On the solutions of some fractional q-differential equations with the Riemann-Liouville fractional q-derivative

This paper is devoted to explicit and numerical solutions to linear fractional q-difference equations and the Cauchy type problem associated with the Riemann-Liouville fractional q-derivative in q-calculus. The approaches based on the reduction to Volterra q-integral equations, on compositional relations, and on operational calculus are presented to give explicit solutions to linear q-difference equations. For simplicity, we give results involving fractional q-difference equations of real order a > 0 and given real numbers in q-calculus. Numerical treatment of fractional q-difference equations is also investigated. Finally, some examples are provided to illustrate our main results in each subsection.

Devising matrix technology for forecasting the dynamics in the operation of a closed military logistics system

There is a tendency of intensive development of a new scientific area aimed at optimizing the processes of comprehensive ensuring the life of society and industrial processes of countries, specifically logistics, and its more important aspect, military logistics. This paper considers typical contradictions between the need and opportunities for additional development of the theory of processes involving this system. On the one hand, the military has important, dynamic, multifaceted processes for the comprehensive provision of their combat operations to analyze, which requires significant intensification of the development of methods and models for quantitative analysis of the effectiveness of the functioning of military logistics systems. On the other hand, there is now limited availability of theoretical developments and the practical application of the necessary, convenient, effective mathematical tools aimed at computerization of solving the problems of providing military scientific and technical problems in real time. Matrix technology for forecasting the dynamics of functioning of closed systems of military logistics of various military purposes is proposed. Matrix calculus makes it possible to obtain intermediate and ultimate results in a compact form and carry out complex and cumbersome calculations using effective algorithms. A method to precisely solve the system of linear differential equations describing processes of arbitrary type has been proposed. The method is based on the use of the operational calculus by Laplace. The possibilities of the method and procedures of forecasting are illustrated by solving practical military tasks that arise during the functioning of military logistics systems of varying complexity. These tasks differ in configuration, different numbers of possible states, and state transitions

Modeling and forecasting the probability of the states of technical support systems for the use of weapons and military equipment

The article discusses a probabilistic model of processes in complex systems of technical support for military vehicles. One of the methods for studying such complex systems is their representation in the form of a set of typical states in which the system can be. Transitions occur between states, the intensities and probabilities of which are assumed to be known. The system is graphically represented using a graph of states and transitions, and the subject of research is the probability of finding the technical support system in these states. The graph of states and transitions is associated with a system of first order linear differential equations with respect to the probabilities of finding the support system in its basic states. To obtain a solution, this system must be supplemented with certain conditions. These are, firstly, the initial conditions that specify the probabilities of all states at the initial moment of time. Second, this is the normalization condition, which states that at any moment in time the sum of the probabilities of all states is equal to unity. An approximate solution to the problem is described in the literature. Such approximate solution is getting more accurate when the sought probabilities depend on time weaker. We propose a method of the exact solution of the above mentioned system of differential equations based on the use of operational calculus. In this case, the system of linear differential equations is transformed into a system of linear algebraic equations for the Laplace images of unknown probabilities. The use of matrix calculus made it possible to write down the obtained results in a compact form and to use effective numerical algorithms of linear algebra for further calculations. The model is illustrated by the example of solving the problem of technical support for the march of a battalion tactical group column, including wheeled and tracked vehicles. The boundaries of the validity of the results of a simpler approximate solution are established.

Phenomenological physical foundations of structural dynamics in technical mechanics

The article deals with the problem of forming dynamic thinking in the course of technical mechanics. This problem covers various forms and levels of professional training, including engineering courses in structural mechanics, postgraduate and doctoral studies. The study of this problem in the article is carried out using dynamic concepts in technical physics and metrology. The main method for analyzing this problem is the method and theory of vibroacoustic computational modeling developed by Prof. Hlystunov M.S. The article presents a comparative analysis of the reaction of a cantilever beam to static and dynamic load. The dynamic characteristics of such a beam, including its AFFC, AFC and FFC, resonances and antiresonances, are considered. Then we consider the reaction of the simplest two-rod frame to a dynamic load and its fundamental difference from the reaction to a static load. The article provides detailed mathematical calculations using the corresponding section of operational calculus. The terms, definitions, and mathematical representations used in the article correspond to similar classical concepts that are widely used in metrology, technical cybernetics, and technical physics. The article is provided with the necessary illustrations for a visual representation of the main dynamic characteristics and properties of building structures.

q-Functions and Distributions, Operational and Umbral Methods

The use of non-standard calculus means have been proven to be extremely powerful for studying old and new properties of special functions and polynomials. These methods have helped to frame either elementary and special functions within the same logical context. Methods of Umbral and operational calculus have been embedded in a powerful and efficient analytical tool, which will be applied to the study of the properties of distributions such as Tsallis, Weibull and Student’s. We state that they can be viewed as standard Gaussian distributions and we take advantage of the relevant properties to infer those of the aforementioned distributions.

Interval Version of the Operational Calculation Method for Solving Ordinary Differential Equations

This article discusses the interval variant of solving ordinary differential equations with given initial conditions, i.e. the Cauchy problem, by the method of operational calculus. This is where the interval version of the operational calculus is motivated and built. As a result, on the basis of the proved theorem in this article, an analytic interval set of solutions is obtained that is guaranteed to contain a real solution to the problem.

General Non-Markovian Quantum Dynamics

A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.