Mathematica Research Articles

Mathematical Modelling of the Oral Administration of Drugs into the Stomach and the Treatment Effect with the Influence of Temperature

This research deals with the mathematical formulation that mimics the oral administration of drug intake into the stomach and the therapeutic effect of temperature increases. We solved the formulated ordinary differential equations (ODEs) using the Laplace method, and the numerical simulation was carried out using Wolfram Mathematica by varying the thermal conductivity and the treatment terms and studying the significant role they play in eliminating the drug from the human body. The study showed that the pertinent parameters influence the drug concentration in the stomach compartment under the influence of temperature. In conclusion, with the help of this simple model, researchers should be able to better understand oral drug administration and provide reliable and appropriate care to patients.

Public Perception of ChatGPT and Transfer Learning for Tweets Sentiment Analysis Using Wolfram Mathematica

Understanding public opinion on ChatGPT is crucial for recognizing its strengths and areas of concern. By utilizing natural language processing (NLP), this study delves into tweets regarding ChatGPT to determine temporal patterns, content features, and topic modeling and perform a sentiment analysis. Analyzing a dataset of 500,000 tweets, our research shifts from conventional data science tools like Python and R to exploit Wolfram Mathematica’s robust capabilities. Additionally, with the aim of solving the problem of ignoring semantic information in the LDA model feature extraction, a synergistic methodology entwining LDA, GloVe embeddings, and K-Nearest Neighbors (KNN) clustering is proposed to categorize topics within ChatGPT-related tweets. This comprehensive strategy ensures semantic, syntactic, and topical congruence within classified groups by utilizing the strengths of probabilistic modeling, semantic embeddings, and similarity-based clustering. While built-in sentiment classifiers often fall short in accuracy, we introduce four transfer learning techniques from the Wolfram Neural Net Repository to address this gap. Two of these techniques involve transferring static word embeddings, “GloVe” and “ConceptNet”, which are further processed using an LSTM layer. The remaining techniques center on fine-tuning pre-trained models using scantily annotated data; one refines embeddings from language models (ELMo), while the other fine-tunes bidirectional encoder representations from transformers (BERT). Our experiments on the dataset underscore the effectiveness of the four methods for the sentiment analysis of tweets. This investigation augments our comprehension of user sentiment towards ChatGPT and emphasizes the continued significance of exploration in this domain. Furthermore, this work serves as a pivotal reference for scholars who are accustomed to using Wolfram Mathematica in other research domains, aiding their efforts in text analytics on social media platforms.

Elementary coupling coefficients for the Wigner supermultiplet symmetry

An algorithm for evaluating Wigner coefficients of U(4) ⊃ SUS(2) ⊗ SUT(2) for the coupling [n14,n24,n34,n44]⊗[1,0,0,0]↓[m14,m24,m34,m44] with arbitrary irreducible representation [n14,n24,n34,n44] is provided based on the algebraic expressions with the expansion coefficients of the U(4)⊃SUS(2)⊗SUT(2) states in terms of those in the U(4) canonical basis and the Clebsch-Gordan coefficients of U(4) in the canonical basis. The state expansion coefficients are evaluated as the components of the null space vectors of the spin-isospin projection matrices. Program summaryProgram Title: Wigner-U4-SU2xSU2CPC Library link to program files:https://doi.org/10.17632/4b7gf2phh2.1Licensing provisions: GPLv3Programming language: Wolfram Mathematica version 9.0.1.0 and beyondNature of problem: Numerical evaluation of the elementary Wigner coefficients (EWCs) of U(4)⊃SUS(2)⊗SUT(2).Solution method: Mathematica implementation of the EWCs of U(4)⊃SUS(2)⊗SUT(2) [1] using the expansion coefficients of U(4)⊃SUS(2)⊗SUT(2) states in terms of those of the U(4) canonical basis determined as components of the null space vectors of the spin-isospin projection matrices [2].

Impurity Bands and Density of State in Doped Silicate Glasses with Metal Oxides (RuO2, CuO, MnO2)

In this article, as a result of double alloying with ruthenium, copper, manganese oxides, the formation of an impurity band in silicate glass and how the density of electron states changes in it was studied by hypothetical and tunneling microprobe spectroscopy. The results of the experiment were processed using Wolfram Mathematica 11 software. It was found that around room temperature, the impurity band touches or merges with the valence band of the glass. At a high temperature (around 1000 K), the impurity band separates from the valence band of the glass, and a pseudo-gap appears between them. As a result, doped silicate glass exhibits metallic conductivity at room temperature, but at high temperatures, it has semiconductor (activation) conductivity.

The Accuracy of Computational Results from Wolfram Mathematica in the Context of Summation in Trigonometry

This article explores the accessibility of symbolic computations, such as using the Wolfram Mathematica environment, in promoting the shift from informal experimentation to formal mathematical justifications. We investigate the accuracy of computational results from mathematical software in the context of a certain summation in trigonometry. In particular, the key issue addressed here is the calculated sum ∑n=044tan⁡1+4n°. This paper utilizes Wolfram Mathematica to handle the irrational numbers in the sum more accurately, which it achieves by representing them symbolically rather than using numerical approximations. Can we rely on the calculated result from Wolfram, especially if almost all the addends are irrational, or must the students eventually prove it mathematically? It is clear that the problem can be solved using software; however, the nature of the result raises questions about its correctness, and this inherent informality can encourage a few students to seek viable mathematical proofs. In this way, a balance is reached between formal and informal mathematics.

Solitary wave solutions and investigation the effects of different wave velocities of the nonlinear modified Zakharov–Kuznetsov model for the wave propagation in nonlinear media

The modified Zakharov-Kuznetsov (mZK) model convey a significant role to analyze the inner mechanism of physical compound phenomenon in the field of two-dimensional discrete electrical lattice, the electrical waves in cold plasmas, plasma physics, nonlinear optics, wave behaviors of deep oceans, etc. In this study, numerous new, advanced, and more general exact solitary travelling wave solutions are explored to the early stated nonlinear model by the aid of newly developed generalized exponential rational function (GERF) method through the traveling wave transformation. The established solutions are in terms of trigonometric functions, rational functions, hyperbolic functions, exponential rational functions, and complex-soliton solutions. The results reveal that the free parameters significantly influence the existence of traveling waves, as well as nature, shape, and stability. The established soliton solutions demonstrate that the method is effective, compatible, scientifically efficient and easily applicable to identify a variety of wave structures of nonlinear evolution models. The mathematical simulations of the results were conducted by sketching 3D and 2D structures for different values of the associated parameters by the aid of the Wolfram Mathematica program.

Математическое моделирование электромеханической модели экзоскелета с тремя активными управляемыми звеньями

Mathematical modeling of an active exoskeleton in the form of an electromechanical model containing three movable, controlled links interconnected by hinges has been carried out. For the considered mathematical model of the active exoskeleton, differential equations of motion are proposed. Numerical methods are used to solve the inverse and direct problems of dynamics in the created software package in the environment of the universal system of computer mathematics. A comprehensive study has been carried out, considering the problems of exoskeleton control, in the form of solving inverse and direct problems of dynamics, in relation to the created mathematical model of three moving parts of the exoskeleton, taking into account electric drives, using modern methods of mathematical modeling. Analytically determined are the angles between the links that define the anthropoid movement. Solving the inverse problem of dynamics, the moments controlling the movement of the links are calculated for each electric drive. The found moments are approximated by stepwise piecewise-constant functions simulating the impulse control of the exoskeleton motion. Dependences of the angular coordinates describing the positions of the links of the active exoskeleton over time are found. A comparative analysis of the numerical solution of the Cauchy problem for the mathematical model of the exoskeleton in the form of differential equations with the initial, given movement of the links is carried out. A good agreement between the results of simulation with impulse control and the original motion is established. The total energy costs have been calculated. Modeling is carried out taking into account the presence of electric drives: dynamic equations for this model are compiled. The Cauchy problem for the system is numerically solved taking into account the presence of electric drives. As a result of applying qualitative, analytical and numerical methods for studying the created mathematical model of three moving parts of the exoskeleton, taking into account the presence of electric drives, the significance of the influence of electric drives on the dynamics of the mechanism was established. The implementation of the electromechanical model of the three links of the exoskeleton was carried out in the environment of the universal system of computer mathematics “Wolfram Mathematica 11.3”.

Numerical simulation for thermal radiative flow of tangent hyperbolic nanofluid due to Riga plate in the presence of joule heating

In this paper, numerical simulation for the two-dimensional flow of tangent hyperbolic fluid in the manifestation of nanoparticles, heat source and joule heating is considered. For motivation of problem, the role of Riga plate is investigated. Chemical reaction and Brownian motion are the parts of the analysis. The set of PDEs of Tangent hyperbolic nanofluids with swimming germs is converted in to couple of ODEs by employing well-known approach namely shooting scheme. The resulting highly nonlinear and coupled ODEs are solved by using ND solve tool of Wolfram Mathematica 11. Numerical outcome of involving parameters of interest like Lewis number Le, Prandtl number Pr, Chemical diffusion k, Peclet number Pe, Weissenberg number We, Bioconvection Lewis number Lb, nonlinear radiation, Biot number Bi, modified Hartmann number Q and magnetic parameter M on tangent hyperbolic nanofluids velocity f″, temperature θ, concentration φ and density of germs distribution are discussed in the form of graphical trend and tables. We professed that the speed of fluid degenerated for the increasing value of power-law index n and magnetic parameter M. The Nusselt number growths for growing values of the thermophoresis parameter Nt, Biot number Bi and radiation parameter Nr.

Solitary waves pattern appear in tropical tropospheres and mid-latitudes of nonlinear Landau–Ginzburg–Higgs equation with chaotic analysis

The objective of this research is to investigate the nonlinear Landau–Ginzburg–Higgs equation, which characterizes nonlinear solitary waves exhibiting distant and feeble scattering interactions among tropical tropospheres and mid-latitudes. Additionally, the study will examine the interchange of mid-latitude Rossby waves and equatorial waves within this context. In this research article, we focus on obtaining exact traveling wave solutions for the Landau–Ginzburg–Higgs equation using a new extended direct algebraic technique. The obtained soliton solutions include various types such as combined and multiple bright-dark, periodic, bright, and multiple bright-periodic. We present these soliton solutions graphically by varying the involved parameters using the advanced software program Wolfram Mathematica. The graphical representations allow us to visualize the behavior of the wave velocity and wave number as the parameters change. Additionally, we conduct a chaotic analysis to examine the wave profiles of the newly designed dynamical framework. The results of this analysis demonstrate the reliability and efficiency of the proposed method, which can be applied to find closed-form traveling wave solitary solutions for a wide range of nonlinear evolution equations.

ANALYSIS OF SOLID SOLUTION DECOMPOSITION IN HIGH-ALLOY AUSTENITIC STEEL

In the field of aviation and rocket engineering, austenitic chrome-nickel steels are widely employed due to their exceptional properties, including high strength, heat resistance, ductility, and phase stability over a broad range of operating temperatures. Nevertheless, the scarcity of chromium and nickel has necessitated the development and utilization of new high-alloy austenitic steels for these industries, which do not contain the aforementioned deficit alloying elements. Modern high-alloy austenitic steels used in aviation and rocket industries are based on Fe – Mn – Al-Calloys. These steels exhibit lower specific weight, approximately 13-15% less than conventional steels, and possess superior plasticity and the ability to strengthen during deformation. It is essential to note that the high strength of these steels is achieved through the precipitation of complex carbide (Fe, Mn)3 Al particles during thermal treatment, with a crystalline lattice period of d = 0.376 nm.Extensive research has been conducted to establish the temperature-time boundaries of the aging process, its stages, and the mechanisms of strengthening during aging for these steels. Geometric models illustrating the changes in mechanical properties during thermal treatment have been constructed utilizing the mathematical package of functional programming, Wolfram Mathematica. The obtained results have facilitated the formulation of practical recommendations for the development of effective aging regimes for these steels. Mechanisms and temperature-time boundaries for the three stages of the aging process of the steel have been identified, opening the possibility of devising more optimal and productive methods for processing these materials.Hence, research in the domain of high-alloy austenitic steels for aviation and rocket construction is a pertinent and significant direction, as it contributes to the enhancement of quality and reliability in structures employing these materials. The findings of these investigations can be invaluable for advancing contemporary technologies in these fields and ensuring the safety and efficiency of aviation and rocket systems.

Machine Learning Algorithm to Perform the American Society of Anesthesiologists Physical Status Classification.

The American Society of Anesthesiologists (ASA) Physical Status (PS) Classification System defines perioperative patient scores ranging from 1 to 6 (healthy to brain dead, respectively).The scoring is performed and used by physician anesthesiologists and providers to classify surgical patients based on co-morbidities and various clinical characteristics. There is potentially a variability in scoring stemming from individual biases.The biases impact the prediction of operating times, length of stay in the hospital, anesthetic management, and billing.This study's purpose was to develop an automated system to achieve reproducible scoring. A machine learning (ML) model was trained on already assigned ASA PS scores of 12,064 patients. The ML algorithm was automatically selected by Wolfram Mathematica (Wolfram Research, Champaign, IL) and tested with retrospective records not used in training. Manual scoring was performed by the anesthesiologist as part of the standard preoperative evaluation.Intraclass correlation coefficient (ICC) in R (version 4.2.2; R Development Core Team, Vienna, Austria) was calculated to assess the consistency of scoring. An ML model was trained on the data corresponding to 12,064 patients. Logistic regression was chosen automatically, with an accuracy of 70.3±1.0% against the training dataset. The accuracy against 1,999 patients (the test dataset) was 69.6±1.0%. The ICC for the comparison between ML and the anesthesiologists' ASA PS scores was greater than 0.4 ("fair to good"). We have shown the feasibility of applying ML to assess the ASA PS score within an oncology patient population. Though our accuracy was not very good, we feel that, as more data are mined, a valid foundation for refinement to ML will emerge.

Precise invariant travelling wave soliton solutions of the Nizhnik–Novikov–Veselov equation with dynamic assessment

The (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation is a significant model that describes the behaviour of conserved scalar nucleons and their connection to neutral scalar masons. This system exhibits self-formation and depends on various free parameters. It is commonly used to study the dynamics of scalar nucleons and neutral scalar masons. This research article used an extended direct algebraic technique to obtain exact travelling wave solutions for the Nizhnik–Novikov–Veselov system. The obtained soliton solutions include various types, such as multi-periodic, periodic, plane, singular, bright, dark, and flat kink-type wave solitons. We present these soliton solutions graphically by varying the involved parameters using the advanced software program Wolfram Mathematica. The graphical representations in 3D, contour, and 2D surfaces allow us to visualise the behaviour of the solutions as the parameters change and the physical interpretation for these solutions is obtained. Additionally, we conduct a sensitivity analysis to examine the wave profiles of the newly designed dynamical framework. The results of this analysis demonstrate the reliability and efficiency of the proposed method, which can be applied to find closed-form travelling wave solitary solutions for a wide range of nonlinear evolution equations.

GENERALIZED FRACTIONAL GAUSSIAN NOISE PREDICTION BASED ON THE WALSH FUNCTIONS

Context. Some of the authors’ recent papers were devoted to the Kolmogorov-Wiener filter for telecommunication traffic prediction in some stationary models, such as the fractional Gaussian noise model, the power-law structure function model, and the GFSD (Gaussian fractional sum-difference) model. Recently, the so-called generalized fractional Gaussian noise model was proposed for stationary telecommunication traffic description in some cases. So, in this paper the theoretical fundamentals of the continuous Kolmogorov-Wiener filter used for the prediction of the generalized fractional Gaussian noise are investigated.
 Objective. The aim of the work is to obtain the filter weight function as an approximate solution of the corresponding Wiener– Hopf integral equation with the kernel equal to the generalized fractional Gaussian noise correlation function.
 Method. A truncated Walsh function expansion is proposed in order to obtain the corresponding solution. This expansion is a special case of the Galerkin method, in the framework of which the unknown function is sought as a truncated series in orthogonal functions. The integral brackets and the results for the mean absolute percentage errors, which are a measure of discrepancy between the left-hand side and the right-hand side of the Wiener-Hopf integral equation, are calculated numerically on the basis of the Wolfram Mathematica package.
 Results. The investigation is made for approximations up to sixty four Walsh functions. Different model parameters are investigated. It is shown that for different model parameters the proposed method is convergent and leads to small mean absolute percentage errors for approximations of rather large numbers of Walsh functions.
 Conclusions. The paper is devoted to a theoretical construction of the continuous Kolmogorov-Wiener filter weight function for the prediction of a stationary random process described by the generalized fractional Gaussian noise model. As is known, this model may give a good description of some actual telecommunication traffic data in systems with packet data transfer. The corresponding weight function is sought on the basis of the truncated Walsh function expansion method. The corresponding discrepancy errors are small and the method is convergent.

THE DEVELOPMENT OF COVID-19 USING OUTBREAK THE SUSCEPTIBLE, INFECTED, AND RECOVERED (SIR) MODEL WITH VACCINATION

The Covid-19 pandemic in 2020 has caused severe problems in Indonesia. The Covid-19 virus epidemic can be modeled using the Susceptible, Infected, and Recovered (SIR) model. This modeling aims to look at the dynamics of Covid-19 to predict when disease-free and endemic disease occurs and to find the basic reproduction number ( ) for policy making in suppressing the spread of Covid-19. In this article, we describe and solve a research result on the SIR model with an assumption. The assumption in the model is that there is vaccination for the population. There are live stages of research conducted. The first is creating the SIR model and determining the equilibrium points on disease-free and disease-endemic. The Second is getting the basic reproduction number. The third is determining the stability around the equilibrium points using the Routh-Hurwitz criteria. Fourth, create a diagram for the subpopulations state at a specific time using Wolfram Mathematica software. As an implementation of the model created, COVID-19 data at the Batanghari Community Health Center Inpatient UPTD was used. Finally, determine the model error percentage with MAPE. The SIR Covid-19 model was made using eight parameters, namely , which are all positive. The results showed that the disease-free and disease-endemic equilibrium points were locally asymptotically stable after being analyzed using the Routh-Hurwitz stability criteria. The model trial using data from UPTD Puskesmas Batanghari obtained a stable condition for up to 100 months with a MAPE of 2.8%. From this study, obtained an . This means that if you want to reduce the rate of spread, then reduce the number of people who are easily infected ( ) and reduce contacts ( ), and increase the healing rate ( ).

Mathematical Modeling of Alcohol-Drug Concentration in the Bloodstream

This research involved the formulation of a system of mathematical models mimicking alcohol concentration in the human body and in the bloodstream, which are subjected to some initial conditions. The formulation starts with the an ideal presentation of the compartment diagram in Figure 1, the figure clearly indicates the two compartment in focus and the diffusion of alcohol from one compartment to the other with different rates including the elimination rate through urination and sweating that were not considered in this research. The aim of the research is to investigate the importance of the various rate constants on the alcohol concentration in the bloodstream, as well as investigating the drug dosage and volume of alcohol concentration impact in the bloodstream. The Laplace method (LM) was adopted to solve the model which represents, firstly, the alcohol concentration in the body where an exact solution was obtained, secondly, the exact solution was also obtained for the model depicting the alcohol concentration in the bloodstream with all pertinent parameters clearly spelt out. Lastly, numerical simulation was carried out using Wolfram Mathematica, version 12, where the pertinent parameters values were varied and the results presented graphically. The graphical results indicate that the various rate constants, dosage increase and the volumetric increase have impact and showed a corresponding effect on the drug concentration in the bloodstream. The novelty of this research is the fact that we’ve been able to mimic a real-life scenario, an alcohol concentration in the bloodstream, mathematically, and solved the system using Laplace methods and performed numerical simulation to show the importance of the pertinent parameters. This system can be recommended for scientists and engineers alike who could be interested in pharmacokinetics and Pharmacodynamics research.

SmeftFR v3 – Feynman rules generator for the Standard Model Effective Field Theory

We present version 3 of SmeftFR, a Mathematica package designed to generate the Feynman rules for the Standard Model Effective Field Theory (SMEFT) including the complete set of gauge invariant operators up to dimension-6 and the complete set of bosonic operators of dimension-8. Feynman rules are generated with the use of FeynRules package, directly in the physical (mass eigenstates) basis for all fields. The complete set of interaction vertices can be derived, including all or any chosen subset of SMEFT operators. As an option, the user can also choose preferred gauge fixing, generating Feynman rules in unitary or Rξ-gauges. The novel feature in version-3 of SmeftFR is its ability to calculate SMEFT interactions consistently up to dimension-8 in EFT expansion (including quadratic dimension-6 terms) and express the vertices directly in terms of user-defined set of input-parameters. The derived Lagrangian in the mass basis can be exported in various formats supported by FeynRules, such as UFO, FeynArts, etc. Initialisation of numerical values of Wilson coefficients of higher dimension operators is interfaced to WCxf format. The package also includes a dedicated Latex generator allowing to print the result in clear human-readable form. The SmeftFR v3 is publicly available at http://www.fuw.edu.pl/smeft. Program summaryProgram Title:SmeftFR v3.0CPC Library link to program files:https://doi.org/10.17632/h4787mfvfz.1Licensing provisions: GPLv3Programming language: Mathematica 12.1 or later (earlier versions were reported to have problems running this code)External routines/libraries: Wolfram Mathematica programSubprograms used:FeynRules v2.3.49 or later packageNature of problem: Automatised generation of Feynman rules in physical (mass) basis for the Standard Model Effective Field Theory with user defined operator subset, gauge fixing and input-parameters scheme selection.Solution method: Expansion of SmeftFR v2 Mathematica package [1]: implementation of the results of Ref. [2] in the FeynRules package [3], including dynamic “model files” generation.

The uncertainty related to the inexactitude of prioritization based on consistent pairwise comparisons.

When the in/consistency in Pairwise Comparisons (PCs) is taken into consideration as the subarea of the Multi Attribute Decision Making (MADM) scientific field, it has many repercussions in various types of research areas including different modelling scenarios e.g. reduction of inconsistency during PCs, deriving appropriate consistency thresholds for inconsistent Pairwise Comparison Matrices (PCMs), completing of incomplete PCMs, aggregating of individual PCMs in relation to Group Decision Making (GDM) aspects, and PCMs in/consistency relation to credibility of Priority Vectors (PV) derived from PCMs with the application of various Priorities Deriving Methods (PDMs). The examination objective in the latter area of research is the uncertainty related to the inexactitude of prioritization based on derived PVs. However, only few research studies examine this problem from the perspective of PCM applicability for credible designation of decision maker's (DM) priorities in the way that leads to minimization of the prioritization uncertainty related to possible, and sometimes very probable, ranking fluctuations. This problem constitutes the primary area of interest for this research paper as no research study was thus far identified that examines this problem from the perspective of consistent PCMs. Hence, a research gap was identified. Thus, the objective of this research paper is to fill in this scientific gap. The research findings have serious repercussions in relation to prioritization quality with the application of PCs methodology, mostly in relation to the interpretation and reliability evaluation of prioritization results. Firstly, the research study outcome changes the perspective of the rank reversal phenomenon, which shed new light on many research studies that have been presented in the subject's literature for many decades. Secondly, the research study results throw new light on the discussion concerning the fuzziness of AHP's results. Last but not least, the effect of the research opens the unique opportunity to evaluate the prioritization outcome obtained within the process of consistent PCs from the well-known perspective of statistical hypothesis testing i.e. the probability designation of the chance that accepted ranking results which were considered as correct due to low probability of change may be incorrect, hence they should be rejected, and the probability designation of the chance that rejected ranking results which were considered as incorrect due to high probability of change may be correct and should be accepted. The paramount finding of the research is the fact that consistent PCMs provide PVs, which elements cannot be considered as established, but only approximated within certain confidence intervals estimated with a certain level of probability. As problems related to heuristics can be analyzed only via a computer simulation process, because they cannot be mathematically determined, the problem examined in this research paper is examined via Monte Carlo simulations, appropriately coded and executed with the application of Wolfram's Mathematica Software. It is believed that this research findings should be very important and useful for all decision makers and researchers during their problems' examinations that relate to prioritization processes with the application of PCs methodology.

Resonances and Periodic Motions of Atwood’s Machine with Two Oscillating Weights

The problem of constructing periodic solutions to the equations of motion of Atwood’s machin in which both weights have the same mass and can oscillate in the vertical plane is discussed. Differential equations governing the motion of this system are derived, and an algorithm for calculating their solutions that determine periodic oscillations of the weights under the condition that the oscillation frequencies are in resonance nω1 = mω2, where n and m are natural numbers, is proposed. These solutions are obtained in the form of series in a small parameter. The comparison of the results with numerical solutions of the equations of motion confirm the validity of the obtained solutions. All computations are performed using the computer algebra system Wolfram Mathematica.

The Implicit Numerical Method for the Radial Anomalous Subdiffusion Equation

This paper presents a numerical method for solving a two-dimensional subdiffusion equation with a Caputo fractional derivative. The problem considered assumes symmetry in both the equation’s solution domain and the boundary conditions, allowing for a reduction of the two-dimensional equation to a one-dimensional one. The proposed method is an extension of the fractional Crank–Nicolson method, based on the discretization of the equivalent integral-differential equation. To validate the method, the obtained results were compared with a solution obtained through the Laplace transform. The analytical solution in the image of the Laplace transform was inverted using the Gaver–Wynn–Rho algorithm implemented in the specialized mathematical computing environment, Wolfram Mathematica. The results clearly show the mutual convergence of the solutions obtained via the two methods.

FeynGrav 2.0

We present a new version of FeynGrav. The present version supports Feynman rules for matter with non-vanishing mass and SU(N) Yang-Mills model. We revisit the gauge fixing procedure for gravity and derive interaction rules valid for an arbitrary gauge fixing parameter. We provide a few simple examples of calculations to illustrate package usage. Program summaryProgram Title: FeynGravCPC Library link to program files:https://doi.org/10.17632/rwnrbbdkkv.1Developer's repository link:github.com/BorisNLatosh/FeynGravLicensing provisions: GPLv3Programming language: Wolfram Mathematica 10 and higherExternal routines/libraries: FeynCalc [1,2]Nature of problem: FeynGrav is a software package that simplifies calculations involving the Feynman rules for gravity. These rules are complex and consist of numerous terms, but FeynGrav makes it possible to work with a specific set of rules within the FeynCalc environment. The package includes interaction rules for regular matter with different spin values, as well as for the SU(N) Yang-Mills model in curved spacetime and general relativity. FeynGrav provides scripts to generate the relevant interaction rules and libraries of pre-generated expressions, making it a powerful tool for researchers in the field of theoretical physics.Solution method: I propose a framework that provides a straightforward way to obtain expressions for the corresponding Feynman rules by efficiently utilizing perturbative expansions. In particular, within the perturbative approach to quantum gravity, certain quantities admit finite perturbative expansions, while others have infinite perturbative expansions. I derive formulas that allow the generation of terms with infinite expansions for any given order of perturbation theory. In all actions describing gravitational interactions, all derivatives can be placed on terms with finite perturbative expansions via integration by parts. In this way, the proposed framework gives a simple algorithm for deriving Feynman rules.Additional comments including restrictions and unusual features: Scripts responsible for generating interaction rules are placed in the “Rules” folder. They can be used separately and only require FeynCalc to run. The “Libs” folder contains the “FeynGravLibrariesGenerator.wl” script and the libraries it generates. “FeynGrav.wl” is the main package file that imports data from the libraries and uses it to work with the interaction rules. The file “FeynGrav_Examples.nb” contains examples of calculations.