Maple Program Research Articles

Modulation instability, and dynamical behavior of solitary wave solution of time M- fractional clannish random Walker's Parabolic equation via two analytic techniques

The theory of solitary waves or solitons is crucial in nonlinear models due to their ability to propagate without distortion, making them essential in fields like physics, biology, engineering, and mathematics. This study explores solitary waves in the time fractional Clannish Random Walker's Parabolic equation using two effective approaches: polynomial expansion technique (PET) and unified technique (UT). This model is particularly significant in areas such as ecology, sociology, and urban planning, where understanding individual interactions within spatial contexts is vital. By solving this equation, we gain insights into group formation and navigation, enhancing our comprehension of emergent patterns and system design. The application of PET and UT allows us to derive various solutions, including exponential, hyperbolic, and trigonometric forms. Utilizing the Maple programming language, we visualize novel phenomena, such as kink waves, anti-kink waves, and periodic lump waves. This research demonstrates that the proposed methodologies can yield precise soliton solutions, contributing valuable insights to nonlinear science and engineering applications.

Electromagnetism and Maxwellian Evolution Equations in terms of Darboux Frame in Minkowski Space with Abnormalities

Abstract Electromagnetic wave propagation is often thought of as the transport of polarised light and this behaviour is well defined by Maxwell’s equations when propagating in an optical fiber. In this paper, we examine the q − direction and n − direction Berry’s phase equation along a Darboux framed optical fibre in Minkowski space. Next, we define q − direction and n − direction for the electromagnetic curves of the Rytov parallel transport laws. And then, the application section, the connections between the Maxwellian evolution of the electromagnetic curve for Maxwell’s equation and the anholonomic coordinates are visualized and illustrated with the MAPLE program.

Integral resonant negative derivative feedback suppression control strategy for nonlinear dynamic vibration behavior model

One of the major problems in robotics research has been developing an actuator system for extremely dynamic-legged robots. High torque density and the capacity to control dynamic physical interactions are two design requirements for high-speed locomotion that are challenging for conventional actuators used in manufacturing applications to meet. To address this system and apply the desired control to reach the best stability position, the robot's foot was simulated with the Van der Pol equations, applied the required control, and studied that application. This work describes the actions of a new novel control mechanism known as the Integral Resonant Negative Derivative Feedback (IRNDF) controller, which reduces the vibration response of a double Van der Pol oscillator subjected to external excitations. This unique controller combines integral resonant control (IRC) and negative derivative feedback (NDF) controllers to provide a new controller effect for double Van der Pol oscillators. The multiple scale perturbation technique (MSPT) has been applied to solve the controlled system analytically. The MATLAB and MAPLE programs have been used to complete and clarify all of the numerical talks. The frequency response curves have been used to study the impact that altering the parameter values had on the amplitude. The controlled system vibration amplitude is governed by frequency-response equations (FREs), which have been constructed. In the vibration system, the IRC, NDF, and IRNDF controllers were compared to see which one was the best. Numerical results show that the unique IRNDF controller is the best at reducing oscillations and decreasing amplitude values. The effects of the effective parameters on the controlled system have been identified. The frequency-response equation that was derived has been used to plot the various response curves for the framework that show the stable and unstable zones when the controller is off and on. Lastly, excellent agreement between the derived numerical findings and the analytical ones was observed. Lastly, utilizing time histories and response curves to compare analytical and numerical solutions was fascinating and significant.

Mathematical Model of Fungal Growth with Hyphae Death and Substrate

One of the most crucial subjects in mathematics education is mathematical modeling, which involves connecting situations from the real world to mathematical formulas and vice versa. Our study will focus on how substrate affects fungal growth and lateral branching behaviors, hyphal death, and tip death due to overcrowding. In order to solve a mathematical model, we will employ the system's three-dimensional of equations (PDEs). using numerical solutions, traveling wave theory, stability, and non-dimensionality in our mathematical approach as well as we use trace and determine to find stable states.Computer programs, such as the Maple program and Pdep code in MATLAB, will also be used to aid in the solution process.

Mathematical Model of FWE Branching Type with Effect on Energy

: Mathematical modeling is one of the most important topics in teaching mathematics and converting real world problems into mathematical formulas in order to solve them. In this research we will take the behavior of lateral branching, tib-tib anostomosis with its effect on the energy of the growth of fungi. We will use the system of equations (PDEs) to solve a mathematical model and we will use mathematical techniques in order to reach the solution using non-dimensionality, stability, traveling wave, and numerical solution and use computer programs to facilitate the solution, including the Maple program, the MATLAB program, use Pdep code and the pplane7 code.

New Exact Solutions for Generalized of Combined with Negative Calogero-Bogoyavlenskii Schiff and Generalized Yu–Toda–Sassa–Fukuyama Equations

In this paper, we present generalized model of combined Calogero-Bogoyavlenskii Schiff and negative-order Calogero-Bogoyavlenskii Schiff G(CBS-nCBS) equation and generalized Yu–Toda–Sassa–Fukuyama g(YTSF) equation. We apply the extended hyperbolic function method, to solve generalized models. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions solutions of these equations from the method with the aid of the computer program Maple.

Theoretical framework for biological control of pest: a mathematical modeling approach.

Crop losses to pests were the main obstacle to food security globally. Pest control was a laudable exercise, but the exercise could be hindered by the inevitable adjustment between pest reductions, operation costs as well as impacts on the environment and human health. The pest could be controlled by many methods, but biological control was the most popular technique because it addressed inevitable trade-offs between costs and side effects. In this paper, a mathematical model was developed to quantify intricate biological procedures in the context of biological control using prey-predator mechanisms. Three equilibrium points (one trivial and two non-trivial) were derived, and the stability of each equilibrium point was examined. The stability results indicated that the adoption of biological control might neutralize pest infestation but the situation might not persist (unstable trivial equilibrium). It was also discovered that pest control through biological means might fail if the predator was wrongly selected or if the population of the predator vanished while the pest remained in existence (unstable non-trivial equilibrium). The analytical results were finally justified by a means of simulation via a computer-in-built maple program.

Hopf Bifurcation of Three-Dimensional Quadratic Jerk System

هذا البحث مخصص لبحث تشعب هوبف لنظام الرعشة التربيعي ثلاثي الأبعاد. تم دراسة الاستقرار للنقاط المنفردة وظهور تشعب هوبف والدورات الحدية للنظام. بالإضافة إلى ذلك، يتم استخدام تقنية كميات ليابانوف لدراسة دورية النظام ومعرفة عدد دورات الحد التي يمكن تشعبها من نقاط هوبف. نظرًا للحمل الحسابي المطلوب لحساب كميات ليابانوف، تم تثبيت بعض المعلمات. حاليًا، يُظهر التحليل أنه يمكن تشعب ثلاث دورات حدية من نقاط هوبف. تم استخدام برمجة Maple للتحقق من جميع النتائج المعروضة في هذه الدراسة.

Using classical analysis and computer technologies to study power-exponential type equations

In the present paper basing on properties of the power and the exponential functions we study power-exponential type equations and establish properties of their roots. Since such equations are transcendental there is no method to find exact roots. But using well known methods of classical calculus and tools of the system of analytical calculations Maple we can establish the boundaries of roots and predict their convergence as the degree tends to infinity. We also give qualitative and quantitative analysis of the difference between the power and the exponential functions for large values of the argument and suggest a convenient formula for approximate but fast calculations. Results of calculations demonstrated in tables and graphs were obtained using Maple program.

Bessel equation and functions in computer mathematics system

The relevance of technical issues of solving problems of mathematical physics is growing every year. The development of computer technologies leads to the use of modern methods for their solution. The application of analytical computing systems is considered as an effective method, which contributes to their productive implementation. The article deals with finding a general solution of the Bessel equation and equations leading to it in the Maple program, graphs of functions are plotted

Simulation of voltage characteristics of vapor-gas shell during electrolyte-plasma treatment

Electrolytic plasma processing (EPT) is a method of surface treatment of materials based on the use of plasma and an electrolytic solution. This abstract discusses the principle of action, main applications and potential benefits of EPТ. The EPТ technique involves immersing the object being treated in an electrolytic solution, after which an electric current is applied, causing decomposition of the solution and the formation of a plasma cloud at the surface of the material being processed. Exposure to plasma and chemically active components of the solution makes it possible to modify the surface of the material, improving its properties, such as adhesion, strength and corrosion resistance. EPТ is widely used in a variety of industries, including metalworking, electronics, medical equipment and food processing. The advantages of the method include high efficiency, the ability to process complex shapes and materials, as well as environmental safety due to the absence of the use of chemically aggressive substances. Electrolytic plasma processing is a promising direction in the field of surface modification of materials with a wide range of potential applications. In this paper, theoretical studies of vapor-gas shell formation in the near-surface region of structural steels in the cathodic heating mode of electrolyte-plasma treatment were considered. The technology of electrolyte-plasma hardening, providing the required mechanical properties of products, which are often subjected to wear, temperature and force effects, was investigated. Based on the results of theoretical studies, mathematical calculations of voltage, current density, and dependence graphs were made to obtain a model of vapor-gas shell formation in the process of cathodic heating. Modeling of calculations of vapor-gas shell formation in the process of cathodic heating was carried out with the help of Maple program.

COMPARATIVE OF RIVEST-SHAMIR-ADLEMAN CRYPTOSYSTEM AND ITS FOUR VARIANTS USING RUNNING TIME AND MEMORY CONSUMPTION ANALYSIS

The Rivest-Shamir-Adleman (RSA) algorithm, known for its slow single-precision multiplication (spm) and overall running time, is not commonly employed to encrypt user data directly. As a result, several researchers have developed various RSA-based cryptosystems to enhance the algorithm's performance while maintaining security. This paper presents a comparative analysis of different variants of the RSA cryptosystem, focusing on five specific cryptosystems: RSA, Somsuk-RSA, Modified-RSA (MRSA), Easy Simple Factoring-RSA (ESF-RSA), and Phony-RSA. The methodology involves evaluating the theoretical running time and memory usage through single-precision multiplication (spm) measurements, while the actual running time is estimated using Maple programming. The research has two primary objectives. Firstly, they examined each algorithm of the RSA variants and analysed them according to the proposed methodology. Secondly, to determine which cryptosystem consumes the most time and memory for key generation, encryption, and decryption. The results indicate that ESF-RSA and RSA are the fastest in terms of key generation, ESF-RSA is the quickest for encryption, and Phony-RSA excels in decryption speed. Additionally, ESF-RSA demonstrates the lowest memory usage, whereas MRSA requires the highest memory allocation for all processes.

Mathematical model written by ordinary differential equations for hysteresis games

In game theory, the hysteresis effect manifests itself in the fact that small differences in one or more parameters lead two systems to opposite stable equilibrium. The mathematical model (called smooth model) of hysteresis game writing by ordinary differential equations with the great parameter K is studied. Estimations of closeness of output functions for smooth and classical models through the continuity module of continuous input function are received. This error can be controlled by increasing the parameter K to a sufficiently large value. The result was conducted by studying of mathematical model and using programs Mathematica, Matlab or Maple. Experiments were carried out, and the obtained results were summarized and presented through evaluations and graphical interpretation.

NGHIÊN CỨU PHÂN BỐ ỨNG SUẤT TẠI BIÊN NGÀM CỦA VỎ TRỤ FGM CHỊU TẢI TRỌNG TĨNH SỬ DỤNG LÝ THUYẾT BIẾN DẠNG CẮT BẬC CAO QUASI-3D

This article presents the results of investigating FGM closed cylindrical shells using quasi-3D higher-order shear deformation theory. The law of volume distribution of materials according to the power function. In this article, the results of the analysis of the stress state at the clamped boundary area of the FGM cylindrical shell subjected to radial local loads distributed according to different laws. The mathematical model and the calculation program in Maple are compared and verified with the published results. The article has shown that at the clamped boundary edge, the values of stress components increase sharply, and at the same time, the strong influence of the load parameters (the variation law, the size of the load area), the power-law index on the values and distribution of stress components in the boundary area is demonstrated.

DERIVATION OF EXPRESSION FOR PHOTOCURRENT DENSITY FOR NON-DESTRUCTIVE TESTING OF 3D PRINTING FILAMENT BY MEANS OF TERAHERTZ SPECTROSCOPY

This report presents a revised expression for the photocurrent density in terahertz spectroscopy, which is a non-destructive testing technique of particular interest to the authors in the context of 3D printed parts. 3D printing, also known as additive manufacturing, involves creating three-dimensional objects based on computer-aided design (CAD) models. The process entails depositing, joining, or solidifying material under computer control, layer by layer.
 
 Defects in 3D printing, such as weak infill, gaps in thin walls, inconsistent extrusion, layer separation, and bed drop, can lead to low printing quality and render some printed parts unfit and unsafe for use. Moreover, the ability to tamper with internal layers without altering the exterior could result in the production of maliciously defective parts without detection. Therefore, it is crucial to test 3D printed details and filaments at each stage of processing using non-destructive methods.
 
 A comprehensive review of the relevant literature indicates the potential for enhancing measurement accuracy through various improvements in terahertz spectrometer models. The mathematical model for the photocurrent involves a convolution integral of the current density and the laser radiation pulse that irradiates the surface of the material under study. The expression within the integral incorporates parameters such as the duration of the optical pulse, carrier lifetime, and momentum relaxation time. By evaluating the integral, the result can be obtained as two terms, each being a product of an exponent and a complementary error function with the same parameters mentioned earlier.
 
 The calculation involves several steps, including a change of variables during integration. Verification using Maple software demonstrates agreement with analytical calculations and suggests a pathway for further refinement of the expression for the photocurrent density. The Maple program influenced the results by means of repeating same calculation with aid of computer and allowing to compare if analytical results are same and true, also it could be use for simulation and example calculation, for results graphical representation. 
 
 The connection between the obtained mathematical expression and its relation to 3D printing (additive manufacturing) exists. The explanation is in that the 3D printer uses filament, filament has defects, defectoscopy of filament in the terahertz domain have models and methods. The research of defectoscopy models and methods is helpful to increase accuracy of measurement of filament defect parameters and account on it and improve the quality of 3D printed details.

Importance of Reflected Solar Energy Loaded with SWCNTs-MWCNTs/EG Darcy Porous Stretched Surface: Midrich Scheme

Saving energy, shortening processing times, maximizing thermal efficiency, and lengthening the life of industrial equipment are all possible outcomes of heating and cooling optimization. In recent years, there has been a rise in interest regarding the development of high-efficiency thermal systems for the purpose of enhancing heat and mass movement. This study presents an investigation on the non-linear flow of a hybrid nanofluid comprising of Multi Walled Carbon Nanotubes (MWCNTs) and Single Walled Carbon Nanotubes (SWCNTs) over an extended surface, considering the effects of Magnetohydrodynamics (MHD) and porosity, with engine oil serving as the base fluid. Also, radiation and Darcy-Forchheimer flow is considered. The problem of regulating flow is transformed into ordinary differential equations (ODEs) by employing similarity variables. The Midrich Scheme is then used to implement a numerical solution to these equations in the program Maple. Through visual representations of fluid velocities and temperatures, the inquiry addresses several important factors, including magnetic parameters, porosity parameters, radiation parameters, Eckert numbers, inertia coefficients, and Biot numbers. The research has important implications in a number of real-world contexts. Due to its exceptional characteristics, such as reduced erosion, reduced compression drops difficulties, and greatly increased heat transfer rates, hybrid nanofluids are frequently used in heat exchangers. For instance, various cooling devices such as electromagnetic cooling systems, as well as heat exchangers including condensers, boilers, chillers, air conditioners, evaporators, coil preheaters, and radiators. Furthermore, it has the potential to be employed in pharmaceutical businesses and the field of biomedical nanoscience.

Relative Entropy Application to Study the Elastoplastic Behavior of S235JR Structural Steel.

The main issue in this work is to study the limit functions necessary for the reliability assessment of structural steel with the use of the relative entropy apparatus. This will be done using a few different mathematical theories relevant to this relative entropy, namely those proposed by Bhattacharyya, Kullback-Leibler, Jeffreys, and Hellinger. Probabilistic analysis in the presence of uncertainty in material characteristics will be delivered using three different numerical strategies-Monte Carlo simulation, the stochastic perturbation method, as well as the semi-analytical approach. All of these methods are based on the weighted least squares method approximations of the structural response functions versus the given uncertainty source, and they allow efficient determination of the first two probabilistic moments of the structural responses including stresses, displacements, and strains. The entire computational implementation will be delivered using the finite element method system ABAQUS and computer algebra program MAPLE, where relative entropies, as well as polynomial response functions, will be determined. This study demonstrates that the relative entropies may be efficiently used in reliability assessment close to the widely engaged first-order reliability method (FORM). The relative entropy concept enables us to study the probabilistic distance of any two distributions, so that structural resistance and extreme effort in elastoplastic behavior need not be restricted to Gaussian distributions.

Detachable connection mechanics of thin-walled cylindrical snap fit docking

A complete structure is composed of different components connected by various links/connections that plays extremely important role in maintaining the integrity of the structure, however, compared to the components, there is relatively less research on links/connections. Based on our accurate angle inversion of a detachable thin-walled cylindrical snap fit docking, we reformulate the elasticity of the snap fit and derive a new disassembly force and frictional energy dissipation of assembly/disassembly process. The studies shows that the irreversibility of frictional energy dissipation is the thermodynamic origin of snap-fit’s symmetry-breaking or asymmetry that is easy to assemble and not easy to disassemble. The results in this study are useful for the design of adjustable mechanical mechanism and/or snap-fit metamaterials. To make it easier for readers to use the formulas in this paper to solve their own problems, we provide a complete Maple program in the appendix.

In biology, solving a problem coming to a differential equation in the maple program

We know that not only the problems of mathematics, but also the mathematical model of a number of processes that occur in nature can been reduced to a differential equation. Most of the quantities found in nature have their own laws. Finding these, laws directly have more complicated matter. Finding the relationship between the quantity in question, its rate of change and acceleration is quite easy by nature. Simple differential equations have formed as a mathematical expression of this connection. It is important and significant to use modern computer programs to find a quick and accurate solution to such equations. Found in Maple.

Information modeling of the normal forms of logical functions in Maple system

The article discusses possible ways to use the Maple computer algebra system to obtain perfect conjunctive and disjunctive normal forms. Computer algebra systems make it possible to expand the possibilities for creating, applying and using mathematical models in the daily activities of engineers, researchers, and also contribute to improving the education of students of various training profiles. In addition, the use of computer technology can help to apply various methods and expand approaches to the representation of the studied objects (Boolean functions) of discrete form, including in education. Computer algebra systems such as Maple, Mathematica, etc. are the most adapted for studying various sections of discrete mathematics and mathematics in general. The possibilities of the Maple program are shown for use at the initial stages of mastering the discrete mathematics section of the university course - Boolean functions, in particular, for work on the construction and study of perfect conjunctive and disjunctive normal forms according to the truth table.