Classical Integer Research Articles

An Exact Solution of the Fractional Transient Electromagnetic Field Inside an Atmospheric Duct

The objective of this paper is to find the exact solution of the fractional electromagnetic (EM) equation within an atmospheric duct (dielectric medium) of the transient EM field caused by a vertical magnetic dipole and discuss the results in both fractional D-dimensional space and integer space. The novelty of this work is the exact solution of the fractional EM equation in terms of Bessel and Mittage-Leffler functions based on Caputo fractional derivative order α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} and the Laplacian operator in D -dimensional fractional space. Using the separation of variables method, the resulting magnetic field can be controlled by D=α1+α2+α3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=\\alpha _1+\\alpha _2+\\alpha _3$$\\end{document} and alpha as spatial and time fractional parameters, respectively. The transient magnetic field behaviour inside the duct is plotted through some of figures depending on fractional parameters D and α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}. The classical integer results in usual space are recovered from fractional solution at D=α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=\\alpha =1$$\\end{document}. The main results highlighted are that spatial frequencies grow at a faster rate during EM wave propagation; thus, the amplitude of the propagated wave in integer space is greater than that in fractional space. Increasing the fractional parameters D and α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} simultaneously increases the amplitude of the EM wave and can be used to control the power and energy of the EM wave.

From Single-Objective to Bi-Objective Maximum Satisfiability Solving

The declarative approach is key to efficiently finding optimal solutions to various types of NP-hard real-world combinatorial optimization problems. Most work on practical declarative solvers—ranging from classical integer programming to finite-domain constraint optimization and maximum satisfiability (MaxSAT)—has focused on optimization under a single objective; fewer advances have been made towards efficient declarative techniques for multi-objective optimization problems. Motivated by significant recent advances in practical solvers for MaxSAT, in this work we develop BiOptSat, an exact declarative approach for finding Pareto-optimal solutions to bi-objective optimization problems, with propositional logic as the underlying constraint language. BiOptSat can be viewed as an instantiation of the lexicographic method. The approach makes use of a single Boolean satisfiability solver that is incrementally employed throughout the entire search procedure, allowing for finding a single Pareto-optimal solution, finding one representative solution for each non-dominated point, and enumerating all Pareto-optimal solutions. We detail several algorithmic instantiations of BiOptSat, each building on recent algorithms proposed for single-objective MaxSAT. We empirically evaluate the instantiations compared to recently-proposed alternative approaches to multi-objective MaxSAT solving on several real-world domains from the literature, showing the practical benefits of our approach.

Electrokinetic properties of NaCl solution via molecular dynamics simulations with scaled-charge electrolytes.

Scaling ionic charges has become an alternative to polarizable force fields for representing indirect charge transfer effects in molecular simulations. In our work, we apply molecular dynamics simulations to investigate the properties of NaCl aqueous solutions in homogeneous and confined media. We compare classical integer- and scaled-charge force fields for the ions. In the bulk, we validate the force fields by computing equilibrium and transport properties and comparing them with experimental data. Integer-charge ions overestimate dielectric saturation and ionic association. Both force fields present an excess in ion-ion correlation, which leads to a deviation in the ionic conductivity at higher ionic strengths. Negatively charged quartz is used to simulate the confinement effect. Electrostatic interactions dominate counter-ion adsorption. Full-charge ions have stronger and more defined adsorption planes. We obtain the electroosmotic mobility of the solution by combining the shear plane location from non-equilibrium simulations with the ionic distribution from equilibrium simulations. From the Helmholtz-Smoluchowski equation, the zeta potential and the streaming potential coupling coefficient are computed. From an atomic-scale perspective, our molecular dynamics simulations corroborate the hypothesis of maximum packing of the Stern layer, which results in a stable and non-zero zeta potential at high salinity. The scaled-charge model representation of both properties is in excellent qualitative and quantitative agreement with experimental data. With our work, we demonstrate how useful and precise simple scaled-charge models for electrolytes can be to represent complex systems, such as the electrical double layer.

The Ruler Sequence Revisited: A Dynamic Perspective

The Ruler function or the Gros sequence is a classical infinite integer sequence that underlies some interesting mathematical problems. In this paper, we provide four new problems containing this type of sequence: (i) demographic discrete dynamical automaton, (ii) the middle interval Cantor set, (iii) construction by duplication of polygons and (iv) the horizontal visibility sequence at the accumulation point of the Feigenbaum cascade. In all of them, the infinite sequence is obtained through a recursive procedure of duplication. The properties of the ruler sequence, in particular, those relating to recursiveness and self-containing, are used to achieve a deeper understanding of these four problems. These new representations of the ruler sequence could inspire new studies in the field of discrete mathematics.

Stability analysis of fractional order breast cancer model in chemotherapy patients with cardiotoxicity by applying LADM

AbstractBreast cancer is the most common type of cancer in women. Chemotherapy is primarily used for patients with stage 2 to 4 breast cancer. Most chemotherapy drugs are effective at destroying rapidly growing and proliferating cancer cells. However, drugs also damage normal, rapidly growing cells, which can lead to serious side effects. Breast cancer treatment with chemotherapy can affect heart health. Side effects of chemotherapy on the heart are called cardiotoxicity. Therefore, we have constructed a mathematical model from the breast cancer patient population. In this article, we utilize the Caputo–Fabrizio fractional order derivative for mathematical modeling of the breast cancer stages in chemotherapy patients. The use of Caputo–Fabrizio fractional derivative provides a more valuable insight into the complexity of the breast cancer model. The stability of the fractional order model is also proven by the $\mathscr{P}$ P -stable approach of the fixed point theorem. Also, the numerical simulations are performed via Laplace Adomian decomposition method to establish the dependence of the breast cancer dynamics on the order of the fractional derivatives. Based on the geometric results in the figures, we can conclude that the magnitude of the fractional order has a considerable impact on the days, which the maximum or minimum of the system solutions are reached, with a shift in the time at which this happens as the fractional order decreases from 1. However, it is obvious that the solutions of Caputo–Fabrizio fractional model approach the relevant results of the classical integer order system, when the fractional order approaches to 1.

A Mixed‐Integer Linear Programming Framework for Optimization of Water Network Operations Problems

AbstractWater distribution systems (WDSs) are critical infrastructure used to convey water from sources to consumers. The mathematical framework governing the distribution of flows and heads in extended period simulations of WDSs lends itself to application in a wide range of optimization problems. Applying the classical mixed integer linear programming (MILP) approach to model WDSs hydraulics within an optimization framework can contribute to higher solution accuracy with lower computational effort. However, adapting WDSs models to conform to a MILP formulation has proven challenging because of the intrinsic non‐linearity of system hydraulics and the complexity associated with modeling hydraulic devices that influence the state of the WDS. This paper introduces MILPNet, an adjustable framework for WDSs that can be used to build and solve an extensive array of MILP optimization problems. MILPNet includes constraints that represent the mass balance and energy conservation equations, hydraulic devices, control rules, and status checks. To conform to MILP structure, MILPNet employs piece‐wise linear approximation and integer programming. MILPNet was implemented and tested using Gurobi Python API. Modeling accuracy was shown to be comparable to EPANET, a public domain software for hydraulic modeling, and sensitivity analyses were conducted to examine the impacts of the modeling assumptions on the performance of MILPNet. Additionally, application of the framework was demonstrated using pump scheduling optimization examples in single and rolling horizon scenarios. Our results show that MILPNet can facilitate the construction and solution of optimization problems for a range of applications in WDSs operations.

Analysis of Age-Structured Mathematical Model of Malaria Transmission Dynamics via Classical and ABC Fractional Operators

Malaria is a complex disease with many factors influencing the transmission dynamics, including age. This research analyzes the transmission dynamics of malaria by developing an age-structured mathematical model using the classical integer order and Atangana–Baleanu–Caputo fractional operators. The analysis of the model focused on several important aspects. The existence and uniqueness of solutions of fractional order were explored based on some fixed-point theorems,such as Banach and Krasnoselski. The Positivity and boundedness of the solutions were also investigated. Furthermore, through mathematical analysis techniques, we analyzed different types of stability results, and the results showed that the disease-free equilibrium point of the model is proved to be both locally and globally asymptotically stable if the basic reproduction number is less than one, whereas the endemic equilibrium point of the model is both locally and globally asymptotically stable if the basic reproduction number is greater than one. The findings from the sensitivity analysis revealed that the most sensitive parameters, essential for controlling or eliminating malaria are mosquito biting rate, density-dependent natural mortality rate, clinical recovery rate, and recruitment rate for mosquitoes. Numerical simulations are also performed to examine the behavior of the model for different values of the fractional-order alpha,and the result revealed that as the value α reduces from 1, the spread of the endemic grows slower. By incorporating these findings, this research helps to clarify the dynamics of malaria and provides information on how to create efficient control measures.

Mathematical analysis of a time-fractional coupled tumour model using Laplace and finite Fourier transforms

A time-fractional diffusive tumour growth model is investigated in this paper. The model assumes two different tumour populations having diverse diffusion and proliferation rates. The classical integer model, coupled with the fractional operators with power-law, Mittag-Leffler and exponential kernels is solved analytically with the help of integral transforms Numerical investigations of the resulting solution are carried out to examine the effect of the different fractional operators. The spherical form of the model is considered, as this describes the tumour more adequately as compared to other coordinates system. From the numerical experiments, it is observed that, the kernel contained in the respective fractional operator could alter the dynamics of the tumour concentration under consideration.

Enhanced general conformable controller based on Lyapunov technique for DC-DC static converters: Application to a solar system

<abstract> <p>To synthesize the proper control signal while guaranteeing the necessary performance indices (speed, resilience, accuracy, etc.), mathematical models were frequently used to represent physical systems. These descriptions were utilized for control, monitoring, and detection in these kinds of systems. Quality and performance of the process may suffer if the model is inaccurate or incomplete. As a result, conformable systems (CS) may be used to make these mathematical models more near to the real world. However, non-power-electronics experts who need to model and simulate complex systems may find the task of modeling static converters to be rather challenging. Researchers have just recently outlined the properties of the general conformable systems (GCS). This innovative approach built upon the principle of the classical integer order systems, employing the same mathematical foundations for its derivation. With the introduction of this novel description of systems, a fresh array of differential equations emerged, specifically tailored for the realm of direct current to direct current (DC-DC) static converters. GCS has been proved to be more flexible and profitable than the traditional integer-order one for representing DC-DC static converters. This advancement paved the way for more effective control techniques based on the Lyapunov method, with practical applications in photovoltaic (PV) systems and beyond.</p> </abstract>

Synthesis of two-mass electromechanical systems with cascade connection of fractional-order controllers

The research focused on two-mass electromechanical systems widely utilized in industry. The challenge addressed in this work was to improve the synthesis of controllers for such systems to simplify it and enhance the quality of transition processes. Traditionally, the synthesis of control system loops for these systems was carried out using integer controllers and standard forms. However, this approach led to the synthesis of complex integer controllers that are difficult to implement. To overcome this issue, an original approach to the synthesis of control system loops based on the fractional characteristic polynomial is proposed. The fractional characteristic polynomial ensures the desired quality of the transition process given the implementation of a specified structure of the fractional controller. A new method of structural-parametric synthesis of fractional-order controllers is developed for the case of their cascade connection in multi-loop two-mass electromechanical systems. Additionally, an algorithm for synthesizing fractional-order controllers for the corresponding control loops is presented. This enabled the structural-parametric synthesis of fractional-order controllers for a two-mass electromechanical system with the cascade connection of controllers. Such an approach provides better quality of transition processes compared to classical integer controllers, simplifies the synthesis, and thereby enhances the quality of the synthesized systems. The impact of the synthesized fractional-order controllers using the proposed approach on the dynamic properties of the two-mass «thyristor converter – motor» system was investigated. The research results demonstrated the practical applicability of fractional controllers designed using the proposed method for the synthesis of automatic control systems of two-mass electromechanical systems in the industry.

Quantum Circuit Design for Integer Multiplication Based on Schönhage–Strassen Algorithm

Quantum arithmetic circuits has attracted extensive attention recently since it plays fundamental roles in many applications of quantum computing. Specifically, quantum circuits for integer multiplication are of great significance to various quantum algorithms including Shor’s integer factorization and discrete logarithm algorithm. In this paper, we design a family of quantum circuits for integer multiplication based on the famous classical integer multiplication algorithm, Schönhage-Strassen algorithm. We have made slight modifications to the algorithm to simplify its quantum circuit implementation. As a result, the quantum circuit we designed has gate depth O(log2n). To the best of our knowledge, this is the first poly-logarithmic depth quantum circuit for integer multiplication which keeps the circuit size and the number of ancillary qubits sub-quadratic. Our design has size O(nlognloglogn) counted by elementary quantum gates which is the same as the time complexity of Schönhage-Strassen algorithm, and it consumes O(nlognloglogn) clean ancillary qubits. In addition, we also utilize a weaker version of Schönhage-Strassen algorithm to give a family of circuits which has depth at the same order O(log2n) but with significantly smaller constants, while still keeping the size and number of ancillary qubits sub-quadratic.

Strictly chordal graphs: Structural properties and integer Laplacian eigenvalues

In this paper, we establish the relation between classic invariants and integer Laplacian eigenvalues of strictly chordal graphs, pointing out how their computation can be efficiently implemented. We review results concerning general graphs showing that the number of universal vertices and the degrees of twins provide integer Laplacian eigenvalues and their respective multiplicities. For strictly chordal graphs, we show new results about integer Laplacian eigenvalues which are directly related to particular vertex sets of the graph.

Analysis of a non-integer order compartmental model for cholera and COVID-19 incorporating human and environmental transmissions

Fractional differential operators have increasingly gained wider applications in epidemiological modelling due to their ability to capture memory effect in their definitions; an attribute which lacks in the concept of classical integer derivatives. In this paper, employing the Caputo fractional operator with singular kernel, the co-dynamical model for cholera and COVID-19 diseases is proposed and analyzed, incorporating both direct and indirect transmission routes for cholera. The necessary conditions for existence of the unique solution of the proposed model are studied. Using the results from fixed point theory, Ulam-Hyers stability analysis of the system is performed. The model is fitted to real data from Pakistan and the optimized order of the fractional derivative for which the system fits well to data is obtained. Other numerical assessments of the model are also executed. Phase portraits of the infected classes with different initial conditions and various order of the fractional derivative are obtained in the cases when the reproduction number and . It is observed that the trajectories for the infected compartments tend towards the infection-free steady state when and the endemic steady state when irrespective of the initial conditions and the order of the fractional derivative. Increment in the COVID-19 vaccine efficacy, keeping the vaccination rate fixed at , resulted in a decline in the COVID-19 disease. Also, increasing the COVID-19 vaccination rate, keeping the vaccine efficacy for COVID-19 fixed at , led to a decline in the COVID-19 associated reproduction number. The simulations also pointed out the impact of COVID-19 and cholera vaccinations, direct and indirect transmissions of cholera.

Numerical assessment of multiple vaccinations to mitigate the transmission of COVID-19 via a new epidemiological modeling approach

Vaccines play a crucial role in mitigating the transmission of COVID-19. The primary objective of COVID-19 vaccination campaigns is to induce immunity in individuals, thereby reducing the risk of infection, severe illness, and subsequent transmission within communities. The present study mainly focuses on examining the impact of multiple vaccinations on the transmission dynamics of COVID-19 using a new epidemic modeling approach. The vaccinated individuals are divided into three subclasses according to their vaccination status. The primary, secondary and booster shots of COVID-19 vaccines are considered in the construction of the model. Initially, the model is constructed using classical integer order differential equations and basic necessary properties are carried out. Furthermore, utilizing a fractional and fractal–fractional epidemiological approach with an exponential kernel, the model is extended with these operators in order to provide insights into the potential benefits of this vaccination strategy in controlling the spread of the disease. The uniqueness and existence criteria of the fractional and fractal–fractional models are presented and an efficient numerical solution is investigated using the modified Adams–Bashforth method. The impact of both fractional (memory index) and fractal parameters is demonstrated graphically for R0<1 and R0>1 showing the converges of solution trajectories to the disease-free and endemic steady states respectively. Moreover, we examine the impact of various vaccination rates (first and second dose, and booster short) on the reduction and mitigation of the disease incidence.

Study of fractional-order alcohol-abuse mathematical model using the concept of piecewise operator

In this paper, we study a novel model of the dynamics of alcohol consumption under induced complications. The mentioned model is considered under some advanced concept of fractional derivative called piecewise fractional-order derivative. Currently, most real-world problems are considered under fractional derivatives because of their stable and global behavior. To deal with and detect the crossover behavior in the dynamics of the proposed model, the piecewise derivative containing the Mittag-Leffler-type kernel combined with classical integer order derivatives in piecewise form is used. First, we will investigate the model for qualitative theory in detail. For qualitative theory, we will use fixed point theory. Additionally, we establish sufficient conditions for Ulam–Hyers (UH) stability. To simulate the model, we develop a numerical algorithm based on the Euler technique. The numerical method is applied to present graphically the approximate solutions of the model. In the final part of the paper, we give a detailed discussion of our numerical results.

Pitch Angle Control of an Airplane Using Fractional Order Direct Model Reference Adaptive Controllers

This paper deals with the longitudinal movement control of an airplane (pitch angle) using fractional order adaptive controllers (FOACs). It shows the improvements achieved in the plane’s behavior, in terms of the minimization of a given performance index. At the same time, less control effort is needed to accomplish the control objectives compared with the classic integer order adaptive controllers (IOACs). In this study, fractional order direct model reference adaptive control (FO-DMRAC) is implemented at the simulation level, and exhibits a better performance compared with the classic integer order (IO) version of the DMRAC (IO-DMRAC). It is also shown that the proposed control strategy for FO-DMRAC reduces the resultant system control structure down to a relative degree 2 system, for which the control implementation is simpler and avoids oscillations during the transient period. Moreover, it is interesting to note that this is the first time that an FOAC with fractional adaptive laws is applied to the longitudinal control of an airplane. A suitable model for the longitudinal movement of the airplane related to the pitch angle θ as the output variable with the lifter angle (δe) as the control variable, is first analyzed and discussed to obtain a reliable mathematical model of the plane. All of the other input variables acting on the plane are considered as perturbations. For certain operating conditions defined by the flight conditions, an FO-DMRAC is designed, simulated, and analyzed. Furthermore, a comparison with the implementation of the classical adaptive general direct control (relative degree ≥ 2) is presented. The boundedness and convergence of all of the signals are theoretically proven based on the new Lemma 3, assuring the boundedness of all internal signals ω(t).

New perspective on the creep characteristic of fiber–dependent shape memory polymers: variable–order fractional constitutive model

With the aim of describing and explaining the creep tensile property of shape memory polymers (SMPs) better, a variable--order fractional creep constitutive model with fewer parameters and good simulation performance was constructed on the basis of existing constitutive models. The consists of a Hookean spring connected in series with a Newtonian dashpot was employed to propose a new variable--order fractional Maxwell model for SMPs. Drawing support from the Laplace transform, we derive the analytical solution of this novel model. The verification of the obtained theoretical results with the creep test data for different fiber weight fractions and stress levels available from the literature reveals their satisfactory agreement. It can be concluded that the developed variable--order fractional constitutive model can better describe the complex viscoelastic relationship of shape memory polymers which is difficult to be described by the classical integer order derivative, and obtain a more accurate constitutive model of shape memory polymers with fewer parameters.

Elucidating the Effects of Ionizing Radiation on Immune Cell Populations: A Mathematical Modeling Approach with Special Emphasis on Fractional Derivatives

Despite recent advances in the mathematical modeling of biological processes and real-world situations raised in the day-to-day life phase, some phenomena such as immune cell populations remain poorly understood. The mathematical modeling of complex phenomena such as immune cell populations using nonlinear differential equations seems to be a quite promising and appropriate tool to model such complex and nonlinear phenomena. Fractional differential equations have recently gained a significant deal of attention and demonstrated their relevance in modeling real phenomena rather than their counterpart, classical (integer) derivative differential equations. We report in this paper a mathematical approach susceptible to answering some relevant questions regarding the side effects of ionizing radiation (IR) on DNA with a particular focus on double-strand breaks (DSBs), leading to the destruction of the cell population. A theoretical elucidation of the population memory was carried out within the framework of fractional differential equations (FODEs). Using FODEs, the mathematical approach presented herein ensures connections between fractional calculus and the nonlocal feature of the fractional order of immune cell populations by taking into account the memory trace and genetic qualities that are capable of integrating all previous actions and considering the system’s long-term history. An illustration of both fractional modeling, which provides an excellent framework for the description of memory and hereditary properties of immune cell populations, is elucidated. The mathematics presented in this research hold promise for modeling real-life phenomena and paves the way for obtaining accurate model parameters resulting from the mathematical modeling. Finally, the numerical simulations are conducted for the analytical approach presented herein to elucidate the effect of various parameters that govern the influence of ionizing irradiation on DNA in immune cell populations as well as the evolution of cell population dynamics, and the results are presented using plots and contrasted with previous theoretical findings.

Quantum Annealing With Integer Slack Variables for Grid Partitioning

Quantum annealing (QA) can be used to efficiently solve quadratic unconstrained binary optimization (QUBO) problems. Grid partitioning (GP), which is a classic NP-hard integer programming problem, can potentially be solved much faster using QA. However, inequality constraints in the GP optimization model are difficult to handle. In this study, a novel solution framework based on QA is proposed for GP problems. The integer slack (IS) and binary expansion methods are applied to transform GP problems into QUBO problems. Instead of introducing continuous variables in traditional slack methods, the proposed IS method can avoid complex iteration processes when using QA. The case study demonstrates that the IS method obtains accurate feasible solutions with less calculation time.

Projections of human papillomavirus vaccination and its impact on cervical cancer using the Caputo fractional derivative.

We propose a fractional order model for human papillomavirus (HPV) dynamics, including the effects of vaccination and public health education on developing cervical cancer. First, we discuss the general structure of Caputo fractional derivatives and integrals. Next, we define the fractional HPV model using Caputo derivatives. The model equilibrium quantities, with their stability, are discussed based on the magnitude of the reproduction number. We compute and simulate numerical solutions of the presented fractional model using the Adams-Bashforth-Moulton scheme. Meanwhile, real data sourced from reports from the World Health Organization is used to establish the parameters and compute the basic reproduction number. We present figures of state variables for different fractional orders and the classical integer order. The impacts of vaccination and public health education are discussed through numerical simulations. From the results, we observe that an increase in both vaccination rates and public health education increases the quality of life, and thus, reduces disease burden and suffering in communities. The results also confirm that modeling HPV transmission dynamics using fractional derivatives includes history effects in the model, making the model further insightful and appropriate for studying HPV dynamics.