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There is a need for further studies on students' learning of Differential Equations (DEs), especially in advanced undergraduate and graduate courses. Research on the mathematical education of engineers shows a conflict between students' demands for practical, contextualized pedagogies and the need for abstract reasoning and appropriate use of mathematical results. Few papers focus on engineering students' interpretation of theorems and their use as tools in argumentation and problem-solving. This paper takes a sociocultural stance on learning and employs dialogical inquiry – a methodology rooted in Bakhtinian theory, newly developed for collaborative inquiry and qualitative data analysis – to investigate the meanings that senior engineering students made while working on a task designed to evaluate their understanding of Existence and Uniqueness Theorems (EUTs) of solutions of DEs. We identified two important epistemological disconnections that explain the difficulties that some of our students faced in making meaning of solutions of DEs and the EUT. • There is a need for more studies on the learning of Differential Equations (DEs). • Few research papers focus on students' meaning-making of theoretical results in DEs. • Dialogical inquiry methodology was used to analyze students' meaning-making processes. • Two epistemological disconnections were found that explain students' difficulties. • The use of warrants in students' dialogue was important in their meaning-making. [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the journal Mathematics titled "Differential Equations of Mathematical Physics and Related Problems of Mechanics." The special issue covers a range of topics related to differential equations in mathematical physics and mechanics, including wave equations, spectral theory, scattering, and inverse problems. The editorial provides a summary of the published papers in the special issue, highlighting their contributions to the field. The document emphasizes the importance of the special issue in covering both applied and fundamental aspects of mathematics, physics, and their applications in various fields. The author expresses gratitude to the authors, reviewers, assistants, associate editors, and editors for their contributions to the special issue. The report does not provide specific details about the content of the papers or the nature of the special issue. [Extracted from the article]

The paper proposes applying an in-house mathematical model of a liquid flat-plate solar collector to calculate the collector time constant. The described model, proposed for the first time in an earlier study, is a one-dimensional distributed parameter model enabling simulations of the collector operation under arbitrarily variable boundary conditions. The model is based on the solution of energy balance equations for all collector components. The formulated differential equations are solved iteratively using an implicit difference scheme. To obtain a stable numerical solution, it is necessary to use appropriate steps of time and spatial division. These were found by comparing the results obtained from the model with the results of the analytical solution available in the literature for the transient state, which constitutes the novelty of the present study. The accuracy of the results obtained from the model was verified experimentally by comparing the measured and calculated history of the fluid temperature at the outlet of the collector. The calculation of the collector time constant is proposed in the paper as an example of the model's practical application. The results of the time constant calculation were compared with the values obtained experimentally on the test stand. This is another novelty of the presented research. The analysed collector instantaneous efficiency was then calculated for selected outdoor conditions. The presented mathematical model can also be used to verify the correctness of the collector operation. By comparing, on an ongoing basis, the measured and calculated values of the fluid temperature at the collector outlet, conclusions can be drawn about the process of solar glass fouling or glycol gelling. The simplicity of the model and the low computational demands enable such comparisons in an online mode. [ABSTRACT FROM AUTHOR]

Reliability analysis is crucial for understanding the performance and failure characteristics of lifetime systems. This paper presents a comprehensive study on the reliability analysis of lifetime systems using the Weibull distribution. The Weibull distribution, known for its flexibility in modeling failure times, provides a versatile framework for capturing diverse failure behaviors. A useful model for redundancy systems is proposed in this paper. The model consists of (n+1) components, where n components serve as spare parts for the main component. The failure rate of the working component is time-dependent, denoted as λ(t), while the failure rates of the non-working components are assumed to be zero. Whenever a component fails, one of the spare parts immediately takes over its role. The failed components in this model are considered non-repairable. To analyze this model, we establish the differential equations that describe the system states. By solving these equations, we calculate important parameters such as system reliability and mean time to failure (MTTF) in real-time scenarios. These parameters provide valuable insights into the performance and behavior of the system under study. By employing the Weibull distribution and the proposed model, this paper contributes to enhancing the understanding of reliability analysis in lifetime systems and enables the estimation of important reliability parameters for practical applications. [ABSTRACT FROM AUTHOR]

A complex dynamical network (CDN) can be considered as the composition system with the nodes subsystem (NS) and the links subsystem (LS), and both subsystems are coupled with each other. In this paper, two vector differential equations (VDE) are used to describe the dynamical behaviours of NS and LS, respectively, in which the dynamical behaviour of NS is considered as the VDE with the second derivative term (SDT). This paper mainly focuses on the dynamics of LS, which is represented as VDE with the intuitive topologic feature of outgoing links, and investigates the design of the tracking controller for NS and the auxiliary tracking objectives (ATO) for LS. Firstly, the dynamical models of NS and LS in CDN are proposed, and the corresponding assumptions are given. Secondly, based on Lyapunov stability theory, the controller of NS and the ATO of LS are designed so that the state of NS can asymptotically track the given reference signal. Finally, the effectiveness of the proposed control strategy in this paper is verified by the numerical simulation example with N two-links robots. Abbreviations: ATO: auxiliary tracking objectives; CDN: complex dynamical network; LS: links subsystem; MDE: matrix differential equation; NS: nodes subsystem; SDT:second derivative term; VDE: vector differential equation; [ABSTRACT FROM AUTHOR]

In this paper, we continue to consider the problem of recovering the port-Hamiltonian structure for an arbitrary system of differential equations. We complement our previous study on this topic by explaining the choice of machine learning algorithms and discussing some details of their application. We also consider the possibility provided by this approach for a potentially new definition of canonical forms and classification of systems of differential equations. [ABSTRACT FROM AUTHOR]

In order to effectively prevent and combat online public opinion crises triggered by major emergencies, this paper explores the dissemination mechanism of uncertain information on online social platforms. According to the decisionmaking behavior of netizens after receiving uncertain information, they are divided into eight categories. Considering that there will be a portion of netizens who clarify uncertain information after receiving it, this paper proposes a SEFTFbTbMR model of uncertain information clarification behavior. The propagation dynamics equations of the model are given based on the theory of differential equations, the basic regeneration number R0 of the model is calculated, and the existence and stability of the equilibrium point of the model are analyzed. The theoretical analysis of the model is validated using numerical simulation software, and sensitivity analysis is performed on the parameters related to R0. In order to reduce the influence caused by uncertain information, the optimal control strategy of the model is proposed using the Hamiltonian function. It is found that the dissemination of uncertain information among netizens can be suppressed by strengthening the regulation of social platforms, improving netizens' awareness of identifying the authenticity of information, and encouraging netizens to participate in the clarification of uncertain information. The results of this work can provide a theoretical basis for future research on the uncertain information dissemination mechanism triggered by major emergencies. In addition, the results can also provide methodological support for the relevant government departments to reduce the adverse effects caused by uncertain information in the future. [ABSTRACT FROM AUTHOR]

Focusing on the bending wave characteristic of plate–shell structures, this paper derives the complex band curve of piezoelectric phononic crystal based on the equilibrium differential equation in the plane stress state using COMSOL PDE 6.2. To ascertain the computational model's accuracy, the computed complex band curve is then cross-validated against real band curves obtained through coupling simulations. Utilizing this model, this paper investigates the impact of structural and electrical parameters on the bandgap range and the attenuation coefficient in the bandgap. Results indicate that the larger surface areas of the piezoelectric sheet correspond to lower center bands in the bandgap, while increased thickness widens the attenuation coefficient range with increased peak values. Furthermore, the influence of inductance on the bandgap conforms to the variation law of the electrical LC resonance frequency, and increased resistance widens the attenuation coefficient range albeit with decreased peak values. The incorporation of negative capacitance significantly expands the low-frequency bandgap range. Visualized through vibration transfer simulations, the vibration-damping ability of the piezoelectric phononic crystal is demonstrated. Experimentally, this paper finds that two propagation modes of bending waves (symmetric and anti-symmetric) result in variable voltage amplitudes, and the average vibration of the system decreases by 4–5 dB within the range of 1710–1990 Hz. The comparison between experimental and model-generated data confirms the accuracy of the attenuation coefficient calculation model. This convergence between experimental and computational results emphasizes the validity and usefulness of the proposed model, and this paper provides theoretical support for the application of piezoelectric phononic crystals in the field of plate–shell vibration reduction. [ABSTRACT FROM AUTHOR]

The human brain operates at multiple levels, from molecules to circuits, and understanding these complex processes requires integrated research efforts. Simulating biophysically-detailed neuron models is a computationally expensive but effective method for studying local neural circuits. Recent innovations have shown that artificial neural networks (ANNs) can accurately predict the behavior of these detailed models in terms of spikes, electrical potentials, and optical readouts. While these methods have the potential to accelerate large network simulations by several orders of magnitude compared to conventional differential equation based modelling, they currently only predict voltage outputs for the soma or a select few neuron compartments. Our novel approach, based on enhanced state-of-the-art architectures for multitask learning (MTL), allows for the simultaneous prediction of membrane potentials in each compartment of a neuron model, at a speed of up to two orders of magnitude faster than classical simulation methods. By predicting all membrane potentials together, our approach not only allows for comparison of model output with a wider range of experimental recordings (patch-electrode, voltage-sensitive dye imaging), it also provides the first stepping stone towards predicting local field potentials (LFPs), electroencephalogram (EEG) signals, and magnetoencephalography (MEG) signals from ANN-based simulations. While LFP and EEG are an important downstream application, the main focus of this paper lies in predicting dendritic voltages within each compartment to capture the entire electrophysiology of a biophysically-detailed neuron model. It further presents a challenging benchmark for MTL architectures due to the large amount of data involved, the presence of correlations between neighbouring compartments, and the non-Gaussian distribution of membrane potentials. Author summary: Our research focuses on cutting-edge techniques in computational neuroscience. We specifically make use of simulations of biophysically detailed neuron models. Traditionally these methods are computationally intensive, but recent advancements using artificial neural networks (ANNs) have shown promise in predicting neural behavior with remarkable accuracy. However, existing ANNs fall short in providing comprehensive predictions across all compartments of a neuron model and only provide information on the activity of a limited number of locations along the extent of a neuron. In our study, we introduce a novel approach leveraging state-of-the-art multitask learning architectures. This approach allows us to simultaneously predict membrane potentials in every compartment of a neuron model. By distilling the underlying electrophysiology into an ANN, we significantly outpace conventional simulation methods. By accurately capturing voltage outputs across the neuron's structure, our method invites comparisons with experimental data and paves the way for predicting complex aggregate signals such as local field potentials and EEG signals. Our findings not only advance our understanding of neural dynamics but also present a significant benchmark for future research in computational neuroscience. [ABSTRACT FROM AUTHOR]

In this paper, the existence of optimal solutions for problems governed by differential equations involving feedback controls is established for when the problem must account for a Volterra-type distributed delay and is subject to the action of impulsive external forces. The problem is reformulated within the class of impulsive semilinear integro-differential inclusions in Banach spaces and is studied by using topological methods and multivalued analysis. The paper concludes with an application to a population dynamics model. [ABSTRACT FROM AUTHOR]

The history of variational calculus dates back to the late 17th century when Johann Bernoulli presented his famous problem concerning the brachistochrone curve. Since then, variational calculus has developed intensively as many problems in physics and engineering are described by equations from this branch of mathematical analysis. This paper presents two non-classical, distinct methods for solving such problems. The first method is based on the differential transform method (DTM), which seeks an analytical solution in the form of a certain functional series. The second method, on the other hand, is based on the physics-informed neural network (PINN), where artificial intelligence in the form of a neural network is used to solve the differential equation. In addition to describing both methods, this paper also presents numerical examples along with a comparison of the obtained results.Comparingthe two methods, DTM produced marginally more accurate results than PINNs. While PINNs exhibited slightly higher errors, their performance remained commendable. The key strengths of neural networks are their adaptability and ease of implementation. Both approaches discussed in the article are effective for addressing the examined problems. [ABSTRACT FROM AUTHOR]

Multifactor uncertain differential equations are powerful tools for studying dynamic systems under multi-source noise. A key challenge in this study is how to accurately estimate unknown parameters based on the framework of uncertainty theory in multi-source noise environments. To address this core problem, this paper innovatively proposes a least-squares estimation method. The essence of this method lies in constructing statistical invariants with a symmetric uncertainty distribution based on observational data and determining specific parameters by minimizing the distance between the population distribution and the empirical distribution of the statistical invariant. Additionally, two numerical examples are provided to help readers better understand the practical operation and effectiveness of this method. In addition, we also provide a case study of JD.com's stock prices to illustrate the advantages of the method proposed in this paper, which not only provides a new idea and method for addressing the problem of dynamic system parameter estimation but also provides a new perspective and tool for research and application in related fields. [ABSTRACT FROM AUTHOR]

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard's iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two. [ABSTRACT FROM AUTHOR]

The present paper dealt with the study of solitons of Lorentzian Kähler space-time manifolds. In this paper, we have discussed different conditions for solitons to be steady, expanding, or shrinking in terms of isotropic pressure, the cosmological constant, energy density, nonlinear equations, and gravitational constant in pseudo-quasi-conformally flat and pseudo-W8 flat Lorentzian Kahler space-time manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we survey selected software packages for the numerical solution of boundary value ODEs (BVODEs), time-dependent PDEs in one spatial dimension (1DPDEs), and initial value ODEs (IVODEs). A unifying theme of this paper is our focus on software packages for these problem classes that compute error-controlled, continuous numerical solutions. A continuous numerical solution can be accessed by the user at any point in the domain. We focus on error-control software; this means that the software adapts the computation until it obtains a continuous approximate solution with a corresponding error estimate that satisfies the user tolerance. The second section of the paper will provide an overview of recent work on the development of COLNEWSC, an updated version of the widely used collocation BVODE solver, COLNEW, that returns an error-controlled continuous approximate solution based on the use of a superconvergent interpolant to the underlying collocation solution. The third section of the paper gives a brief review of recent work on the development of a new 1DPDE solver, BACOLIKR, that provides time- and space-dependent event detection for an error-controlled continuous numerical solution. In the fourth section of the paper, we briefly review the state of the art in IVODE software for the computation of error-controlled continuous numerical solutions. [ABSTRACT FROM AUTHOR]

The collocation method is a widely used numerical method for science and engineering problems governed by partial differential equations. This paper provides a comprehensive review of collocation methods and their applications, focused on elasticity, heat conduction, electromagnetic field analysis, and fluid dynamics. The merits of the collocation method can be attributed to the need for element mesh, simple implementation, high computational efficiency, and ease in handling irregular domain problems since the collocation method is a type of node-based numerical method. Beginning with the fundamental principles of the collocation method, the discretization process in the continuous domain is elucidated, and how the collocation method approximation solutions for solving differential equations are explained. Delving into the historical development of the collocation methods, their earliest applications and key milestones are traced, thereby demonstrating their evolution within the realm of numerical computation. The mathematical foundations of collocation methods, encompassing the selection of interpolation functions, definition of weighting functions, and derivation of integration rules, are examined in detail, emphasizing their significance in comprehending the method's effectiveness and stability. At last, the practical application of the collocation methods in engineering contexts is emphasized, including heat conduction simulations, electromagnetic coupled field analysis, and fluid dynamics simulations. These specific case studies can underscore collocation method's broad applicability and effectiveness in addressing complex engineering challenges. In conclusion, this paper puts forward the future development trend of the collocation method through rigorous analysis and discussion, thereby facilitating further advancements in research and practical applications within these fields. [ABSTRACT FROM AUTHOR]

In this paper, we collect the basic information data of online rumors and highly topical public opinions. In the research of the propagation model of online public opinion rumors, we use the improved SCIR model to analyze the characteristics of online rumor propagation under the suspicion mechanism at different propagation stages, based on considering the flow of rumor propagation. We analyze the stability of the evolution of rumor propagation by using the time-delay differential equation under the punishment mechanism. In this paper, the evolution of heterogeneous views with different acceptance and exchange thresholds is studied, using the standard Deffuant model and the improved model under the influence of the media, to analyze the evolution process and characteristics of rumor opinions. Based on the above results, it is found that improving the recovery rate is better than reducing the deception rate, and increasing the eviction rate is better than improving the detection rate. When the time lag τ < 110, it indicates that the spread of rumors tends to be asymptotic and stable, and the punishment mechanism can reduce the propagation time and the maximum proportion of deceived people. The proportion of deceived people increases with the decrease in the exchange threshold, and the range of opinion clusters increases with the decline in acceptance. [ABSTRACT FROM AUTHOR]

This paper is concerned with the oscillation of solutions to a class of first-order differential equations with variable coefficients and a general delay argument. New oscillation criteria are established, which improve and extend many known results reported in the literature. A couple of illustrative examples are given to show the efficiency of the newly obtained results. In particular, it is shown that our criteria partially fulfill a remaining gap in a recent sharp result by Pituk et al. [31]. [ABSTRACT FROM AUTHOR]

This research paper presented a novel investigation into an implicit coupled system of fractional variable order, which has not been previously studied in the existing literature. The study focused on establishing and developing sufficient conditions for the existence and uniqueness of solutions, as well as the Ulam-Hyers stability, for the proposed coupled system without using semigroup property. By extending the existing conclusions examined for the Atangana-Baleanu-Caputo (ABC) operator, we contributed to advancing the understanding of variable-order fractional differential equations. The paper provided a solid theoretical foundation for further analysis, numerical simulations, and practical applications. The obtained results have implications for designing and controlling systems modeled using fractional variable order equations and serve as a basis for addressing a wide range of dynamical problems. The transformation techniques, qualitative analysis, and illustrative examples presented in this work highlight its unique contributions and potential to serve as a foundation for future research. [ABSTRACT FROM AUTHOR]

In this two-paper series, we prove the invariance of the Gibbs measure for a threedimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In this paper, we focus on the dynamical aspects of our main result. The local theory is based on a paracontrolled approach, which combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. The main contribution, however, lies in the global theory. We develop a new globalization argument, which addresses the singularity of the Gibbs measure and its consequences. [ABSTRACT FROM AUTHOR]

Copyright of Kirkuk Journal of Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The basic reproduction, or reproductive number, is a useful index that indicates whether or not there will be an epidemic. However, it is also very important to determine whether an epidemic will eventually decrease and disappear or persist as an endemic. Different infectious diseases have different behaviors and mathematical models used to simulated them should capture the most important processes; however, the models also involve simplifications. Influenza epidemics are usually short-lived and can be modeled with ordinary differential equations without considering demographics. Delays such as the infection time can change the behavior of the solutions. The same is true if there is permanent or temporary immunity, or complete or partial immunity. Vaccination, isolation and the use of antivirals can also change the outcome. In this paper, we introduce several new models and use them to find the effects of all the above factors paying special attention to whether the model can represent an infectious process that eventually disappears. We determine the equilibrium solutions and establish the stability of the disease-free equilibrium using various methods. We also show that many models of influenza or other epidemics with a short duration do not have solutions with a disappearing epidemic. The main objective of the paper is to introduce different ways of modeling immunity in epidemic models. Several scenarios with different immunities are studied since a person may not be re-infected because he/she has total or partial immunity or because there were no close contacts. We show that some relatively small changes, such as in the vaccination rate, can significantly change the dynamics; for example, the existence and number of the disease-free equilibria. We also illustrate that while introducing delays makes the models more realistic, the dynamics have the same qualitative behavior. [ABSTRACT FROM AUTHOR]

This paper provides a comparison between the fractal calculus of fractal sets and fractal curves. There are introduced the analogues of the Riemann–Liouville and Caputo integrals and derivatives for fractal curves, which are non-local derivatives. Moreover, the concepts analogous to the fractional Laplace operator to address fractal non-local differential equations on fractal curves are defined. Additionally, in the paper it is introduced the fractal local Mellin transform and fractal non-local transform as tools for solving fractal differential equations. The results are supported with tables and examples to demonstrate the findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider the mass-critical nonlinear Schrödinger equation in one dimension. Ogawa-Tsutsumi [Proc. Amer. Math. Soc. 111 (1991), no. 2, 487-496] proved a blow-up result for negative energy solution by using a scaling argument for initial data. In general, a equation with a linear potential does not have a scale invariant, so the method by Ogawa-Tsutsumi cannot be used directly to that. In this paper, we prove a blow-up result for the equation with the linear potential by modifying the argument of Ogawa-Tsutsumi. [ABSTRACT FROM AUTHOR]

Purpose: Chemical reaction effects are added to the governing equation. This paper aims to get the solution by converting the partial differential equation into an ordinary differential equation and solve using a perturbation scheme and applying the boundary conditions. Design/methodology/approach: In this paper, the authors discussed the chemical reaction effects of heat and mass transfer on megnato hydro dynamics free convective rotating flow of a visco-elastic incompressible electrically conducting fluid past a vertical porous plate through a porous medium with suction and heat source. The authors analyze the effect of time dependent fluctuating suction on a visco-elastic fluid flow. Findings: Using variable parameters of the fluid, the velocity, temperature and concentration of the fluid are analyzed through graphs. Originality/value: The velocity profile reduces by increasing the values of thermal Grashof number (Gr), mass Grashof number (Gc) and the magnetic parameter (M). On the other hand, the velocity profile gets increased by increasing the permeability parameter (K). The temperature profile decreases by raising the value of Prandtl number (Pr) and frequency of oscillation parameter (ω). However, the source parameter (S) has the opposite effect on the temperature profile. The concentration profile reduces in all points by raising the chemical reaction parameter Kl, Schmidt number Sc, frequency of oscillation ω and the time t. [ABSTRACT FROM AUTHOR]

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence of random mild solutions for a general class of second-order abstract random differential equations with state-dependent delay. The technique used is a generalization of the classical Darbo fixed point theorem for Fréchet spaces associated with the concept of measures of noncompactness. An application related to partial random differential equations with state-dependent delay is presented. [ABSTRACT FROM AUTHOR]

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point 0 ≤ c < d and fixed points at the end of the interval [0, d]. For i = 0, 1, the boundary conditions are as follows: ai, bi > 0, a0p(c) = -b0p(d), a1p'(c) = -b1p'(d). Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixedpoint theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved. [ABSTRACT FROM AUTHOR]

The aim of this paper is to numerically solve the Fokker-Planck-Kolmogorov fractional-time differential equations using the Legendre wavelet. Also, we analyzed the convergence of function approximation using Legendre wavelets. Introduced the absolute value between the exact answer and the approximate answer obtained by the given numerical methods, and analyzed the error of the numerical method. This method has the advantage of being simple to solve. The results revealed that the suggested numerical method is highly accurate and effective. The results for some numerical examples are documented in table and graph form to elaborate on the efficiency and precision of the suggested method. The simulation was carried out using MATLAB software. In this paper and for the first time, the authors presented results on the numerical simulation for classes of time-fractional differential equations. The authors applied the reproducing Legendre wavelet method for the numerical solutions of nonlinear Fokker-Planck-Kolmogorov time-fractional differential equation. The method presented in the present study can be used by programmers, engineers, and other researchers in this field. [ABSTRACT FROM AUTHOR]

Copyright of HETS Online Journal is the property of Hispanic Educational Technology Services, Inc. (HETS) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The inconsistency between the whitening differential equation and the grey basic form of the non-homogeneous continuous grey model CFNGM(1,1,k,c) will result in internal errors. Thus this paper proposes a CFNGMA(1,1,k,c) model with adjustable parameters, which improves the accuracy of the CFNGM(1,1,k,c). This paper first elucidates reasons for the internal errors generated by the continuous grey model CFNGM(1,1,k,c), and explains the classic method, the discrete grey forecasting model, of eliminating internal errors. On the basis of an in-depth analysis of the modeling mechanism of CFNGM(1,1,k,c) model, a new parameter adjustable grey forecasting model is proposed by introducing parameter adjustment factors to modify model's parameters. Finally, the new model is applied to explore the gross regional product of Chengdu and Deyang in the Chengdu metropolitan area. The calculation results indicate that the newly proposed model can obtain more accurate results. [ABSTRACT FROM AUTHOR]

The paper is concerned with the derivation of asymptotically consistent equations governing the long-wave flexural response of a two-layered rectangular plate with high-contrast elastic properties. In the general case, the plate is under dynamic and variable surface, volume, and edge forces. Performing the asymptotic integration of the three-dimensional (3D) elasticity equations in the transverse direction and satisfying boundary conditions on the faces and interface, we derived the sequence of two-dimensional (2D) differential equations with respect to required functions in the first two approximations. The eight independent restraints for the generalized displacements and stress resultants are considered to formulate the 16 independent variants of boundary conditions. One of the main results of the paper is the Timoshenko–Reissner type equation capturing the effect of the softer layer and taking into account the in-plane deformation induced by the edge forces. Comparative calculations of natural frequencies were carried out based on our and alternative models. [ABSTRACT FROM AUTHOR]

The dynamic analysis of the out-of-plane response of masonry walls, with different reinforcing systems and hit by sequences of pulses alternating in sign, representative of the seismic action, is the aim of the paper. The analysis is done in the context of the Heyman model of the no tension masonry structures. We consider the wall of the last story of a simple masonry house, commonly reinforced at its head by a ring beam, usually in a reinforced concrete. It has also been considered the retrofitting of the wall by applying, on both its sides, vertical bands in FRP, or similar. The dynamics of the wall, activated by sequences of pulses, is analyzed in detail, considering the change of geometry occurring in the mechanism during the motion. The corresponding differential equation is integrated in a closed form. It is shown that, under the alternating sequence of shocks, the dynamical collapse of the wall takes place immediately after the occurrence of an accumulation of out-of-plane deformation of the wall that drags it into the configuration of zero strength, where any resistance, due to the weight, has been lost. The collapse takes place right after when the subsequent incoming pulse pushes the wall to go further beyond this configuration. The check of the seismic safety of the masonry wall concludes the paper. The smallest limit static strength of the wall required in order the wall, having pulsation p, could be able to sustain, at the limit of the dynamical collapse, the action of the asymptotic pulse sequence with acceleration intensity PGA and specific duration p TE/2 of the expected earthquake. It is shown that the wall, reinforced by a top ring beam, with the addition of vertical bands of glass or similar, of suitable length, pasted with lime mortar, can reach the required level of seismic safety. [ABSTRACT FROM AUTHOR]

This paper introduces a new approach for analyzing the dynamics of DC–DC converters. Currently, the primary widely accepted method for examining dynamic processes is the Small Signal Analysis technique. However, when applied to modern complex converters, this method poses additional challenges in formulating and solving systems of differential equations. The method proposed in this paper is based on its application to the analysis of dynamic modes of energy functions—Lagrangians. These functions make it possible to define simple criteria to describe the course of dynamic processes, and in the end define an equivalent (approximating) conventional converter identical to the original one with respect to the course of dynamics. If the magnetic and electrical energies in the Lagrangians of both the converters are equal, the outcome is practically identical transient processes. These findings were confirmed by both theoretical analysis and experimentally modelling the dynamics of the initial converter and an equivalent to it in the Matlab–Simscape program. An additional possibility of using the transfer functions of a conventional boost converter for the theoretical analysis of the converters of much greater orders is also discussed. The authors' experiments confirm the correctness of their theoretical conclusions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to determine the eigenvalue intervals of μk; 1 < k < n for which an iterative systems of a class of fractional-order differential equations with parameterized integral boundary conditions (BCs) has at least one positive solution by means of standard fixed point theorem of cone type. To the best of our knowledge, this will be the first time that we attempt to reach such findings for the topic at hand in the literature. The obtained results in the paper are illustrated with an example for their feasibility. [ABSTRACT FROM AUTHOR]

In this paper, a semi-analytical technique based on Taylor's series method namely DTM has been used to solve the differential equation which governs the motion of three types of annular FGM plates. The differential equation has been obtained using Hamilton principle and classical plate theory. The mechanical properties of the plate (Young's modulus and density) are considered to be graded in thickness direction and vary following the power-law. The behaviour of volume fraction index and radii ratio has been investigated onto first three modes of frequency parameter for all three plates. Moreover, the novelty of this paper is the application of the versatile technique DTM to study the effect of radii ratio and volume fraction index on three different types of annular FGM plates. A comparison has been made between the obtained numerical results and the results are available in the literature. A good agreement of the results verifies the accuracy of the present technique. Three-dimensional mode shapes for all three plates are also presented. [ABSTRACT FROM AUTHOR]

This paper explores the existence and stability of implicit neutral Caputo fractional qdifference equations within four distinct classes, incorporating various delay types such as finite, infinite, and state-dependent delays. To establish the existence of solutions, we utilize the fixed point theorem of Krasnoselskii in Banach spaces. The concluding section provides illustrative examples that highlight the obtained results. [ABSTRACT FROM AUTHOR]

Purpose: In order to solve the problem that the existing models can't accurately reproduce the mechanical properties of metallic plates under complex working conditions, and the accuracy and efficiency can't be satisfied at the same time. The analysis of metallic plates by different refined finite elements is presented in this paper. The working efficiency and accuracy of the higher-order model in engineering applications are studied. Methods: The refined plate elements are based on several series expansion, and applied to the modeling and analysis of plate structures. The Carrera unified formulation is introduced to express the plate displacement field, the theoretical model of plate thickness expansion is established by using Taylor series expansion and Lagrange series expansion. The governing differential equations of metallic plate are established by using the principle of virtual displacements, the mass matrix and stiffness matrix of plate elements are deduced simultaneously. Finally, the shear locking phenomenon of the plate models is considered, tensor component mixed interpolation (MITC4) is used to revise the model. The accuracy and the reliability of the refined plate models are verified by comparing several order models and solid models generated in the commercial software ANSYS. Results and Conclusion: In this paper, the higher-order model has very low degree of freedoms (DOFs) on the premise of ensuring accuracy. And this modeling method can be used not only for thin plate analysis, but also for medium-thick plate analysis. Meanwhile, the refined plate model has high working efficiency and wide application range, which provides a new modeling method for the research of metallic plates. [ABSTRACT FROM AUTHOR]

This paper delves into the asymptotic and oscillatory behavior of all classes of solutions of fourth-order nonlinear neutral delay differential equations in the noncanonical form with damping terms. This research aims to improve the relationships between the solutions of these equations and their corresponding functions and derivatives. By refining these relationships, we unveil new insights into the asymptotic properties governing these solutions. These insights lead to the establishment of improved conditions that ensure the nonexistence of any positive solutions to the studied equation, thus obtaining improved oscillation criteria. In light of the broader context, our findings extend and build upon the existing literature in the field of neutral differential equations. To emphasize the importance of the results and their applicability, this paper concludes with some examples. [ABSTRACT FROM AUTHOR]

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called 'diffusion-wave-type solutions'. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties. [ABSTRACT FROM AUTHOR]

This research investigates the double tracking control problem for the complex dynamic network (CDN) with an unavailable link state. Firstly, from the angle of a large-scale system, the dynamical model of CDN is described by the vector differential equations, which consists of node dynamic subsystem (NDS) and link dynamic subsystem (LDS), in which the weighted-values of links are regarded as the state variables of LDS. Secondly, to realise the double tracking control (DT-Control) of CDN, the presented DT-Control scheme in this paper includes the synthesis of controller for NDS and the coupling term in LDS, which can ensure that the two subsystems track the given reference targets. The tracking of NDS contains the synchronisation of nodes as the special case, and the tracking of LDS shows that the eventual topologic structure of CDN will be determined only by the given link reference signal. Due to the economic and technological limitations of sensors in the practice applications, this paper assumes that the state variables of LDS are unavailable in the DT-Control scheme. Finally, the engineering simulation example is given to verify the validity of DT-Control scheme proposed in this paper. [ABSTRACT FROM AUTHOR]

The supersaturation equation for a vertically moving adiabatic cloud parcel is analyzed. The effects of turbulent updrafts are incorporated in the shape of a stochastic Lagrangian model, with spatial and time correlations expressed in terms of turbulent kinetic energy. Using the Fokker–Planck equation, the steady-state probability distributions of supersaturation are analytically computed for a number of approximations involving the time-scale separation between updraft fluctuations and phase relaxation, and droplet or ice particle size fluctuations. While the analytical results are presented in general for single-phase clouds, the calculated distributions are used to compute mixed-phase cloud properties—mixed fraction and mean liquid water content in an initially icy cloud—and are argued to be useful for generalizing and constructing new parameterization schemes. Significance Statement: Supersaturation is the fuel for the development of clouds in the atmosphere. In this paper, our goal is to better understand the supersaturation budget of clouds embedded in a turbulent environment by analyzing the basic equations of cloud microphysics. It is found that the turbulent characteristics of an air parcel substantially affect the cloud's supersaturation budget and hence its life cycle. This is also shown in the context of mixed-phase clouds where, depending on the turbulent regime, different liquid-to-ice ratios are found. Consequently, the theoretical approach of this paper is crucial to develop tools to parameterize small-scale atmospheric features, like clouds, into global circulation models to improve climate projections for the future. [ABSTRACT FROM AUTHOR]

In this paper, we study oscillatory properties of a fourth-order differential equation and spectral properties of a corresponding differential operator. These properties are established by first proving a new second-order Hardy-type inequality, where the weights are the coefficients of the equation and the operator. This new inequality, in its turn, is established for functions satisfying certain boundary conditions that depend on the boundary behavior of one of its weights at infinity and at zero. [ABSTRACT FROM AUTHOR]

Several research papers have attempted to describe the dynamics of COVID-19 based on systems of differential equations. These systems have taken into account quarantined or isolated cases, vaccinations, control measures, and demographic parameters, presenting propositions regarding theoretical results that often investigate the asymptotic behavior of the system. In this paper, we discuss issues that concern the theoretical results proposed in the paper "An Extended SEIR Model with Vaccination for Forecasting the COVID-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter". We propose detailed explanations regarding the resolution of these issues. Additionally, this paper focuses on extending the local stability analysis of the disease-free equilibrium, as presented in the aforementioned paper, while emphasizing the derivation of theorems that validate the global stability of both epidemic equilibria. Emphasis is placed on the basic reproduction number R 0 , which determines the asymptotic behavior of the system. This index represents the expected number of secondary infections that are generated from an already infected case in a population where almost all individuals are susceptible. The derived propositions can inform health authorities about the long-term behavior of the phenomenon, potentially leading to more precise and efficient public measures. Finally, it is worth noting that the examined paper still presents an interesting epidemiological scheme, and the utilization of the Kalman filtering approach remains one of the state-of-the-art methods for modeling epidemic phenomena. [ABSTRACT FROM AUTHOR]

This paper is devoted to a thin-walled sandwich plate with an individual functionally graded core. The nonlinear shear deformation theory of a straight normal line is applied. A system of three differential equations of equilibrium of this plate is obtained, based on the principle of stationary potential energy, which is reduced to two differential equations and solved analytically. The critical load of the rectangular sandwich plate is determined. A detailed analytical study is carried out for selected exemplary plates. Moreover, a numerical FEM model of this plate is developed. The results of these calculations are compared with each other. [ABSTRACT FROM AUTHOR]

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The primary objective of this paper is to create a novel infinite Toeplitz matrix by leveraging Tetranacci numbers. This matrix serves as the foundation for defining new sequence spaces denoted as c 0 ⁢ ( G ) {c_{0}(G)} , c ⁢ ( G ) {c(G)} , ℓ ∞ ⁢ ( G ) {\ell_{\infty}(G)} , and ℓ p ⁢ ( G ) {\ell_{p}(G)} , where 1 ≤ p < ∞ {1\leq p<\infty} . By utilizing this newly constructed matrix, the paper also explores and examines various algebraic and topological properties inherent to the sequence spaces c 0 ⁢ ( G ) {c_{0}(G)} , c ⁢ ( G ) {c(G)} , ℓ ∞ ⁢ ( G ) {\ell_{\infty}(G)} , and ℓ p ⁢ ( G ) {\ell_{p}(G)} for values of p within the range of 1 ≤ p < ∞ {1\leq p<\infty} . At last, we also prove existence theorem with example for infinite systems of differential equations in ℓ p ⁢ ( G ) {\ell_{p}(G)} . [ABSTRACT FROM AUTHOR]

In this paper, the reliability characteristics of a parallel system are investigated. The parallel system under consideration is made up of three active units that run in parallel, with two of them having to be operational in order for the system to work. The main purpose of this study is to quantify/examine the effect of online and offline preventive maintenance. Preventive maintenance is carried out on the systems in two ways: online and offline preventive maintenance. After the first unit of each system fails, online preventive maintenance is performed. Following the failure of the second unit of each system, offline preventive maintenance is performed. Partial and complete failures are the two types of failures that may occur. Both systems can undergo exponential failure and repair. Using supplementary variable technique, Laplace transform, and Copula repair approach, the system of first-order differential equations associated with system effectiveness, which are crucial to this research, is established and resolved. Tables and graphs are used to illustrate the important findings based on assumed numerical values. System designers, programmers, and maintenance supervisors will be able to create and maintain more crucial systems with the assistance of this research paper. [ABSTRACT FROM AUTHOR]

This paper deals with the probabilistic analysis of a class of compartmental models formulated via a system of linear differential equations with time‐dependent non‐homogeneous terms. For the sake of generality, we assume that initial conditions and rates between compartments are random variables with arbitrary distributions while the source terms defining the flows entering the compartments are stochastic processes. We then take extensive advantage of the so‐called random variable transformation (RVT) technique to determine the first probability distribution of the solution of such a randomized model under very general hypotheses. In the simplest but relevant case of time‐independent source terms, we also obtain the probability distribution of the equilibrium, which is a random vector. Furthermore, we first particularize the aforementioned results for different types of integrable source terms that permit obtaining an explicit solution of the compartmental model: random constants, a train of Dirac delta impulses with random intensities, and a Wiener process. Secondly, we show an alternative approach based on the Liouville–Gibbs equation, which is useful when dealing with source terms that do not admit a closed‐form primitive. All the previous theoretical results are first illustrated through several numerical examples and simulations where a wide range of different probability distributions are assumed for the model parameters. The paper concludes by applying a compartmental model that describes the dynamics of oral drug administration through multiple chronologically spaced doses using synthetic data generated according to pharmacological references. [ABSTRACT FROM AUTHOR]