Algebraic System Research Articles

Numerical solutions of stochastic delay integro-differential equations by block pulse functions

This paper presents an efficient numerical method for solving nonlinear stochastic delay integro-differential equations based on block pulse functions. Firstly, the equation is transformed into an algebraic system by the integral delay operator matrixes of block pulse functions. Then, error analysis is conducted on the method. Finally, some numerical examples are provided to validate the method. This work provides numerical solutions for the stochastic delay integro-differential equations by global approximation method. This method has the advantages of simple calculation and higher error accuracy.

Application of fractional shifted vieta-fibonacci polynomials in nonlinear reaction diffusion equation with variable order time-space fractional derivative

Abstract In this article, an accurate optimization algorithm based on new polynomials namely generalized shifted Vieta-Fibonacci polynomials (GSVFPs) is employed to solve the nonlinear variable order time-space fractional reaction diffusion equation (NVOTSFRDE). The algorithm combines GSVFPs, new variable order fractional operational matrices in the Caputo sense, and the Lagrange multipliers to achieve the optimal solution. First, the solution of the NVOTSFRDE is approximated as a series of GSVFPs with unknown coefficients and parameters. Then, the Lagrange multipliers method is adopted so that the NVOTSFRDE can be transformed into a class of nonlinear algebraic system of equations and we solve these equations using MATLAB and MAPLE software. Solving this system and substituting the coefficients and parameters into the approximation of the guessed functions, the solution of the NVOTSFRDE is obtained. The convergence analysis of the approach are discussed. The accuracy of the algorithm is verified through error analysis and mathematical examples. The accuracy of the new method is higher than that of the exciting method. The reconstruction results demonstrate that the proposed optimization algorithm is efficient for the NVOTSFRDE, and the algorithm is also convergent.

The Building of Mathematical Concepts When Solving Mathematical Tasks in Traditional Way and Using Graphing Calculators

This article aims to understand how students learn mathematical concepts related to the study of functions and differential calculus using graphing calculators with the computer algebra system (GCC) and traditional methods. With a qualitative methodology, we show that students can make the transition to mathematical meanings of concepts using both processes.

SMARANDACHE ANTI ZERO DIVISORS

In this paper, we study and discuss the concept of Smarandache anti zero divisor (SAZD) element of the ring and the group ring , where is a cyclic group of order generated by . Moreover, we introduce and discuss the concept of SAZD ideal of the ring . Some results related to the given concepts are proved in detail. Accordingly, a Computer Algebra System (GAP) is used to verify the results of this study

A study of the receding contact problem for functionally graded piezoelectric materials

We investigated a receding contact problem for the functionally graded piezoelectric coating (FGPC) materials. Both the elastic constants and piezoelectric and dielectric constants of the FGPC are considered to vary exponentially along the gradient direction. This contact system degenerates into a system of singular integral equations by applying a standard Fourier integral transform to the governing equations of the FGPC and the elastic substrate. Then, the Gauss–Chebyshev polynomials are introduced to discrete this singular integral system into an algebraic system, in which the unknowns (i.e., the contact radius and contact stress at both interfaces) can be solved by our proposed iterative algorithm using the given boundary conditions. Numerical examples are performed to verify the accuracy of this iterative algorithm, and the influence of the gradient parameters and applied loads on the top surface of FGPC on the contact radius and contact stress at both interfaces are further analyzed.

Cadabra and python algorithms in general relativity and cosmology I: Generalities

The aim of this work is to present a series of concrete examples which illustrate how the computer algebra system Cadabra can be used to manipulate expressions appearing in General Relativity and other gravitational theories. We highlight the way in which Cadabra's philosophy differs from other systems with related functionality. The use of various new built-in packages is discussed, and we show how such packages can also be created by end-users directly using the notebook interface. The current paper focuses on fairly generic applications in gravitational theories, including the use of differential forms, the derivation of field equations and the construction of their solutions.

Developing of neural network computing methods for solving inverse elasticity problems

This paper examines the use of neural network methods to solve inverse problems in the mechanics of elastic materials. The aim is to design physics-informed neural networks that can predict the parameters of structural components, and the physical properties of materials based on a specified displacement distribution. A key feature of the specified neural networks is the integration of differential equations and boundary conditions into the loss function calculation. This approach ensures that the error in approximating unknown functions has a direct impact on optimizing the network's weights. As a result, the resulting neural network approximations of unknown functions comply with the differential equations and boundary conditions. To test the capabilities of the designed neural networks, inverse problems involving the bending of plates and beams have been solved, focusing on determining one or two unknown parameters. Comparison of predicted and exact values demonstrates high accuracy of the constructed neural network models, with a relative prediction error of less than 3 % across all cases. Unlike analytical methods for solving inverse problems, the primary advantage of physics-informed neural networks is their flexibility when addressing both linear and nonlinear problems. For instance, the same network architecture can be employed to solve various boundary-value problems without modification. Compared to classical numerical methods, the parallelization capability of neural networks is inherently supported by modern software libraries. Therefore, the application of physics-informed neural networks for solving inverse elasticity problems of plates and beams is effective, as evidenced by the achieved relative errors and the computational robustness of the method. In practice, the proposed solution can be used for relevant calculations during the design of structural elements. The developed software code can also be integrated into automated design systems or computer algebra systems

Exact behavior of the critical Kauffman model with connectivity one

The critical Kauffman model with connectivity one is the simplest class of critical Boolean networks. Nevertheless, it exhibits intricate behavior at the boundary of order and chaos. We show that the model is equivalent to a deceptively simple algebraic system of polynomials which count the number and length of cycles. The polynomial for multiple loops is the product of the polynomials for individual loops. Using this perspective, we prove that the number of cycles scales as 2m, where m is the number of nodes in loops—as fast as possible and faster than previously believed. Published by the American Physical Society 2024

Algorithm xxx: ellipFor , a Fortran software library for Legendre elliptic integrals and Jacobi elliptic functions with generalized input arguments

Legendre elliptic integrals and Jacobi elliptic functions arise in multiple applications within the physical sciences, including oscillations, celestial mechanics, and geodynamics. In this study, we describe the Fortran library ellipFor capable of evaluating the following for generalized input values: 1) the complete Legendre elliptic integrals of the first and second kinds, 2) the incomplete Legendre elliptic integrals of the first and second kinds, and 3) the principal Jacobi elliptic functions. Our software builds upon previously developed Fortran routines, which were designed with restrictions on input parameters that may be limiting in applications. Our routines apply multiple transformations to allow for more general input values, such as elliptic moduli greater than unity for points 1–3, arbitrary real Jacobi amplitudes for points 1–2, and complex first arguments for point 3. In addition, our routines are thread-safe, allowing for parallel computations. Our routines were compared with values from the computer algebra system SageMath over a wide range of input parameters. Values from ellipFor and SageMath agreed to within tolerances commensurate with the limitations of floating-point arithmetic used for the elliptic integrals and Jacobi elliptic functions listed in points 1, 2, and 3 above for generalized input arguments.

The Modelling of Light Absorption and Reflection in a SiOx/Si Structure with Al Nanoparticles for Solar Cells

In this study, the light propagation in a structure consisting of SiOx on Si substrate with Al nanoparticles regularly placed in the SiOx layer is considered. Numerical modelling is performed by solving the Maxwell equations for the electromagnetic waves. In distinction from the well-known finite-difference time-domain (FDTD) simulation technique, we do not solve time-dependent wave equations here; rather, we propose a new numerical technique. This technique allows us to determine the stationary amplitudes of the electromagnetic oscillations directly from the linear algebraic equation system. The obtained results apply to silicon solar cells with an SiOx + Al top layer to maximise their efficiency. We found that 26 nm and 39 nm diameters of spherical Al nanoparticles are nearly optimal for a λ = 435.8 nm wavelength of the incident light. In addition, we evaluated the (nearly) optimal parameters of their placement in the SiOx layer. The results show the possibility of increasing the efficiency of solar cells by increasing the light absorption inside the active Si layer from ≈60% to ≈80%. Future perspectives on the proposed method and its possible applications are discussed.

The Economic Potential Impact of Climate Change on The Gambia’s Agriculture Sector: A Multi-Market Model Analysis

Given the global climate challenges, countries must comprehend how climate change will impact agricultural commodity prices, production and consumption, agricultural trade, and food security. The study used the General Algebraic Modelling System (GAMS), a multi-market model approach, with the base year of 2015, to assess the impact of climate change on the agriculture sector, given three scenarios—hydro-meteorological impact, agricultural productivity impact and their combined total effect. The results show a decline in demand for food across The Gambia. Seed demand will increase, whilst animal feed demand will decline over time. Crop yields for groundnuts (the main cash crop) and rice (the main staple crop) by 2040 will decline by 11.09% and 13.32%, respectively, given the combined total effect. The Gambia will continue to depend on the importation of basic necessities, given its food trade deficits over time. Average food prices will increase by 16.2%; consequently, The Gambia will not be able to attain food self-sufficiency; rather, food insecurity will continually be acute, leading to increased prices and decreased production and consumption. Thus, the government should increase investment in the agriculture sector, especially in crops and livestock that are resilient to adverse climate conditions.

CAN SYMBOLIC COMPUTATION AND FORMALIST SYSTEMS ENHANCE MATH EDUCATION WITH ARTIFICIAL INTELLIGENCE?

In recent years, a solution developed using deep learning methods has been used to solve difficult problems in a field. The capability of deep learning models is that they require large and heavily sampled data sets. Computer Algebra Systems developed over time have made significant progress, especially in the field of symbolic mathematics solutions solved by machine learning. It is a persistent problem how appropriate it is to use such formal systems in some aspects of algorithmic decision-making. In this paper, we discussed the suitability of artificial intelligence applications to formal propositions by evaluating a deep learning study conducted especially in the field of symbolic mathematics and Math education. Symbolic computation systems have a strong potential for enhancing math education. Furthermore, within the framework of the Incompleteness Theorem, to show why the construction of a mathematical grammar is not a complete solution for Mathematics education systems.

Perturbation Methods in Solving the Problem of Two Bodies of Variable Masses with Application of Computer Algebra

The classical many-body problem is not integrable, so perturbation theory based on an exact solution to the two-body problem is usually applied to investigate the dynamics of planetary systems. However, in the case of variable masses, the two-body problem is not integrable, in general, and application of perturbation theory is required to investigate it, as well. In the present paper, we use the perturbation theory to derive the differential equations determining the orbital elements of the relative motion of one body around the other. Two models of the perturbed aperiodic motion on conic and quasi-conic sections are considered and compared. Special attention is paid to the practically important case of small eccentricities, when the perturbing forces may be replaced by the corresponding power series expansions. The differential equations of the perturbed motion are averaged over the mean anomaly, and the evolutionary equations describing the behavior of the orbital elements over long periods of time are obtained for two models. Comparing the corresponding solutions to the evolutionary equations, we have shown that both models demonstrate similar behavior with regard to the secular perturbations of the orbital elements. However, the second model, based on the aperiodic motion on a quasi-conic section, is more appropriate for generalization to the many-body problem with variable masses. All the relevant symbolic and numerical calculations are performed with the computer algebra system Wolfram Mathematica.

On a Model for Solving Mixed Fractional Integro Differential Equation

In this work, the mixed fractional integro differential equation (MfrIo-DE) of the second kind, under certain condition is considered, in the space L2(−1, 1)× C [0, T]; T < 1 T is the time. The position kernel k (|x−y|) of IE has a singularity. After integrating and using the properties of fractional integral, we have a MIE in position and time, where the kernel of position takes the singular form k (|x−y|), and the kernel of time takes the singular Abel form (t −τ)α−1 , 0 < α < 1. Then, using separation of variable method, under certain substitution, we obtain FIE in position, with variable fractional coefficients in time. Using the Toeplitz matrix method (TMM), we have a nonlinear algebraic system (NAS). Moreover, numerical results are obtained and discussed, especially when 0 < α < 1. Also, the solutions of the mixed equation are considered when α = 0, α = 1. Finally, the error estimate, in each case, is computed.

Practical Canonical Labeling of Multi-Digraphs via Computer Algebra

Practical algorithms for computing canonical forms of multi-digraphs do not exist in the literature. This paper proposes two practical approaches for finding canonical forms, from the perspective of nD symbolic computation. Initially, the approaches turn the problem of finding canonical forms of multi-digraphs into computing canonical forms of indexed monomials in computer algebra. Then, the first approach utilizes the double coset representative method in computational group theory for canonicalization of indexed monomials and shows that finding the canonical forms of a class of multi-digraphs in practice has polynomial complexity of approximately O((k+p)2) or O(k2.1) by the computer algebra system (CAS) tool Tensor-canonicalizer. The second approach verifies the equivalence of canonicalization of indexed monomials and finding canonical forms of (simple) colored tripartite graphs. It is found that the proposed algorithm takes approximately O((k+2p)4.803) time for a class of multi-digraphs in practical implementation, combined with one of the best known graph isomorphism tools Traces, where k and p are the vertex number and edge number of a multi-digraph, respectively.

Differential Transform Method (DTM) and Physics-Informed Neural Networks (PINNs) in Solving Integral–Algebraic Equation Systems

Integral–algebraic equations and their systems are a common description of many technical and engineering problems. Often, such models also describe certain dependencies occurring in nature (e.g., ecosystem behaviors). The integral equations occurring in this problem may have two types of domains—symmetric or asymmetric. Depending on whether such symmetry exists in the system describing a given problem, we must choose the appropriate method to solve this system. In this task, the absence of symmetry is more advantageous, but the presented examples demonstrate how one can approach cases where symmetry is present. In this paper, we present the application of two methods for solving such tasks: the analytical Differential Transform Method (DTM) and Physics-informed Neural Networks (PINNs). We consider a wide class of these types of equation systems, including Volterra and Fredholm integrals (which are also in a single model). We demonstrate that despite the complex nature of the problem, both methods are capable of handling such tasks, and thus, they can be successfully applied to the issues discussed in this article.

Numerical solution of fractional integro-differential equations using ultraspherical polynomials

In this work we will discuss the fractional integro-differential (FIDE) equations of the Fredholm and Volterra type, where we begin by presenting some special concepts about the definition of the fractional integral and the fractional derivative of Riemann-Liouville and Caputo, while addressing the Gegenbauer (Ultraspherical) polynomials and its special differential equation, mentioning the primary polynomials, in addition to presenting some theories such as the Krasnoselskii fixed point theorem, which guarantees the existence and uniqueness of solutions to fractional integro-differential equations (FIDE). To find the approximate solution of fractional integro-differential equations, we relied on orthogonal polynomials of the ultraspherical type (Gegenbauer), which have a great advantage in solving integral equations and integro-differential equations. These polynomials, with the application of fractional Caputo derivatives with the series method, in addition to fixing the values and changing the degree of the polynomials used with fixing the values of , lead to the equation becoming a linear algebraic system of equations. By solving this linear system, the approximate solution is obtained in the form of a Gegenbauer (ultraspherical) series. The change in the degree of the polynomials used will inevitably show the effectiveness and speed of the method through the examples that will be presented, where the approximate solutions obtained are compared with the exact solutions.

Multiscale mortar mixed finite element methods for the Biot system of poroelasticity

We develop a mixed finite element domain decomposition method on non-matching grids for the Biot system of poroelasticity. A displacement–pressure vector mortar function is introduced on the interfaces and utilized as a Lagrange multiplier to impose weakly continuity of normal stress and normal velocity. The mortar space can be on a coarse scale, resulting in a multiscale approximation. We establish existence, uniqueness, stability, and error estimates for the semi-discrete continuous-in-time formulation under a suitable condition on the richness of the mortar space. We further consider a fully-discrete method based on the backward Euler time discretization and show that the solution of the algebraic system at each time step can be reduced to solving a positive definite interface problem for the composite mortar variable. A multiscale stress–flux basis is constructed, which makes the number of subdomain solves independent of the number of iterations required for the interface problem, and weakly dependent on the number of time steps. We present numerical experiments verifying the theoretical results and illustrating the multiscale capabilities of the method for a heterogeneous benchmark problem.

Helmholtz preconditioning for the compressible Euler equations using mixed finite elements with Lorenz staggering

AbstractImplicit solvers for atmospheric models are often accelerated via the solution of a preconditioned system. For block preconditioners, this typically involves the factorisation of the (approximate) Jacobian resulting from linearization of the coupled system into a Helmholtz equation for some function of the pressure. Here we present a preconditioner for the compressible Euler equations with a flux‐form representation of the potential temperature on the Lorenz grid using mixed finite elements. This formulation allows for spatial discretisations that conserve both energy and potential temperature variance. By introducing the dry thermodynamic entropy as an auxiliary variable for the solution of the algebraic system, the resulting preconditioner is shown to have a similar block structure to an existing preconditioner for the material‐form transport of potential temperature on the Charney–Phillips grid. This new formulation is also shown to be more efficient and stable than both the material‐form transport of potential temperature on the Charney–Phillips grid and a previous Helmholtz preconditioner for the flux‐form transport of density‐weighted potential temperature on the Lorenz grid for a 1D thermal bubble configuration. The new preconditioner is verified further against standard two‐dimensional test cases in a vertical slice geometry.

Boundary element method for buckling analysis of elastically connected double-beam system

Studies of Double-beam systems have received significant attention from researchers due to their wide applications in civil and mechanical engineering. Commonly, finite element method or exact solutions have been employed to solve double-beam systems, with few others considering buckling of axially loaded double-beam systems with classical boundary conditions. This paper presents a novel formulation of the Boundary Element Method (BEM) to determine the critical buckling load of the double-beam system elastically connected by a Winkler elastic layer with generalized boundary conditions. Based on the Euler-Bernoulli beam theory, the double-beam system is composed of two identical beams. This paper provides a detailed discussion of each step involved in the BEM, including the fundamental solution, boundary integral equations, and algebraic system. Examples considering different boundary conditions, material properties, and load cases are done and compared to corresponding analytical solution. The results show excellent agreement between the BEM approach and the analytical solution, confirming the accuracy and effectiveness of the technique.