Algebraic Method Research Articles

Analyzing solitary wave solutions of the nonlinear Murray equation for blood flow in vessels with non-uniform wall properties

Solitary wave solutions are of great interest to bio-mathematicians and other scientists because they provide a basic description of nonlinear phenomena with many practical applications. They provide a strong foundation for the development of novel biological and medical models and therapies because of their remarkable behavior and persistence. They have the potential to improve our comprehension of intricate biological systems and help us create novel therapeutic approaches, which is something that researchers are actively investigating. In this study, solitary wave solutions of the nonlinear Murray equation will be discovered using a modified extended direct algebraic method. These solutions represent a uniform variation in blood vessel shape and diameter that can be used to stimulate blood flow in patients with cardiovascular disease. These solutions are newly in the literature, and give researchers an important tool for grasping complex biological systems. To see how the solitary wave solutions behave, graphs are displayed using Matlab.

Killing vector fields of locally rotationally symmetric Bianchi type V spacetime

The classification of locally rotationally symmetric Bianchi type V spacetime based on its killing vector fields is presented in this paper using an algebraic method. In this approach, a Maple algorithm is employed to transform the Killing’s equations into a reduced evolutive form. Subsequently, the integration of the Killing’s equations is carried out subject to the constraints provided by the algorithm. The algorithm demonstrates that there exist fifteen distinct metrics that could potentially possess Killing vector fields. Upon solving the Killing equations for all fifteen metrics, it is observed that seven out of the fifteen metrics exclusively exhibit the minimum number of Killing vector fields. The remaining eight metrics admit a varying number of Killing vector fields, specifically 6, 7, or 10. The Kretschmann scalar has been computed for each of the obtained metrics, revealing that all of them possess a finite Kretschmann scalar and thus exhibit regular behavior.

Algebraic Multigrid Methods for Metric-Perturbed Coupled Problems

Algebraic Multigrid Methods for Metric-Perturbed Coupled Problems

Approximate analytical solutions to nonlinear oscillations via semi-analytical method

A modified algebraic method is proposed in this study as an effective semi-analytical method for obtaining analytical solutions for two types of nonlinear differential equations. The trial solution is assumed to be a polynomial, and the algebraic approach is used to find the unknown coefficients. To enhance the solution, first, we apply the Laplace transformation on the series solution and then create the Padé approximations of the resultant equation. Finally, for the nonlinear problems under discussion, the inverse Laplace transformation is used to obtain periodic solutions.

An Algebraic Method for the Synthesis of Error-Free Binary Neural Network

An Algebraic Method for the Synthesis of Error-Free Binary Neural Network

Surface chemistry of ion beam modified native titania/Ti interfaces examined using X-ray photoelectron spectroscopy

It is often assumed in X-ray Photoelectron Spectroscopy that binding energy shifts are synonymous with changes in the chemical state of an atom and the chemical state can be described in terms of oxidation state and stoichiometry of the elements in a material that correlates with photoemission peaks. However, when an atom is bonded into a crystal lattice, the shapes and binding energy of photoemission may not match the expected stoichiometry when measured by XPS, even though shifts in binding energy suggest new oxidation states. In this work, a set of experiments is presented, in which Ar+ and He+ ions of different energies modify the native oxide on a titanium foil which yields XPS spectra that are not easily open to analysis by conventional peak models. In the course of analyzing these complex photoemission data by linear algebraic methods, the prospect emerged that XPS is suggesting that sputtering a native oxide may be providing insight into structural perturbation in addition to the stoichiometry-type changes to the native oxide.

Analytical solutions for hydrodynamic responses of arrays of floating truncated cylinders using multi-term Galerkin method and its application to a new wave energy converter device

In the context of linear water wave theory, the analytical solutions for the diffraction and radiation of a truncated cylinder array are developed in the presence of ambient incident waves. Each cylinder in the array can oscillate with five degrees of freedom (DOFs), i.e., surge, sway, heave, roll, and pitch. This paper adopts the multi-term Galerkin method to expand the fluid velocity at the interface of different regions into a set of basis functions containing Gegenbauer polynomials, which accurately and efficiently characterizes the cube root singularity of the fluid velocity near the edges of the truncated cylinders. Using the dynamic equilibrium equations, the amplitudes of each DOF of the cylinders in the array are solved. The analytical solution presented in this paper converges rapidly, and high-precision hydrodynamic response results can be obtained using just a few truncated terms (e.g., the upper bounds of m0 = 5 and p0 = 22 can yield results of five-figure accuracy). For the 4-cylinder array, under the same accuracy conditions (the error less than 1%), the computation time of the conventional method developed by Zeng et al. [“Hydrodynamic interactions between waves and cylinder arrays of relative motions composed of truncated floating cylinders with five degrees of freedom,” J. Fluids Struct. 115, 103785 (2022d)] based on the exact algebraic method [Kagemoto and Yue, “Interactions among multiple three-dimensional bodies in water waves: An exact algebraic method,” J. Fluid Mech. 166, 189–209 (1986)] is 3.9 times longer than that of the present method. As the number of cylinders increases, the advantage of the present method in terms of convergence speed becomes more apparent, e.g., for the 16-cylinder array, the conventional solution takes 6.3 times longer than the present solution. To extract wave energy more efficiently, a new 5DOF wave energy converter (WEC) device that can extract energy in 5DOFs is proposed. The present method is adopted to investigate the hydrodynamic performance of the 5DOF WEC arrays. Compared with the traditional 1DOF (heave) WEC, the 5DOF WEC can significantly improve the energy capture performance of arrays, especially in the high-frequency wave region.

Investigating dispersive optical solitons with the generalized stochastic perturbed Schrödinger–Hirota equation incorporating power-law nonlinearity and multiplicative white noise

This article aims to introduce the generalized stochastic perturbed Schrödinger–Hirota equation, which incorporates multiplicative white noise in the Itô sense. The study focuses on investigating the stochastic optical soliton solutions related to the governing model to understand the characteristics and properties of these soliton solutions. We can achieve this by utilizing the enhanced Kudryashov method and the enhanced direct algebraic approach. The study stands out for its innovative approach to examining the effects of multiplicative white noise on stochastic optical soliton solutions through the lens of the generalized stochastic perturbed Schrödinger–Hirota equation. Utilizing the enhanced Kudryashov method alongside the novel enhanced direct algebraic method, this research effectively identifies and delineates a range of soliton solution types, such as bright, dark, singular, and straddled solitons. This thorough investigation not only underscores the efficacy and efficiency of these methods but also sheds light on the impact of noise on the structural and dynamic properties of soliton solutions, supported by illustrative visualizations. A notable finding is that all solutions derived using the enhanced direct algebraic method feature a nonzero coefficient for the highest-order term of the auxiliary equation. The diversity of solutions obtained through these approaches includes bright, dark, singular, and straddled solitons. Additionally, the study reveals Weierstrass and Jacobi elliptic doubly periodic solutions, which can transition into various soliton types under specific parameter conditions, offering a broader understanding of soliton behavior and properties.

ПАКЕТ GAP ДЛЯ РОЗЩЕПЛЕННЯ ЛІНІЙНИХ СИСТЕМ

There are a number of problems, the solution of which requires the breakdown of the initial system of equations using algebraic decoupling methods. This means reducing the matrix of coefficients to block-diagonal (or block-triangular) form by means of substitution of variables. The main computational tasks when using such methods are finding the centralizer of several matrices or compiling the algebra generated by these matrices. For calculations, it is convenient to use the GAP computer algebra system because the system itself is designed for discrete algebra calculations. The problem is that the GAP program does not support calculations with real numbers. For practical problems you can try to replace them (with some accuracy) by rational numbers. At the same time, the decision may turn out to be excessively cumbersome. On the other hand, the advantage of GAP is the complete absence of rounding errors.

Impressive Exact Solitons to the Space-Time Fractional Mathematical Physics Model via an Effective Method

A new class of truncated M-fractional exact soliton solutions for a mathematical physics model known as a truncated M-fractional (1+1)-dimensional nonlinear modified mixed-KdV model are achieved. We obtain these solutions by using a modified extended direct algebraic method. The obtained results consist of trigonometric, hyperbolic trigonometric and mixed functions. We also discuss the effect of fractional order derivative. To validate our results, we utilized the Mathematica software. Additionally, we depict some of the obtained kink, periodic, singular, and kink-singular wave solitons, using two and three dimensional graphs. The obtained results are useful in the fields of fluid dynamics, nonlinear optics, ocean engineering and others. Furthermore, these employed techniques are not only straightforward, but also highly effective when used to solve non-linear fractional partial differential equations (FPDEs).

Study of traveling soliton and fronts phenomena in fractional Kolmogorov-Petrovskii-Piskunov equation

The present research work presents the modified Extended Direct Algebraic Method (m-EDAM) to construct and analyze propagating soliton solutions for fractional Kolmogorov-Petrovskii-Piskunov equation (FKPPE) which incorporates Caputo’s fractional derivatives. The FKPPE has significance in various disciplines such as population growth, reaction-diffusion mechanisms, and mathematical biology. By leveraging the series form solution, the proposed m-EDAM determines plethora of travelling soliton solutions through the transformation of FKPPE into Nonlinear Ordinary Differential equation (NODE). These soliton solutions shed light on propagation processes in the framework of the FKPPE model. Our study also offers some graphical representations that facilitate the characterization and investigation of propagation processes of the obtained soliton solutions which include kink, shock soliton solutions. Our work advances our understanding of complicated phenomena across multiple academic disciplines by fusing insights from mathematical biology and reaction-diffusion mechanisms.

Investigation of space-time dynamics of perturbed and unperturbed Chen-Lee-Liu equation: Unveiling bifurcations and chaotic structures

In this paper, the fractional space-time nonlinear Chen-Lee-Liu equation has been considered using various methods. The investigation of the transition from periodic to quasi-periodic behavior has been conducted using a saddle-node bifurcation approach. The paper reports the conditions of multi-dimensional bifurcations of dynamical solutions. Additionally, a direct algebraic method has been used to calculate various 2D and 3D solitonic structures of the equation, and an analysis of their accuracy and effectiveness has been conducted. Furthermore, the Galilean transformation has been used to convert the equation into a planar dynamical system, which is further utilized to obtain bifurcations and chaotic structures. Chaotic structures of perturbed dynamical system are observed and detected through chaos detecting tools such as 2D-phase portrait, 3D-phase portrait, time series analysis, multistability and Lyapunov exponents over time. Further, sensitivity behavior for a range of initial conditions, both perturbed and unperturbed. The results suggest that the investigated equation exhibits a higher degree of multi-stability.

About the Kyiv school mathematicians contribution to the extrema of functions of many variables theory

The subject of this article is the study of Kyiv school mathematicians’ contribution to the many variable extrema function theory. The purpose of this article is to study the work of Kyiv mathematician Professor M. Ye. Vashchenko-Zakharchenko. Task: for the first time, to carry out a detailed analysis of the results obtained in the article by M. Ye. Vashchenko-Zakharchenko «Signs of the highest and lowest value of functions». The research method is a historical and scientific analysis of the original source, which allows the scientific results obtained in the late 19th and early 20th centuries to be estimated from the viewpoint of modern mathematical analysis. The following results were obtained. Thanks to the analysis of this article, it was found out that here M. Ye. Vashchenko-Zakharchenko first used the methods of linear algebra and applied D. Sylvester’s criterion of positive (negative) definiteness of quadratic form to obtain sufficient conditions for the existence of an extremum of functions of many variables. The problem is considered in general for functions of many variables and for special cases of functions of two and three variables. Conclusions. Comparing the results obtained from the Kyiv mathematician, Professor M. Ye. Vashchenko-Zakharchenko with the presentation of this topic in modern textbooks on higher mathematics and mathematical analysis, we can conclude that they are included in these textbooks in virtually the same form. This is what determines the relevance of this topic: methods for solving problems for the extremum of functions of many variables, obtained at the turn of the 19th and 20th centuries, are used in modern optimization problems. The breadth of the scientific erudition of M. Ye. Vashchenko-Zakharchenko allowed him to immediately perceive new methods obtained by foreign scientists and immediately find their application to obtain sufficient conditions for the existence of an extremum of functions of many variables. This shows the high scientific level of the state of mathematics at Kyiv University in the second half of the 19th century. Further research should focus on the contribution of other mathematicians of the Kyiv school to the development of the theory of extrema of functions of many variables.

Abundant new optical solitary waves of paraxial wave dynamical model with kerr media via new extended direct algebraic method

Abundant new optical solitary waves of paraxial wave dynamical model with kerr media via new extended direct algebraic method

On calculating partial sums of multiple numerical series by methods of Computer Algebra

A method to calculate partial sums of some multiple numerical series arising when searching for the resultant of a polynomial and an entire function is proposed. One can apply a symbolic algorithm that uses recurrent Newton formulas to find power sums of roots included in this formula without finding the very roots of the system. The algorithm that implements the proposed approach to calculate partial sums of multiple numerical series is implemented in Maple. Examples of using this algorithm to find partial sums of some classes of multiple numerical series are given.

Solving rician data analysis problems: theory and numerical modeling using computer algebra metods in Wolfram Mathematica

This paper considers theoretical foundations and mathematical methods of data analysis under the conditions of the Rice statistical distribution. The problem involves joint estimation of the signal and noise parameters. It is shown that this estimation requires the solution of a complex system of essentially nonlinear equations with two unknown variables, which implies significant computational costs. This study is aimed at mathematical optimization of computer algebra methods for numerical solution of the problem of Rician data analysis. As a result of the optimization, the solution of the system of two nonlinear equations is reduced to the solution of one equation with one unknown variable, which significantly simplifies algorithms for the numerical solution of the problem, reduces the amount of necessary computational resources, and enables the use of advanced methods for parameter estimation in information systems with priority of real-time operation. Results of numerical experiments carried out using Wolfram Mathematica confirm the effectiveness of the developed methods for two-parameter analysis of Rician data. The data analysis methods considered in this paper are useful for solving many scientific and applied problems that involve analysis of data described by the Rice statistical model.

Applying computer algebra systems to study Chaundy-Bullard identities for the vector partition function with weight

An algorithm for obtaining the Chaundy-Bullard identity for a vector partition function with weight that uses computer algebra methods is proposed. To automate this process in Maple, an algorithm was developed and implemented that calculates the values of the vector partition function with weight by finding non-negative solutions of systems of linear Diophantine equations that are used to form the identities involved. The algorithm’s input data is represented by the set of integer vectors that form a pointed lattice cone and by some point from this cone, and the Chaundy-Bullard identity for the vector partition function with weight is its output. The code involved is stored in the depository and is ready-to-use. An example demonstrating the algorithm’s operation is given.

A Discrete Dynamic Conflict Model

Relevance. Conflicts play a significant role in the development of society, the state and all of humanity. They arise in interpersonal relationships, economics, organizational activities, social problems, and world politics. The study of the essence of conflicts, the causes of their emergence and ways to resolve them is carried out by conflictology, which was formed as an interdisciplinary field of knowledge in the middle of the 20th century. Currently, it has become obvious that the conflict itself is not a negative phenomenon, but a competently resolved conflict situation can benefit all parties to the conflict. As a result, it became necessary to consider conflicts not only from a descriptive, philosophical point of view, but also to be able to predict a possible conflict situation, describe its development over time and calculate acceptable ways out of it. This approach to the study of conflicts is mathematical modeling, based on mathematical methods for describing conflict situations, which allow analyzing conflicts and predicting their outcome.The purpose is to describe a general methodology for constructing a conflict scheme of any origin within the framework of the mathematical theory of conflicts. The objectives is to consider and analyze a discrete dynamic model of conflicts using the example of a childparent conflict.Methodology. The work uses algebraic methods and methods of discrete mathematics.Results. Within the framework of the mathematical theory of conflicts, a conflict with a discrete mismatch is considered the example of a child-parent model conflict. A general method for constructing a conflict situation diagram and analyzing it has been developed. Possible ways of influencing the conflicting parties on each other in any conflict situation are discussed. When analyzing possible scenarios for the development of a conflict situation, the possible types of impacts of the conflicting parties on each other are considered.Conclusions. Based on the developed dynamic discrete model, it is shown that the mathematical description of conflicts is reduced to the problem of combinatorial optimization. The developed scheme of mathematical description of conflicts can be applied to a wide class of conflict situations in various subject areas.

Gaussian elimination for flexible systems of linear inclusions

Flexible systems are linear systems of inclusions in which the elements of the coefficient matrix are external numbers in the sense of nonstandard analysis. External numbers represent real numbers with small, individual error terms. Using Gaussian elimination, a flexible system can be put into a row-echelon form with increasing error terms on the right-hand side. Then parameters are assigned to the error terms and the resulting system is solved by common methods of linear algebra. The solution set may have indeterminacy not only in terms of linear spaces, but also of modules. We determine maximal robustness for flexible systems.

Control and physical verification of 6-DOF manipulator for power inspection robots based on expert PID algorithm

To enhance the performance of power inspection robots in intricate nuclear power stations, it is necessary to improve their response speed and accuracy. This paper uses the manipulator of the power inspection robot as the primary research object, and unlike previous control algorithm research, which only remained in the software simulation stage, we constructed a set of physical verification platforms based on CAN communication and physically verified the robotic arm’s control algorithm. First, the forward motion model is established based on the geometric structure of the manipulator and D-H parameter method, and the kinematic equation of the manipulator is solved by combining geometric method and algebraic method. Secondly, in order to conduct comparison tests, we designed PID controllers and expert PID controllers by utilising the expertise of experts. The results show that compared with the traditional PID algorithm, the expert PID algorithm has a faster response speed in the control process of the manipulator. It converges quickly in 0.75 s and has a smaller overshoot, with a maximum of only 6.9%. This confirms the expert PID algorithm’s good control effect on the robotic arm, allowing the six-degree-of-freedom robotic arm to travel more accurately and swiftly along the trajectory of the target point.