Operational Calculus Research Articles

Inverse problem for a multi-parameters space-time fractional diffusion equation with nonlocal boundary conditions: operational calculus approach

Inverse problem for a multi-parameters space-time fractional diffusion equation with nonlocal boundary conditions: operational calculus approach

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.

Calculus of multilinear differential operators, operator L∞-algebras and IBL∞-algebras

We propose an operadic framework suitable for describing algebraic structures with operations being multilinear differential operators of varying orders or, more generally, formal series of such operators. The framework is built upon the notion of a multifiltration of a linear operad generalizing the concept of a filtration of an associative algebra. We describe a particular way of constructing and analyzing multifiltrations based on a presentation of a linear operad in terms of generators and relations. In particular, that allows us to observe a special role played in this context by Lie, Lie-admissible and Lie∞-structures. As a main application, and the original motivation for the present work, we show how a certain generalization of the well-known big bracket construction of Lecomte–Roger and Kosmann-Schwarzbach encompassing the case of homotopy involutive Lie bialgebras can be obtained.

A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.

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.

A New Method for Calculating Energy of Matter

An approximate calculation of the spatial characteristics on finite range is required, so one quantitative continuum represents the accumulation of infinite great quantities is artificially divided it into smaller and camparable parts in which calculus operation can be applied .This operation is defined as Theorem 1 in which infinity is not involved, there is a camparable finity is constantly (forever) approaching and not reaching infinity, and only staying within a finite range. Theorem 1 can exist in this paper as a new mathematical basis for physics. Because the essence of all physical quantities is size comparison, and the size comparison relation of matter can only be space/time, so relation formula space/time is the only expression of the concept of matter, all physical quantities are applicable to this expression, each different physical quantity is a multi-dimensional representation of this expression. A new mass energy formula is aslo derived from this paper.

Thermoelastic diffusion phase-lag model for a layer with internal heat and mass sources

This paper analyses a problem of unsteady bulk thermoelastic diffusion perturbations propagation in a multi-component layer. The local equilibrium model describes one-dimensional physical processes in the continuum. It includes the coupled equations of elastic medium motion, heat transfer and mass transfer with cross-diffusion effects. The model also considers phase-lag effects in a continuum with internal heat and mass sources. The unknown functions of displacement, temperature and concentration increments are investigated in the Green's functions and the boundary conditions integral convolution form. We use the integral Laplace transform to time and the Fourier expansion into series by a spatial coordinate to find Green's functions. It allows us to reduce the problem to a system of linear algebraic equations. The Fourier series coefficients originals are found using known theorems and operational calculus tables. The numerical algorithms used is minimised, and Green's functions are analytically found. Calculation examples are represented and analysed.

The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. This equation is partially elliptic in the velocity direction and degenerates in the spatial variable. We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity. Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.

Photovoltaic Energy All-Day and Intra-Day Forecasting Using Node by Node Developed Polynomial Networks Forming PDE Models Based on the L-Transformation

Forecasting Photovoltaic (PV) energy production, based on the last weather and power data only, can obtain acceptable prediction accuracy in short-time horizons. Numerical Weather Prediction (NWP) systems usually produce free forecasts of the local cloud amount each 6 h. These are considerably delayed by several hours and do not provide sufficient quality. A Differential Polynomial Neural Network (D-PNN) is a recent unconventional soft-computing technique that can model complex weather patterns. D-PNN expands the n-variable kth order Partial Differential Equation (PDE) into selected two-variable node PDEs of the first or second order. Their derivatives are easy to convert into the Laplace transforms and substitute using Operator Calculus (OC). D-PNN proves two-input nodes to insert their PDE components into its gradually expanded sum model. Its PDE representation allows for the variability and uncertainty of specific patterns in the surface layer. The proposed all-day single-model and intra-day several-step PV prediction schemes are compared and interpreted with differential and stochastic machine learning. The statistical models are evolved for the specific data time delay to predict the PV output in complete day sequences or specific hours. Spatial data from a larger territory and the initially recognized daily periods enable models to compute accurate predictions each day and compensate for unexpected pattern variations and different initial conditions. The optimal data samples, determined by the particular time shifts between the model inputs and output, are trained to predict the Clear Sky Index in the defined horizon.

An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes: Exactness, Poincaré Inequalities, and Consistency

In this paper, we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of text {L}^2-products, it can be used to design schemes for partial differential equations that benefit from the exactness of the sequence but, unlike classical (e.g., Raviart–Thomas–Nédélec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: exactness properties, uniform Poincaré inequalities, as well as primal and adjoint consistency. We also show how this DDR complex enables the design of a numerical scheme for a magnetostatics problem, and use the aforementioned results to prove stability and optimal error estimates for this scheme.

On a left-α-ψ-hyperholomorphic Bergman space

Given a structural set , some papers [M. Shapiro, V. Vasilevski, (Complex Var Theory Appl. 1995;27:17–46 and 1995;27:67–96)] study the quaternionic ψ-Fueter operator calculus, where the Fueter-type operator is The aim of this paper is to present the function theory and the Bergman theory induced by the ψ-Fueter-type operator: where a is a quaternionic constant and f is a continuously differentiable function. Particularly, this work shows the Stokes and Borel–Pompieu formulas associated to and presents several results of the Bergman theory such as the existence of the Bergman kernel, Bergman projection and their conformal covariant properties.

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.

Certain fractional formulas of the extended k-hypergeometric functions

In this article, we aim to investigate various formulae for the (p,k)-analogues of Gauss hypergeometric functions, including the integral transforms and the operators of fractional calculus. All the outcomes presented here are of general attractiveness and can yield a number of previous works as special cases.

An Introductory Overview of Fractional-Calculus Operators Based Upon the Fox-Wright and Related Higher Transcendental Functions

This survey-cum-expository review article is motivated essentially by the widespread usages of the operators of fractional calculus (that is, fractional-order integrals and fractional-order derivatives) in the modeling and analysis of a remarkably large variety of applied scientific and real-world problems in mathematical, physical, biological, engineering and statistical sciences, and in other scientific disciplines. Here, in this article, we present a brief introductory overview of the theory and applications of the fractional-calculus operators which are based upon the general Fox-Wright function and its such specialized forms as (for example) the widely- and extensively investigated and potentially useful Mittag-Leffter type functions.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

On fractional calculus with analytic kernels with respect to functions

Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important classes of fractional-calculus operators are the fractional integrals and derivatives with respect to functions (dating back to the 1970s) and those with general analytic kernels (introduced in 2019). To cover both of these settings in a single study, we can consider fractional integrals and derivatives with analytic kernels with respect to functions, which have never been studied in detail before. Here we establish the basic properties of these general operators, including series formulae, composition relations, function spaces, and Laplace transforms. The tools of convergent series, from fractional calculus with analytic kernels, and of operational calculus, from fractional calculus with respect to functions, are essential ingredients in the analysis of the general class that covers both.

On Laplace transforms with respect to functions and their applications to fractional differential equations

An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called Ψ‐fractional calculus. The operational calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.

ON A CLASS OF FRACTIONAL HARDY-TYPE INEQUALITIES

In this paper, we study a new class of Hardy-type inequalities involving fractional calculus operators. We derive the Hardy-type inequalities for the variant of Riemann–Liouville fractional calculus operators and [Formula: see text]-Hilfer fractional derivative operator. The obtained inequalities involving fractional operators are more general as compared to some existing results in the literature.

Properties of fractional operators with fixed memory length

In this present paper, we discuss some properties of fractional operators with fixed memory length (Riemann–Liouville fractional integral, Riemann–Liouville and Caputo fractional derivatives). Some observations and examples are discussed during the article, in order to make the results well defined and clear. Furthermore, we consider the fundamental theorem of calculus for fractional operators with fixed memory length.

Exact density matrix elements for a driven dissipative system described by a quadratic Hamiltonian

For a prototype quadratic Hamiltonian describing a driven, dissipative system, exact matrix elements of the reduced density matrix are obtained from a generating function in terms of the normal characteristic functions. The approach is based on the Heisenberg equations of motion and operator calculus. The special and limiting cases are discussed.