and Computational Methods

doi: 10.18698/2309-3684-2020-4-111119

Comparison of two approaches to calculate values of Chebishev polynomials via recurrent procedures is realized. At that, first approach is based upon recursion upwards with respect to the index starting from the least index value. Second approach is based upon recursion downwards starting from evident asymptotical expressions of the functions with high values of index.

Апельцин В.Ф., Краснов И.К. Сравнение методов вычисления значений специальных функций математической физики. Математическое моделирование и численные методы, 2020, № 4, с. 111–119

doi: 10.18698/2309-3684-2017-4-316

The article presents a mathematical model of the alloys nanoparticles structure rearrangement dynamics after the instantaneous thermal influence (heating or cooling). The model is based on using the method of molecular dynamics of multicomponent alloys with the Lennard-Jones and Morse interatomic potentials as well as the initial conditions of momentary expansion or compression of the alloy nanoparticle regular crystalline structure. We computationally investigate the regularities of rearranging the initially regular atomic structure of a nanoparticle over time. It is shown that depending on the number of atoms in a nanoparticle various finite settled forms of the alloys nanoparticle are possible, both amorphous and new crystalline structures different from the alloy original crystalline nanostructure. We provide numerical results for the titanium nanoparticles and the titanium-nickel alloy (nitinol).

Krasnov I.K., Mozzhorina T.Yu., Balanin A.N. Numerical modeling of alloys nanostructure rearrangement by means of molecular dynamics methods. Маthematical Modeling and Computational Methods, 2017, №4 (16), pp. 3-16

doi: 10.18698/2309-3684-2018-1-4154

The problem of the electrophysical parameter recovery of layered media, which is the inverse problem of mathematical physics, is solved on the basis of the electromagnetic field measurement results. Various optimization methods for its solution are formulated. The mathematical model is proposed for a horizontally layered medium with specified parameters consistent with real values. The algorithm is developed for solving the direct problem allowing finding an analytical solution for various environmental parameter values. For solving inverse problems the complete enumeration and Hook - Jeeves methods as well as the developed modified method of complete enumeration are used. According to the results of solving the direct problem, the characteristic features of the medium are found for various values of the electrophysical parameters. When solving the inverse problem using various optimization methods, the features of each algorithm are described.

Краснов И.К., Зубарев К.М., Иванова Т.Л. Численное решение задачи восстановления электрофизических параметров по результатам зондирования переменным током. Математическое моделирование и численные методы, 2018, № 1, с. 41-54

doi: 10.18698/2309-3684-2017-3-7182

The article considers the application of the direct statistical simulation method to the problems of gas dynamics in a rarefied region. An analytical method for assignment and taking into account complex boundary conditions associated with the geometry of the body located in the computational domain is proposed. An algorithm for the rational description of a body streamlined by a gas is developed.

Krasnov I.K.,Mozzhorina T.Yu., Dzhus D.V. Numerical statistical simulation of the process of rarefied gas flow over an aircraft .Маthematical Modeling and Computational Methods, 2017, №3 (15), pp. 71–82

doi: 10.18698/2309-3684-2019-1-4353

The inverse problem of restoring the thermal conductivity coefficient of a nonlinear parabolic equation by the final temperature distribution, which serves as a mathematical model for the problem of determining structural defects, is investigated. An algorithm for numerical solution of the problem is proposed. A numerical example is considered.

Грибов А.Ф., Жидков Е.Н., Краснов И.К. О численном решении обратной зада-чи теплопроводности с излучением. Математическое моделирование и численные методы, 2019, № 1, с. 43–53.