problems.

The use of eddy-resolving approaches to solving problems on arbitrary unstructured grids is investigated. The applications of such approaches requires the use of low dissipation numerical schemes, which can lead to numerical oscillations of the solution on unstructured grids. Numerical oscillations typically occur in domains with large gradients of velocities, in particular, in the near-wall layer. It is proposed to single out the boundary layer and use a numerical scheme with increased numerical dissipation in it. The algorithm for singling out the boundary layer uses a switching function to change the parameters of the numerical scheme. This algorithm is formulated based on the BCD scheme from the family NVD. Its validity and advantages are investigated using the zonal RANS–LES approach for solving some problems of turbulent flow of incompressible fluids.

Heat and mass transfer effects in the three-dimensional mixed convection flow of a viscoelastic fluid with internal heat source/sink and chemical reaction have been investigated in the present work. The flow generation is because of an exponentially stretching surface. Magnetic field normal to the direction of flow is considered. Convective conditions at the surface are also encountered. Appropriate similarity transformations are utilized to reduce the boundary layer partial differential equations into the ordinary differential equations. The homotopy analysis method is used to develop the solution expressions. Impacts of different controlling parameters such as ratio parameter, Hartman number, internal heat source/sink, chemical reaction, mixed convection, concentration buoyancy parameter and Biot numbers on the velocity, temperature and concentration profiles are analyzed. The local Nusselt and Sherwood numbers are sketched and examined.