Definite Integrals Research Articles

Some summation theorems and transformations for hypergeometric functions of Kampé de Fériet and Srivastava

Abstract Owing to the remarkable success of the hypergeometric functions of one variable, the authors present a study of some families of hypergeometric functions of two or more variables. These functions include (for example) the Kampé de Fériet-type hypergeometric functions in two variables and Srivastava’s general hypergeometric function in three variables. The main aim of this paper is to provide several (presumably new) transformation and summation formulas for appropriately specified members of each of these families of hypergeometric functions in two and three variables. The methodology and techniques, which are used in this paper, are based upon the evaluation of some definite integrals involving logarithmic functions in terms of Riemann’s zeta function, Catalan’s constant, polylogarithm functions, and so on.

Comparison of the integration policies of selected European countries concerning holders of international protection

The following study aims to compare the policies of European countries related to the treatment of international protection holders, especially in the initial moments of the integration process. The object of the analysis is the policy frameworks for integrating international protection holders and how these frameworks are reflected in the relevant policy documents. We compare 12 policies through the lens of Hartmut Esser’s theoretical model of social integration. In the study, we identify four main findings: (1) the EU definition of international protection holder integration is paradoxically best matched by the public policies of non-EU countries; (2) the labor market is still perceived as a key area in the integration process; (3) despite the declared two-way principle, the topic of removing barriers to integration is neglected; (4) the term community is used exclusively in relation to the local, neighborhood context, not in relation to (ethnic) “origin”.

New results of unified Chebyshev polynomials

<abstract><p>This paper presents a new approach for the unified Chebyshev polynomials (UCPs). It is first necessary to introduce the three basic formulas of these polynomials, namely analytic form, moments, and inversion formulas, which will later be utilized to derive further formulas of the UCPs. We will prove the basic formula that shows that these polynomials can be expressed as a combination of three consecutive terms of Chebyshev polynomials (CPs) of the second kind. New derivatives and connection formulas between two different classes of the UCPs are established. Some other expressions of the derivatives of UCPs are given in terms of other orthogonal and non-orthogonal polynomials. The UCPs are also the basis for additional derivative expressions of well-known polynomials. A new linearization formula (LF) of the UCPs that generalizes some well-known formulas is given in a simplified form where no hypergeometric forms are present. Other product formulas of the UCPs with various polynomials are also given. As an application to some of the derived formulas, some definite and weighted definite integrals are computed in closed forms.</p></abstract>

Numerical treatment of the fractional Rayleigh-Stokes problem using some orthogonal combinations of Chebyshev polynomials

<p>This work aims to provide a new Galerkin algorithm for solving the fractional Rayleigh-Stokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples.</p>

On generalized Hermite polynomials

<p>This article is devoted to establishing new formulas concerning generalized Hermite polynomials (GHPs) that generalize the classical Hermite polynomials. Derivative expressions of these polynomials that involve one parameter are found in terms of other parameter polynomials. Some other important formulas, such as the linearization and connection formulas between these polynomials and some other polynomials, are also given. Most of the coefficients are represented in terms of hypergeometric functions that can be reduced in some specific cases using some standard formulas. Two applications of the developed formulas in this paper are given. The first application is concerned with introducing some weighted definite integrals involving the GHPs. In contrast, the second is concerned with establishing the operational matrix of the integer derivatives of the GHPs.</p>

A class of definite integrals involving generalized hypergeometric functions

Recently Masjed-Jamei and Koepf obtained generalizations of various classical summation theorems for the 2F1, 3F2, 4F3, 5F4 and 6F5 generalized hypergeometric series. We evaluate a new class of integrals involving generalized hypergeometric function by employing the results given by Masjed- Jamei and Koepf and MacRobert integral, and we give several special cases.

α-FRACTAL FUNCTION WITH VARIABLE PARAMETERS: AN EXPLICIT REPRESENTATION

In this paper, new results on the [Formula: see text]-fractal function with variable parameters are presented. The Weyl–Marchaud variable order fractional derivative of an [Formula: see text]-fractal function with variable parameters is examined by imposing certain conditions on the scaling factors. Following the investigation of fractional derivative, the definite integral of the [Formula: see text]-fractal function with variable parameters is evaluated for various intervals in the prescribed domain. Finally, an explicit structure for the [Formula: see text]-fractal function is provided using the base [Formula: see text] representation of numbers.

New formulas of convolved Pell polynomials

<abstract><p>The article investigates a class of polynomials known as convolved Pell polynomials. This class generalizes the standard class of Pell polynomials. New formulas related to convolved Pell polynomials are established. These formulas may be useful in different applications, in particular in numerical analysis. New expressions are derived for the high-order derivatives of these polynomials, both in terms of their original polynomials and in terms of various well-known polynomials. As special cases, connection formulas linking the convolved Pell polynomials with some other polynomials can be deduced. The new moments formula of the convolved Pell polynomials that involves a terminating hypergeometric function of the unit argument is given. Then, some reduced specific moment formulas are deduced based on the reduction formulas of some hypergeometric functions. Some applications, including new specific definite and weighted definite integrals, are deduced based on some of the developed formulas. Finally, a matrix approach for this kind of polynomial is presented.</p></abstract>

Generalization of Beckmann’s Transformation for Traffic Assignment Models with Asymmetric Cost Functions

An optimization model is developed to solve the deterministic traffic assignment problem under congested transport networks with cost functions that have an asymmetric Jacobian. The proposed formulation is a generalization of Beckmann’s transformation that can incorporate network links with multivariate vector cost functions to capture the asymmetric interactions between the flows and costs of the different links. The objective function is built around a line integral that generalizes the simple definite integral in Beckmann’s transformation and is parameterised to ensure the solution of the new problem satisfies Wardrop’s first principle of network equilibrium. It is shown that this method is equivalent to the variational inequality approach. Our new approach could be extended to supply‐demand equilibria models in other markets than transportation, with complementary or substitute goods/services in which there are asymmetric interactions between prices.

Understanding and Defining Hindu and Integrative Spiritual Care

Spiritual care holds significant promise for the future of work in organizations. However, it lacks a workable ontological foundation in the management context due to its heterogeneity. This article investigates the nature of spiritual care in contemporary contexts and identifies its key elements from an integrative perspective using the holistic framework of Hinduism. The definition of spiritual care was derived by applying design science research and a novel integrative approach adapted from the Tarka method of the Hindu philosophical system. The proposed definition was validated by spiritual-care providers in the field. The proposed integrative definition will allow scholars to develop their own comprehensive definitions for practice and for contributions to the field. The expansion of the scope of spiritual care into domains like business will support the recognition of the positive outcomes of spiritually based interventions in the workplaces of the future.

Can We Understand Leadership Better? A Framework for Shaping an Integrative Definition of Leadership

Despite hundreds of existing definitions, defining leadership remains a thorny issue today. We attempted to tackle this issue and summarise here a framework being developed to shape a renewed, simplified definition of leadership. The paper first discusses several critical philosophical and linguistic principles for defining a concept. The paper then integrates these principles to build a three-step process (namely, naturisation) to trace back to the nature of the leadership phenomenon by reducing its meanings from abstraction to simplicity. As a result, and using 12 existing definitions, we found a renewed, simplified definition of leadership, which is the use of resources. This definition was finally verified by relating it to diverse bodies of knowledge and differentiating it from management. Beyond the definition and the framework, our paper potentially contributes to moderating the leadership-management distinction debate and offers some premises of leadership for future explorations. In addition, the paper proposes the ‘right’ nature of leadership to augment other approaches examining the trait, the behavioural, and the relational natures of leadership.

Information Modelling of Extrusion-based 3D Printing Process for Digital Twins

This paper focuses on the information modelling of the extrusion-based 3D printing process. It aims to facilitate the creation and management of a digital twin (DT) through a comprehensive understanding of the manufacturing process. The workflow of extrusion-based 3D printing, spanning from the CAD model to the printed part, is analysed and modelled using the Integration Definition for Process Modelling 0 (IDEF0) method. Building upon the IDEF0 model and relevant literature, a Petri net is developed to identify specific information and logic inherent to the printing process. The identified elements are subsequently filtered by the specific purpose of the DT and categorized into three classes: Component, Activity, and Property. Specifically, this study explores the material extrusion (MEX) process as implemented by the Prusa i3 MK3 printer. Derived from functional flow block diagrams and mathematical formulas, the information model consists of 21 components, 25 activities and 83 properties. This model can be considered as an information repository, which enables DTs to consider, select and synchronise specific physical information elements in their digital representation within the context of the MEX process.

Solution of Integral of the Fourth Power of a Finite-Length Exponential Fourier Series

For a periodic function in the form of a finite-length exponential Fourier series (i.e., a discrete finite Fourier transform), this work derives an analytical solution to the definite integral of the fourth power of the function across its periodic interval.

On integration with respect to filter

This article is devoted to the study of one generalization of the Riemann integral. Namely, in the paper, it was observed that the classical definition of the Riemann integral over a finite segment as a limit of integral sums, when the diameter of the division of the segment tends to zero, can be replaced by a limit of integral sums over a filter of sets, which can be described in a certain "good way". This idea was continued, and in the work we propose a new concept - the integral of a function over a filter on the set of all tagged partitions of a segment. Using of filters is a very good method in questions related to convergence or some of its analogues in general topological vector spaces. Namely, if the space is non-metrizable, then the concept of convergence is introduced precisely with the help of filters. Also, using filters, you can formulate the concept of completeness and its analogues. The completeness of spaces is one of the central concepts of the theory of topological vector spaces, since Banach spaces are complete. That is, using a generalization of the completeness of spaces constructed using filters, we can explore various generalizations of Banach spaces. We study standard issues related to integration. For example, does the integrability of the filter function imply its boundedness? The answer to this question is affirmative. Namely: the concept of filter boundedness of a function is introduced, and it is shown that if a function is integrable over filter, then its integral sums are bounded over the filter, and this function itself is bounded in the classical sense. Next, we showed that the filter integral satisfies the linearity property, namely, the integral over filter of the sum of two functions is the sum of the filter integrals of these functions. In addition, we can to subtract the constant factor from the sign of the integral over filter. We introduce the concept of an exactly tagged filter, and with the help of such filters we study the filter integrability of unbounded functions on a segment. We give an example of a specific unbounded function and a specific filter under which this function is integrable. Next, we prove a theorem that describes unbounded filter-integrable functions on a segment. The last section of the article is devoted to the integration of functions relative to the filter on a subsegment of this segment.

A novel refined Caputo kernel and constitutive concepts for semi-exact nonlinear dynamic and creep analyses of suddenly pressurized hollow fractional-order visco-hyperelastic cylinders

Some reports have confirmed that discretizing the mass, damping, and stiffness characteristics of a continuum can result in remarkable deviations from the experimental results; as all these properties are intertwined and coupled within a single material. The viscoelastic constitutive laws with fractional-order time derivatives can model this integrity much more accurately, due to their capability of using a set of an infinite number of orders. Here, an innovative constitutive model with a nonlinear hyperelastic memory element that treats simultaneously the fractional-order viscoelasticity and Mooney-Rivlin hyperelasticity of the continuum is proposed to track the suppression of the short-term dynamic stresses and displacements and the redistribution of the long-term creep stress and deformations of the suddenly pressurized thick cylinders. While the spatial variations of the quantities of the incompressible cylinder are treated by an exact analytical approach, the resulting time-dependent system of equations is converted by a novel numerical solution scheme to an integrodifferential system that contains integer-order partial derivatives and a time-growing number of integral terms. An enhancement to Caputo's singular-kernel integral definition of the fractional-order derivatives is proposed here to replace these derivatives with equivalent integrals whose integer derivatives are computed based on second-order Taylor's expansion. The integer-order and fractional-order time derivatives are then computed numerically using the second-order Runge-Kutta and accumulation-based trapezoidal techniques, respectively. Finally, comparisons are made among the dissipative pattern and time variations of the dynamic/vibration stresses and large-deformations of the hyperelastic and fractional-order visco-hyperelastic cylinders in addition to the assessment of the effects of the constitutive parameters of the fractional-order model. Results show that while the larger magnitudes of both the nonlinear viscosity coefficients and fractional-order time derivatives diminish the resulting deformations and stresses, they respectively lead to smaller and larger frequencies of oscillation.

Spiral model development of retrofitted robot for tele-operation of conventional hydraulic excavator

This paper describes the effect of applying spiral model to the development process of robot system for a new entrant company. The robot system was developed to remotely control a conventional hydraulic excavator in order to improve the safety of operators in disaster emergency restoration. The issues of development are the definition of requirements and integration for a practical system in a real environment by a new entrant company. The constraints to the new entry of smaller companies are the following three points. (1) Lack of industry knowledge and data to define requirements (2) Lack of on-site environment and machinery for investigation and testing (3) Lack of experience in robot development To solve the problems under these constraints, the spiral model divides the development based on the prototype into 4-steps, and repeats this series of processes. This method was applied to clarify the necessary functions and performance of the robot step by step, and to construct a system with robustness in a real environment. As a result, this robot system has been successfully utilized in emergency disaster recovery tasks due to landslides, and removing debris in the Fukushima Daiichi Nuclear Power Plant, reducing the mental and physical burden on the operators.

Application of Cauchy’s Residue Theorem in Several Improper Integrals

Calculating definite integrals in complex functions requires the Cauchy's residue theorem, which is a key concept in the complex variables. It is based on several ideas, including the isolated singular points theory, the Laurent theorem, and the Cauchy integral theorem. Nevertheless, it is highly challenging to calculate closed curves using the conventional integrals method since numerous formulas must be substituted. The singular points must all be isolated in the case of a holomorphic function that belongs to an enclosing contour and is positively oriented, at which time the contour integral can be used to resolve the issue. Additionally, the problem can be solved by using visual aids to help people understand them. By doing so, the notion of solving the problem will be made clearer and some calculating steps will be cut short. This study exhibits the residue theorem's potent efficiency in computing improper integrals and other difficult problems.

Parallel scalability of sparse solvers for sparsified fractional diffusion problems discretized with FEM

Anomalous diffusion is a non-local process that describes a wide spectrum of natural processes and phenomena with many applications in science and technology. It is described mathematically by the fractional Laplace operator. In this work we examine the integral definition of the fractional Laplacian modeled with the Riesz potential and discretized with the finite element method. The thus obtained system of linear algebraic equations is dense due to the non-local nature of the fractional Laplacian and is computationally complex to solve. With LU factorization, for example, solving the problem has an O(N 3) computational complexity, where N is the number of unknowns. However, it can be observed that a large amount of the off-diagonal coefficients have very small absolute values compared to the diagonal coefficients. Those small off-diagonal coefficients can be lumped (set to zero and added to the diagonal coefficient) without significant loss of accuracy. In this work we employ a direct sparse parallel solver to the resultant sparse matrix. We analyze the parallel performance and speed-up, as well as the accuracy, varying the fractional power and the lumping threshold.

Novel Definitions of α-Fractional Integral and Derivative of the Functions

An α-fractional integral and derivative of real function have been introduced in new definitions and then, they compared with the existing definitions. According to the properties of these definitions, the formulas demonstrate that they are most significant and suitable in fractional integrals and derivatives. The definitions of α-fractional derivative and integral coincide with the existing definitions for the polynomials for 0 ≤ α < 1. Furthermore, if α = 1, the proposed definitions and the usual definition of integer derivative and integral are identical. Some of the properties of the new definitions are discussed and proved, as well, we have introduced some applications in the α- fractional derivatives and integrals. Moreover, α-power series and α–rule of integration by parts have been proposed and implemented in this study.

Arithmetic Mean Derivative-Based Quartet Midpoint Rule

A definite integral that is difficult to solve analytically can be calculated using the numerical integration methods. The midpoint rule is a prominent rule for approximating definite integrals. This article discusses a version of the quartet midpoint rule that includes the derivative of the arithmetic mean . The proposed rule increases precision over the previous rules. Furthermore, the error term is obtained by using the concept of precision between quadrature and exact values. Finally, the proposed rule is more effective than the present rule, according to numerical simulation results.