Volume 59, Nº 11 (2019)

The problem of solving overdetermined, underdetermined, singular, or ill conditioned SLAEs using matrix canonization is considered. A modification of an existing canonization algorithm based on matrix decomposition is proposed. Formulas using LU decomposition, QR decomposition, LQ decomposition, or singular value decomposition, depending on the properties of the given matrix, are obtained. A method for evaluating the condition number of the canonization problem is proposed. It is based on computing the norm of the matrices obtained as a result of canonization; this method does not require the original matrix to be inverted. A general step-by-step matrix canonization algorithm is described and implemented in MATLAB. The implementation is tested on a set of 100 000 randomly generated matrices. The testing results confirmed the validity and efficiency of the proposed algorithm.

For a system of nonlinear fractional differential equations, the problem of reconstruction of an unknown input action is considered. An algorithm for its solution, stable to information interference and computational errors, is proposed. This algorithm is based on regularization methods and construction of the dynamic inversion theory. Dynamic reconstruction of the input action is carried out using the residual method, which does not require introducing auxiliary model systems.

The possibility of finding soliton solutions of a nonintegrable generalization of the coupled Volterra system is studied. This generalization is a system of two equations each of which includes terms that take into account the spatial dependence. At the first stage, the continual limit of the generalization is studied. At the second stage, soliton solutions for the continual limit are sought. At the third, final, step, soliton solutions of the nonintegrable generalization are sought.

Regularized equations for binary mixtures of viscous compressible gases (in the absence of chemical reactions) are considered. Two new simpler systems of equations are constructed for the case of a homogeneous mixture, when the velocities and temperatures of the components coincide. In the case of moderately rarefied gases, such a system is obtained by aggregating previously derived general equations for binary mixtures of polyatomic gases. In the case of relatively dense gases, the regularizing terms in these equations are subjected to a further substantial modification. For both cases, balance equations for the total mass, kinetic, and internal energy and new balance equations for total entropy are derived from the constructed equations; additionally, the entropy production is proved to be nonnegative. As an example of successful use of the new equations, the two-dimensional Rayleigh–Taylor instability of relatively dense gas mixtures is numerically simulated in a wide range of Atwood numbers.

A linearly implicit (Rosenbrock-type) numerical method for the integration of three-dimensional Navier–Stokes equations for compressible fluid with respect to time is proposed. The method has four stages and third order of accuracy with respect to time. As the benchmark, the Cauchy problem on a 3D torus is solved. The computed distributions are compared with the solution specified by the ABC flow.

The survival probability of an insurance company in a collective pension insurance model (so-called dual risk model) is investigated in the case when the whole surplus (or its fixed fraction) is invested in risky assets, which are modeled by a geometric Brownian motion. A typical insurance contract for an insurer in this model is a life annuity in exchange for the transfer of the inheritance right to policyholder’s property to the insurance company. The model is treated as dual with respect to the Cramér–Lundberg classical model. In the structure of an insurance risk process, this is expressed by positive random jumps (compound Poisson process) and a linearly decreasing deterministic component corresponding to pension payments. In the case of exponentially distributed jump sizes, it is shown that the survival probability regarded as a function of initial surplus defined on the nonnegative real half-line is a solution of a singular boundary value problem for an integro-differential equation with a non-Volterra integral operator. The existence and uniqueness of a solution to this problem is proved. Asymptotic representations of the survival probability for small and large values of the initial surplus are obtained. An efficient algorithm for the numerical evaluation of the solution is proposed. Numerical results are presented, and their economic interpretation is given. Namely, it is shown that, in pension insurance, investment in risky assets plays an important role in an increase of the company’s solvency for small values of initial surplus.