Discrete Calculus Research Articles

Unified Generalizations of Hardy-Type Inequalities Through the Nabla Framework on Time Scales

This research investigates innovative extensions of Hardy-type inequalities through the use of nabla Hölder’s and nabla Jensen’s inequalities, combined with the nabla chain rule and the characteristics of convex and submultiplicative functions. We extend these inequalities within a cohesive framework that integrates elements of both continuous and discrete calculus. Furthermore, our study revisits specific integral inequalities from the existing literature, showcasing the wide-ranging relevance of our results.

Fuzzy discrete fractional calculus and fuzzy fractional discrete equations

This paper aims to highlight certain limitations in the study of fuzzy fractional discrete equations (FFDEs) based on the generalized Hukuhara difference (gH-difference) in the previous papers. In general, the equivalence between FFDEs and the associated fuzzy discrete fractional sum equations (FDFSEs) is not achieved, requiring the introduction of an appropriate hypothesis to establish this equivalence. Furthermore, this paper introduces the fundamental theory of fuzzy fractional discrete calculus through granular arithmetic operations between fuzzy intervals to address restrictions in the formerly mentioned approaches involving the generalized Hukuhara difference. These operations are constructed based on the concept of the horizontal membership function (HMF) utilized in multidimensional fuzzy arithmetic (MFA). Additionally, the paper proposes the application of fractional discrete calculus to two types of time-discretization diffusion equations with non-zero right-hand sides. Finally, several numerical examples are provided to validate the main results.

Mini-Workshop: Multivariate Orthogonal Polynomials: New synergies with Numerical Analysis

Multivariate polynomials and, in particular, multivariate orthogonal polynomials (MOPs) are research areas within the fields of special functions, Lie groups, quantum groups, computer algebra to name only some of them. However, there are many important areas in the field of numerical analysis where multivariate polynomials (of high order) play a crucial role: approximation by spectral methods and finite elements, discrete calculus, polynomial trace liftings, exact sequence properties, sparsity, efficient and stable recursions, analysis of the geometry of the zeros. The miniworkshop brought together experts from the fields of MOPs and numerical analysis of partial differential equations.

Analysis on the behavior of the logistic fixed effort harvesting model through the difference equation under uncertainty

ABSTRACT In this article, a logistic fixed effort harvesting model is architected in an imprecise, discrete dynamical frame of mathematical logic. The fuzzy difference equation explores the philosophy behind the computational structure that represents the underflowing discrete behaviour and uncertainty associated with the modelling through discrete calculus and fuzzy decision-making mechanisms. The nonlinear fuzzy difference equations with different initial conditions and coefficients as fuzzy numbers are manifested to recognize the model. Interestingly, the fuzzy difference equations identified in this article can be imparted into a system of crisp difference equations by the characterization theorem. The equilibrium points are traced, and their corresponding stability criteria are analyzed considering different fuzzy cases. The merits and applicability of the proposed theory have been validated through numerical simulation and graphical visualization.

Generalized Polynomials and Their Unification and Extension to Discrete Calculus

In this paper, we introduce a comprehensive and expanded framework for generalized calculus and generalized polynomials in discrete calculus. Our focus is on (q;h)-time scales. Our proposed approach encompasses both difference and quantum problems, making it highly adoptable. Our framework employs forward and backward jump operators to create a unique approach. We use a weighted jump operator α that combines both jump operators in a convex manner. This allows us to generate a time scale α, which provides a new approach to discrete calculus. This beneficial approach enables us to define a general symmetric derivative on time scale α, which produces various types of discrete derivatives and forms a basis for new discrete calculus. Moreover, we create some polynomials on α-time scales using the α-operator. These polynomials have similar properties to regular polynomials and expand upon the existing research on discrete polynomials. Additionally, we establish the α-version of the Taylor formula. Finally, we discuss related binomial coefficients and their properties in discrete cases. We demonstrate how the symmetrical nature of the derivative definition allows for the incorporation of various concepts and the introduction of fresh ideas to discrete calculus.

Exploring Generalized Hardy-Type Inequalities via Nabla Calculus on Time Scales

In this research, we aim to explore generalizations of Hardy-type inequalities using nabla Hölder’s inequality, nabla Jensen’s inequality, chain rule on nabla calculus and leveraging the properties of convex and submultiplicative functions. Nabla calculus on time scales provides a unified framework that unifies continuous and discrete calculus, making it a powerful tool for studying various mathematical problems on time scales. By utilizing this approach, we seek to extend Hardy-type inequalities beyond their classical continuous or discrete settings to a more general time scale domain. As specific instances of our discoveries, we have the integral inequalities previously established in the existing literature.

Finite Time Stability Results for Neural Networks Described by Variable-Order Fractional Difference Equations

Variable-order fractional discrete calculus is a new and unexplored part of calculus that provides extraordinary capabilities for simulating multidisciplinary processes. Recognizing this incredible potential, the scientific community has been researching variable-order fractional discrete calculus applications to the modeling of engineering and physical systems. This research makes a contribution to the topic by describing and establishing the first generalized discrete fractional variable order Gronwall inequality that we employ to examine the finite time stability of nonlinear Nabla fractional variable-order discrete neural networks. This is followed by a specific version of a generalized variable-order fractional discrete Gronwall inequality described using discrete Mittag–Leffler functions. A specific version of a generalized variable-order fractional discrete Gronwall inequality represented using discrete Mittag–Leffler functions is shown. As an application, utilizing the contracting mapping principle and inequality approaches, sufficient conditions are developed to assure the existence, uniqueness, and finite-time stability of the equilibrium point of the suggested neural networks. Numerical examples, as well as simulations, are provided to show how the key findings can be applied.

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.

Solving Partial Differential Equations for Physical and Chemical Problems Using Physics-Informed Neural Networks

Numerous physical and chemical problems at a high school level can be described by ordinary differential equations (ODEs) and partial differential equations (PDEs). However, the underlying equations troubled high school students because they often lack advanced mathematical skills, such as discrete calculus. Our goal is not to elaborate on those skills, but to offer a shortcut to the solution. In this paper, we demonstrated the use of Physics-Informed Neural Networks (PINNs), a neural network which solves the PDEs by incorporating the PDEs into the loss functions. The heat transfer equation and second order chemical kinetics are the two chosen model problems for high school seniors. Using PINNs, we were able to solve these two problems without recurring to university math. Hence, we strongly recommend peers to employ this method for physical or chemical problems for high school students and beyond.

Applications of Differential-Difference Algebra in Discrete Calculus

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

Deep Learning Discrete Calculus (DLDC): a family of discrete numerical methods by universal approximation for STEM education to frontier research

The article proposes formulating and codifying a set of applied numerical methods, coined as Deep Learning Discrete Calculus (DLDC), that uses the knowledge from discrete numerical methods to interpret the deep learning algorithms through the lens of applied mathematics. The DLDC methods aim to leverage the flexibility and ever-increasing resources of deep learning and rich literature on numerical analysis to formulate a general class of numerical methods that can directly use data with uncertainty to predict the behavior of an unknown system as well as elevate the speed and accuracy of numerical solution of the governing equations for known systems. The article is structured into two major sections. In the first section, the building blocks of the DLDC methods are presented and deep learning structures analogous to traditional numerical methods such as finite difference and finite element methods are constructed with a view to incorporate these techniques in Science, Technology, Engineering, Mathematics syllabus for K-12 students. The second section builds upon the building blocks of the previous discussion and proposes new solution schemes for differential and integral equations pertinent to multiscale mechanics. Each section is accompanied by a mathematical formulation of the numerical methods, analogous DLDC formulation, and suitable examples.

A review of definitions of fractional differences and sums

<p style='text-indent:20px;'>Given the increasing importance of discrete fractional calculus in mathematics, science engineering and so on, many different concepts of fractional difference and sum operators have been defined. In this paper, we mainly reviews some definitions of fractional differences and sum operators that emerged in the fields of discrete calculus. Moreover, some properties of those operators are also analyzed and compared with each other, including commutation rules, linearity, Leibniz rules, etc.</p>

Brachistochrone of off-centered cylinders

We consider the problem of finding paths of shortest transit time between two points (popularly known as brachistochrone) for cylinders with off-centered center of mass, rolling down without slip, subject solely to the force of gravity. This problem is set up using principles of classical rigid body dynamics and the desired path function is solved for numerically using the method of discrete calculus of variations. We discover a distinct array of brachistochrone trajectories for off-centered cylinders, demonstrate a critical dependence of such paths on the initial location and orientation of cylinders’ centers of mass and bring new insights into the family of brachistochrone problems and solutions.

Modulated Harmonic Wave in Series‐Connected Discrete Josephson Transmission Line: The Discrete Calculus Approach

The modulated harmonic wave in the discrete series‐connected Josephson transmission line (JTL) is considered. The approach to the modulation problems for discrete wave equations based on discrete calculus is formulated. The approach is checked by applying it to the Fermi–Pasta–Ulam–Tsingou (FPUT) ‐type problem. Applying the approach to the discrete JTL, the equation describing the modulation amplitude is obtained, which turns out to be the defocusing nonlinear Schrödinger (NLS) equation. The profile of the single soliton solution of the NLS is compared with that of the soliton obtained in the previous publication.

ANDERSON’S ∇- INTEGRAL INEQUALITY

Basically, time scale calculus is the theory of unification of traditional calculus with that calculus of difference i.e. discrete calculus. Time Scale Calculus is a field of discussion in the area of traditional analysis of mathematics. It focuses on the dynamic system which has a lot of applications in various fields of life. Calculus of time scales is a valuable field due to numerous applications in covid-19 disease cases. Notably, Time scale calculus has a long relation with mathematical inequalities that can be discussed with fractional calculus. The Anderson Integral Inequality, which provides a lower constraint for the integration of convex mapping in the form of the averages of each constituent, is described in this research paper on ∇- time-scale calculus. On ∇-time scale we formulated Anderson’s integral inequality as given below: if φ_j (j=1,….,α) accomplish some appropriate cases.

Some New Anderson Type h and q Integral Inequalities in Quantum Calculus

The calculus in the absence of limits is known as quantum calculus. With a difference operator, it substitutes the classical derivative, which permits dealing with sets of functions that are non-differentiations. The theory of integral inequality in quantum calculus is a field of mathematics that has been gaining considerable attention recently. Despite the fact of its application in discrete calculus, it can be applied in fractional calculus as well. In this paper, some new Anderson type q-integral and h-integral inequalities are given using a Feng Qi integral inequality in quantum calculus. These findings are highly beneficial for basic frontier theories, and the techniques offered by technology are extremely useful for those who can stimulate research interest in exploring mathematical applications. Due to the interesting properties in the field of mathematics, integral inequalities have a tied correlation with symmetric convex and convex functions. There exist strong correlations and expansive properties between the different fields of convexity and symmetric function, including probability theory, convex functions, and the geometry of convex functions on convex sets. The main advantage of these essential inequalities is that they can be converted into time-scale calculus. This kind of inevitable inequality can be very helpful in various fields where coordination plays an important role.

Integral transform approach to mimetic discrete calculus

We introduce an integral transform that maps differential equations and special functions of standard continuous calculus onto finite difference equations and deformed special functions of mimetic discrete calculus, or h-calculus. We show that our procedure leads to insightful reformulations of several problems in mathematics and physics where discrete equations play a significant role, such as in solving finite difference equations, in applying discrete versions of integral transforms, such as the h-Laplace transform, in solving master equations of stochastic physics, in developing a discrete version of H theory of multiscale complex hierarchical phenomena and in finding lattice Green’s functions for describing quantum charge transport through phase coherent systems. We believe that our integral transform technique, or mimetic map, will help systematize the connections through analogy between discrete calculus and standard continuous calculus.

Asynchronous finite differences in most probable distribution with finite numbers of particles

For a discrete function fx on a discrete set, the finite difference can be either forward and backward. If fx is a sum of two such functions fx=f1x+f2x, the first order difference of Δfx can be grouped into four possible combinations, in which two are the usual synchronous ones Δff1x+Δff2x and Δbf1x+Δbf2x, and other two are asynchronous ones Δff1x+Δbf2x and Δbf1x+Δff2x, where Δf and Δb denote the forward and backward difference respectively. Thus, the first order variation equation δfx=0 for this function fx gives at most four different solutions which contain both true and false one. A formalism of the discrete calculus of variations is developed to single out the true one by means of comparison of the second order variations, in which the largest value in magnitude indicates the true solution, yielding the exact form of the distributions for Boltzmann, Bose and Fermi system without requiring the numbers of particle to be infinitely large. When there is only one particle in the system, all distributions reduce to be the Boltzmann one.

Hardy–Leindler, Yang and Hwang Inequalities for Functions of Several Variables via Time Scale Calculus

In this paper, Hardy–Leindler, Hardy–Yang and Hwang type inequalities are extended on time scales calculus. These extensions are depending upon use of symmetric multiple delta integrals. The target is achieved by utilizing some inequalities in literature along with mathematical induction principle and Fubini’s theorem on time scales. The obtained inequalities are discussed in discrete, continuous and quantum calculus in search of applications. Particular cases of proved results include Hardy, Copson, Hardy–Littlewood, Levinson and Bennett-type inequalities for symmetric sums.

Formula omitted]-Choquet integral on time scales with applications

The fundamental purpose of this work is to analyze Δ-Choquet integrals on time scales which is a special case of Choquet integral on abstract fuzzy (non-additive) measure space. We first present a Δ-Choquet integral with respect to non-additive Δ-measure or more precisely a distorted Lebesgue Δ-measure on an arbitrary time scale. Consequently, we come up with a more general integral than the standard Choquet integral of continuous and discrete calculus. Its use can be seen as convenient in economics, decision making, artificial intelligence, and many more. Particularly, in economics, most of the models are dynamic models (continuous and/or discrete), and those can be easily studied on time scales. Further, some basic essential results and properties of the general integral are studied. For instance, we discuss translation, homogeneity, linearity, and many more with respect to the functions and measures of the integral.Then, after that, we present some theorems for computing the integral. The findings agree to unify and extend a number of well-known results reported in the literature to a broader calculus, including continuous, discrete, and quantum calculus, among others. We also evaluate the integral on an invariant under the translation of time scales. Besides, a short note on Δ-Choquet integral with the Caputo-Fabrizio fractional derivative on the time scales is given. The significance of the outcomes is also further enhanced by a variety of interesting examples.Moreover, eventually, we stop findings after discussing an another way to calculate the Δ-Choquet integral on the time scales. To do this, we define Stieltjes distorted types-I and II Lebesgue Δ-measures on time scales which are accomplished with the help of distorted Lebesgue Δ-measure.