Rubric: "1.2.2. Mathematical modeling, numerical methods and software packages (physical and mathematical sciences)"
Oblakova T. V. (Bauman Moscow State Technical University), Pham Q. (-)
doi: 10.18698/2309-3684-2024-3-120139
Application of generalized decomposition of polynomial chaos (RPH) in problems of quantitative estimation of uncertainty is considered. A program code has been implemented to study the influence of the input data generation scheme on the quality of the model whose coefficients are calculated by the least squares method. Standard error and sliding control values were used as quality criteria. Along with the classical method of filling the space of the input features on the scheme of the Latin hypercube, two variants of modelling coherent-optimal sample are considered: using the Markov chain and with additional thinning on the D-criterion. While the Latin hypercube sample evenly distributes points across the whole space of random variables, coherent optimum methods aim to distribute samples more densely in areas with greater variance and more rarely in areas with small variance. This approach allows for a better integration of information about the real model, which leads to a reduction in the number of samples in the planning of the experiment and as a result save costly CPU time. The implemented methods were compared on the Ishigami model function and the farm design with random values of physical characteristics. As a result of comparative modeling, it is established that in case of small range of change of random parameters, when their gradients slowly change, the design of the Latin hypercube shows the lowest values of error and sliding control. At the same time, in the case of strong non-linearity, the application of coherent-optimal design leads to a more stable and efficient model, and additional thinning according to the criterion of D-optimality gives the best result and is the most sustainable. It has also been shown that both the planning algorithms of the experiment are unstable and incorrect if there are insufficient samples.
Облакова Т.В., Фам Куок Вьет. Об оптимальной конструкции моделирования разложения полиномиального хаоса в задачах количественной оценки неопределенности. Математическое моделирование и численные методы, 2024, № 3, с. 120–139.
517.929+517.95 Analytical solutions of nonlinear transport equations with delay
Sorokin V. G. (Ishlinsky Institute for Problems in Mechanics)
doi: 10.18698/2309-3684-2024-3-140167
Nonlinear transport equations with constant delay are considered. The introduction provides a brief overview of publications that study transport mathematical models with delay and develop numerical methods for solving the corresponding problems. The main sections of the article describe more than forty transport equations with constant delay and various transfer coefficients, which allow exact analytical solutions. The kinetic functions of all considered equations contain free parameters or arbitrary functions. Additive, multiplicative, generalized, and functional separable solutions, as well as traveling-wave and self-similar solutions are obtained. Many solutions are expressed in terms of elementary functions. For some types of equations, theorems on the “multipli-cation” of solutions are formulated. The described equations and their solutions can be used to evaluate the accuracy of numerical methods for integrating the corresponding nonlinear transport problems with delay.
Сорокин В.Г. Аналитические решения нелинейных уравнений с запаздыванием, используемых при математическом моделировании процессов переноса. Математическое моделирование и численные методы, 2024, № 3, с. 140–167.