Operational Calculus Research Articles

Generalized Fock space and contextuality.

This paper is devoted to linear space representations of contextual probabilities-in generalized Fock space. This gives the possibility to use the calculus of creation and annihilation operators to express probabilistic dynamics in the Fock space (in particular, the wide class of classical kinetic equations). In this way, we reproduce the Doi-Peliti formalism. The context-dependence of probabilities can be quantified with the aid of the generalized formula of total probability-by the magnitude of the interference term. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.

Originals of operating images for generalized problems of unsteady heat conductivity

A series of operating (Laplace) non-standard images, the originals of which are absent in well-known reference books on operational calculus, are considered. By reducing one of the basic images to the Riemann-Mellin contour integral for the modified Bessel functions and analyzing the corresponding inversion formula using the approaches of the complex variable function theory, an analytical form of the original original is found, which is abrupt in nature with a break point. It is shown that analytical solutions of the corresponding mathematical models using the found originals have a wave character, which is expressed by the presence of the Heaviside step function in the solutions. The latter means that at any time there is a region of physical disturbance to the point of discontinuity and an unperturbed area after the point of discontinuity. The images studied are included in the operational solutions of mathematical models in many areas of applied mathematics. physics, thermomechanics, thermal physics, in particular in the theory of thermal shock of viscoelastic bodies, in the study of the thermal reaction of solids based on the classical Maxwell-Cattaneo-Lykov-Vernott phenomenology, taking into account the final rate of heat propagation. These models are needed to study the thermal reaction of relatively new consolidated structurally sensitive polymeric materials in structures exposed to high-intensity external influences. The analytical relations obtained for the originals and the original improper integrals resulting from them, containing combinations of Bessel functions, can be used in the general methodology of constructing and applying various mathematical models in a wide range of external influences on materials in many fields of science and technology.

Quantum Calculus and Ideals in the Algebra of Compact Operators

One of the goals of noncommutative geometry is to translate the basic notions of analysis into the language of Banach algebras. This translation is based on the quantization procedure. The arising operator calculus is called, following Connes, the quantum calculus. In this paper we give several assertions from this calculus concerning the interpretation of Schatten ideals of compact operators in a Hilbert space in terms of function theory. The main focus is on the case of Hilbert-Schmidt operators.

Novel technique of Atangana and Baleanu for heat dissipation in transmission line of electrical circuit

Abstract The new idea of Atangana and Baleanu introduced recently for fractional derivatives has received tremendous attention from researchers of various fields. However, this idea has not been applied for heat dissipation in transmission line of electrical circuit. Although the phenomenon of heat dissipation potentially damages the electrical devices which leads to catastrophic failure and shortens the life expectancy of costly electrical components. This manuscript signifies an electro-analog model of fractional diffusion process which enables to simulate heat dissipation in circuit board. The modeling of the problem has been established through modern operators of fractional calculus via the governing differential equation. By invoking Laplace transform on fractional governing equation of heat dissipation in transmission line of circuit, the analytic solutions are investigated through the exponential and Mittage Leffler kernel of non-integer order differentiations. The duality of analytic solutions has been compared with Caputo–Fabrizio and Atangana–Baleanu fractional differentiations graphically using embedded pertinent parameters. The comparison of both approaches of fractional differentiations has disclosed the heat transfer of circuit board and diffusion process with dissipation vividly.

The influence of thermal contact between medium on the surface waves propagation in a prestressed thermoelastic layered half-space

In the framework of the linearized theory of the propagation of thermoelastic waves, a dynamic mixed coupled problem on the oscillations of an inhomogeneous medium under the action of a thermal load oscillating on its surface is considered. The medium is a thermoelastic prestressed layer rigidly coupled with a thermoelastic prestressed half-space. At the boundary of the medium, two types of thermal conditions are considered: ideal thermal contact and heat insulation. Using operational calculus methods, the boundary value problem with mixed thermal boundary conditions is reduced to an integral equation of the first kind with respect to the unknown distribution function of the heat flux in the contact zone. We studied the distribution of the poles of the Green’s function of a medium, taking into account the effect of initial strains and preheating. It is shown that thermal conditions at the interface between the medium have little effect on the distribution of the vertical displacement of the layer surface in the natural state. At the same time, the condition of thermal insulation between the layer and half-space allows to compensate for the effect of body pre-heating on the formation of the surface wave field.

Harmonic univalent functions defined by post quantum calculus operators

Abstract We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion and covering theorems were also obtained. The extreme points of the family and a radius result were also obtained. The results obtained include several known results as special cases.

Abstraction and operator characterization of fractal ladder viscoelastic hyper-cell for ligaments and tendons

This study investigates the viscoelastic behaviour of soft bio-fibres in association with their fractal structures. A spring-dashpot fractal network with the self-similar topology, named the Π-type fractal ladder hyper-cell (FLHC), is abstracted from the micro/nano-structure of ligaments and tendons (LTs). Its constitutive operator is derived by the Heaviside operational calculus, which is of intrinsic fractional order. In terms of this operator, the long-term viscoelastic relaxation of bio-fibres arising from the fractal ladder topology is expounded. In addition, the fractional-order viscoelastic constitutive equation is obtained based on the FLHC of LTs, and its results are consistent with those of available human knee and spinal LT relaxation experiments. Results on the constitutive equation of FLHCs are formulated into two propositions. The multidisciplinary invariance and implications from the fractal ladder pattern of bio-fibres are also discussed.

Operational calculus for the solution of fractional differential equations with noncommensurate orders

We present an operational method suitable for solving fractional differential equations with two or more noncommensurate orders. The method can be applied in a recursive way in the sense that one can add or remove pseudo‐poles or pseudo‐zeros and recompute the output from pre‐existing solutions. In order to illustrate how the method is applied and its efficiency and simplicity, we present several examples.

Neighborhoods and Partial Sums of a New Class of Meromorphic Multivalent Functions Defined by Fractional Calculus

In this paper, we introduce and study a new class A^* (λ,μ,ν,η,p,ζ,τ) of meromorphic multivalent functions defined by fractional calculus operator of the punctured unit disc U^*. On this class we obtain several results like, coefficient inequality, modified Hadamard product, (N,δ)- neighborhood, partial sums, convex linear combination and integral operator.

Dual Numbers and Operational Umbral Methods

Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus.

Extended Mittag-Leffler function and associated fractional calculus operators

Abstract Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

Umbral calculus in Ore extensions

The aim of the paper is to show the existence of some ingredients for an umbral calculus on some Ore extensions, in a manner analogous to Rota's classical umbral calculus which deals with a univariate polynomial ring on a field of characteristic zero. For that, we introduce the notion of a quasi-derivation in order to specify Ore extensions on which building up this umbral calculus is possible. This allows in particular to define an action of the Ore extension on tensor products of modules. We develop also a Pincherle calculus for operators and we define a coalgebra structure on the Ore extension.

Exponential Sequence in the Operational Calculus Model for the n TH-Order Forward Difference

Abstract In the paper, there has been determined an exponential element in the discrete model of the non-classical Bittner operational calculus for the n th-order forward difference.

From positive PDL to its non-classical extensions

AbstractWe provide a complete binary implicational axiomatization of the positive fragment of propositional dynamic logic (PDL). The intended application of this result are completeness proofs for non-classical extensions of positive PDL. Two examples are discussed in this article, namely, a paraconsistent extension with modal De Morgan negation and a substructural extension with the residuated operators of the non-associative Lambek calculus. Informal interpretations of these two extensions are outlined.

Krawtchouk transforms and convolutions

We present an operator calculus based on Krawtchouk polynomials, including Krawtchouk transforms and corresponding convolution structure which provides an inherently discrete alternative to Fourier analysis. This approach is well suited for applications such as digital image processing. This paper includes the theoretical aspects and some basic examples.

Injective stabilization of additive functors, I. Preliminaries

This paper is the first one in a series of three dealing with the concept of injective stabilization of the tensor product and its applications. Its primary goal is to collect known facts and establish a basic operational calculus that will be used in the subsequent parts. This is done in greater generality than is necessary for the stated goal. Several results of independent interest are also established. They include, among other things, connections with satellites, an explicit construction of the stabilization of a finitely presented functor, various exactness properties of the injectively stable functors, a construction, from a functor and a short exact sequence, of a doubly-infinite exact sequence by splicing the injective stabilization of the functor and its derived functors. When specialized to the tensor product with a finitely presented module, the injective stabilization with coefficients in the ring is isomorphic to the 1-torsion functor. The Auslander-Reiten formula is extended to a more general formula, which holds for arbitrary (i.e., not necessarily finite) modules over arbitrary associative rings with identity. Weakening of the assumptions in the theorems of Eilenberg and Watts leads to characterizations of the requisite zeroth derived functors.The subsequent papers, provide applications of the developed techniques. Part II deals with new notions of torsion module and cotorsion module of a module. This is done for arbitrary modules over arbitrary rings. Part III introduces a new concept, called the asymptotic stabilization of the tensor product. The result is closely related to different variants of stable homology (these are generalizations of Tate homology to arbitrary rings). A comparison transformation from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be epic.

On Joint Functional Calculus for Ritt Operators

In this paper, we study joint functional calculus for commuting $n$-tuple of Ritt operators. We provide an equivalent characterisation of boundedness for joint functional calculus for Ritt operators on $L^p$-spaces, $1< p<\infty$. We also investigate joint similarity problem and joint bounded functional calculus on non-commutative $L^p$-spaces for $n$-tuple of Ritt operators. We get our results by proving a suitable multivariable transfer principle between sectorial and Ritt operators as well as an appropriate joint dilation result in a general setting.

A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

In this paper, we develop a fractional integro‐differential operator calculus for Clifford algebra‐valued functions. To do that, we introduce fractional analogues of the Teodorescu and Cauchy‐Bitsadze operators, and we investigate some of their mapping properties. As a main result, we prove a fractional Borel‐Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge‐type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann‐Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.

Solving Fractional – Time Convection Diffusion Equations with Shifting Coefficients Using Legendre Wavelet Method

In the proposed paper, Legendre wavelet approach relies on the matrices of fractional operational calculus is suggested to solve fractional – time convection-diffusion equations (FTCDEs) with shifting coefficients. The propounded numerical mechanism is worthy, suitable, meticulous and accurate for solving such Dirichlet’s IBV problems since these conditions are considered instinctively. Numerical clarifications are manifested to establish evidence for the validity, accuracy, rationality, and efficiency of the propound mechanism.

Some Schemata for Applications of the Integral Transforms of Mathematical Physics

In this survey article, some schemata for applications of the integral transforms of mathematical physics are presented. First, integral transforms of mathematical physics are defined by using the notions of the inverse transforms and generating operators. The convolutions and generating operators of the integral transforms of mathematical physics are closely connected with the integral, differential, and integro-differential equations that can be solved by means of the corresponding integral transforms. Another important technique for applications of the integral transforms is the Mikusinski-type operational calculi that are also discussed in the article. The general schemata for applications of the integral transforms of mathematical physics are illustrated on an example of the Laplace integral transform. Finally, the Mellin integral transform and its basic properties and applications are briefly discussed.