Terms Of Special Functions Research Articles

A mathematical model for two solutes transport in a poroelastic material and its applications

Using well-known mathematical foundations of the elasticity theory, a mathematical model for two solutes transport in a poroelastic material (soft tissue is a typical example) is suggested. It is assumed that molecules of essentially different sizes dissolved in fluid and are transported through pores of different sizes. The stress tensor, the main force leading to the material deformation, is taken not only in the standard linear form but also with an additional nonlinear part. The model is constructed in 1D space and consists of five nonlinear equations. It is shown that the governing equations are integrable in stationary case, therefore all steady-state solutions are constructed. The obtained solutions are used in an example for healthy and tumour tissue, in particular, tissue displacements are calculated and compared for parameters taken from experimental data in cases of the linear and nonlinear stress tensors. Since the governing equations are non-integrable in non-stationary case, the Lie symmetry analysis is used in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with non-trivial Lie symmetries are identified. As a result, multiparameter families of exact solutions are constructed including those in terms of special functions(hypergeometric and Bessel functions). A possible application of the solutions obtained is demonstrated.

A Fractional Study of MHD Casson Fluid Motion With Thermal Radiative Flux and Heat Injection/Suction Mechanism Under Ramped Wall Condition: Application of Rabotnov Exponential Kernel

Abstract The primary objective of this research is to extend the concept of fractionalized Casson fluid flow. In this study, a comprehensive analysis of magnetohydrodynamic (MHD) natural convective flow of Casson fluid is conducted, focusing on obtaining analytical solutions using the non-integer-order derivative known as the Yang–Abdel-Aty–Cattani (YAC) operator. The YAC operator utilized in this research possesses a more generalized exponential kernel. The fluid flow is examined in the vicinity of an infinitely vertical plate with a characteristic velocity denoted as u 0. The mathematical modelling of the problem incorporates partial differential equations, incorporating Newtonian heating and ramped conditions. To facilitate the analysis, a suitable set of variables is introduced to transform the governing equations into a dimensionless form. The Laplace transform (LT) is then applied to the fractional system of equations, and the obtained results are presented in series form and also expressed in terms of special functions. The study further investigates the influence of relevant parameters, such as α, β, Pr , Q, Gr, M, Nr and K, on the fluid flow to reveal interesting findings. A comparison of different approaches reveals that the YAC method yields superior results compared to existing operators found in the literature. Graphs are generated to illustrate the outcomes effectively. Additionally, the research explores the limiting cases of the Casson and viscous fluid models to derive the classical form from the YAC fractionalized Casson fluid model.

Charged dust in Einstein–Gauss–Bonnet models

We investigate the influence of the higher order curvature terms on the static configuration of a charged dust distribution in EGB gravity. The EGB field equations for such a fluid are generated in higher dimensions. The governing equation can be written as an Abel differential equation of the second kind, or a second order linear differential equation. Exact solutions are found to these equations in terms of special functions, series and polynomials. The Abel differential equation of the second kind is reducible to a canonical differential equation; three new families of solutions are found by constraining the coefficients of the canonical equation. The charged dust model is shown to be physically well behaved in a region at the centre, and dust spheres can be generated. The higher order curvature terms influence the dynamics of charged dust and the gravitational behaviour which is distinct from general relativity.

An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals

In recent years, differential equations have become the method of choice to compute multi-loop Feynman integrals. Whenever they can be cast into canonical form, their solution in terms of special functions is straightforward. Recently, progress has been made in understanding the precise canonical form for Feynman integrals involving elliptic polylogarithms. In this article, we make use of an algorithmic approach that proves powerful to find canonical forms for these cases. To illustrate the method, we reproduce several known canonical forms from the literature and present examples where a canonical form is deduced for the first time. Together with this article, we also release an update for INITIAL, a publicly available Mathematica implementation of the algorithm.

Downlink power control in C-RAN enabled full duplex cellular networks

Full duplex (FD) in cloud radio access networks (C-RAN) has improved the overall system throughput at most to double the half duplex (HD) counterpart. Although there is an overall improvement in the throughput, it is mainly due to the downlink (DL) rate, while the uplink (UL) rate is highly degraded due to the substantial DL interference. In this paper, we showed how by employing a downlink power control (DLPC), the overall capacity could be improved along with UL rate improvement. In DL, we could achieve 2.9 times the HD rate and 1.5 times that of FD without DLPC. In UL, 1.6 times the HD rate and the improvement up to 3.49 bps/Hz from rates of the order 10−4 bps/Hz of FD without DLPC. We have also shown how by appropriately selecting DL and UL power control parameters, we can reduce the cluster size requirement in the FD C-RAN system and the fronthaul capacity requirements. We use Matern Cluster Process (MCP) to model the network. We provide coverage and rate as integral expressions in terms of special functions and algebraic terms. We also derive closed-form approximations for asymptotic rates and reflect few insights into the system design.

The sixth Painlevé equation as isomonodromy deformation of an irregular system: monodromy data, coalescing eigenvalues, locally holomorphic transcendents and Frobenius manifolds

The sixth Painlevé equation PVI is both the isomonodromy deformation condition of a 2-dimensional isomonodromic Fuchsian system and of a 3-dimensional irregular system. Only the former has been used in the literature to solve the nonlinear connection problem for PVI, through the computation of invariant quantities . We prove a new simple formula expressing the invariants p jk in terms of the Stokes matrices of the irregular system, making the irregular system a concrete alternative for the nonlinear connection problem. We classify the transcendents such that the Stokes matrices and the p jk can be computed in terms of special functions, providing a full non-trivial class of 3-dim. examples such that the theory of non-generic isomonodromy deformations of Cotti et al (2019 Duke Math. J. 168 967–1108) applies. A sub-class of these transcendents realises the local structure of all the 3-dim Dubrovin–Frobenius manifolds with semisimple coalescence points of the type studied in Cotti et al (2020 SIGMA 16 105). We compute all the monodromy data for these manifolds (Stokes matrix, Levelt exponents and central connection matrix).

Hypergeometric structures in Feynman integrals

For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

Two-component Yang–Baxter maps and star-triangle relations

It is shown how Yang–Baxter maps may be directly obtained from classical counterparts of the star-triangle relations and quantum Yang–Baxter equations. This is based on reinterpreting the latter equation and its solutions which are given in terms of special functions, as a set-theoretical form of the Yang–Baxter equation whose solutions are given by quadrirational Yang–Baxter maps. The Yang–Baxter maps obtained through this approach are found to satisfy two different types of Yang–Baxter equations, one that is the usual equation involving a single map, and another equation that involves a pair of maps, which is a case of what is also known as an entwining Yang–Baxter equation. Apart from the elliptic case, each of these Yang–Baxter maps are quadrirational, but only maps that solve the former type of Yang–Baxter equation are reversible. The Yang–Baxter maps are expressed in terms of two-component variables, and two-component parameters, and have a natural QRT-like composition of separate maps for each component. Through this approach, sixteen different Yang–Baxter maps are derived from known solutions of the classical star-triangle relations.

Static and free vibration analysis of functionally graded annular plates using stress-driven nonlocal theory

This study presents analytical and numerical solutions of the bending and the free vibration analysis of a functionally graded annular nanoplate based on the stress-driven nonlocal theory. The nonlocal equation is obtained using the classical plate theory; the power law distribution is assumed to model changes in material properties throughout the thickness. The governing differential equation is analytically solved for different types of edge supports. The explicit analytical solution is obtained in terms of special functions, namely, hypergeometric and Meijer functions. Furthermore, based on the Galerkin technique, a finite element method is introduced to figure out the nanoplate’s flexural deformation and natural frequencies. The analytical and numerical solutions are compared with those available in the literature. The size effects in the nonlocal theory and FGM’s gradient properties have been discussed, considering clamped and simply supported boundary conditions. According to the stress-driven nonlocal model, size-dependent flexural responses of the plate demonstrate stiffening behavior with increasing nonlocal parameters. Which, in general, leads to a decrease in the value of transversal deformation and an increase in natural frequencies when compared to the plate’s local solutions.

Moduli of quadrilaterals and quasiconformal reflection

We study the interior and exterior moduli of polygonal quadrilaterals. The main result is a formula for a conformal mapping of the upper half plane onto the exterior of a convex polygonal quadrilateral. We prove this by a careful analysis of the Schwarz-Christoffel transformation and obtain the so-called accessory parameters and then the result in terms of the Lauricella hypergeometric function. This result enables us to understand the dissimilarities of the exterior and interior of a convex polygonal quadrilateral. We also give a Mathematica algorithm for the computation. Then we study the special case of an isosceles trapezoidal polygon L and obtain two-sided estimates for the coefficient of quasiconformal reflection over L in terms of special functions and geometric parameters of L.

Weak similarity orbit of (log)‐self‐similar Markov semigroups on the Euclidean space

AbstractWe start by identifying a class of pseudo‐differential operators, generated by the set of continuous negative definite functions, that are in the weak similarity (WS) orbit of the self‐adjoint log‐Bessel operator on the Euclidean space. These WS relations turn out to be useful to first characterize a core for each operator in this class, which enables us to show that they generate a class, denoted by , of non‐self‐adjoint ‐contraction positive semigroups. Up to a homeomorphism, includes, as fundamental objects in probability theory, the family of self‐similar Markov semigroups on . Relying on the WS orbit, we characterize the nature of the spectrum of each element in that is used in their spectral representation which depends on analytical properties of the Bernstein‐gamma functions defined from the associated negative definite functions, and, it is either the point, residual, approximate or continuous spectrum. We proceed by providing a spectral representation of each element in which is expressed in terms of Fourier multiplier operators and valid, at least, on a dense domain of a natural weighted ‐space. Surprisingly, the domain is the full Hilbert space when the spectrum is the residual one, something which seems to be noticed for the first time in the literature. We end up the paper by presenting a series of examples for which all spectral components are computed explicitly in terms of special functions or recently introduced power series.

Heat and mass flux analysis of magneto-free-convection flow of Oldroyd-B fluid through porous layered inclined plate

The present work examines the analytical solutions of the double duffusive magneto free convective flow of Oldroyd-B fluid model of an inclined plate saturated in a porous media, either fixed or moving oscillated with existence of slanted externally magnetic field. The phenomenon has been expressed in terms of partial differential equations, then transformed the governing equations in non-dimensional form. On the fluid velocity, the influence of different angles that plate make with vertical is studied as well as slanted angles of the electro magnetic lines with the porous layered inclined plate are also discussed, associated with thermal conductivity and constant concentration. For seeking exact solutions in terms of special functions namely Mittag–Leffler functions, G-function etc., for Oldroyd-B fluid velocity, concentration and Oldroyd-B fluid temperature, Laplace integral transformation method is used to solve the non-dimensional model. The contribution of different velocity components are considered as thermal, mass and mechanical, and analyse the impacts of these components on the fluid dynamics. For several physical significance of various fluidic parameters on Oldroyd-B fluid velocity, concentration and Oldroyd-B fluid temperature distributions are demonstrated through various graphs. Furthermore, for being validated the acquired solutions, some limiting models such as Newtonian fluid in the absence of different fluidic parameters. Moreover, the graphical representations of the analytical solutions illustrated the main results of the present work and studied various cases regarding the movement of plate.

On uniform asymptotic approximations of whispering gallery modes propagating along curved penetrable interfaces

While in the classical theory of whispering gallery waves it is assumed that they propagate along a cylindrical perfectly reflecting wall, in many real situations it is important to consider similar waves propagating along penetrable interfaces or curvilinear level sets of the refractive index. An example of waves of the latter kind are whispering gallery modes formed in a shallow-water waveguide due to the horizontal refraction. Such modes are counterpart of quasistationary states in quantum mechanics, as they do not exponentially decay at infinity, and their respective eigenvalues are complex. Recently it was shown that Maslov canonical operator (MCO) theory can be used to compute uniform asymptotic expansions of the solutions to various wave propagation problems. Such approximations admit convenient analytical representation in terms of special functions that can be used for investigating their properties and for other purposes. The MCO theory is applicable only in cases where the solution exponentially decays at infinity, e.g., for computing bound states of a given quantum system. Nevertheless, the technique outlined here allows to obtain analytical expressions for the radial profiles of whispering gallery modes propagating along a penetrable wall or along a curvilinear level set of the refractive index.

On the properties of some inversion methods of the Laplace transform

The problem of inversion of the integral Laplace transform, which belongs to the class of ill-posed problems, is considered. Integral equations are reduced to ill-conditioned systems of linear algebraic equations, in which the unknowns are either the coefficients of the series expansion in terms of special functions, or the approximate values of the desired original at a number of points. Described method of inversion using special quadrature formulas of the highest degree of accuracy and the characteristics of the accuracy and stability of this method are indicated. Quadrature inversion formulas are constructed, which are adapted for the inversion of long-term and slow processes of linear viscoelasticity. A method of deformation of the integration contour in the Riemann-Mellin inversion formula is proposed, which leads the problem to the calculation of certain integrals and allows obtaining error estimates. A method is described for determining the possible breakpoints of the original and calculating the magnitude of the jump at these points.

A Quadruple Integral Involving the Legendre Function Pn(x) of the First Kind: Derivation and Evaluation

A closed form expression of a quadruple integral involving the Legerndre polynomial Pn(x) is derived. Special cases are expressed in terms of special functions and fundamental constants. All the results in this work are new.

Conservation laws for dispersive optical solitons with Radhakrishnan–Kundu–Lakshmanan model having quadrupled power-law of self-phase modulation

This paper obtains conservation laws for the newest version of Radhakrishnan–Kundu–Lakshmanan model that comes with quadrupled form of nonlinearity. The multiplier approach makes this retrieval possible. The conserved quantities are derived from the bright 1-soliton that has been recently reported. The results are in terms of special functions.

Generalized boundary value problem for a linear ordinary differential equation with a discretely distributed fractional differentiation operator

This paper formulates and solves a generalized boundary value problem for a linear ordinary differential equation with a discretely distributed fractional differentiation operator. The fractional derivative is understood as the Gerasimov–Caputo derivative. The boundary conditions are given in the form of linear functionals, which makes it possible to cover a wide class of linear local and non-local conditions. A representation of the solution is found in terms of special functions. A necessary and sufficient condition for the solvability of the problem under study is obtained, as well as conditions under which the solvability condition is certainly satisfied. The theorem of existence and uniqueness of the solution is proved.

A Quadruple Integral Involving the Hermite polynomial Hn(x): Derivation and Evaluation

A closed form expression of a quadruple integral involving the Hermite polynomial Hn(x) is derived. Special cases are expressed in terms of special functions and fundamental constants. All the results in this work are new.

Extended Apelblat Integrals for Fractional Calculus

A quadruple integral involving the logarithmic, exponential, polynomial and Gamma functions is derived in terms of the Hurwitz-Lerch zeta function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Hurwitz-Lerch zeta functions have an asymmetrical zero-distribution. The majority of the results in this work are new.

A Quadruple Integral Involving the Hermite Polynomial Hn(x): Derivation and Evaluation

A closed form expression of a quadruple integral involving the Hermite polynomial $H_{n}(x)$ is derived. Special cases are expressed in terms of special functions and fundamental constants. All the results in this work are new.