This article describes the solution of two problems. First, based on the fractional diffusion equation, a boundary problem with arbitrary values of derivative indicators was formulated and solved, describing more general cases than existing solutions. Secondly, from the consideration of the probability schemes of transitions between states of the process, which can be observed in complex systems, a fractional-differential equation of the telegraph type with multiples is obtained (in time: β, 2β, 3β, … and state: α, 2α, 3α, …) using orders of fractional derivatives and its analytical solution for one particular boundary problem is considered. In solving edge problems, the Fourier method was used. This makes it possible to represent the solution in the form of a nested time series (one in time t, the second in state x), each of which is a function of the Mittag-Leffler type. The eigenvalues of the Mittag-Leffler function for describing states can be found using boundary conditions and the Fourier coefficient based on the initial condition and orthogonality conditions of the eigenfunctions. An analysis of the characteristics of time series of changes in the emotional color of users’ comments on published news in online mass media and the electoral campaigns of the US presidential elections showed that for the mathematical expectation of amplitudes of deviations of series levels from the size of the amplitude calculation interval (“sliding window”), a root dependence of fractional degree was observed; for dispersion, a power law with a fractional index greater than 1.5 was observed; and the behavior of the excess showed the presence of so-called “heavy tails”. The obtained results indicate that time series have unsteady non-locality, both in time and state. This provides the rationale for using differential equations with partial fractional derivatives to describe time series dynamics.

В статье предложена модель кластеризации коллекций новостных текстовых сообщений, а также соответствующий алгоритм кластеризации «bubble trap». Суть предлагаемого подхода заключается в разделения всего векторного пространства текстовых документов на оболочки смысловых кластеров с минимальными ограничениями на признаки отбора таким образом, что объем смыслового кластера и положение его центра остаются неизменными в процессе добавления в него новых векторов, а критерием соотнесения является заданная постоянная метрика точности. The article proposes a clustering model for collections of news text messages, as well as the corresponding bubble trap clustering algorithm. The essence of the proposed approach is to divide the entire vector space of text documents into shells of semantic clusters with minimal restrictions on the selection criteria in such a way that the volume of the semantic cluster and the position of its center remain unchanged in the process of adding new vectors to it, and the criterion of affiliation is a given constant accuracy metric.

The paper considers the solution of a fractional differential boundary value problem, that is, a diffusion-type equation with arbitrary values of the derivative orders on an infinite axis. The difference between the obtained results and other authors’ ones is that these involve arbitrary values of the derivative orders. The solutions described in the literature, as a rule, are considered in the case when the fractional time derivative β lies in the range: 0 < β ≤ 1, and the fractional state derivative α (the variable describing the state of the process) is in the range: 1 < α ≤ 2. The solution presented in the article allows us to consider any ranges for α and β, if the inequality 0 < β/α ≤ 0.865 is satisfied in the range β/α. In order to solve the boundary value problem, the probability density function of the observed state x of a certain process (for example, the magnitude of the deviation of the levels of a time series) from time t (for example, the time interval for calculating the amplitudes of the deviation of the levels of a time series) can be captured.