Class Of Special Functions Research Articles

Solution of the Vector Three-Dimensional Inverse Problem on an Inhomogeneous Dielectric Hemisphere Using a Two-Step Method

This work is devoted to the development and implementation of a two-step method for solving the vector three-dimensional inverse diffraction problem on an inhomogeneous dielectric scatterer having the form of a hemisphere characterized by piecewise constant permittivity. The original boundary value problem for Maxwell’s equations is reduced to a system of integro-differential equations. An integral formulation of the vector inverse diffraction problem is proposed and the uniqueness of the solution of the first-kind integro-differential equation in special function classes is established. A two-step method for solving the vector inverse diffraction problem on the hemisphere is developed. Unlike traditional approaches, the two-step method for solving the inverse problem is non-iterative and does not require knowledge of the exact initial approximation. Consequently, there are no issues related to the convergence of the numerical method. The results of calculations of approximate solutions to the inverse problem are presented. It is shown that the two-step method is an efficient approach to solving vector problems in near-field tomography.

Bridging the p-Special Functions between the Generalized Hyperbolic and Trigonometric Families

Here, we study the extension of p-trigonometric functions sinp and cosp family in complex domains and p-hyperbolic functions sinhp and the coshp family in hyperbolic complex domains. These functions satisfy analogous relations as their classical counterparts with some unknown properties. We show the relationship of these two classes of special functions viz. p-trigonometric and p-hyperbolic functions with imaginary arguments. We also show many properties and identities related to the analogy between these two groups of functions. Further, we extend the research bridging the concepts of hyperbolic and elliptical complex numbers to show the properties of logarithmic functions with complex arguments.

Going Next after “A Guide to Special Functions in Fractional Calculus”: A Discussion Survey

In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions pΨq and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and pFq-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the H¯-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and H¯-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of Γ-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and H¯-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems.

Valued-inverse dombi fuzzy graphs

Fuzzy graphs as labeled graphs (fuzzy vertex labeling and fuzzy edge labeling) have many applications in real life such as complex networks, coding theory, medical sciences, communication networks, and management sciences. Also, triangular norms as a special class of functions, have many applications in fuzzy set theory, probability and statistics, and other areas. This paper considers the notations of an inverse fuzzy graph and triangular norms to introduce the new type of graphs as valued-inverse Dombi fuzzy graphs. The valued-inverse Dombi fuzzy graphs are a generalization of inverse fuzzy graphs and are dual to Dombi fuzzy graphs. For any given greater than or equal to one real number, we construct a type of Dombi inverse fuzzy graph and investigate some conditions such that the product and union of Dombi inverse fuzzy graphs be a Dombi inverse fuzzy graph.

The Multi-Index Mittag-Leffler-Le Roy Functions as I- and H¯-Functions and New Fractional Calculus Operators

In the last decade, the interest in a special function named after the French mathematician Le Roy has provoked several authors to introduce and study its various extensions. Historically, the Le Roy function appeared almost in same time and in similar way of goals like the Mittag-Leffler function that has important role in Fractional Calculus. The Le Roy type functions are also “fractional exponentials” but the fractionalizing parameters appear as power indices of the involved Gamma functions. In our recent works, we have introduced rather general multi-index Le Roy type functions, studied their analytical properties and proposed their association to the class of Special Functions of Fractional Calculus.A newly developed idea is to show that the Le Roy type functions can be represented in terms of the I-functions of Rathie and H-functions of Inayat-Hussain that are further extensions of the Fox H¯-functions and Fox-Wright pΨq-functions. It happens that other important mathematical functions as the polylogarithms, Riemann Zeta function and its extensions, also belong to this more general class of special functions. We have solved, at least in some simpler case, the problem to find integral and differential-like operators for which the Le Roy type functions are eigenfunctions. This lead us to a new class of operators with I-functions kernels that can be considered as further extensions of the operators of generalized fractional calculus.

On the convergence properties of generalized Szász–Kantorovich type operators involving Frobenious–Euler–Simsek-type polynomials

<p>This work focuses on the study of approximation properties of functions by Szász type operators involving Frobenius–Euler–Simsek-type polynomials, which have become more popular recently because of their special characteristics and functional organization. The convergence properties such as uniformly convergence and pointwise convergence in terms of modulus of continuity and Peetre-$ \it K $ functional are investigated with the help of these sequences of operators in depth. This paper also includes the estimation of the error of the approximation of these sequences of operators to some particular class of functions. The estimates are depicted using the Maple scientific computing program and presented in tables.</p>

Mathematical approximator based on basic spline approximation

The article considers the problem of constructing a mathematical piecewise linear approximator based on approximation by basis splines. An algorithm has been developed designed to implement a class of special functions and create parallel microprogramming for the algorithm for calculating spline functions and, on their basis, parallel operation of a microprogram automaton with associative memory.

О решении смешанной задачи для уравнения колебаний движущегося вязкоупругого полотна

A model initial boundary value problem of small transverse oscillations of a viscoelastic moving web with a hinged condition of fastening is considered. The vibrations of such a canvas are described by a linear differential equation of the 5th order in a spatial variable with constant coefficients. It is worth noting that the equation includes mixed derivatives of the desired function both with respect to the spatial variable and with respect to time. The paper describes a technique for constructing a solution in the form of a functional series based on a system of basis functions. To solve the initial-boundary value problem under the additional condition of conservation of energy, a condition is obtained that ensures the uniqueness of the solution. A special class of functions for which the uniqueness theorem holds is explicitly described.

On special solutions to the Ermakov–Painlevé XXV equation

In this paper, we study a nonlinear second-order ordinary differential equation which we call the Ermakov–Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced either to the Ermakov or the Painlevé XXV equation. The Ermakov–Painlevé XXV equation arises from a generalized Riccati equation and the related third-order linear differential equation via the Schwarzian derivative. The generalized Riccati equation has two families of Riccati solutions and we study the corresponding solutions to the Ermakov–Painlevé XXV equation. We show that one of these families appears only in the Ermakov case. We give examples of the Ermakov–Painlevé XXV equations and show how to construct their solutions expressed in terms of elementary or in terms of the classical special functions.

Two-well linearization for solid-solid phase transitions

In this paper we consider nonlinearly elastic, frame-indifferent, and singularly perturbed two-well models for materials undergoing solid-solid phase transitions in any space dimensions, and we perform a simultaneous passage to sharp-interface and small-strain limits. Sequences of deformations with equibounded energies are decomposed via suitable Caccioppoli partitions into the sum of piecewise constant rigid movements and suitably rescaled displacements. These converge to limiting partitions, deformations, and displacements, respectively. Whereas limiting deformations are simple laminates whose gradients only attain two values, the limiting displacements belong to the class of special functions with bounded variation ( SBV ). The latter feature elastic contributions measuring the distance to simple laminates, as well as jumps associated to two consecutive phase transitions having vanishing distance, and thus undetected by the limiting deformations. By \Gamma - convergence we identify an effective limiting model given by the sum of a quadratic linearized elastic energy in terms of displacements along with two surface terms. The first one is proportional to the total length of interfaces created by jumps in the gradient of the limiting deformation. The second one is proportional to twice the total length of interfaces created by jumps in the limiting displacement, as well as by the boundaries of limiting partitions. A main tool of our analysis is a novel two-well rigidity estimate which has been derived in [Calc. Var. Partial Differential Equations 59, art. 44 (2020)] for a model with anisotropic second-order perturbation.

Coupled strain and temperature gradient analysis in curved surfaces

AbstractA common method in experimental mechanics is 3D digital image correlation (DIC) for determining deformations of curved surfaces, whereby the motion of material points is measured. The temperatures induced by the deformation of surfaces or by other heat sources are of particular interest as well. In this context, surface temperatures can be measured using infrared thermography systems. However, the disadvantage of this approach is that a thermographic camera only provides the two‐dimensional temperature distribution (plane images) of three‐dimensionally curved surfaces. Thus, the temperature and temperature gradients on the curved surface assigned to material points cannot be specified. However, for particular problems the temperature and the motion of material points are of particular interest. In the literature, planar problems are studied and simplifying assumptions are made so that a simple coupling of thermography and DIC is possible. In contrast, this contribution presents a method where 3D DIC is coupled with thermography in such a way that the temperature, temperature gradient, as well as displacement and strain distributions at material points of curved surfaces can be determined over temporal processes. The method also allows a continuous strain and temperature gradient determination over the entire domain, since a special class of functions named radial basis functions is used for interpolating the measured data.

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained.

Outlier-free spline spaces for isogeometric discretizations of biharmonic and polyharmonic eigenvalue problems

We present outlier-free isogeometric Galerkin discretizations of eigenvalue problems related to the biharmonic and the polyharmonic operator in the univariate setting. These are Galerkin discretizations in certain spline subspaces that provide accurate approximations of all eigenfrequencies and eigenmodes, without the occurrence of spurious outliers. The theoretical cornerstone is the use of spline subspaces that are optimal in the sense of L2 Kolmogorov n-widths for special function classes related to the eigenvalue problems under investigation. This work is a continuation and extension of a similar study, recently presented for the Laplace operator, towards higher-order problems. As for the Laplace operator, the considered optimal spline spaces are identified by additional homogeneous boundary conditions and by specific sequences of break points. For fourth- and higher-order problems, however, optimal spline spaces are not known for all degrees and the optimal break points are not explicitly given. A careful analysis of the properties of optimal spaces allows us to determine spline subspaces of practical interest that are outlier-free. In particular, in the biharmonic case, we are able to construct such outlier-free spline spaces, for any odd and even degree, by considering break points that are uniform, or just a minor modification of the uniform ones. This improves upon the (numerical) results available in the literature. The theoretical findings are validated by a selection of numerical tests.

Multiscale matrix pencils for separable reconstruction problems

The nonlinear inverse problem of exponential data fitting is separable since the fitting function is a linear combination of parameterized exponential functions, thus allowing to solve for the linear coefficients separately from the nonlinear parameters. The matrix pencil method, which reformulates the problem statement into a generalized eigenvalue problem for the nonlinear parameters and a structured linear system for the linear parameters, is generally considered as the more stable method to solve the problem computationally. In Section 2 the matrix pencil associated with the classical complex exponential fitting or sparse interpolation problem is summarized and the concepts of dilation and translation are introduced to obtain matrix pencils at different scales. Exponential analysis was earlier generalized to the use of several polynomial basis functions and some operator eigenfunctions. However, in most generalizations a computational scheme in terms of an eigenvalue problem is lacking. In the subsequent Sections 3–6 the matrix pencil formulation, including the dilation and translation paradigm, is generalized to more functions. Each of these periodic, polynomial or special function classes needs a tailored approach, where optimal use is made of the properties of the parameterized elementary or special function used in the sparse interpolation problem under consideration. With each generalization a structured linear matrix pencil is associated, immediately leading to a computational scheme for the nonlinear and linear parameters, respectively from a generalized eigenvalue problem and one or more structured linear systems. Finally, in Section 7 we illustrate the new methods.

On symmetric solutions of the fourth q-Painlevé equation

The Painlevé equations possess transcendental solutions y(t) with special initial values that are symmetric under rotation or reflection in the complex t-plane. They correspond to monodromy problems that are explicitly solvable in terms of classical special functions. In this paper, we show the existence of such solutions for a q-difference Painlevé equation. We focus on symmetric solutions of a q-difference equation known as or and provide their symmetry properties and solve the corresponding monodromy problem.

Generalized Mixed Type Bernoulli-Gegenbauer Polynomials

The generalized mixed type Bernoulli-Gegenbauer polynomials of order α >−1/2 are special polynomials obtained by use of the generating function method. These polynomials represent an interesting mixture between two classes of special functions, namely generalized Bernoulli polynomials and Gegenbauer polynomials. The main purpose of this paper is to discuss some of their algebraic and analytic properties.

Research Spotlights

What surprising patterns do fluid dynamics and neurobiological networks have in common? In this issue's first Research Spotlights article, “Bump Attractors and Waves in Networks of Leaky Integrate-and-Fire Neurons,” a connection is made between the waves at the onset of pipe turbulence and the localized traveling waves in spiking neural networks. Authors Daniele Avitabile, Joshua L. Davis, and Kyle Wedgwood consider a so-called “discrete integrate-and-fire model” to describe the dynamics of a connected neuron firing via a system of piecewise linear ordinary differential equations. Constructing a solution to this system effectively amounts to determining firing times of the neurons. One can then witness “bump” patterns, in this case characterized by a core of neurons that fire frequently, while neurons away from the core do not fire. Such a model's deterministic chaotic behavior can then be studied around a firing set and used to construct waves and analyze their stability. Simulations and analysis demonstrate oscillatory and nested branching of these waves, and such behavior can accumulate in a bump attractor and wandering wave. The authors show that such complex spatiotemporal patterns, and their conditions for stability, are not unlike those witnessed in the transition to turbulence in a pipe. The paper concludes with a challenge to future researchers: can further dynamical systems characterizations of such event-driven models uncover links between localized waves and bumps in complex spatial networks? The second Research Spotlights article addresses the problem of identifying a system from noisy observations of dynamical data. In “Learning Dynamical Systems with Side Information” authors Amir Ali Ahmadi and Bachir El Khadir present a way to incorporate system knowledge to improve the learning process beyond what is possible using data alone. Such knowledge, or “side information,” arises when one has contextual information about the unknown dynamics without knowing the dynamics precisely. Examples of side information covered by the authors' framework include known trajectories (e.g., when one has knowledge of some equilibrium points), group symmetry, and coordinatewise properties (e.g., nonnegativity or monotonicity). Restricting attention to a particular class of functions to be learned and particular side information allows the learning process to be posed as a semidefinite optimization formulation amenable to efficient solution. This approach is demonstrated numerically on systems modeling the diffusion of a contagion, behavior of a simple pendulum, and time evolution of a cancerous tumor, as well as on the classical Lorenz system. In each case, imposing side information via semidefinite programming allows the authors to learn the unknown dynamics from far fewer observations than otherwise possible and provides a potential road map for addressing even broader classes of systems and side information.

On the weighted Bojanov-Chebyshev problem and the sum of translates method of Fenton

Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and a very general external field function. Due to our very general setting the minimax, equioscillation and characterization results obtained extend those of Bojanov, Fenton, Hardin, Kendall, Saff, Ambrus, Ball and Erdélyi. Moreover, we discover a surprising intertwining phenomenon of interval maxima, which provides new information even in the most classical extremal problem of Chebyshev. Bibliography: 25 titles.

On the shape-preserving properties of λ-Bernstein operators

We investigate the shape-preserving properties of λ-Bernstein operators B_{n,lambda } ( f;x ) that were recently introduced Bernstein-type operators defined by a new Beziér basis with shape parameter lambda in [ -1,1 ] . For this purpose, we express B_{n,lambda } ( f;x ) as a sum of a classical Bernstein operator and a sum of first order divided differences of f. Using this new representation, we prove that B_{n,lambda } ( f;x ) preserves monotonic functions for all lambda in [ -1,1 ] . However, we show by a counter example that B_{n,lambda } ( f;x ) does not preserve convex functions for some lambda in [ -1,1 ] . We present a weaker result for the case lambda in [ 0,1 ] for a special class of functions. Finally, we analyze the monotonicity of λ-Bernstein operators with n and show that B_{n,lambda } ( f;x ) is not monotonic with n for some λ if 1/2 <lambda leq 1.

Block dense weighted networks with augmented degree correction

AbstractDense networks with weighted connections often exhibit a community-like structure, where although most nodes are connected to each other, different patterns of edge weights may emerge depending on each node’s community membership. We propose a new framework for generating and estimating dense weighted networks with potentially different connectivity patterns across different communities. The proposed model relies on a particular class of functions which map individual node characteristics to the edges connecting those nodes, allowing for flexibility while requiring a small number of parameters relative to the number of edges. By leveraging the estimation techniques, we also develop a bootstrap methodology for generating new networks on the same set of vertices, which may be useful in circumstances where multiple data sets cannot be collected. Performance of these methods is analyzed in theory, simulations, and real data.