Mathematica Package Research Articles

Olsson.wl & ROC2.wl: Mathematica packages for transformations of multivariable hypergeometric functions & regions of convergence for their series representations in the two variables case

We present the Olsson.wlMathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson [1] (1964) to derive the analytic continuations of the Appell F1 double hypergeometric series using the linear transformations of the Gauss F12 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples that are presented in the text and in some ancillary Mathematica notebooks. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6−2ϵ. We also provide a companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl. Program summaryProgram Title:Olsson.wlCPC Library link to program files:https://doi.org/10.17632/gc63xzwzz5.1Licensing provisions: GNU General Public License v3.0.Programming language: Wolfram Mathematica version 11.3 and beyond.Nature of problem: To find the transformation formulas of multivariable hypergeometric series appearing in Feynman integral calculus.Solution method: Mathematica implementation of the method of Olsson [1]. The method uses the transformation theory of lower variable hypergeometric functions to find the transformation formulas of higher variable hypergeometric functions.Companion package:ROC2.wl, which is a Mathematica package used inside Olsson.wl and can also be used as a standalone package. The package is used to find the region of convergence of double hypergeometric functions, using Horn's theorem.

Viscous Dissipation and Mixed Convection Effects on the Induced Magnetic Field for Peristaltic Flow of a Jeffrey Nanofluid

The issue of Jeffrey nanofluid peristaltic flow in an asymmetric channel being affected by an induced magnetic field was studied. In addition, mixed convection and viscous dissipation were considered. Under the supposition of a long wave length and a low Reynolds number, the problem was made simpler. The system and corresponding boundary conditions were solved numerically by using the built-in package NDSolve in Mathematica software. This software ensures that the boundary value problem solution is accurate when the step size is set appropriately. It computes internally using the shooting method. Axial velocity, temperature distribution, nanoparticle concentration, axial induced magnetic field, and density distribution were all calculated numerically. An analysis was conducted using graphics to show how different factors affect the flow quantities of interest. The results showed that when the Jeffrey fluid parameter is increased, the magnitude of axial velocity increases at the upper wall of the channel, while it decreases close to the lower walls. Increasing the Hartmann number lads to increases in the axial velocity near the channel walls and in the concentration of nanoparticles. Additionally, as the Brownian motion parameter is increased, both temperature and nanoparticle concentration grow.

Exact wave solutions of new generalized Bogoyavlensky–Konopelchenko model in fluid mechanics

In this paper, we examine a novel generalized [Formula: see text]-dimensional Bogoyavlensky–Konopelchenko (gBK) model analytically via application of five mathematical methods, namely called Extended Simple Equation (ESE) method, Modified Extended Auxiliary equation mapping (MEAEM) method, [Formula: see text]-expansion method, [Formula: see text]-expansion method and modified F-expansion method, respectively, to obtain different types solitary wave solutions in the form of trigonometric, hyperbolic, rational and exponential functions as compared to solutions exist in [S. T. Chen and W. X. Ma, Math. China 13 (2018) 525; A. Sonia, A. Jamshad, S.-U. Rehman and A. Ali, Int. J. Appl. Comput. Math. 9 (2023)]. The research focuses on physical systems, physiology, mathematical applications, engineering fields, and the chemical processes of species in the porous nanoparticles, examining the impact of various parameters on the nonlinear model. The graphs of some solutions are plotted in the form of two-dimensional and three-dimensional by assigning the particular values to the paraments with the assistance of mathematica software which illustrated the physical behavior of the concern model. The derived solution has numerous applications in fluid dynamics and the interaction of Riemann waves and long waves.

Non-perturbative Symmetries of Little Strings and Affine Quiver Algebras

We consider Little String Theories (LSTs) that are engineered by N parallel M5-branes probing a transverse ℤM geometry. By exploiting a dual description in terms of F-theory compactified on a toric Calabi-Yau threefold XN,M, we establish numerous symmetries that leave the BPS partition function \U0001d4b5N,M invariant. They furthemore act in a non-perturbative fashion from the point of view of the low energy quiver gauge theory associated with the LST. We present different group theoretical organisations of these symmetries, thereby generalising the results of [1] to the case of generic M ≥ 1. We also provide a Mathematica package that allows to represent them in terms of matrices that act linearly on the Kähler parameters of XN,M. From the perspective of dual realisations of the LSTs the symmetries found here act in highly nontrivial ways: as an example, we consider a formulation of \U0001d4b5N,M in terms of correlation functions of a vertex operator algebra, whose commutation relations are governed by an affine quiver algebra. We show the impact of the symmetry transformations on the latter and discuss invariance of \U0001d4b5N,M from this perspective for concrete examples.

Significance of hafnium nanoparticles in hydromagnetic non-Newtonian fluid-particle suspension model through divergent channel with uniform heat source: thermal analysis

Abstract This theoretical analysis provided the exact solution of a steady flow of Casson rheological fluid in fluid-particle suspension models through a divergent channel with consideration of porous medium, electric, and magnetic fields, and slip boundary conditions. The thermal transport analysis is also observed with the consideration of viscous dissipation and uniform heat source. The suitable transformation is used to reduce the partial differential equation into ordinary differential equations and obtain the exact solution by adopting the mathematical software MATHEMATICA 12.0. The momentum and thermal profiles are decreasing functions of the magnetic field parameter. The number of streamlines is increased and covers more parts of the channel for increasing the Darcy force and velocity slip parameters. The computational results of this study will help to understand the momentum and thermal analysis in the fluid-particle suspension model. The results of the current study are useful to increase the oil recovery system, in thermal transport energy, energy production, cooling and heating systems, etc. The current model can be useful in renewable energy to store thermal energy by using the hafnium nanoparticles. The present analysis is original and has not been submitted nor published before.

Fourth-grade nanofluid model with dissipative and nonlinear radiative properties transported peristaltically via a flexible diverging duct holding a porous media under the influence of concentration and heat convection in an induced magnetic field

The major goal of this work is to give a thorough examination of the effect of double diffusion convection (DDC) in addition to a generated magnetic field on peristaltic movement of fourth-grade nanofluid across a vertical sophisticated asymmetrical microchannel using a nondeformable porous media as a basis for intricate pumping systems inspired by biological processes for hazardous waste. The mathematical formulas pertaining to flow, heat/mass transfer under the influence of viscous dissipation, nonlinear heat radiation, and Joule heating were developed using Buongiorno’s framework for nanofluids with combining the thermophoresis and Brownian motion characteristics. Mathematical analysis has been conducted under the suppositions of an extended wavelength and a relatively small Reynolds number. Magnetic field induced axially, density of current, magnetic force function, thermal characteristics, nanoparticles proportion gradient, an additional stress tensor, pressure gradient, and stream function are all given explicit formulas. The constructed function (ND Solve function) within the Wolfram software (Mathematica) is employed to computationally resolve the ensuing system of coupled nonlinear differential equations. Numerical and pictorial evidence is presented to highlight the significance of different physiological characteristics of flow volumes. Further, contour visualizations and circulation bolus have been used to highlight the trapping phenomena, one of among the most noteworthy peristaltic motion occurrences. The main results showed that, despite the dissolvent concentration and the volume percentage of nanoparticles having the opposite effects, the resistance of a substance to heat is shown to climb as the Soret and Dufour numbers rise. At larger levels of the electromagnetic Reynolds number, Strommer’s number, electric field parameter, and thermal Grashof number, stronger axial induced magnetic fields (IMFs) are also provided.

Entropy optimization in bio-convective chemically reactive flow of micropolar nanomaterial with activation energy and gyrotactic microorganisms

The main objective of this research paper is to scrutinize the entropy generation(EG) in the bioconvective flow of micropolar fluid by porous surface of a stretched sheet. Mixed convection effects are considered. Thermal radiation, internal fluid friction, Joule heating, and heat generation aspects are accounted for in the energy relation. Mass concentration relation is developed because of the binary chemical reactions associated with Arrhenius kinetics. The governing equations for the micropolar fluid are developed based on Buongiorno's model with the help of boundary layer restrictions. By invoking the transformation procedure, dimensional model is made dimensionless. The well-known shooting technique RKF-45 in Mathematica software is implemented to solve the transformed equations. Performance of micropolar fluid velocities, thermal field, entropy generation, mass concentration, Bejan number, and motile density versus influential variables are highlighted. Furthermore, the intensity of skin friction, Nusselt quantity, motile density, and Sherwood number is computed and analyzed. It is observed that the rate of EG improves while the Bejan number diminishes versus higher Brinkman number, and diffusion parameter due to gyrotactic microorganisms. Magnitude of heat transfer rate boosts through higher Γ, Qs, Ec, Ha and Rd at an average percentage of 44, 82, 189, 156 respectively.

New optical wave solutions of M-fractional nonlinear Schrödinger’s complex hyperbolic model by an efficient method

In this paper, we succeed to obtain the new optical wave solutions to the truncated M-fractional (2+1)-dimensional nonlinear Schrödinger’s complex hyperbolic model by utilizing the Sardar sub-equation method. The obtained solutions are in the form of trigonometric, hyperbolic and mixed form. These solutions have many applications in nonlinear optics, fiber optics, and other areas of physics and engineering where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by two-dimensional, three-dimensional and contour plots. The gained solutions are helpful for the further development of concerned model. Finally, this technique is simple, fruitful and reliable to handle the nonlinear PDEs.

A study on Chlamydia transmission in United States through the Haar wavelet technique

In this study, a novel method known as the Haar wavelet collocation method (HWCM) is used to analyze the mathematical model of Chlamydia transmission in the United States. We use dependent variables with various parameters, such as birth rate, mortality rate, and recovery rate, etc., to explore the nature of the Chlamydia disease in susceptible unvaccinated individuals, susceptible vaccinated individuals, infected, treated, and recovered individuals. The ordinary differential equations (ODEs) in this model are coupled nonlinear. In this method, the Chlamydia model is converted into a system of non-linear algebraic equations using properties of the operational matrix of Haar wavelets. Later, the Newton–Raphson approach is used to extract the unknown coefficients. Mathematica software has been used for all calculations. Tables and graphs contain tabulated results of calculations. As a result, it can be seen that the current approach is more precise than those used in the literature. Also, we discuss the effect of different parameters on susceptible unvaccinated individuals, susceptible vaccinated individuals, infected, treated, and recovered individuals through graphs.

Pulsatile flow of Casson hybrid nanofluid between ternary-hybrid nanofluid and nanofluid in an inclined channel with temperature-dependent viscosity

This study presents a mathematical model for layered non-miscible liquid flow in an inclined channel subject to a pulsatile pressure gradient. The Casson hybrid nanofluid (hnf) is sandwiched between the layers of Newtonian ternary hybrid nanofluid (t-hnf) and nanofluid (nf). Additionally, the impact of temperature-dependent viscosity is taken into consideration. Multiwall carbon nanotubes, molybdenum disulfide, and silver are considered nanoparticles, with water and kerosene oil as base fluids. The governing partial differential equations are transformed into coupled non-linear ordinary differential equations (ODEs) using the perturbation technique. The resulting ODEs are solved numerically using MATHEMATICA software with the help of the shooting technique associated with the Runge-Kutta fourth-order scheme. The behaviors of numerous sundry parameters against velocity, temperature, concentration, and shear stress are depicted graphically and in tabular form. The outcomes of the study reveal that if the viscosity is held constant, then the fluid’s velocity is initially low and gradually increases as the viscosity variation parameter rises. The velocity of fluid flow is higher when a Newtonian fluid occupies the middle region of the channel than when Casson fluid does. The increase in the nanoparticle shape factor, at both boundaries, results in an increase in the rate of heat transfer.

Kerr-like oblate heterogeneous primaries in PCRFB problem with variable mass infinitesimal body

The dynamical motion properties of the variable mass infinitesimal body are investigated under the effects of the gravitational forces of the kerr-like oblate heterogeneous primaries which are situated at the vertices of an equilateral triangle. The effects of the coriolis as well as centrifugal forces are also considered. Under the effects of the above perturbations we evaluate the equations of motion, and then we perform the important dynamical properties of motion such as the locations of parking points, their stability, first return map (i.e. Poincaré surfaces of section), regions of motion by evaluating quasi-Jacobian integral and trajectory allocations by well-known software Mathematica. This study will show the effects of perturbations on these motion properties in various cases such as classical case and for different values of perturbations. One can compare these results from the results available in the literature.

Analysis of Torsional Vibration in a Fractured Poroelastic Half-Space Coated with Metal Foam and Sliding Interfaces

Metal foams are highly useful in industries because of their lightweight, energy and vibration absorption properties. This study investigated the propagation of torsional waves in an elastic layer over a fluid-saturated fractured poroelastic half-space with a metal foam coated layer. It is assumed that the interfaces are in sliding contact with two different sliding parameters. The coated layer is closed-cell aluminium foam. We use the separation variable technique and the Bessel function to solve the equation of motion in different layers. The displacement components are written in terms of the second kind Whittaker functions. Using an asymptotic formulation of the Whittaker function and appropriate boundary conditions, the dispersion equation is derived in terms of the determinant. The control of the vibration due to the metal foam-coated layer is one of the important goals of this study. Also, numerical and graphical analyses have been done with the help of Mathematica software to see the effect of different parameters on torsional wave propagation. It is found that the presence of coated metal foam layer decreases the phase velocity of the torsional wave propagation. The work may be helpful in the seismology, automobile, and aerospace industries.

¬¬Chemical Reaction and Nonlinear Rosseland Approximation Effects On Double-Diffusive MHD Sisko Nanofluid Flow Over a Nonlinear Stretching Sheet in A Porous Medium with Concentration-Dependent Internal Heat Source

This study investigates the two-dimensional magnetohydrodynamics laminar flow of a Sisko nanofluid over a nonlinear stretching sheet in a porous medium, considering chemical reactions, nonlinear Rosseland approximation, and an internal heat source based on concentration. The governing boundary layer equations and associated boundary conditions are transformed into non-dimensional form using appropriate variables. The resulting non-dimensional equations are solved using the modified Adomian decomposition technique, implemented with the symbolic computing software MATHEMATICA. The influence of various parameters, including Prandtl number, thermophoresis parameter, Brownian motion parameter, chemical reaction parameter, Sisko fluid parameters, heat source parameter, radiation parameter, Darcy number, mass Grashof number, magnetic field, and Grashof number, on the flow fields of velocity, temperature, and nanoparticles volume fractions is depicted through graphical representations. The study reveals that the velocity profile is accelerated by the first value of the Sisko fluid parameter and Darcy number, but it diminishes in the presence of a magnetic field and the second Sisko fluid parameter. The temperature profile decreases with an increment in the Prandtl number, while it improves in proportion to increases in the heat source, thermophoresis, Brownian motion, and radiation parameters respectively. Additionally, an increase in the chemical reaction parameter leads to an increase in the concentration profile of the fluid.

An alternative numerical solution for the Orr–Sommerfeld problem

A simple tutorial for the numerical solution of the Orr–Sommerfeld problem is presented. The aim is the determination of the neutral stability threshold to the linear instability for the Poiseuille flow within a plane parallel channel. Such an instability is hydrodynamic in nature as it is driven by the advection term in the local momentum balance equation. The numerical solution is achieved by a spectral method. The basis functions are chosen as Chebyshev polynomials multiplied by a suitable overall factor meant to allow the fulfilment of the boundary conditions. The details of the numerical code implemented via the Mathematica software environment are also provided. A comparison with other numerical solutions of the same problem implemented via the tau-method is presented.

A Spectral Collocation Method for Solving Caputo-Liouville Fractional Order Fredholm Integro-differential Equations

In this paper, a numerical method for solving the fractional order Fredholm integro-differential equations via the Caputo-Liouville derivative is presented. The method uses the well-known shifted Chebyshev expansion and a truncated series to represent the unknown function. It also incorporates numerical integration techniques like the Trapezoidal, Simpson’s 1/3, and Simpson’s 8/3 methods. The paper also provides an approximation for the derivative of an integer. The procedure converts the provided problem into a system of algebraic equations using shifted Chebyshev coefficients and collocation points. The coefficients are found by solving this system using well-known techniques like Newton’s method. Numerical results are presented graphycally to illustrate the applicability, efficacy, and accuracy of the approach presented in this work. All calculations in this study were performed using the MATHEMATICA software program.

The automation of SMEFT-assisted constraints on UV-complete models

The ongoing Effective Field Theory (EFT) program at the LHC and elsewhere is motivated by streamlining the connection between experimental data and UV-complete scenarios of heavy new physics beyond the Standard Model (BSM). This connection is provided by matching relations mapping the Wilson coefficients of the EFT to the couplings and masses of UV-complete models. Building upon recent work on the automation of tree-level and one-loop matching in the SMEFT, we present a novel strategy automating the constraint-setting procedure on the parameter space of general heavy UV-models matched to dimension-six SMEFT operators. A new Mathematica package, match2fit, interfaces MatchMakerEFT, which derives the matching relations for a given UV model, and SMEFiT, which provides bounds on the Wilson coefficients by comparing with data. By means of this pipeline and using both tree-level and one-loop matching, we derive bounds on a wide range of single- and multi-particle extensions of the SM from a global dataset composed by LHC and LEP measurements. Whenever possible, we benchmark our results with existing studies. Our framework realises one of the main objectives of the EFT program in particle physics: deploying the SMEFT to bypass the need of directly comparing the predictions of heavy UV models with experimental data.

Impact of Porous Media on Peristaltic Transport of Tangent Hyperbolic Nanofluid in Asymmetric Channel

The purpose behind this paper is to discuss nanoparticles effect, porous media, radiation and heat source/sink parameter on hyperbolic tangent nanofluid of peristaltic flow in a channel type that is asymmetric. Under a long wavelength and the approaches of low Reynolds number, the governing nanofluid equations are first formulated and then simplified. Associated nonlinear differential equations will be obtained after making these approximations. Then the concentration of nanoparticle exact solution, temperature distribution, stream function, and pressure gradient will be calculated. Eventually, the obtained results will be illustrated graphically via MATHEMATICA software.

Energy spectrum of selected diatomic molecules (H2, CO, I2, NO) by the resolution of Schrodinger equation for combined potentials via NUFA method.

To determine the properties of diatomic molecules, studying their chemical bond energy spectrum is essential since it enables the assessment of their characteristics. This research presents diatomic molecules spectroscopic characteristics and rovibrational energy (H2, CO, I2, NO). The Schrodinger equation is solved to determine these energies, considering the presence of a combination of two distinct potentials. the inverse quadratic Yukawa potential in combination with the screened modified Kratzer. This work used the Greene-Aldrich assumption and the Nikiforov-Uvarov functional analysis approach as analytical tools to solve the Schrodinger equation and determine the energy spectrum of diatomic molecules (H2, CO, I2, NO). The use of Mathematica software allows for the calculation of the eigenvalues of energy of the previously mentioned diatomic molecules (H2, CO, I2, NO) based on their rovibrational energies in the final equation. By comparing the eigenvalue findings with previous research, it was seen that the technique yielded the expected and desirable outcomes.

Carrera Unified Formulation (CUF) for the composite plates and shells of revolution. Layer-wise models

Here, higher order layer-wise models of elastic composite multilayer plates and shells of revolution are developed using the variational principle of virtual power for the 3-D linear anisotropic theory of elasticity and generalized series in the thickness coordinates. Following the Unified Carrera Formulation (CUF), the stress and strain tensors, as well as the displacement vector, were expanded into series in terms of the coordinates of the shell thickness. The higher-order rectangular plate and cylindrical shell supported on the edges under sinusoidal loading, are considered and solved analytically using a Navier close form solution method. Also, composite axisymmetric conical, spherical, elliptical and catenoidal shell fixed at the ends are considered. Numerical calculations were performed using the computer algebra software Mathematica. The resulting equations can be used for theoretical analysis and calculation of the stress–strain state, as well as for modeling thin-walled structures used in science, engineering, and technology. The results of calculation can be used as benchmark examples for finite element analysis of the higher order composite laminate shells.

Exact solutions of the (3+1)-generalized fractional nonlinear wave equation with gas bubbles

In this manuscript, we implement the travelling wave solutions of the fractional (3+1) generalized computational nonlinear wave equation with gas bubbles via application of five mathematical methods. Liquids with gas bubbles primarily arise in various applications like science, engineering, and mathematical physics. The obtained solitary waves solutions have fruitful applications in engineering, science, life, nature and physics. Several novel soliton solutions of concerned model are established in the form of hyperbolic, trigonometric, exponential and rational functions. To handle all calculations and verification of obtained results, computational software Mathematica 12.1 is used. For the demonstration of the physical behaviour of concern model, some solutions are plotted graphical in 2-dimensional and 3-dimensional by imparting specific values to the parameters under constrain conditions. Finally, we intrigue both two and three dimensional to explain the physical behavior of the model.