Symbolic Computation Research Articles

Numerical and analytical solutions of new Blasius equation for turbulent flow

The Blasius equation for laminar flow comes from the Prandtl boundary layer equations. In this article, we establish a new and generic Blasius equation for turbulent flow derived from the turbulent boundary layer equation that can be used for turbulent as well as laminar flow. The analytical and numerical solutions have been investigated under specific conditions to the developed new Blasius equation. The analytical and numerical results have been compared through tables and graphs to validate the established model. In fluid dynamics, analytical solutions to complicated systems are tedious and time-consuming. Changing one or more constraints can introduce new challenges. In this case, symbolic computation software provides an easier and more flexible solution for fluid dynamical systems, even if boundary conditions are adjusted to explain reality. Therefore, the MATLAB code is used to investigate the new third-order Blasius equation. The comparison and graphical representations demonstrate that the achieved results are encouraging.

DEVELOPMENT OF ALGORITHMS AND SOFTWARE FOR MODELING CONTROLLED DYMAMIC SYSTEMS USING SYMBOLIC COMPUTATIONS AND STOCHASTIC METHODS

The development of software for synthesizing and analyzing models of controlled systems taking into account their deterministic and stochastic description is an important direction of research. Results of the development of software for modeling dynamic systems the behavior of which can be described by onestep processes are presented. Models of population dynamics are considered as an example. The software uses a deterministic description of the model at its input to obtain a corresponding stochastic model in symbolic form and also analyze the model in detail (calculate trajectories in the deterministic and stochastic cases, find control functions, and visualize the results). An important aspect of the development is the use of computer algebra for analyzing the model and synthesizing controls. Methods and algorithms based on deterministic and stochastic Runge–Kutta methods, stability and control theory, methods for designing self-consistent stochastic models, numerical optimization algorithms, and artificial intelligence are implemented. The software was developed using high-level programming languages Python and Julia. As the basic tools, high-performance libraries for vector–matrix computations, symbolic computation libraries, libraries for the numerical solution of ordinary differential equations, and libraries of global optimization algorithms are used.

Bifurcation analysis, stationary optical solitons and exact solutions for generalized nonlinear Schrödinger equation with nonlinear chromatic dispersion and quintuple power-law of refractive index in optical fibers

In this paper, the bifurcation, stationary optical solitons and exact solutions in optical fiber propagation are studied. The propagation model is described by a generalized nonlinear Schrödinger equation with Kudryashov’s quintuple power law of refractive index. First, the equation is transformed into a second-order nonlinear ordinary differential equation by an appropriate transformation. Then, the bifurcation of the phase diagram of the obtained ordinary differential equation is established using planar dynamic system theory, and the types of optical solitons and exact solutions are determined. What is more, the stationary optical solitons and exact solutions are obtained by the polynomial complete discriminant system, and are demonstrated in 3D and 2D graphics by using the symbolic computation software Maple. Finally, the obtained stationary optical solitons and exact solutions include the Jacobi elliptic function solution, hyperbolic function solution, rational function solution, trigonometric function solution and exponential function solution. Moreover, the results provide a method to further study stationary optical solitons and exact solutions of propagation pulses in optical fibers.

SYMBOLIC-NUMERICAL IMPLEMENTATION OF THE GALERKIN METHOD FOR APPROXIMATE SOLUTION OF THE WAVEGUIDE DIFFRACTION PROBLEM

In this paper, we construct a symbolic-numerical implementation of the Galerkin method for approximate solution of the waveguide diffraction problem at the junction of two open planar three-layer waveguides. The Gelerkin method is implemented in the Maple computer algebra system by symbolic manipulations; its software implementation is based on the scprod symbolic-numerical procedure, which enables the numerical calculation of scalar products for the Galerkin method based on symbolic expressions. The use of symbolic manipulations makes it possible to speed up the calculation of integrals in the Galerkin method owing to single-run symbolic calculation of integrals typical for the problem, rather than multiple numerical integration.

Jupyter Notebooks for the study of advanced topics in Fluid Mechanics

AbstractIn recent years, Jupyter Notebooks have become a very useful free and open‐source tool in teaching, as they allow you to combine text, images, mathematical expressions, links and code into a single document. This gives students an interactive document with which they can experiment and learn with the help of high‐level mathematical calculus. In Fluid Mechanics, it is very common for students to deal with complex computations that take away attention from the Mechanic, especially in advanced topics such as Rheology, Turbulence, or Boundary Layer. The subject “Advanced Fluid Mechanics” is an elective one of the last year of the Bachelor's degree in Industrial and Aerospace Technology Engineering at the Terrassa School of Industrial, Aerospace and Audiovisual Engineering at the Universitat Politècnica de Catalunya. This subject has three ECTS credits and has been taught since the academic year 2020–2021 This subject complements the compulsory subject Fluid Mechanics and is developed in 6 weeks with 5 h of class each week. This work presents Fluid Mechanics modules with Jupyter Notebooks that complement the syllabus given in the compulsory subject. An elective subject is presented where subjects of Fluid Mechanics per week are studied independently, using different Python tools: symbolic calculation, modeling of experimental data, statistical analysis, numerical calculation, and so forth. The main goal is for the student to focus on mechanical concepts and actively learn to use the tools available, especially open source, to do the associated mathematical calculations.

Novel rational solitons and generalized breathers for (1+1)-dimensional longitudinal wave equation

This paper deals with the evaluation of multiwave, homoclinic breather, M-shaped rational solitons and interaction of M-shaped rational solitons with single and double kinks for [Formula: see text]-dimensional longitudinal wave equation via the logarithmic transformation and symbolic computation with ansatz functions method. We will obtain two types of M-shaped rational solitons and their dynamics are revealed in figures by choosing the distinct values of involved parameters. In addition, two forms of interaction of M-shaped rational soliton with kink wave will also be evaluated. Furthermore, we will compute kink cross-rational solutions, periodic cross-rational solutions, generalized breathers and Akhmediev breathers for governing model. The solitons behaviors via different values of parameters in the produced solutions will be analyzed.

Stability Analysis of the Rational Solutions, Periodic Cross-Rational Solutions, Rational Kink Cross-Solutions, and Homoclinic Breather Solutions to the KdV Dynamical Equation with Constant Coefficients and Their Applications

We explore various analytical rational solutions with symbolic computation using the ansatz transformation functions. We gain a variety of rational solutions such as M-shaped rational solutions (MSRs), periodic cross-rationals (PCRs), multi-wave solutions, rational kink cross-solutions (RKCs), and homoclinic breather solutions (HBs), and by using the appropriate values for the relevant parameters, their dynamics are visualized in figures. Additionally, two different types of interactions between MSRs and kink waves are analyzed. Furthermore, we examine the stability of the obtained solutions and create a corresponding table. We analyze the stability of these solutions and the movement role of the wave by making graphs as two-dimensional, three-dimensional and density graphs as well as contour visual and stream plots.

Toward Network Intelligence.

This article proposes a conceptual framework to guide research in neural computation by relating it to mathematical progress in other fields and to examples illustrative of biological networks. The goal is to provide insight into how biological networks, and possibly large artificial networks such as foundation models, transition from analog computation to an analog approximation of symbolic computation. From the mathematical perspective, I focus on the development of consistent symbolic representations and optimal policies for action selection within network settings. From the biological perspective, I give examples of human and animal social network behavior that may be described using these mathematical models.

Forecasting the dynamics of the model of cold bosonic atoms in a zig-zag optical lattice by symbolic computation

The work aims to explore the exponential rational function technique and the generalized Kudryashov procedure to investigate one of the most important equations which is the equation of cold bosonic atoms in a zig-zag optical lattice. The considered equation is reduced to a supreme equation by using the continuum approximation which describes the soliton’s dynamics with the implication pulse variables. The adopted techniques are accurate, simple, straightforward, and succinct to compute.

New modified tanh-function method for nonlinear evolution equations

The aim of this paper is to calculate the Traveling waves solutions by using a new technique which is called modified tanh-function method, which are successfully performed to get analytical solutions for Korteweg-deVries (KdV)–Burgers’ equation and (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. As a result, when the equation parameters are taken as special values, some new solitary wave solution are obtained. Moreover we find in this work that the modified tanh-function method give some new results which are easier and faster to compute by the help of a symbolic computation system. The results obtained were compared with standard tanh-function method.

Bifurcation analysis and optical soliton solutions for the fractional complex Ginzburg–Landau equation in communication systems

The nonlinear fractional complex Ginzburg–Landau system is a classical equation which has been developed vigorously in the fields of the combustion theory, nonlinear optics, signal processing, statistical mechanics and so on. This paper mainly concentrates on the bifurcations and single dispersive optical solitons for the fractional complex Ginzburg–Landau equation in communication systems. Starting with the traveling wave transformations and some other suitable transformations, the fractional complex Ginzburg–Landau system is converted into an equivalent ordinary differential system. Secondly, the bifurcation phase diagram and Hamiltonian of fractional complex Ginzburg–Landau system are also given. In addition to, the dispersive optical solitons and traveling wave solutions are derived via the dynamical theory method, the polynomial complete discriminant system technique and symbolic computation. It is notable that the obtained optical soliton solutions may improve or complement the corresponding results in the literature (Akram et al., 2022, Ouahid et al., 2021). Finally, in order to further understand the propagation of the fractional complex Ginzburg–Landau equation in nonlinear optics, three-dimensional and two-dimensional diagrams are given.

Interaction solutions for the second extended (3+1)-dimensional Jimbo–Miwa equation

Based on the Hirota bilinear method, the second extended (3+1)-dimensional Jimbo–Miwa equation is established. By Maple symbolic calculation, lump and lump-kink soliton solutions are obtained. The interaction solutions between the lump and multi-kink soliton, and the interaction between the lump and triangular periodic soliton are derived by combining a multi-exponential function or trigonometric sine and cosine functions with quadratic functions. Furthermore, periodic-lump wave solution is derived via the ansatz including hyperbolic and trigonometric functions. Finally, 3D plots, 2D curves, density plots, and contour plots with particular choices of the suitable parameters are depicted to illustrate the dynamical features of these solutions.

How to hunt wild constants

Abstract There are now several comprehensive web applications, stand-alone computer programs and computer algebra functions that, given a floating point number such as 6.518670730718491, can return concise nonfloat constants such as 3 arctan 2 + ln 9 + 1, that closely approximate the float. This is analogous to unscrambling an egg. Such software includes AskConstants, Inverse Symbolic Calculator, the Maple identify function, MESearch, OEIS, Ordner, RIES, and WolframAlpha. Usefully often such a result is the exact limit as the float is computed with increasing precision. Therefore these program results are candidates for proving an exact result that you could not derive or conjecture without the program. Moreover, candidates that are not the exact limit can be provable bounds, or convey qualitative insight, or suggest series that they truncate, or provide sufficiently close efficient approximations for subsequent computation. This article describes some of these programs, how they work, and how best to use each of them. Almost everyone who uses or should use mathematical software can benefit from acquaintance with several such programs, because these programs differ in the sets of constants that they can return.

Benchmarking tools for a priori identifiability analysis.

The theoretical possibility of determining the state and parameters of a dynamic model by measuring its outputs is given by its structural identifiability and its observability. These properties should be analysed before attempting to calibrate a model, but their a priori analysis can be challenging, requiring symbolic calculations that often have a high computational cost. In recent years, a number of software tools have been developed for this task, mostly in the systems biology community. These tools have vastly different features and capabilities, and a critical assessment of their performance is still lacking. Here, we present a comprehensive study of the computational resources available for analysing structural identifiability. We consider 13 software tools developed in 7 programming languages and evaluate their performance using a set of 25 case studies created from 21 models. Our results reveal their strengths and weaknesses, provide guidelines for choosing the most appropriate tool for a given problem and highlight opportunities for future developments. https://github.com/Xabo-RB/Benchmarking_files.

SYMBA: symbolic computation of squared amplitudes in high energy physics with machine learning

The cross section is one of the most important physical quantities in high-energy physics and the most time consuming to compute. While machine learning has proven to be highly successful in numerical calculations in high-energy physics, analytical calculations using machine learning are still in their infancy. In this work, we use a sequence-to-sequence model, specifically, a transformer, to compute a key element of the cross section calculation, namely, the squared amplitude of an interaction. We show that a transformer model is able to predict correctly 97.6% and 99% of squared amplitudes of quantum chromodynamics and quantum electrodynamics processes, respectively, at a speed that is up to orders of magnitude faster than current symbolic computation frameworks. We discuss the performance of the current model, its limitations and possible future directions for this work.

The applications of symbolic computation to exact wave solutions of two HSI-like equations in (2+1)-dimensional

It is renowned that Hirota–Satsuma–Ito (HSI) equation is widely used to study wave dynamics of shallow water. In this work, two new HSI-like equations are investigated which could be extended to diversify problems in natural phenomena and give admirable contributions by applying the generalized exponential rational function method (GERFM). With the aid of symbolic calculations, various constraints on the free parameters are given, while classes of wave solutions are explicitly constructed from the coefficients of the combined non-linear and dissipative terms. After specifying values for free parameters, singular, periodic singular and anti-kink waves are demonstrated in 3D figures to exhibit different kinds of wave propagations. The fact that parameters directly influence the wave amplitude and speed of traveling waves is illustrated. The derived results are innovative and have important applications in the current field of mathematical physics research. Eventually, we show that generalized exponential rational function method is effective and straightforward to solve higher-order and high-dimensional non-linear evolution equations.

Increasing beam stability zone in synchrotron light sources using polynomial quasi-invariants

The objective of this article is to propose a scheme to increase the stability zone of a charged particles beam in synchrotrons using a suitable objective function that, when optimized, inhibits the resonances onset in phase space and the dynamic aperture of electrons in storage rings can be improved. The proposed technique is implemented by constructing a quasi-invariant in a neighborhood of the origin of the phase space, then, by using symbolic computation software, sets of coupled differential equations for functions involved in nonlinear dynamics are obtained and solved numerically with periodic boundary conditions. The objective function is constructed by proposing that the innermost momentum solution branch of the polynomial quasi-invariant approaches to the corresponding ellipse of the linear dynamics. The objective function is optimized using a genetic algorithm, allowing the dynamic aperture to be increased. The quality of results obtained with this scheme are compared with particle tracking simulations performed with available software in the field, showing good agreement. The scheme is applied to a synchrotron light source model that can be classified as third generation due to its emittance.

Lump solutions to an integrable (3 + 1)-dimensional Boussinesq equation and its dimensionally reduced equations in shallow water

This study uses the Hirota bilinear method and Maple, a symbolic computation program, to derive lump solutions for a new integrable (3 + 1)-dimensional Boussinesq equation and its dimensionally reduced equations. Furthermore, lump solutions with free parameters have been constructed using the dimensionally reduced new form of the (3 + 1)-dimensional Boussinesq equation. The derived lump solutions show it has two trough positions and one crest position. The amplitudes and shapes of lump waves don't vary during propagation but they change their positions. By making three-dimensional, two-dimensional, and density plots for specific values of the relevant free parameters, the propagations of the obtained lump wave solutions are displayed. They also demonstrate how the trough and crest positions of a lump wave change over time with constant velocity. The phase shifts, propagation directions, and energy distributions can be seen from the graphical outputs that the free parameters of the model play a significant role in changing the shapes and amplitudes of the waves. The resulting solutions and their physical characteristics may help to understand how the waves propagate in shallow water in oceanography.

Exact Solution of ( 4 + 1 )-Dimensional Boiti–Leon–Manna–Pempinelli Equation

Based on the Hirota bilinear method, using the heuristic function method and mathematical symbolic computation system, various exact solutions of the ( 4 + 1 )-dimensional Boiti–Leon–Manna–Pempinelli equation including the block kink wave solution, block soliton solution, periodic block solution, and new composite solution are obtained. Upon selection of the appropriate parameters, three-dimensional and contour diagrams of the exact solution were generated to illustrate their properties.

Symbolic computation to construct new soliton solutions and dynamical behaviors of various wave structures for two different extended and generalized nonlinear Schrödinger equations using the new improved modified generalized sub-ODE proposed method

In this work, we propose a new improved modified generalized sub-ODE method for constructing new solitons and traveling wave solutions, and also show the dynamical behaviors of various wave structures to the extended nonlinear Schrödinger equation with higher-order odd and even terms, as well as a generalized nonlinear Schrödinger equation of fourth-order, using symbolic computerized work via Mathematica. This newly proposed method improves and modifies the Li method (Li-Hua and Jin-Yu, 2009) The improved method presented in this paper can be used to solve other nonlinear equations with nonlinear terms of any order of physical systems in order to obtain many solitary wave solutions and other traveling wave solutions for such nonlinear models in a unified manner. The resulting soliton solutions and other forms of solutions are very useful and advantageous in many branches of mathematical physics and nonlinear sciences such as ocean engineering, optical fibers, plasma physics, and fluid dynamics.