Mathematica Code Research Articles

Using Laplace Residual Power Series Method in Solving Coupled Fractional Neutron Diffusion Equations with Delayed Neutrons System

In this paper, a system of coupled fractional neutron diffusion equations with delayed neutrons was solved efficiently by using a combination of residual power series and Laplace transform techniques, and the anomalous diffusion was considered by taking the non-Gaussian case with different values of fractional parameter α. The Laplace residual power series method (LRPSM) does not require differentiation, conversion, or discretization for the assumed conditions, so the approach is simple and suitable for solving higher-order fractional differential equations. To assure the theoretical results, two different neutron flux initial conditions were presented numerically, where the needed Mathematica codes were performed using essential nuclear reactor cross-section data, and the results for different values of times were tabulated and graphically figured out. Finally, it must be noted that the results align with the Adomian decomposition method.

Bernstein spectral method for quasinormal modes of a generic black hole spacetime and application to instability of dilaton–de Sitter solution

We present the improved Mathematica code which computes quasinormal frequencies with the help of the Bernstein spectral method for a general class of black holes, allowing for asymptotically flat, de Sitter or anti-de Sitter asymptotic. The method is especially efficient when searching for purely imaginary and unstable modes and here it is used for detecting the instability region of a charged scalar field in the background of the charged asymptotically de Sitter dilatonic black hole. We show that the instability has superradiant nature and the dilaton field essentially influences the region of instability.

Computer-aided design of offshore S-lay fiber-reinforced polymer pipeline installation

It is well known that the installation loads during offshore pipeline installation are usually higher than the operation loads. Fibre-reinforced plastic (FRP) materials for offshore pipeline applications seem to be very promising due to their high strength, light weight and excellent corrosion and fatigue resistance. However, during S-lay installation, high bending stresses that may cause material failure or buckling take place. Due to the anisotropic nature of multi-layered FRP pipelines, the permissible bending curvature is difficult to estimate. Few published works have proposed a framework for bending stress calculations and those that have provide complex equations that are not convenient for use. In this study, an open numerical code in Mathematica is offered for the first time. Numerical results for S-lay installation of representative pipelines are provided and commended.

Analyzing Electric Circuits with Computer Algebra

This report shows how starting from classic electric circuits embodying commonly electric components we have reached semi-complicated circuits embodying the same components that analyzing the signal characteristics requires a Computer Algebra System. Our approach distinguishes itself from the electrical engineers’ (EE) approach that relies on utilizing commercially available software. Our approach step-by-step shows how Kirchhoff’s rules are applied conducive to the needed circuit information. It is shown for the case at hand the characteristic information is a set of coupled differential equations and that with the help of Mathematica numeric solutions are sought. Our report paves the research road for unlimited creative similar circuits with any degree of complications. Occasionally, by tweaking the circuits we have addressed the “what if” scenarios widening the scope of the investigation. Justification of the accuracy of our analysis for the generalized circuits is cross-checked by arranging the components symmetrizing the circuit leading to an intuitively predictable reasonable result. Mathematica codes are embedded assisting the interested reader in producing and extending our results.

Copula-based probability for occurrence of the event due to one of dependent NHPPs before the other, past a truncated time

The bivariate Gumbel-Hougaard copula is considered for two dependent non-homogenous Poisson processes, The purpose of this article is to provide a computational tool for estimation of the probability that the next event arrival after a truncated time τ, is due to a specific process. There are many situations in which non-homogenous Poisson processes could be dependent, such as failure times of a two-component repairable system in which the failure of the system at a given time is due to only one of the components. Power Law is used as the intensity function for both non-homogenous Poisson Processes. Maximum likelihood and Bayes estimators for the probability based on gamma priors are derived. It is verified through simulations that Bayes estimator outperforms maximum likelihood estimator with regard to their accuracy. Two examples for illustration of computation via Mathematica code (in the Appendix) are provided.

Non-freeness of Groups Generated by Two Parabolic Elements with Small Rational Parameters

Let q∈C, let a=1011,bq=1q01, and let Gq< SL2(C) be the group generated by a and bq. In this paper, we study the problem of determining when the group Gq is not free for |q|<4 rational. We give a robust computational criterion, which allows us to prove that if q=s/r for |s|≤27, then Gq is non-free with the possible exception of s=24. In this latter case, we prove that the set of denominators r∈N for which G24/r is non-free has natural density 1. For a general numerator s>27, we prove that the lower density of denominators r∈N for which Gs/r is non-free has a lower bound 1−(1−11 s)∏n=1∞(1−4 s2n−1). Finally, we show that for a fixed s, there are arbitrarily long sequences of consecutive denominators r such that Gs/r is non-free. The proofs of some of the results are computer assisted, and Mathematica code has been provided together with suitable documentation.

The Bifurcation Curves of a Category of Dirichlet Boundary Value Problems

We study the Dirichlet boundary value problemu″t+λfut=0,−1&lt;t&lt;1,u−1=u1=0,generally and develop a schema for determining the relationship between the values of its parameters and the number of positive solutions. Then, we focus our attention on the special cases whenfu=σ−uexp−K/1+uandfu=∏i=1mai—u, respectively. We prove first that all positive solutions of the first problem are less than or equal toσ, obtain more specific lower and upper bounds for these solutions, and compute a curve in theσK-plane with accuracy up to10−6, below which the first problem has a unique positive solution and above which it has exactly three positive solutions. For the second problem, we determine its number of positive solutions and find a formula for the value ofλthat separates the regions ofλ, in which the problem has different numbers of solutions. We also computed the graphs for some special cases of the second problem, and the results are consistent with the existing results. Our code in Mathematica is available upon request.

Neumann's principle based eigenvector approach for deriving non-vanishing tensor elements for nonlinear optics.

Physical properties are commonly represented by tensors, such as optical susceptibilities. The conventional approach of deriving non-vanishing tensor elements of symmetric systems relies on the intuitive consideration of positive/negative signflipping after symmetry operations, which could be tedious and prone to miscalculation. Here, we present a matrix-based approach that gives a physical picture centered on Neumann's principle. The principle states that symmetries in geometric systems are adopted by their physical properties. We mathematically apply the principle to the tensor expressions and show a procedure with clear physical intuition to derive non-vanishing tensor elements based on eigensystems. The validity of the approach is demonstrated by examples of commonly known second and third-order nonlinear susceptibilities of chiral/achiral surfaces, together with complicated scenarios involving symmetries such as D6 and Oh symmetries. We then further applied this method to higher-rank tensors that are useful for 2D and high-order spectroscopy. We also extended our approach to derive nonlinear tensor elements with magnetization, which is critical for measuring spin polarization on surfaces for quantum information technologies. A Mathematica code based on this generalized approach is included that can be applied to any symmetry and higher order nonlinear processes.

Stability of Time-Invariant Extremum Seeking Control for Limit Cycle Minimization

This article presents a time-invariant extremum seeking controller (ESC) for nonlinear autonomous systems with limit cycles. For this time-invariant ESC, we propose a method to prove the closed-loop system has an asymptotically stable limit cycle. The method is based on a perturbation theorem for maps, and, unlike existing techniques that use averaging and singular perturbation tools, it is not limited to weakly nonlinear systems. We use a typical example system to show that our method does indeed establish asymptotic stability of the limit cycle with minimal amplitude. Utilizing the example, we provide a general guide for analytic computations that are required to apply our method. The corresponding Mathematica code is available as supplementary material.

Reading the lattice QCD form factors of the Λ b → Λ c transition using a C-code

The process has been discussed in the literatures as a probe for new physics. This process receives contributions in new physics models from scalar, vector, and tensor hadronic currents. The form factors of these currents can be obtained from the quark model or the lattice QCD. In this work, we present a c-code to read the scalar, vector, and tensor lattice QCD form factors. The code is published on github (https://github.com/darkfiresmith96/Lattice_QCD). A Mathematica code was used in reference Datta et al (2017 J. High Energy Phys. 8 131) to read the form factors. Our c-code is much faster than the Mathematica code where the ratio of the wall clock time for the c-code compared to the Mathematica code is 1:64.2 per data point. A web interface for user inputs is introduced.

High-order post-Newtonian expansion of the redshift invariant for eccentric-orbit nonspinning extreme-mass-ratio inspirals

We calculate the eccentricity dependence of the high-order post-Newtonian (PN) series for the generalized redshift invariant $⟨{u}^{t}{⟩}_{\ensuremath{\tau}}$ for eccentric-orbit extreme-mass-ratio inspirals on a Schwarzschild background. These results are calculated within first-order black hole perturbation theory using Regge-Wheeler-Zerilli (RWZ) gauge. Our Mathematica code is based on a familiar procedure, using PN expansion of the Mano-Suzuki-Takasugi analytic function formalism for $l$ modes up to a certain maximum and then using a direct general-$l$ PN expansion of the RWZ equation for arbitrarily high $l$. We calculate dual expansions in PN order and in powers of eccentricity, reaching 10PN relative order and ${e}^{20}$. Detailed knowledge of the eccentricity expansion at each PN order allows us to find within the eccentricity dependence numerous closed-form expressions and multiple infinite series with known coefficients. We find leading logarithm sequences in the PN expansion of the redshift invariant that reflect a similar behavior in the PN expansion of the energy flux to infinity. A set of flux terms and special functions that appear in the energy flux, like the Peters-Mathews flux itself, are shown to reappear in the redshift PN expansion.

Ranked hesitant fuzzy sets for multi-criteria multi-agent decisions

This paper introduces and investigates ranked hesitant fuzzy sets, a novel extension of hesitant fuzzy sets that is less demanding than both probabilistic and proportional hesitant fuzzy sets. This new extension incorporates hierarchical knowledge about the various evaluations submitted for each alternative. These evaluations are ranked (for example by their plausibility, acceptability, or credibility), but their position does not necessarily derive from supplementary numerical information (as in probabilistic and proportional hesitant fuzzy sets). In particular, strictly ranked hesitant fuzzy sets arise when no ties exist, i.e., when for any fixed alternative, each submitted evaluation is either strictly more plausible or strictly less plausible than any other submitted evaluation. A detailed comparison with similar models from the literature is performed. Then in order to produce a natural strategy for multi-criteria multi-agent decisions with ranked hesitant fuzzy sets, canonical representations, scores and aggregation operators are designed in the framework of ranked hesitant fuzzy sets. In order to help implementation of this model, Mathematica code is provided for the computation of both scores and aggregators. The decision-making technique that is prescribed is tested with a comparative analysis with four methodologies based on probabilistic hesitant fuzzy information. A conclusion of this numerical exercise is that this methodology is reliable, applicable and robust. All these evidences show that ranked hesitant fuzzy sets are an intuitive extension of the hesitant fuzzy set model designed by V. Torra, that can be implemented in practice with the aid of computationally assisted algorithms.

Generalized forms of fractional Euler and Runge–Kutta methods using non-uniform grid

Abstract In this article, we propose generalized forms of three well-known fractional numerical methods namely Euler, Runge–Kutta 2-step, and Runge–Kutta 4-step, respectively. The new versions we provide of these methods are derived by utilizing a non-uniform grid which is slightly different from previous versions of these algorithms. A new generalized form of the well-known Caputo-type fractional derivative is used to derive the results. All necessary analyses related to the stability, convergence, and error bounds are also provided. The precision of all simulated results is justified by performing multiple numerical experiments, with some meaningful problems solved by implementing the code in Mathematica. Finally, we give a brief discussion on the simulated results which shows that the generalized methods are novel, effective, reliable, and very easy to implement.

Moments of Markovian growth–collapse processes

AbstractWe apply general moment identities for Poisson stochastic integrals with random integrands to the computation of the moments of Markovian growth–collapse processes. This extends existing formulas for mean and variance available in the literature to closed-form moment expressions of all orders. In comparison with other methods based on differential equations, our approach yields explicit summations in terms of the time parameter. We also treat the case of the associated embedded chain, and provide recursive codes in Maple and Mathematica for the computation of moments and cumulants of any order with arbitrary cut-off moment sequences and jump size functions.

Two Complementary Methods of Inferring Elastic Symmetry

An elastic map mathbf {T} describes the strain-stress relation at a particular point mathbf {p} in some material. A symmetry of mathbf {T} is a rotation of the material, about mathbf {p}, that does not change mathbf {T}. We describe two ways of inferring the group mathcal {S} _{ mathbf {T} } of symmetries of any elastic map mathbf {T}; one way is qualitative and visual, the other is quantitative. In the first method, we associate to each mathbf {T} its “monoclinic distance function” on the unit sphere. The function is invariant under all of the symmetries of mathbf {T}, so the group mathcal {S} _{ mathbf {T} } is seen, approximately, in a contour plot of . The second method is harder to summarize, but it complements the first by providing an algorithm to compute the symmetry group mathcal {S} _{ mathbf {T} }. In addition to mathcal {S} _{ mathbf {T} }, the algorithm gives a quantitative description of the overall approximate symmetry of mathbf {T}. Mathematica codes are provided for implementing both the visual and the quantitative approaches.

Coulomb and Higgs branches from canonical singularities. Part I. Hypersurfaces with smooth Calabi-Yau resolutions

Compactification of M-theory and of IIB string theory on threefold canonical singularities gives rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. The resolutions and deformations of the singularities encode salient features of the SCFTs and of their moduli spaces. In this paper, we build on Part 0 of this series [1] and further explore the physics of SCFTs arising from isolated hypersurface singularities. We study in detail these canonical isolated hypersurface singularities that admit a smooth Calabi-Yau (crepant) resolution. Their 5d and 4d physics is discussed and their 3d reduction and mirrors (the magnetic quivers) are determined in many cases. As an explorative tool, we provide a Mathematica code which computes key quantities for any canonical isolated hypersurface singularity, including the 5d rank, the 4d Coulomb branch spectrum and central charges, higher-form symmetries in 4d and 5d, and crepant resolutions.

2-Group symmetries and their classification in 6d

We uncover 2-group symmetries in 6d superconformal field theories. These symmetries arise when the discrete 1-form symmetry and continuous flavor symmetry group of a theory mix with each other. We classify all 6d superconformal field theories with such 2-group symmetries. The approach taken in 6d is applicable more generally, with minor modifications to include dimension specific operators (such as instantons in 5d and monopoles in 3d), and we provide a discussion of the dimension-independent aspects of the analysis. We include an ancillary mathematica code for computing 2-group symmetries, once the dimension specific input is provided. We also discuss a mixed 't Hooft anomaly between discrete 0-form and 1-form symmetries in 6d

A Computer Code for Wave Propagation in Flexible Pipelines Under Fluid Hammer Conditions

Sudden or quick reduction of fluid flow in subsea or onshore polymeric pipelines for oil or water transportation, hydroelectric power plants, water supply networks, turbomachinery etc., has usually catastrophic consequences. Apart from the hydrostatic-induced mechanical stresses, the pipeline designers should take into account the dynamic pressure under unsteady loading conditions. Interaction of the fluid hammer-induced impact pressure with the elastic properties of a flexible Fiber Reinforced Polymeric (FRP) pipeline causes a radial displacement wave propagation in the pipe wall. Since the loading term in the governing equation contains the discontinuous Dirac Delta function to simulate the impactful nature of the pressure surge, the analytical solution is not an easy task. Because of the lack of tools for fluid hammer-induced pipeline deformation, the development of a computer code for simulation of the dynamic deformation of a flexible pipeline is the aim of the present work. For the solution of the motion equation, the adopted methodology employs integral transforms and generalized functions. An original code in Mathematica® platform for the solution of the model is proposed and results for the wave propagation in a FRP pipeline are provided and discussed.

Practical numerical solutions of Wassermann–Wolf equations

In this paper, we present a complete method for determining the two surfaces of an aspheric singlet by numerical integration based on the classical Wassermann–Wolf equations (hereinafter WW). Although aplanatic optics is mainly treated in WW, we also show some other examples that can be determined by the WW. This indicates that surface shape determination by optimization software such as Zemax or CodeV is not necessary for some aspherical singlet designs, including the aplanatic case. A link to an executable Mathematica code for this purpose is also provided.

Entropy generation in Sutterby nanomaterials flow due to rotating disk with radiation and magnetic effects

Entropy generation is a novel potential in various thermodynamic processes and presents dynamic applications in thermal polymer processing optimization. The significance of entropy generation is observed in heat exchangers, combustion, turbine systems, thermal systems, porous media, nuclear reactions etc. In view of such thermal applications, the prime objective of present analysis is to scrutinize the entropy optimized hydromagnetic flow of Sutterby nanofluid due to a stretchable rotating disk. Thermal radiation and heat source effects are considered. Effects of Brownian movement and thermophoresis diffusion are considered. Physical description of entropy generation is also addressed. Adopting procedure of transformations the non-linear PDEs are converted to ODEs. The obtained system is solved by ND-solve code in Mathematica. Influence of different parameters involved in velocity, temperature, concentration and entropy generation is discussed via graphs. Skin friction coefficient, Nusselt number and Sherwood number are discussed through Tables. Larger magnetic variable has decaying effect on velocity. An amplification in temperature distribution and entropy rate is observed for radiation variable. Larger approximation of radiation variable leads to improved thermal field and entropy rate. An opposite effect is noticed for temperature against Prandtl and heat generation variable. Variation in thermal ratio variable improves the entropy rate and temperature. Higher stretching parameter leads to produce more drag force at surface of disk. The main results are listed at the end.