Concepts Of Calculus Research Articles

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.

A note on the post quantum-Sheffer polynomial sequences

Abstract In this article, the post quantum analogue of Sheffer polynomial sequences is introduced using concepts of post quantum calculus. The series representation, recurrence relations, determinant expression and certain other properties of this class are established. Further, the 2D-post quantum-Sheffer polynomials are introduced via generating function and their properties are established. Certain identities and integral representations for the 2D-post quantum-Hermite polynomials, 2D-post quantum-Laguerre polynomials, and 2D-post quantum-Bessel polynomials are also considered.

An Overview of Strategies for Conceptualizing Derivative and their Applications in Daily Life for Secondary-Level Mathematics Students

Derivative is a foundational concept in calculus but can be challenging for secondary students to grasp conceptually. This paper provides an overview of research-backed strategies for developing a strong conceptual understanding of derivative. Visual and intuitive approaches are discussed, such as relating the derivative to real-world con-texts involving speed, growth, and modeling phenomena. The importance of multiple linked representations and building connections between graphical, numeric, verbal and symbolic perspectives is emphasized. Strategies for motivating learning through real-world applications and simulations relating concepts to students' lives are out-lined. Procedural expertise and conceptual mastery are cultivated together. The benefits of a conceptual learning, problem solving abilities, and STEM-related fields are noted. Curricular recommendations focus on conceptual exploration prior to for-mal definitions. Thus, the paper highlights best practices for conceptualizing deriva-tive through visual-intuitive, multi-representational and application-based ap-proaches to promote flexible, adaptive understanding and lay the groundwork for calculus success.

On fractional Bullen-type inequalities with applications

<p>Integral inequalities in mathematical interpretations are a substantial and ongoing body of research. Because fractional calculus techniques are widely used in science, a lot of research has recently been done on them. A key concept in fractional calculus is the Caputo-Fabrizio fractional integral. In this work, we focus on using the Caputo-Fabrizio fractional integral operator to build a multi-parameter fractional integral identity. Using the obtained integral identity, certain generalized estimates of Bullen-type fractional inequalities have been generated. By establishing certain inequalities, this study advances the fields of fractional calculus and convex function research. Both graphical and numerical statistics are provided to show the correctness of our results. We finally provide applications to modified Bessel functions, $ \mathfrak{h} $-divergence measures, and probability density functions.</p>

The Role of Calculus in Powering Technological Advancements

The mathematical concept of differential and integral calculus has a range of applications in almost every aspect of our daily lives and is an essential part of complicated technology, spanning across from the business sector to the scientific world. Initially, the concept was used as a part of the mathematical protocol but as it deals with the rate of change the extent of application is exponential.

UNBOXING UNDERSTANDING: SELF-EXPLANATION PROMPTING AS A KEY TO UNLOCKING DEEP LEARNING IN CALCULUS

Deep learning in calculus often poses challenges for students, requiring a nuanced approach to foster comprehension. This study explores the efficacy of self-explanation prompting as a key strategy for enhancing deep learning in calculus. The research investigates the impact of guided self-explanation prompts on students' understanding and retention of calculus concepts. By employing a carefully designed intervention, we aim to uncover the mechanisms through which self-explanation facilitates meaningful learning in calculus. The results highlight the potential of self-explanation as a powerful tool in the educational toolkit, shedding light on how it can unlock a deeper understanding of calculus concepts and improve overall learning outcomes.

New results on dynamic output state feedback stabilization of some class of time-varying nonlinear Caputo derivative systems

In modern control theory applications, input–output feedback often arises and plays a crucial role in stabilizing any target control systems. However, in many practical applications, the knowledge of full information on the state of systems may not be available in general due to measurements. In order to control such types of systems, this paper uses a universal concept of real order calculus to stabilize by the method of dynamic output feedback control. Is it possible to design a dynamic output feedback stabilization for time-varying real order control systems? This work develops a new dynamic output feedback control method and proposes linear homogeneous time-varying control law to stabilize incommensurate nonlinear time-varying real order systems under the action of Caputo derivative with an addendum of ultimate random initial-time. We utilize the ideas of the comparison method and establish order-dependent asymptotic results associated with uniform Lipschitz continuous functions that provide stabilization by means of the dynamic output state feedback control method. We provide three examples that include a practical system and show the importance of some theoretical results in the demonstration to stabilize the problems by means of proposed control method.

Deep Q Network Based on a Fractional Political–Smart Flower Optimization Algorithm for Real-World Object Recognition in Federated Learning

An imperative application of artificial intelligence (AI) techniques is visual object detection, and the methods of visual object detection available currently need highly equipped datasets preserved in a centralized unit. This usually results in high transmission and large storage overheads. Federated learning (FL) is an eminent machine learning technique to overcome such limitations, and this enables users to train a model together by processing the data in the local devices. In each round, each local device performs processing independently and updates the weights to the global model, which is the server. After that, the weights are aggregated and updated to the local model. In this research, an innovative framework is designed for real-world object recognition in FL using a proposed Deep Q Network (DQN) based on a Fractional Political–Smart Flower Optimization Algorithm (FP-SFOA). In the training model, object detection is performed by employing SegNet, and this classifier is effectively tuned based on the Political–Smart Flower Optimization Algorithm (PSFOA). Moreover, object recognition is performed based on the DQN, and the biases of the classifier are finely optimized based on the FP-SFOA, which is a hybridization of the Fractional Calculus (FC) concept with a Political Optimizer (PO) and a Smart Flower Optimization Algorithm (SFOA). Finally, the aggregation at the global model is accomplished using the Conditional Autoregressive Value at Risk by Regression Quantiles (CAViaRs) model. The designed FP-SFOA obtained a maximum accuracy of 0.950, minimum loss function of 0.104, minimum MSE of 0.122, minimum RMSE of 0.035, minimum FPR of 0.140, maximum average precision of 0.909, and minimum communication cost of 0.078. The proposed model obtained the highest accuracy of 0.950, which is a 14.11%, 6.42%, 7.37%, and 5.68% improvement compared to the existing methods.

Lobachevsky's formula and its applications to several definite integrals

The definite integral is a fundamental concept in calculus that has many applications in various fields such as physics, engineering, and economics. However, integration can be difficult and requires a variety of skills such as substitutions and partial integration. In this paper, Lobachevskys formula is explored, which provides a new way to evaluate definite integrals. It should be noted that Lobachevskys formula can only be applied in specific cases where the integrand is even and -periodic. However, it is demonstrated to be an effective method in these cases. In this paper, the proof of the theorem is given, and a variety of examples are solved by virtue of this method. Hence, this paper may serve as a reference for relevant research in the field of calculus and provide insights into the applications of Lobachevskys formula.

An Exploration of Grade 12 Learners’ Misconceptions on Solving Calculus Problem: A Case of Limits

This case study explored Grade 12 learners’ misconceptions in limits of functions. The study aimed at understanding problems that learners face in learning the concepts of limit by analyzing misconceptions that learners have and identifying the possible sources of these misconceptions to take remedial action. An exploratory research design was adopted, and purposive sampling employed to select 35 Grade 12 learners (21 females and 14 males) who wrote a test on limits. From the 35 learners who wrote the test, five learners were interviewed guided by their written responses. Theoretically, the study was guided by the constructivism theory and used a systematic error conceptual framework to categories the learners` misconceptions according to extrinsic and intrinsic calculus misconceptions. The study sought to answer the questions: (a) What type of misconceptions do Grade 12 learners exhibit in responding to limit questions? (b) What are the possible sources of these misconceptions? The study found that Learners exhibited misconceptions on the limit concept and related symbolism. Learners who had a flawed understanding of algebraic concepts struggled to apply limits correctly. The weak foundation on algebraic skills impacted negatively on their learning of new concepts on limits. The study recommends that, educators should always check and make sure that learners have appropriate prior knowledge before the teaching of new concepts in calculus. It is recommended that that teaching and learning approaches need to be adjusted to give equal weight to both the procedural and the conceptual understanding of mathematical knowledge in learners.

Fuzzy Caputo [formula omitted]-fractional linear equations on the time scale [formula omitted

In this paper, we explore various concepts in q-fractional calculus and extend them to the fuzzy domain. Specifically, we define fuzzy q-exponential and fuzzy q-Mittag-Leffler functions using r-cuts. Additionally, we evaluate the fuzzy Caputo q-derivatives of the fuzzy q-exponential functions and introduce the concept of fuzzy q-Laplace transform. Furthermore, we provide a formula for the fuzzy q-Laplace transform of the fuzzy Caputo q-fractional derivative.We also tackle the problem of solving fuzzy Caputo q-fractional linear differential equations under the generalized Hukuhara differentiability conditions, with a focus on specific forcing functions. The key technique employed for solving these equations is the fuzzy q-Laplace transform. To illustrate the practicality of our theoretical results, we present several numerical examples.This paper contributes to the development of q-fractional calculus in the context of fuzzy mathematics and provides valuable insights into solving fuzzy differential equations using the proposed techniques.

Discovering stochastic partial differential equations from limited data using variational Bayes inference

We propose a novel framework for discovering Stochastic Partial Differential Equations (SPDEs) from data. The proposed approach combines the concepts of stochastic calculus, variational Bayes theory, and sparse learning. We propose the extended Kramers–Moyal expansion to express the drift and diffusion terms of an SPDE in terms of state responses and use Spike-and-Slab priors with sparse learning techniques to efficiently and accurately discover the underlying SPDEs. The proposed approach has been applied to three canonical SPDEs, (a) stochastic heat equation, (b) stochastic Allen–Cahn equation, and (c) stochastic Nagumo equation. Our results demonstrate that the proposed approach can accurately identify the underlying SPDEs with limited data. This is the first attempt at discovering SPDEs from data, and it has significant implications for various scientific applications, such as climate modeling, financial forecasting, and chemical kinetics.

An Overview of Strategies for Conceptualizing Derivative and their Applications in Daily Life for Secondary-Level Mathematics Students

Derivative is a foundational concept in calculus but can be challenging for secondary students to grasp conceptually. This paper provides an overview of research-backed strategies for developing a strong conceptual understanding of derivative. Visual and intuitive approaches are discussed, such as relating the derivative to real-world contexts involving speed, growth, and modeling phenomena. The importance of multiple linked representations and building connections between graphical, numeric, verbal and symbolic perspectives is emphasized. Strategies for motivating learning through real-world applications and simulations relating concepts to students' lives are outlined. Procedural expertise and conceptual mastery are cultivated together. The benefits of a conceptual learning, problem solving abilities, and STEM-related fields are noted. Curricular recommendations focus on conceptual exploration prior to formal definitions. Thus, the paper highlights best practices for conceptualizing derivative through visual-intuitive, multi-representational and application-based approaches to promote flexible, adaptive understanding and lay the groundwork for calculus success.

Novel Techniques for Classifying Exotic Spheres in High Dimensions

Discrete calculus deals with developing the concepts and techniques of differential and integral calculus in a discrete setting, often using difference equations and discrete function spaces. This paper explores how differential-difference algebra can provide an algebraic framework for advancing discrete calculus. Differential-difference algebra studies algebraic structures equipped with both differential and difference operators. These hybrid algebraic systems unify continuous and discrete analogues of derivatives and shifts. This allows the development of general theorems and properties that cover both settings. In particular, we construct differential-difference polynomial rings and fields over discrete function spaces. We define discrete derivatives and shifts algebraically using these operators. We then study integration, summation formulas, fundamental theorems, and discrete analogues of multivariate calculus concepts from an algebraic perspective. A key benefit is being able to state unified theorems in differential-difference algebra that simultaneously yield results for both the continuous and discrete cases. This provides new tools and insights for discrete calculus using modern algebraic techniques. We also discuss applications of representing discrete calculus problems in differential-difference algebras. This allows bringing to bear algebraic methods and software tools for their solution. Specific examples are provided in areas such as numerical analysis of discrete dynamical systems defined through difference equations. The paper aims to demonstrate the capabilities of differential-difference algebra as a unifying framework for further developing the foundations and applications of discrete calculus. Broader connections to algebraic modeling of discrete physical systems are also discussed.

Nonlocal geometric fracture theory: A nonlocal‐local asymptotically compatible framework for continuous–discontinuous deformation of solid materials

SummaryOne of the fundamental but unsolved problems in fracture mechanics of solid materials is how to describe the continuous and discontinuous deformation of continuum in a unified manner. This paper aims to develop a novel nonlocal geometric fracture theory attempting to address this essential issue. Using the concepts of nonlocal calculus, a new nonlocal equation for the deformation of continuum is obtained by minimizing a nonlocal Lagrangian. This is done to ensure that the proposed nonlocal model can be used to describe the continuous and discontinuous deformation of a continuum in a unified geometric manner, as well as to be asymptotically compatible with the classical continuum theory. Inspired by the phase field fracture theory, a nonlocal geometric crack surface functional is proposed to approximate the Griffith fracture energy in continuum. Note, however, that a distinguished feature of the proposed functional is that it enables strong discontinuities appear in the crack phase fields, which is difficult for the phase field fracture theory. After obtaining the nonlocal fracture energy, a nonlocal elastic energy is phenomenologically decomposed into volumetric and deviatoric parts, and then a system of nonlocal geometric fracture equations is derived to describe the damage nucleation, cracks initiation and propagation, which covers the whole physical process of fracture in solid materials. A staggered discontinuous Galerkin method is developed to reach an accurate and stable numerical solution. Numerical examples demonstrate the proposed theory can well capture the transition from the continuous to discontinuous deformation of the continuum and accurately describe the topological evolution of cracks in a unified framework.

Linear and nonlinear fractional order diffusion equations with initial and bowndary conditions

The usual differentiation and integration are expanded to any non-integer order in fractional calculus. The topic predates the development of differential calculus by Leibnitz and Newton and is therefore as old as classical theory. The concept of fractional calculus has generated interest not only among mathematicians but also among physicists and engineers. This concept is calculated in the same way as the classical methods of differential and integral calculus, and also dates back to the time when Leibniz and Newton invented differential calculus. The idea of calculating fractional order is of interest not only among individual mathematicians, but also among physicists and engineers. The method of upper and lower solutions has been extended to FDEs using these minimum-maximum principles, and various existence results have been established. In this paper, a one-dimensional subdiffusion equation is investigated using the principle of the maximum of the Riemann-Liouville derivative of fractional order. It is proved that for fractional order diffusion equations in linear and nonlinear time there is a unique classical solution to the initial boundary value problem and that the solution continuously depends on the initial and boundary conditions.

The Properties of Meromorphic Multivalent q-Starlike Functions in the Janowski Domain

Many researchers have defined the q-analogous of differential and integral operators for analytic functions using the concept of quantum calculus in the geometric function theory. In this study, we conduct a comprehensive investigation to identify the uses of the Sălăgean q-differential operator for meromorphic multivalent functions. Many features of functions that belong to geometrically defined classes have been extensively studied using differential operators based on q-calculus operator theory. In this research, we extended the idea of the q-analogous of the Sălăgean differential operator for meromorphic multivalent functions using the fundamental ideas of q-calculus. With the help of this operator, we extend the family of Janowski functions by adding two new subclasses of meromorphic q-starlike and meromorphic multivalent q-starlike functions. We discover significant findings for these new classes, including the radius of starlikeness, partial sums, distortion theorems, and coefficient estimates.

Error estimates of Hermite‐Hadamard type inequalities with respect to a monotonically increasing function

Fractional calculus is used to examine and enhance the concept of calculus in diverse fields of science. In this paper, we establish Hermite‐Hadamard inequalities for composite ‐convex function. The generalized identities are established for Riemann‐type fractional integrals. The explored identities are used to examine error estimates of Hermite‐Hadamard inequalities for ‐convex function concerning a strictly monotone function. Some results that exist in literature are obtained as special cases to our general results. The conclusions of this article may be useful in determining the uniqueness of partial differential equations and fractional boundary value problems.

Some basic concepts of algebraic calculus in tensors

Some basic concepts of algebraic calculus in tensors

A theory of calculus for A-correlated fuzzy processes

This article presents a new concept of calculus for fuzzy number-valued functions based on Zadeh's extension principle. More precisely, we propose a calculus theory for A-correlated processes which correspond to functions of the form fˆ(t,A), where fˆ denotes the Zadeh extension of a function of f(t,⋅) at a fixed fuzzy number A. The notions of derivatives and integrals for these functions are defined in natural way by means of the Zadeh extension of the derivatives and integral operators of the function f. We show that these definitions coincide with the notions of derivative and integral based on arithmetic operations with interactive operands. In addition, we apply the proposed calculus to study certain fuzzy differential equations. We argued that the proposed fuzzy calculus theory is adequate to deal with certain fuzzy processes that represent epistemic states of single-valued variables.