We propose an efficient method for solving the Cauchy problem for an ordinary differentialequation with multiple poles of integer order. The method allows one to carry out an end-to-endcalculation through a pole both for a single pole and in the case of a chain of poles. The methoduses a special algorithm for finding the multiplicity of each pole. This multiplicity is used to definea generalized reciprocal function for which the $$K$$ -tuple pole of theoriginal function is a simple zero. The calculation of such a zero is not difficult, and therefore, theproposed method allows obtaining high accuracy even near the poles. After passing this zero, thecalculation of the original function is resumed. The application of this method on a sequence ofpoles permits one to find a numerical solution simultaneously with an a posteriori estimate of theerror. The method is illustrated by test examples.

Optical properties of dense plasma depend on electric fields inside it (the so-called microfield). A critical survey is made of existing theoretical models of plasma microfields. The most important models are verified using experiments in which dense plasma was created by powerful laser radiation. It is shown that the Quasi-Independent Particle model (QUIP) provides the best theoretical verification and agreement with experiments.

To compute controlled fusion problems, four nuclear reactions are conventionally used. In the paper, cross sections of many thermonuclear reactions between the lightest elements are compared. We show that, along with the traditional four reactions, a tangible contribution can be made by the reaction $${\text{T}} + {\text{T}} \to 2n + \,{}^{{\text{4}}}{\text{He}}$$. At low temperatures, the reactions $${\text{D}} + p \to \gamma + {}^{3}{\text{He}}$$ and $${\text{T}} + p \to \gamma + {}^{4}{\text{He}}$$ make a considerable contribution and it appears that the rest of the reactions can be neglected. For the outlined reactions, we perform a more thorough processing of the experimental data and construct high-order accurate approximations. The accuracy of finding the S-factor is 2–6% and that of the reaction rate is 3–4% for the new reactions.

Optical properties of plasmas are determined by presence of fluctuating microscopic electric field. The present work provides thorough analysis of contemporary models and points out their shortcomings. To overcome the latter, we take the QUIP (QUasi-Independent Particles) model derived ab initio. We provide generalization of the model allowing to account for microfield heterogeneity up to octupole term. We investigate convergence of the multipole series and show that higher order terms can be neglected. The model does not require laborious computations because all formulae are rather simple and are given in explicit form. This fact is an advantage of the proposed model compared to other contemporary models. We perform verification of the model via comparison with experiments. We emphasize that the comparison should be made with respect to the number of observed lines because this number strongly depends on the selected model. We outline experiments suitable for such testing. These are the experiments on Ar+Kr radiating plasma heated by laser radiation. In these experiments, the entire Ar+16 spectral series is observed. The QUIP model correctly describes the number of observed lines of the series, so its adequateness is justified.

A new method for joint processing of experimental data from various laboratories based on their approximation by the generalized Arrhenius law is proposed. The method is based on the construction of a system of functions that are orthogonal on a given set of points with arbitrary weights. As a result, the confidence intervals of the approximation coefficients can be estimated and the number of terms required for the approximation can be correctly determined. The performance of the method is demonstrated as applied to reactions of hydrogen combustion in air that are important at T < 1000 K. High accuracy of reaction rate approximation is achieved.

For the numerical solution of the Cauchy problem with multiple poles, we propose a reciprocal function method. In the case of first-order poles, it makes it possible to continue the solution through the poles and to determine the solution and the pole positions with good accuracy. The method allows one to employ conventional explicit and implicit schemes, for example, explicit Runge–Kutta schemes. A test problem with multiple poles is computed as an example. The proposed method is useful for construction of software for direct computation of special functions.

The problem of generating sequences of uniformly distributed pseudorandom numbers is considered. A simple visual test for estimating the randomness of numbers in a sequence is used. The test shows that the most popular modern random number generators, such as the Mersenne Twister, linear congruential sequence, and others, yield unsatisfactory results. Accordingly, the generation of good generators remains an open problem, and results of computing stochastic processes (molecular dynamics method, etc.) have to be treated with caution.

We consider a linear ill-posed problem for the Fredholm equation of the first kind. For its regularization, Tikhonov’s stabilizer is implemented. To solve the problem, we use the mesh method, in which we replace integral operators by the simplest quadratures; and the differential ones, by the simplest finite differences. We investigate experimentally the influence of the regularization parameter and mesh thickening on the algorithm’s accuracy. The best performance is provided by the zeroth-order regularizer. We explain the reason of this result. We use the proposed algorithm for an applied problem of the recognition of two closely situated stars if the telescope instrument function is known. In addition, we show that the stars are clearly distinguished if the distance between them is ~0.2 of the instrumental function’s width and the values of brightness differ by 1–2 stellar magnitudes.

The notion of stiffness of a system of ordinary differential equations is refined. The main difficulties encountered when solving the Cauchy problem for stiff systems are indicated. The advantages of switching to a new argument, the integral curve arc length, are demonstrated. Various mesh step selection criteria are discussed, and the integral curve curvature criterion is recommended. The most reliable implicit and explicit schemes suitable for solving stiff problems are presented. A strategy permitting an asymptotically accurate computation of the error of a numerical solution simultaneously with the solution itself is described. An analysis of chemical kinetics of hydrogen combustion in oxygen with allowance for 9 components and 50 reactions between them is provided as an illustration.

Оптические свойства плазмы определяются наличием в ней флуктуирующего микроскопического электрического поля. В работе на основе первых принципов построена простая модель плазменного микрополя, впервые учитывающая его неоднородность по октупольный член включительно. Сравнение с экспериментами показало, что только эта модель микрополя правильно описывает наблюдаемое число линий в спектральных сериях.

The optical properties of plasma are determined by the presence of a fluctuating microscopic electric field. A simple ab initio model of a plasma microfield accounting for its inhomogeneity up to an octupole term has been constructed for the first time. A comparison with experiments has shown that only this model correctly describes the observed number of spectral lines.

An explicit method for solving stiff Cauchy problems is proposed. The method relies on explicit schemes and a step size selection algorithm based on the curvature of an integral curve. Closed-form formulas are derived for finding the curvature. For Runge–Kutta schemes with up to four stages, the corresponding sets of coefficients are given. The method is validated on a test problem with a given exact solution. It is shown that the method is as accurate and robust as implicit methods, but is substantially superior to them in efficiency. A numerical example involving chemical kinetics computations with 9 components and 50 reactions is given.

A new method for automatic step size selection in the numerical integration of the Cauchy problem for ordinary differential equations is proposed. The method makes use of geometric characteristics (curvature and slope) of an integral curve. For grids generated by this method, a mesh refinement procedure is developed that makes it possible to apply the Richardson method and to obtain a posteriori asymptotically precise estimate for the error of the resulting solution (no such estimates are available for traditional step size selection algorithms). Accordingly, the proposed methods are more robust and accurate than previously known algorithms. They are especially efficient when applied to highly stiff problems, which is illustrated by numerical examples.

We propose a new mathematical technique for processing of experimental data measured with large errors. The method is applied to all available experiments on 4 major thermonuclear reactions taken into account in simulations of deuterium and deuterium-tritium fusion targets. We present new approximations for the dependence of the cross sections on energy and for the dependence of the reaction rates on temperature. Along with these approximations, we propose a procedure allowing to estimate their confidence belts. Such estimations were not known before. New approximations provide error similar to 0.3% for the cross sections and similar to 4% for the reaction rates. The present data are up to similar to 5 times more accurate than reported in literature.