Concepts Of Calculus Research Articles

RAMP summer bridge: Room acoustics: Connecting research, academics and mentoring

The Research, Academics and Mentoring Pathways (RAMP) to Success summer bridge program at the University of Massachusetts Lowell prepares new engineering students from all majors for an early transition from high school to college. Over the last five years, RAMP has engaged 107 students, the majority being women and underrepresented minorities, in a six-week research, academics and mentoring experience. During RAMP, students are enrolled in Calculus and Introduction to Engineering courses, earning six credits towards their degree pathway. Groups of 5–6 students take on research projects with engineering faculty while also meeting weekly with professionals from industry. The structuring of a room acoustics research project is presented. A project-based learning model helps build skills in collaborative problem solving while also supporting each student to find authentic interest in the research. The project integrates experiments on measuring room impulse responses and spatial variation of source pressure amplitude in rooms of different sizes with a model-driven analysis of the data collected. Concepts of reverberation time, backward integration and room constants reinforce calculus concepts being studied. Coding experiments on rendering speech and music through measured impulse responses and filters created interest across a range of acoustics topics from concert halls to virtual audio.

Fractional Derivative Gradient-Based Optimizers for Neural Networks and Human Activity Recognition

In this paper, fractional calculus principles are considered to implement fractional derivative gradient optimizers for the Tensorflow backend. The performance of these fractional derivative optimizers is compared with that of other well-known ones. Our experiments consider some human activity recognition (HAR) datasets, and the results show that there is a subtle difference between the performance of the proposed method and other existing ones. The main conclusion is that fractional derivative gradient descent optimizers could help to improve the performance of training and validation tasks and opens the possibility to include more fractional calculus concepts to neural networks applied to HAR.

Students' Mathematical Understanding Ability Through the M-Apos Approach on Derivative Materials

This study aimed to improve students' mathematical understanding skills. The understanding ability is students' ability to understand the concept of the material, not just memorize the material. When students understand the concepts of differential calculus, students can quickly solve problems in differential calculus. The students' understanding gained in the differential calculus material will be the basis for understanding the integral and advanced calculus material. However, the current conditions where the Coronavirus that has hit the world, including Indonesia, have resulted in students having to study from home with a learning process that is carried out online. Students cannot access books in the library and limited credit to access teaching materials from the internet because the learning materials owned by students are limited. It makes it difficult for students to learn, so students' mathematical understanding abilities are low. The M-APOS learning approach was used in this study to answer this problem. The study results found that learning with this approach has succeeded in increasing students' mathematical understanding abilities on derivative material. This research is an experimental study with a pretest-posttest control group research design.

Constructing Calculus Concepts through Worksheet Based Problem-Based Learning Assisted by GeoGebra Software

This study aims to produce a valid, practical, and effective calculus learning worksheet using a problem-based learning method assisted by GeoGebra software in improving problem-solving skills. This worksheet serves as a guide to students in constructing the calculus concepts through the provided instructions according to the Problem-Based Learning syntax. Furthermore, the construction process was carried out using GeoGebra software. Students are given scaffolding to construct concepts. This research is development research. This study employs a Plomp model, which consists of three stages, namely the preliminary study, development or prototyping, and assessment phases. The subjects are 32 students in class XI.2 at the State Senior High School of 14 Padang. The results showed that the worksheet produced was valid, practical, and effective in improving students' problem-solving abilities. This implies that it would assist students in constructing calculus concepts. Furthermore, it showed that GeoGebra is capable of visualizing abstract calculus concepts, which enables students to understand higher mathematical thinking on the main concept of calculus. In conclusion, the use of GeoGebra in mathematics learning is able to help students construct mathematical concepts and improve their problem-solving abilities. Doi: 10.28991/HIJ-2022-03-03-04 Full Text: PDF

The Effect of GeoGebra on Students’ Abilities to Study Calculus

The study aimed to investigate the effect of GeoGebra on students’ ability to learn calculus. Calculus can be a challenging subject to teach. Moreover, students have problems, especially in connecting the concepts of calculus with the real world. Concerning gender, the study looked at the impact of utilizing GeoGebra Mathematical software on students’ calculus proficiency and ability to utilize GeoGebra software to learn calculus. The study developed a cycle model that posits nine steps to promote the teaching and learning process with the help of GeoGebra to improve the learning process. A quantitative research methodology was employed to achieve the goal of the study. A quasi-experiment with a pretest and posttest design was used in the quantitative part of the study. Students learning calculus were the subjects of the study, which took place at a university in Ethiopia. The quantitative data were analyzed using SPSS version 27. The results show that there is a significant difference between pretest and posttest in students’ performance (ability) when using GeoGebra mathematics software, indicating that students performed better after the intervention (F (1,64) = 10.495,p=0.002 < 0.05). The treatment benefited both high- and low-ability pupils in their brain-based learning, although students in the experimental group, both female and male, outperformed those in the control group. While the GeoGebra-oriented learning approach to calculus has the potential to improve competency, it is still necessary that it be structured (cycle model) to address a specific deficiency.

Power and delay optimization based uplink resource allocation for wireless networks with device-to-device communications

In this paper, the resource allocation problem for all devices is investigated in a multi-sharing uplink scenario of mmWave cyclic prefix-orthogonal frequency division multiplexing (CP-OFDM) based wireless networks with device-to-device (D2D) communications. More specifically, a power and delay optimization based uplink resource allocation (PDO-URA) algorithm is proposed divided into two steps. At first, network resources are allocated to cellular user equipments (CUEs) in terms of power and transmission rate through a proposed approach that intends to maximize throughput. Next, idle resources are allocated by considering delay minimization. To this end, we propose to apply an algorithm based on delay optimization where idle resources are shared with D2D user equipments (DUEs) in the network considering the formed conflicts graphs and the delay information estimated through Network Calculus concepts such as envelope process and service curve. Computational simulations are carried out considering a communication scenario with 5G characteristics such as millimeter waves at frequencies above 6 GHz, comparing also the performance of other algorithms from the literature in terms of quality of service (QoS) parameters such as throughput and delay. The results confirm that the proposed algorithm presents superior performance in terms of throughput and delay for the mmWave CP-OFDM wireless networks with D2D communications than the other considered algorithms.

Languaging and conceptual understanding in engineering mathematics

The ability to apply mathematical concepts and procedures in relevant contexts in engineering subjects sets the fundamental basis for the mathematics competencies in engineering education. Among the plethora of digital techniques and tools arises a question: Do the students gain a deep and conceptual enough understanding of mathematics that they are able to apply mathematical concepts in engineering studies? This paper introduces the use of languaging exercises in the engineering mathematics course ‘Differential Calculus’ during the spring semester 2020, at Tampere University of Applied Sciences, TAMK. In this study, the students’ conceptual understanding and learning of differential calculus is researched. In the learning process, the languaging method is used to deepen the conceptual understanding of the concepts of differential calculus. Pre-test/post-test setup was used to see the possible gain in conceptual understanding. During the course, students did online assignments, which included languaging exercises. Students described the concepts of differential calculus using natural language, pictures, or a combination of them. The students were also asked to fill in a self-evaluation form to collect their perception of their own knowledge of mathematical skills. Mid-term and final exams summarized the acquired knowledge. The study aimed to enhance the learning outcomes and to gain a deeper understanding of mathematical concepts by exploiting the languaging method.

Multivalent Functions and Differential Operator Extended by the Quantum Calculus

We used the concept of quantum calculus (Jackson’s calculus) in a recent note to develop an extended class of multivalent functions on the open unit disk. Convexity and star-likeness properties are obtained by establishing conditions for this class. The most common inequalities of the proposed functions are geometrically investigated. Our approach was influenced by the theory of differential subordination. As a result, we called attention to a few well-known corollaries of our main conclusions.

Fractional memetic computing paradigm for reactive power management involving wind-load chaos and uncertainties

Optimal reactive power dispatch (ORPD) with integration of renewable energy resources has a growing interest in research community due to its utmost requirements during the operation, planning and design of the modern electrical power networks. The objective of ORPD is to improve the performance of power network by means of reducing the losses in transmission line, improving the voltage profile, and decreasing the overall cost of operation through optimal tuning of the operational variables such as tap position of transformers, generator output voltages and capacitor banks. However, the nonlinear, non-stationary and complex nature of power network, presence of load uncertainties, and dynamic behavior of wind generation introduces a complex optimization task which cannot be readily solved in an efficient manner. In this research work, a new fractional memetic computing paradigm, i.e., the fractional particle swarm optimization gravitational search algorithm with entropy evolution (FPSOGSA-EE), is designed to solve the ORPD problems in power system adopting wind power plants (WPPs) and load uncertainties. The proposed optimization framework FPSOGSA-EE integrates the concept of fractional calculus and Shannon entropy to strengthen the optimization characteristics of canonical algorithm. The exhaustive experimentation endorse the efficacy of FPSOGSA-EE by providing minimum gauge of fitness evaluation function, namely, the line loss and voltage deviation index minimization, in IEEE 30 and 57 bus networks. The stability, consistency and reliability of proposed FPSOGSA-EE is ascertained through statistical interpretations by means of boxplots, probability measures for cumulative distribution function, and histogram illustrations.

Concept and application of interval-valued fractional conformable calculus

This paper introduces a new concept of Caputo type interval-valued fractional conformable calculus. Based on this, some theorems and properties related to fractional conformable calculus are presented. As an application, this paper attempts to investigate an initial value problem of functional integro-differential equations with Caputo type interval-valued fractional conformable derivatives. We not only prove the existence of at least one solution, the extremal solutions and the unique solution of the above problem, but also establish some monotone iterative sequences converging to the extremal solutions. More meaningful and important, some comparison principles and generalized Gronwall inequality related to the proposed problem are also proved. Two examples are presented to illustrate the main results.We firmly believe that the concept of interval-valued fractional conformable calculus, new comparison principles and generalized Gronwall inequality introduced here can be conveniently applied to qualitative analysis of a variety of new interval-valued fractional conformable biological models, economic models, etc., which will better describe and explain various complex phenomena derived from the real world.

RETRACTED: On novel application of piece-wise fractional operators for a predator–prey model

Humans have observed complex behaviors presented by nature. They have observed behaviors resembling power-law, to model these problems; they have used fractional differential operators with a power-law kernel. They have observed behaviors resembling a declining process, to model these problems; they used fractional differential operators based on exponential decay kernel. They have observed crossover behaviors from fading process to power-law, to model these problems; they used fractional derivatives based on the generalized Mittag-Leffler function. However, they have also observed a passage from randomness to power-law or vice-versa, or even from deterministic to randomness, these behaviors could not be depicted using above mentioned operators. To overcome these problems, the concepts of piece-wise differential and integral calculus have been introduced. In this paper, we adopt such a concept to model the dynamical interaction of predator–prey, several scenarios are considered and simulations are provided for each case. According to the results, it seems that the idea of using piece-wise operators seems provides a better understanding of the behavior of dynamic systems.

On the Calculus of Functionally Graded Plates

Abstract This paper presents some of our results regarding calculus of the functionally graded plates (FGPs). Such plates are made of Functionally Graded Materials (FGMs), which represent a new material class belonging to the composite materials. Our paper presents some material laws in a comparative way. But the main purpose of this paper is to provide calculus concepts and calculus methodologies, based on the means available in scientific research of mechanical engineering field, for the calculation of plates made of FGMs. Thus, the authors use the concepts of multilayer plate and equivalent plate, using both analytical and numerical calculus. The numerical method used are the Finite Element Method (FEM).The calculation aims both to determine the displacements and the stresses of the plates under statical loads, as well as to determine the free vibration frequencies. The research methodology is based on combining analytical calculation, where and when possible, with numerical simulation. The validation of our results is done by comparison with the analytical solution and the comparative analysis of the methods.

About the Teaching and Learning of Differentiability for Piecewise Functions in Science Degrees’ First-Year Calculus Courses

In this work, we present an issue we have observed over the last several years in the first year of Spanish science degrees’ (such as mathematics, physics or engineering) calculus courses. It is related to the study of differentiability of piecewise one real variable real functions at a point. We have analyzed how students study the differentiability of piecewise functions at a point, explaining the students’ work and reasoning and commenting about the common misunderstandings we have found. Then, we have researched about how to help the students in their learning process related to this calculus concept, and we have also used several activities or ways to work: student meeting groups to talk about their worries and misunderstandings, mathematical definitions, explanations and theory and also working out specific examples.

Analysis of the meaning of continuity and differentiation: A study on graph problems

Continuity and differentiability are concepts that students must master to effectively learn and solve problems in a variety of contexts. Thus, this study sought to elicit students' interpretations of the relationship between continuity and differentiability, particularly in graphic problems. Through a qualitative approach, this study involved 195 third-year undergraduate students from various Indonesian universities. Ten of them agreed to an in-depth interview for exploration and clarification. Thematic analysis was conducted to deduce patterns from the responses of participants based on the findings. This study discovered three types of meanings that students construct when they solve problems: 1) physical meaning; 2) analytical meaning, and 3) covariational meaning. The three findings could serve as a conceptual framework for future learning processes that emphasize continuity and differentiability. Additionally, our research revealed that Indonesian undergraduate students are unfamiliar with the graphical problems associated with the two concepts. Thus, future research will focus on developing learning strategies that incorporate a variety of representations to improve students' conceptual understanding of calculus concepts.

On the Efficiency of Staggered C-Grid Discretization for the Inviscid Shallow Water Equations from the Perspective of Nonstandard Calculus

This paper provides a rationale for the commonly observed numerical efficiency of staggered C-grid discretizations for solving the inviscid shallow water equations. In particular, using the key concepts of nonstandard calculus, we aim to show that the grid staggering of the primitive variables (surface elevation and normal velocity components) is capable of dealing with flow discontinuities. After a brief introduction of hyperreals through the notion of infinitesimal increments, a nonstandard rendition of the governing equations is derived that essentially turns into a finite procedure and permits a convenient way of modeling the hydraulic jumps in open channel flow. A central result of this paper is that the discrete formulations thus obtained are distinguished by the topological structures of the solution fields and subsequently provide a natural framework for the staggered discretization of the governing equations. Another key of the present study is to demonstrate that the discretization naturally regularizes the solution of the inviscid flow passing through the hydraulic jump without the need of non-physical dissipation. The underlying justification is provided by analytically studying the distributions of the flow variables across an infinitesimal thin hydraulic jump along with the use of hyperreal Heaviside step functions. This main finding is shown to be useful to comprehend the importance of the application of staggered finite difference schemes to accurately predict rapidly varying free-surface flows. A numerical experiment is provided to confirm this result.

The Hermite–Hadamard–Mercer Type Inequalities via Generalized Proportional Fractional Integral Concerning Another Function

In order to be able to study cosmic phenomena more accurately and broadly, it was necessary to expand the concept of calculus. In this study, we aim to introduce a new fractional Hermite–Hadamard–Mercer’s inequality and its fractional integral type inequalities. To facilitate that, we use the proportional fractional integral operators of integrable functions with respect to another continuous and strictly increasing function. Moreover, we establish some new fractional weighted φ -proportional fractional integral Hermite–Hadamard–Mercer type inequalities. Furthermore, in this article, we are keen to present some special cases related to our current study compared to the previous work of the inequality under study.

An explorative digital tool as a pathway to meaning: the case of the inflection point

AbstractThe learning of calculus concepts is considered challenging for students. This claim is actual for calculus in general and for specific concepts in particular. In this paper, we focus on the concept of the inflection point. We argue that one of the roots of this problem is the lack of a useful and productive meaning of the concept—the understanding of the inflection point as the point where the behaviour of a curve (graph of function) changes in relation to the tangent line. With the purpose of helping students to construct this meaning we developed a specific digital tool: a teaching unit based on the interactive diagrams framework. Does this tool help students to achieve this meaning (i.e., to construct and consolidate new knowledge)? To answer this question, we conducted an empirical feasibility experiment (in the form of a case study with two first-year students from the Industrial Engineering College) and analysed the gathered data using the framework of abstraction in context as the theoretical and methodological basis. Our findings show that the designed tool (the interactive digital teaching unit) has potential for helping students to make the above-mentioned meaning for this mathematical concept and can serve as a useful basis to continue the investigation of designing tools that support the meaning-making of advanced mathematical concepts.

Crank Nicholson scheme to examine the fractional-order unsteady nanofluid flow of free convection of viscous fluids.

Fractional fluid models are usually difficult to solve analytically due to complicated mathematical calculations. This difficulty in considering fractional model further increases when one considers nth order chemical reaction. Therefore, in this work an incompressible nanofluid flow as well as the benefits of free convection across an isothermal vertical sheet is examined numerically. An nth order chemical reaction is considered in the chemical species model. The specified velocity (wall's) is time-based, and its motion is translational into mathematical form. The fractional differential equations are used to express the governing flow equations (FDEs). The non-dimensional controlling system is given appropriate transformations. A Crank Nicholson method is used to find solutions for temperature, solute concentration, and velocity. Variation in concentration, velocity, and temperature profiles is produced as a result of changes in discussed parameters for both Ag-based and Cu-based nanofluid values. Water is taken as base fluid. The fractional-order time evaluation has opened the new gateways to study the problem into a new direction and it also increased the choices due to the extended version. It records the hidden figures of the problem between the defined domain of the time evaluation. The suggested technique has good accuracy, dependability, effectiveness and it also cover the better physics of the problem specially with concepts of fractional calculus.

Girsanov theorem for multifractional Brownian processes

ABSTRACT In this article, we will present a new perspective on the variable-order fractional calculus, which allows for differentiation and integration to a variable order. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the past 20 years. We develop a multifractional differential operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional differential operator, we prove a Girsanov's theorem for multifractional Brownian motions of Riemann–Liouville type. As an application, we show how Girsanov's theorem can then be applied to prove the existence of a unique strong solution to stochastic differential equations where the drift coefficient is merely of linear growth, and the driving noise is given by a non-stationary multifractional Brownian motion with a Hurst parameter as a function of time. The Hurst functions we study will take values in a bounded subset of . The application of multifractional calculus to SDEs is based on a generalization of the works of D. Nualart and Y. Ouknine [Regularization of differential equations by fractional noise, Stoch Process Appl. 102(1) (2002), pp. 103–116].

Inequalities for fractional Riemann\u2013Liouville integrals of certain class of convex functions

Fractional calculus operators play a very important role in generalizing concepts of calculus used in diverse fields of science. In this paper, we use Riemann–Liouville fractional integrals to establish generalized identities, which are further applied to obtain midpoint and trapezoidal inequalities for convex function with respect to a strictly monotone function. These inequalities reproduce midpoint and trapezoidal inequalities for convex, harmonic convex, p-convex, and geometrically convex functions. Also, some new inequalities can be generated via specific strictly monotone functions.