Operational Calculus Research Articles

Jackson and Hahn Difference Models as Discrete Bittner Operational Calculus Representations

Abstract In the paper, there have been constructed discrete models of the non-classical Bittner operational calculus that are related to the notions of Jackson and Hahn derivatives, both known from the quantum calculus. To achieve it, we have used the Bittner operational calculus representation for the forward difference.

Berezin symbols on the unit sphere of ℂn

ABSTRACTWe describe the symbolic calculus of operators on the unit sphere in the complex n-space defined by the Berezin quantization. In particular, we derive an explicit formula for the composition of Berezin symbols and thus a non-commutative star product. In order to do so, it was necessary to introduce holomorphic spaces that admit a reproducing kernel in the form of generalized hypergeometric series.

A new view of some operators and their properties in terms of the Non-Newtonian Calculus

Abstract Differentiation and integration are basic operations of calculus and analysis. Indeed, they are in- finitesimal versions of substraction and addition operations on numbers, respectively. From 1967 till 1970 Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, converting the roles of substraction and addition into division and multiplication, respectively and thus established a new calculus, called Non-Newtonian Calculus. So in this paper, it is investigated to a new view of some operators and their properties in terms of Non-Newtonian Calculus. Then we compare with the Newtonian and Non-Newtonian Calculus.

Preliminaries of Bigraphical Calculus for Context-awareness

Context-awareness has become one of the active topics in ubiquitous computing, cloud computing and so on, recently. In this paper, preliminaries of bigraph based context-aware calculus are presented for context-aware system. We define the syntax and semantics of process, context and context expressions, as well as the semantics of operations in the bigraphical calculus of context-awareness.

The Fourier Transform on the Group mathrm {GL}_2({mathbb R }) and the Action of the Overalgebra mathfrak {gl}_4

We define a kind of ’operational calculus’ for the Fourier transform on the group mathrm {GL}_2({mathbb R }). Namely, mathrm {GL}_2({mathbb R }) can be regarded as an open dense chart in the Grassmannian of 2-dimensional subspaces in {mathbb R }^4. Therefore the group mathrm {GL}_4({mathbb R }) acts in L^2 on mathrm {GL}_2({mathbb R }). We transfer the corresponding action of the Lie algebra mathfrak {gl}_4 to the Plancherel decomposition of mathrm {GL}_2({mathbb R }), the algebra acts by differential-difference operators with shifts in an imaginary direction. We also write similar formulas for the action of mathfrak {gl}_4oplus mathfrak {gl}_4 in the Plancherel decomposition of mathrm {GL}_2({mathbb C }).

Analytical and numerical solutions of the unsteady 2D flow of MHD fractional Maxwell fluid induced by variable pressure gradient

The nonlocal property of the fractional derivative can supply more precise mathematical models for depicting flow dynamics of complex fluid which cannot be modelled appropriately by normal integer order differential equations. This paper studies the analytical and numerical methods of unsteady 2D flow of Magnetohydrodynamic (MHD) fractional Maxwell fluid in a rectangular pipe driven by variable pressure gradient. The governing equation is formulated with Caputo time dependent fractional derivatives whose orders are distributed in interval (0, 2). A challenge is to firstly obtain the exact solution by combining modified separation of variables method with Mikusiński-type operational calculus. Meanwhile, the numerical solution is also obtained by the implicit finite difference method whose validity has been confirmed by the comparison with the exact solution constructed. Different to the most classical works, both the stability and convergence analysis of two-dimensional multi-term time fractional momentum equation are derived. Based on numerical analysis, the results show that the velocity increases with the rise of the fractional parameter and relaxation time. While an increase in the values of Hartmann number leads to a slower velocity in the rectangular pipe.

Study on A Combined Optimal Rational Approximation Method of Fractional-order Calculus Operator and Its Application

Background: Because the fractional-order calculus operator is an irrational function of the complex variable, it is required to deal with rational approximation. Objective: It is difficult to directly implement the fractional-order calculus operator in the numerical simulation and practical application. Methods: So the minimum amplitude frequency approximation error and phase frequency approximation error of rational approximation function are comprehensively considered in the same rational function class. And the rational approximation function construction method and combined optimal rational approximation method of fractional-order calculus operator based on the function approximation theory and logarithmic frequency characteristic asymptote idea are described in detail. Then, an approximation method of fractional-order calculus operator based on the combined optimal rational approximation is proposed in this paper. The approximation effectiveness of fractional-order calculus operator is analyzed by using s-0.6 . And the combined optimal rational approximation method is applied to the design and digital realization of fractional-order controller. In order to simplify the controller design, the controller structure of parallel connection mode is used and the simplified rational approximation expression and transform of fractionalorder integral operator in the different order is obtained. Results: The analyzed results show that the combined optimal rational approximation method of fractional-order calculus operator is consistent with the ideal c in amplitude frequency domain and has smallest error for the phase frequency characteristic. And the applied results show that the proposed combined optimal rational approximation method can conveniently control and simulate the practical computer control system by analyzing the simplified rational approximation expression and z transform of fractional-order integral operator. Conclusion: So the combined optimal rational approximation method can provide better engineering applications of fractional-order controller in the practical control system. Keywords: Fractional-order calculus, rational approximation of operator, combined optimal rational approximation, discretization, fractional-order PIα Dβ controller, parallel connection structure.

Mixed Order Fractional Differential Equations

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

Elastic Neumann–Poincaré Operators on Three Dimensional Smooth Domains: Polynomial Compactness and Spectral Structure

We prove that the elastic Neumann--Poincar\'e operator defined on the smooth boundary of a bounded domain in three dimensions, which is known to be non-compact, is in fact polynomially compact. As a consequence, we prove that the spectrum of the elastic Neumann-Poincar\'e operator consists of three non-empty sequences of eigenvalues accumulating to certain numbers determined by Lam\'e parameters. These results are proved using the surface Riesz transform, calculus of pseudo-differential operators and the spectral mapping theorem.

Математическое моделирование влияния релаксационных процессов на температурные поля в упругом полупространстве

The article examines the relaxation processes that occur in an elastic solid medium when it is heated and cooled, especially their influence on the temperature field. Besides, we considered in this paper the heat equation of parabolic type arising in the theory of thermal conductivity for different modes of heating the border. We present a solution of the boundary value problem of nonstationary heat conduction for an infinite plate with the following regimes of loading the boundaries: a single slow temperature change at the border, a single instantaneous temperature variation at the border, and, finally, multiple instantaneous changes of temperature at the border. In order to solve these three heat problems, they were brought to a dimensionless form. Then the operational calculus method was applied. The essence of the method consists in the following. According to the obtained analytical solutions three-dimensional graphics characterizing the relaxations processes were built in the computer algebra system Wolfram Mathematica for different ranges of the Fourier criterion.

Fractional powers of quaternionic operators and Kato’s formula using slice hyperholomorphicity

In this paper we introduce fractional powers of quaternionic operators. Their definition is based on the theory of slice-hyperholomorphic functions and on the $S$-resolvent operators of the quaternionic functional calculus. The integral representation formulas of the fractional powers and the quaternionic version of Kato's formula are based on the notion of $S$-spectrum of a quaternionic operator. The proofs of several properties of the fractional powers of quaternionic operators rely on the $S$-resolvent equation. This equation, which is very important and of independent interest, has already been introduced in the case of bounded quaternionic operators, but for the case of unbounded operators some additional considerations have to be taken into account. Moreover, we introduce a new series expansion for the pseudo-resolvent, which is of independent interest and allows to investigate the behavior of the $S$-resolvents close to the $S$-spectrum. The paper is addressed to researchers working in operator theory and in complex analysis.

Foundation of symbol theory for analytic pseudodifferential operators, I

A new symbol theory for pseudodifferential operators in the complex analytic category is given. Here the pseudodifferential operators mean integral operators with real holomorphic microfunction kernels. The notion of real holomorphic microfunctions had been introduced by Sato, Kawai and Kashiwara by using sheaf cohomology theory. Symbol theory for those operators was partly developed by Kataoka and by the first author and it has been effectively used in the analysis of operators of infinite order. However, there was a missing part that links the symbol theory and the cohomological definition of operators, that is, the consistency of the Leibniz–Hörmander rule and the cohomological definition of composition for operators. This link has not been established completely in the existing symbol theory. This paper supplies the link and provides a cohomological foundation of the symbolic calculus of pseudodifferential operators.

An Operational Calculus Model for the nTH – Order Forward Difference

Abstract In this paper, there has been constructed such a model of a non-classical Bittner operational calculus, in which the derivative is understood as a forward difference Δn{x(k)}:={x(k+n)-x(k)}. Next, considering the operation Δn,b{x(k)}:={x(k+n)-bx(k)}, the presented model has been generalized.

A symbol calculus for foliations

The classical Getzler rescaling theorem of [15] is extended to the transverse geometry of foliations. More precisely, a Getzler rescaling calculus, [15], as well as a Block–Fox calculus of asymptotic pseudodifferential operators (A‰ \Psi DOs), [10], is constructed for all transversely spin foliations. This calculus applies to operators of degree m globally times degree \ell in the leaf directions, and is thus an appropriate tool for a better understanding of the index theory of transversely elliptic operators on foliations [13]. The main result is that the composition of A‰ \Psi DOs is again an A \Psi ‰DO, and includes a formula for the leading symbol. Our formula is more complicated due to its wide generality but its form is essentially the same, and it simplifies notably for Riemannian foliations. In short, we construct an asymptotic pseudodifferential calculus for the “leaf space” of any foliation. Applications will be derived in [5,6] where we give a Getzler-like proof of a local topological formula for the Connes–Chern character of the Connes–Moscovici spectral triple of [20], as well as the (semi-finite) spectral triple given in [5], yielding an extension of the seminal Atiyah–Singer L^2 covering index theorem, [2], to coverings of “leaf spaces” of foliations.

Discrete versus continuous operational calculus in antagonistic stochastic games

We consider a class of antagonistic stochastic games in real time between two players A and B, formalized by two marked point processes. The players attack each other at random times with random impacts. Either player can sustain casualties up to a fixed threshold. A player is defeated when its underlying threshold is crossed. Upon that time (referred to as the first passage time), the game is over. We introduce a joint functional of the first passage time, along with the status of each player upon this time. The latter are the cumulative magnitudes of casualties to each player upon the end of the game, obtained in an analytically tractable form. We then use discrete and continuous operational calculus for the transform inversion. We demonstrate in a special case that the discrete operational calculus is more efficient, allowing us to avoid numerical inversion. It leads to explicit formulas for the joint distribution of associated random variables (first passage time and the status of cumulative casualties to the players upon the end of the game).

Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials.

Operator calculus approach for route optimizing and enhancing wireless sensor network

Route optimization is one of important feature in wireless sensor networks in order to enhancing the life time of WSNs. Since Centrality is one of the greatest challenges in computing and estimating the important node metrics of a structural graph, it is necessary to calculate and determine the importance of a node in a network. This paper proposes an alternative way to optimizing the route problems which is based on multi-constrained optimal path (MCOP) and operator calculus approach. A novel routing protocol called Path Operator Calculus Centrality (POCC) is proposed as a new method to determine the main path which contains high centrality nodes in a wireless sensor network deployment. The estimation of centrality is using the operator calculus approach based on network topology which provides optimal paths for each source node to base station. Some constraints such as energy level and bit error rate (BER) of node are considered to define the path centrality in this approach. The simulation evaluation shows improved performance in terms of energy consumption and network lifetime.

Hypocoercive estimates on foliations and velocity spherical Brownian motion

By further developing the generalized $Γ$-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose generator is a step-3 generating hypoelliptic Hörmander's type operator. To prove hypocoercivity in that case, the key point is to show the existence of a convenient Riemannian foliation associated to the diffusion. We will then deduce, under suitable geometric conditions, the convergence to equilibrium of the diffusion in $H^1$ and in $L^2$.

A study of ∇-discrete fractional calculus operator on the radial Schrödinger equation for some physical potentials

The fractional calculus includes concepts of integrals and derivatives of any complex or real order. The fractional calculus is as old as the usual calculus. Recently, many scientists have been studying on this field to provide the development and applicability to various areas of mathematics, physics, engineering and other sciences. The discrete fractional calculus has an important position in this field. Some ordinary diffrential equations can be solved by means of nabla (∇) discrete fractional calculus operator. In this paper, we aim to apply this operator to the radial Schrödinger equation for some physical potentials such as pseudoharmonic and Mie-type potentials.

Determination of high power synchronous generator subtransient reactances based on the waveforms for a steady state two-phase short-circuit

In the paper, there is presented an analysis of the waveforms at the stator winding terminals of a high power synchronous generator for a steady state two-phase short-circuit. The analysis was performed by the finite element method based on a 2D generator model taking into account higher spatial and time harmonics of the magnetic field as well as the impact of eddy currents in the rotor damping circuits. The used FE model was experimentally verified. For the steady state two-phase short-circuit, the waveforms determined by the FE method were compared with the approximated waveforms calculated using analytical formulas. The analytical formulas were derived from the machine equations written in the stator fixed coordinate system (α, β, 0), using normalized transformation of coordinates and the Heaviside operational calculus. The higher time harmonics obtained based on the FFT analysis of the waveforms of the short-circuit current and open phase voltage were used for determining subtransient reactances in the d and q axes of a 200MW generator. The thus determined generator reactances were compared with the values determined by other methods.