Concepts Of Calculus Research Articles

Invention in Bergson's Integral Empiricism: Differential Matter and Integral Memory

Abstract Bergson's integral empiricism is a philosophical approach that aims to harmonize the scientific study of nature with the experience of consciousness. Bergson drew heavily on the basic concepts of calculus in framing his philosophy of intuition. He did not merely use calculus metaphorically to describe processes and organizational patterns of consciousness. Instead, he studied the thought processes actually involved in calculus and showed that they are intimately connected to how we think about change more generally. I propose a distinction between derivative empiricism, i.e., reductionism, and integral empiricism that incorporates the study of the mind by the mind. The method involves the invention of a new kind of knowledge that is not based on generalization or abstraction but rather on the integration of experience. This is predicated on a study of the act of integration in consciousness that underpins our ability to think and do philosophy.

The Game of Dominoes as Teaching-learning Method of Basic Concepts of Differential Calculus

This paper will present teaching-learning mathematics strategies, specifically differential calculus in a variable, through a fun and competitive space in which action is manifested by the student and not only the repetition of information by the teacher. This action refers to motivate, problematise, summarise, and coordinate a dominoes game whose thematic cards are designed around the basic and main contents of differential calculus. The strategies for teaching this area are diverse and precisely the game of dominoes is one of the most used strategies in the practice of mathematics because it stimulates logical reasoning and mental skills. The objective is to identify the way in which the game of dominoes affects the learning and understanding of fundamental concepts of differential calculus in a variable by means of experimentation carried out on students of the first semester of the School of Engineering and Sciences of Technological of Monterrey. Finally, the results of this study will be presented and the use of this strategy in other topics around mathematics will be recommended to facilitate logical and meaningful learning in students.

A COMPUTATIONAL AND FRACTIONAL MATHEMATICAL MODEL FOR ASSESSING HEPATITIS B WITH OPTIMAL THERAPY

This paper presents a mathematical study applied to Hepatitis B disease utilizing fractional calculus and numerical methods. We begin by discussing mathematical fundamentals, focusing on fractional calculus concepts and relevant numerical aspects for disease modeling. We then formulate the problem, describing a nonlinear fractional differential system that models hepatitis B virus (HBV) infection and its response to drug therapy. A detailed theoretical analysis explores system properties and biological implications. We provide a qualitative analysis of the model, discussing important dynamic behaviors such as stability and parameter sensitivity. Subsequently, numerical analysis is performed to validate the model and estimate unknown parameters from patient data. Our findings include significant insights into biological parameters and their impact on therapeutic response. We conclude that the proposed fractional-order model is effective, offering optimal therapeutic strategies for hepatitis B.

Analysis of a class of fractal hybrid fractional differential equation with application to a biological model

Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. Different problems of real world processes have been investigated by using the concepts of fractional calculus and important and applicable outcomes were obtained. Because, there has been a lot of interest in fractional differential equations. It is brought on by both the extensive development of fractional calculus theory and its applications. The use of linear and quadratic perturbations of nonlinear differential equations in mathematical models of a variety of real-world problems has received a lot of interest. Therefore, motivated by the mentioned importance, this research work is devoted to analyze in detailed, a class of fractal hybrid fractional differential equation under Atangana- Baleanu- Caputo ABC derivative. The qualitative theory of the problem is examined by using tools of non-linear functional analysis. The Ulam-Hyer’s (U-H) type stability criteria is also applied to the consider problem. Further, the numerical solution of the model is developed by using powerful numerical technique. Lastly, the Wazewska-Czyzewska and Lasota Model, a well-known biological model, verifies the results. Several graphical representations by using different fractals fractional orders values are presented. The detailed discussion and explanations are given at the end.

Servo Control of a Current-Controlled Attractive-Force-Type Magnetic Levitation System Using Fractional-Order LQR Control

Recent research on fractional-order control laws has introduced the fractional calculus concept into the field of control engineering. As described herein, we apply fractional-order linear quadratic regulator (LQR) control to a current-controlled attractive-force-type magnetic levitation system, which is a strongly nonlinear and unstable system, to investigate its control performance through experimentation. First, to design the controller, a current-controlled attractive-force-type magnetic levitation system expressed as an integer-order system is extended to a fractional-order system expressed using fractional-order derivatives. Then, target value tracking control of a levitated object is achieved by adding states, described by the integrals of the deviation between the output and the target value, to the extended system. Next, a fractional-order LQR controller is designed for the extended system. For state-feedback control, such as fractional-order servo LQR control, which requires the information of all states, a fractional-order state observer is configured to estimate fractional-order states. Simulation results demonstrate that fractional-order servo LQR control can achieve equilibrium point stabilization and enable target value tracking. Finally, to verify the fractional-order servo LQR control effectiveness, experiments using the designed fractional-order servo LQR control law are conducted with comparison to a conventional integer-order servo LQR control.

ϕ−Multiplicative Calculus

In this paper, we present a novel mathematical framework termed “ϕ-multiplicative calculus”, which serves as a Golden Fibonacci calculus to fundamental concepts in multiplicative calculus. This innovative calculus introduces a parameter ϕ (Golden ratio), offering a nuanced extension of traditional calculus. Our work encompasses the establishment of ϕ-multiplicative calculus and the demonstration of essential theorems concerning derivatives, integrals, and their operation properties within this mathematical framework. The paper contributes to the academic discourse by providing a comprehensive exploration of the proposed ϕ-multiplicative calculus, presenting a robust foundation for further investigations in this specialized mathematical domain.

Focusing on foundational Calculus ideas to understand the derivative concept via problem-solving tasks that involve the use of a Dynamic Geometry System

The aim of this paper is to review recently calculus curriculum reforms and research studies that document what types of understanding students develop in their precalculus courses. We argue that it is important to characterize what difficulties students experience to solve tasks that include the use of foundational calculus concepts and to look for possible ways for students to develop a way of reasoning to work on problems that involve variational phenomena. Thus, we identified tasks in which calculus students exhibit limited understanding of essential concepts to approach and solve those tasks. The purpose is to illustrate and discuss how the systematic use of a Dynamic Geometry System (DGS) could provide a set of affordances for students to develop ways of thinking to grasp calculus foundational ideas and to study the derivative concept. Here, we relied on Thurston’s seminal work that emphasizes the relevance for learners to identify, connect, and coordinate different dimensions and meanings (intuitive, symbolic, algorithmic, geometric, physical, and formal) to construct, understand, and apply the concept of derivative.

Torricelli’s Law in Fractal Space–Time Continuum

A new formulation of Torricelli’s law in a fractal space–time continuum is developed to compute the water discharge in fractal reservoirs. Fractal Torricelli’s law is obtained by applying fractal continuum calculus concepts using local fractional differential operators. The model obtained can be used to describe the behavior of real flows, considering the losses in non-conventional reservoirs, taking into account two additional fractal parameters α and β in the spatial and temporal fractal continuum derivatives, respectively. This model is applied to the flows in reservoirs with structures of three-dimensional deterministic fractals, such as inverse Menger sponge, Sierpinski cube, and Cantor dust. The results of the level water discharge H(t) are presented as a curve series, showing the impact and influence of fluid flow in naturally fractured reservoirs that posses self-similar properties.

Ftdho Zfnet: Block chain based Fractional Tasmanian Devil Harris optimization enabled deep learning using attack detection and mitigation

Block chain technology is regarded for enhancing the characteristics of security because of decentralized design, safe distributed storage, and privacy. However, in recent times the present situation of block chain technology has experienced some crisis that may delay the quick acceptance and utilization in real-time applications. To conquer this subdues, a blockchain based system for attack detection and mitigation with Deep Learning (DL) named Fractional Tasmanian Devil Harris Optimization_Zeiler and Fergus network (FTDHO_ZFNet) is introduced. In this investigation, the entities utilized are owner, block chain, server, trusted authority and user. Here, authentication phase is done by means of Ethereum block chain by Key Exchange module and privacy preserved data sharing and communication is also done. Then, recorded log file creation is executed by the below mentioned stages. At first, a log file is generated with the basis of communication to record the events. After wards, the features are extracted by BoT-IoT database. Then, feature fusion is done by overlap coefficient utilizing Deep Q-Network (DQN). Moreover, data augmentation (DA) is doneusing bootstrapping method. At last, attack detection is observed by ZFNet tuned by FTDHO. Here, FTDHO is unified by Fractional Tasmanian Devil Optimization (FTDO) and Harris Hawks Optimization (HHO). Additionally, FTDO is integrated by Fractional Calculus (FC) concept and Tasmanian devil optimization (TDO). Furthermore, attack mitigation is performed. The performance measures applied for FTDHO_ZFNet are accuracy, and True Negative rate (TNR), observed supreme values with 92.9%, 93.8% and 92.9%.

Bergson and the Metaphysical Implications of Calculus

Abstract Henri Bergson's philosophy is centered on forming a concept of lived time or durée, which he saw as a process of continuous variation and flux. He believed that the study of time should be the foundation of philosophy. By studying time, we find an integration of concrete, infinite, qualitative multiplicity within consciousness that we should use to understand the essence of reality. I show that his insights into the reality of duration come directly from a metaphysical or phenomenological interpretation of integral and differential calculus. Drawing on the recently published lectures from 1902–1905, I show how An Introduction to Metaphysics schematizes the methodology of Matter and Memory and Time and Free Will by means of this mathematical analogy. To Bergson, the concepts of calculus are not mere metaphors; they unleash certain metaphysical implications that problematize the mind's relation to movement, life, consciousness, time, and consequently, reality itself.

On a Novel Fractional Calculus and its Applications to Well-known Problems

The objective of this study is to introduce three novel and comprehensive models utilizing the concept of conformable fractional calculus. These models encompass the fractional population growth model (FPGM), the fractional body cooling model (FBCM), and the fractional heat differential equation (FHDE). Firstly, using new results, a type of new conformable fractional derivative introduced in \cite{5}:\begin{center}$\left(\mathcal{D}^\alpha H\right)(y)=\lim _{z \longrightarrow 0} \frac{H\left(y+z e^{(\alpha-1) y}\right)-H(y)}{z},$\end{center}where $\alpha \in(0,1]$ and $H$ is a function. We obtain exact solutions of the considered models. The results feature excellent agreement of exact solutions. Finally, it is shown that the proposed method provides a more powerful mathematical tool for solving models in mathematical physics.

Empowering Students’ Comprehension in Calculus with eCALculator

Calculus is a fundamental subject in STEM (Science, Technology, Engineering, and Mathematics) study programs. Despite the introduction of basic calculus concepts as early as secondary school, many undergraduate students still encounter difficulties in comprehending and solving differentiation and integration problems. As calculus is fundamental to fields like science and engineering, possessing a solid mathematical and calculus foundation becomes essential for STEM students. Struggling to grasp these fundamental concepts not only impacts performance in calculus but also poses challenges when tackling advanced related topics. Recognizing this challenge, the eCALculator is introduced as a comprehensive and accessible solution. This innovative tool takes the form of a step-by-step e-manual, guiding students in determining a function’s derivatives and integrals using a calculator. The e-manual can be accessed via a flipbook, encompasses instructional videos and exercises. The eCALculator was tested to 40 engineering students enrolled in Calculus courses at UiTM Cawangan Pulau Pinang. Survey results indicate that the eCALculator not only helps improving students’ comprehension and accuracy in calculus problem-solving but also increases motivation and interest in learning, particularly among low-achieving mathematics students who require additional support.

Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function

Abstract There are several problems in physics, such as kinetic energy equation, wave equation, anomalous diffusion process, and viscoelasticity that are described well in the fractional differential equation form. Therefore, the solutions with elementary solution method cannot be solved and described deliberately with detailed physics of the problems, so these problems are solved with the help of special operators such as Mittag–Leffler (M–L) functions equipped with Riemann–Liouville (R–L) fractional operators. Hence, keeping in view the above-mentioned problems in physics in the current study, the generalized properties are derived M–L functions connected with R–L fractional operators that are investigated in the generalized form. These extended special operators will be used for the solutions of generalized kinetic energy equation. The M–L function is a fundamental special function with a wide range of applications in mathematics, physics, engineering, and various scientific disciplines. Ayub et al. gave the definition of newly extended M–L ( p , s , k ) \left(p,s,k) function. Also, they gave its convergence condition and found several results relevant to that. The purpose of this study is to investigate newly extended M–L function and study its elementary properties and integral transforms such as Whittaker transform and fractional Fourier transform. The R–L fractional operator is a fundamental concept in fractional calculus, a branch of mathematics that generalizes differentiation and integration to non-integer orders. In this study, we discuss the relation of M–L ( p , s , k ) \left(p,s,k) -function and R–L fractional operators. In some cases, fractional calculus is used to describe kinetic energy equations, particularly in systems where fractional derivatives are more appropriate than classical integer-order derivatives. The M–L function can appear as a solution or as a part of the solution to these fractional kinetic energy equations. Also, we gave the generalization of kinetic energy equation and its solution in terms of newly extended M–L function.

Role of Mathematics in Data Science - Machine learning

The mathematical concepts help with identifying patterns and algorithmic development. Mathematics is an important part of many modern scientific areas. It is also a field of major significance in data science. Mainly because of the concepts they propose and the tools they allow, statistics and arithmetic have such an essential role in data science. A specific type of mathematical reasoning is needed for analysing data, in addition to the basic concepts of calculus, discrete mathematics, and linear algebra. To understand how to apply such algorithms in data science, one must have a solid understanding of many different statistical and probability theory ideas. Modern data science methods have a strong mathematical basis for almost all of their techniques, including machine learning.

The strategic prize game Le Her

AbstractLe Her is a strategic two-person prize game the classic version of which uses a deck of cards. Despite its simple rules, the analysis of the game is surprisingly complex. I describe a modern variant of Le Her that is amenable to elementary analysis, yet retains and highlights all central aspects of the classical game. This elementary analysis combines naturally central concepts of calculus (partial derivatives, multidimensional optimization, double integrals, fixed and saddle points) with major probabilistic concepts (random variable, event, independence) and basic notions in decision theory (Nash equilibrium). Due to its stimulating context as a strategic prize game Le Her offers an attractive illustration and an appealing application of learning goals and curricular contents which are otherwise often perceived as remote and abstract.

An argumentation experience regarding concepts of calculus with ChatGPT

ABSTRACT ChatGPT, an artificial intelligence-supported chatbot, has become a resource in the field of education for students across various disciplines. The conversation that unfolds based on the user-generated questions and ChatGPT’s responses implies the emergence of an argumentation between the student and ChatGPT. In this study, the argumentation between a university student and ChatGPT, conducted to examine situations where a tangent line cannot be drawn to a function’s graph, was analyzed using Toulmin’s model enriched with Habermas’ construct of rationality. The findings indicate that ChatGPT offers rational support to the warrant in student’s reasoning in the of context epistemic rationality regarding the definitions of continuity, derivative, and tangent concepts. However, ChatGPT provides problematic responses in the context of epistemic rationality regarding the relationship between continuity and derivative, as well as in the context of communicative rationality concerning the geometric interpretation of the derivative and the definition of the tangent line. ChatGPT has not provided reliable and stable support to the student in the reasoning process. It can be concluded that ChatGPT, which is considered to have the potential to assist learning processes, requires expertise in calculus concepts for its use in mathematical reasoning processes. Otherwise, it may be confusing and/or misleading.

Eksplorasi Etnomatematika pada Kerajinan Gerabah dan Penerapannya dalam Pembelajaran Kalkulus Integral

Ethnomathematics is a study that links culture with mathematics. Ethnomathematics can bridge students' learning difficulties in understanding mathematics. Ethnomathematics objects can be found around the environment, for example, pottery. This research aims to find mathematical elements in the typical Kulon Progo pottery culture so that it can be used to facilitate understanding of the concept of integral calculus, precisely the volume of a solid of revolution. This research is qualitative descriptive research with an ethnographic approach. The subjects of this research are pottery craftsmen in Bumirejo village, Lendah, Kulon Progo. Data collection was carried out by observation and interviews. Next, literature studies and data analysis were carried out to conclude. The research results found that pottery crafts in the form of mortars, jars, cauldrons, and flowerpots can be linked to learning the volume of rotating objects through rotating curves that resemble the shape of the pottery crafts. Hopefully, this research can help educators find fun learning alternatives and increase students' understanding of integral material. Keywords: ethnomathematics, pottery crafts, integral

A historical analysis of the independent development of calculus by Newton and Leibniz

This paper undertakes a historical investigation of the separate and independent development of calculus by Isaac Newton and Gottfried Leibniz in the late 17th century. Through analysis of primary sources and historiographical perspectives, it explores the differences in notation, methods, and applications used by each mathematician to formulate foundational concepts of calculus. The research demonstrates that Newton relied more on geometric intuition, developing calculus concepts like fluxions and fluents rooted in kinematic problems. His 1687 Philosophiae Naturalis Principia Mathematica synthesized many calculus innovations. Meanwhile, Leibniz approached calculus from an algebraic mindset, utilizing infinitesimal differentials and comprehensively explaining integral and differential calculus in publications like Nova Methodus pro Maximis et Minimis. Evaluation of letters and documents from the 1670s and 1680s shows no direct collaboration or communication about calculus between Newton and Leibniz. This lack of transmission, coupled with the disparities in their notation and calculus techniques, provides evidence for independent creation. However, Newton and Leibniz shared key insights regarding rates of change, derivatives and integrals, hinting at a broader zeitgeist in early modern mathematics and science. Thus, this dual achievement illustrates how the Scientific Revolution facilitated conceptual convergence despite geographic separation between great thinkers. Investigating this case study offers perspective on the interplay between individual genius and wider social contexts in driving scientific progress. This paper concludes by assessing the legacy of the Newton-Leibniz debate over priority and analyzing work that paved the way for modern unified calculus notation and applications.

Investigating undergraduate students ability in solving calculus problems using microsoft maths solver application

This research work investigated the ability of 100 level undergraduate students in Novena University Ogume, Delta State. Two hundred (200) students from four departments were selected using the stratified random sampling techniques. Frequency counts and percentages were used to obtain a total picture of student ability towards calculus. Also, a student’s t-test was used to check whether there was a significant difference between the means. The observed difference between the mean is significant at the .01 level. Results of the research showed that the use of Microsoft Math Solver aided learners in understanding basic concepts in calculus, as compared with the traditional tutorial approach.

Study of radioactive decay process of uranium atoms via fractals-fractional analysis

This manuscript is devoted to derive a mathematical formulation for radioactivity process of uranium to radium which decays to produce radon. Radon which further decays to produce polonium atoms. The said interesting phenomenon is investigated using the concept of fractals-fractional calculus. The concerned differential operator plays significant role to describe the complex and irregular geometry of real world problems. Also, we deduce the existence theory of such process through mathematical analysis using fractal type Laplace transform. Finally, a computational algorithm is established to simulate the corresponding results for various fractals and fractional orders. In this regard, computational software Matlab-16 is used to simulate our theoretical results. Various graphical presentations are displayed for the computed theoretical results. A comparison with classical traditional order model has also done.