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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]

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]

It is a fundamental problem to understand the complexity of high-accuracy sampling from a strongly log-concave density π on ℝd. Indeed, in practice, high-accuracy samplers such as the Metropolis-adjusted Langevin algorithm (MALA) remain the de facto gold standard; and in theory, via the proximal sampler reduction, it is understood that such samplers are key for sampling even beyond log-concavity (in particular, for sampling under isoperimetric assumptions). This article improves the dimension dependence of this sampling problem to \(\widetilde{O}(d^{1/2})\). The previous best result for MALA was \(\widetilde{O}(d)\). This closes the long line of work on the complexity of MALA and, moreover, leads to state-of-the-art guarantees for high-accuracy sampling under strong log-concavity and beyond (thanks to the aforementioned reduction). Our starting point is that the complexity of MALA improves to \(\widetilde{O}(d^{1/2})\) , but only under a warm start (an initialization with constant Rényi divergence w.r.t. π). Previous algorithms for finding a warm start took O(d) time and thus dominated the computational effort of sampling. Our main technical contribution resolves this gap by establishing the first \(\widetilde{O}(d^{1/2})\) Rényi mixing rates for the discretized underdamped Langevin diffusion. For this, we develop new differential-privacy-inspired techniques based on Rényi divergences with Orlicz–Wasserstein shifts, which allow us to sidestep longstanding challenges for proving fast convergence of hypocoercive differential equations. [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]

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]

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and j th largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes. [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 develops a novel set of displacement temperature potential functions to solve the thermoelastodynamic problems in functionally graded transversely isotropic media subjected to thermal source. For this purpose, three-dimensional heat and wave equations are considered to obtain the displacement temperature equations of motion for functionally graded materials. In the present study, a systematic method is used to decouple the elasticity and heat equations. Hence one sixth-order differential equation and two second-order differential equations are obtained. Completeness of the solution is proved using a retarded logarithmic Newtonian potential function for functionally graded transversely isotropic domain. To verify the obtained solution, in a simpler case, potential functions are generated for homogeneous transversely isotropic media that coincide with respective equations. Presented potential functions can be used to solve the problems in various media like infinite and semi-infinite space, beams and columns, plates, shells, etc., with arbitrary boundary conditions and subjected to arbitrary mechanical and thermal loads. [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]

This paper points out the interval form deficiencies in the recent paper "An application of parametric approach for interval differential equation in inventory model for deteriorating items with selling-price-dependent demand" by Rehman et al. (Neural Comput Appl 32(17):14069–14085, 2020). The comments in this paper pertain mainly to the drawbacks of interval arithmetic (SIA) presented in Section 3 of Rehman et al. (2020) referencing the convention of converting an interval-valued model into a parametric form. Proposed corrections to the overestimation of the interval form through examples are presented. [ABSTRACT FROM AUTHOR]

This work explores a well-established eighth-algebraic-order numerical method belonging to the explicit Numerov-type family. To enhance its efficiency, we integrated a cost-effective algorithm for adjusting the step size. After each step, the algorithm either maintains the current step length, halves it, or doubles it. Any off-step points required by this technique are calculated using a local interpolation function. Numerical tests involving diverse problems demonstrate the significant efficiency improvements achieved through this approach. The method is particularly effective for solving differential equations with oscillatory behavior, showcasing its ability to maintain high accuracy with fewer function evaluations. This advancement is crucial for applications requiring precise solutions over long intervals, such as in physics and engineering. Additionally, the paper provides a comprehensive MATLAB-R2018a implementation, facilitating ease of use and further research in the field. By addressing both computational efficiency and accuracy, this study contributes a valuable tool for the numerical analysis community. [ABSTRACT FROM AUTHOR]

This paper focuses on the finite-time tracking control problem of shield tunneling systems in the presence of constraints on the states and control input. By modeling the system based on the LuGre friction model, an effective method of tracking control in finite time is designed to overcome these actual constraints at the same time. First, the constraint on the system state is transformed into a symmetric constraint on the tracking error, and the constraint on control input is handled by designing an auxiliary differential equation. Then, radial basis function (RBF) neural networks are introduced to approximate the uncertainties. Next, using an adaptive finite-time backstepping method and choosing a logarithmic barrier Lyapunov function (BLF), a finite-time controller is designed to realize the finite-time stability of the closed-loop system. Finally, a simulation example is given to verify the correctness and validity of the theoretical results. [ABSTRACT FROM AUTHOR]

Nonlinearity presents a significant challenge in developing quantum algorithms involving differential equations, prompting the exploration of various linearization techniques, including the well-known Carleman Linearization. Instead, this paper introduces the Koopman Spectral Linearization method tailored for nonlinear autonomous ordinary differential equations. This innovative linearization approach harnesses the interpolation methods and the Koopman Operator Theory to yield a lifted linear system. It promises to serve as an alternative approach that can be employed in scenarios where Carleman Linearization is traditionally applied. Numerical experiments demonstrate the effectiveness of this linearization approach for several commonly used nonlinear ordinary differential equations. [ABSTRACT FROM AUTHOR]

In this work, we use two unique methodologies, the homotopy perturbation transform method and Yang transform decomposition method, to solve the fractional nonlinear seventh-order Kaup–Kupershmidt (KK) problem. The physical phenomena that arise in chemistry, physics, and engineering are mathematically explained in this equation, in particular, nonlinear optics, quantum mechanics, plasma physics, fluid dynamics, and so on. The provided methods are used to solve the fractional nonlinear seventh-order KK problem along with the Yang transform and fractional Caputo derivative. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approaches and the fractional operator to develop satisfactory approximations to the offered problem. To solve the fractional KK equation, we first use the Yang transform and fractional Caputo derivative. He's and Adomian polynomials are useful to manage nonlinear terms. It is shown that the suggested approximate solution converges to the exact one. In these approaches, the results are calculated as convergent series. The key advantage of the recommended approaches is that they provide highly precise results with little computational work. The suggested approach results are compared to the precise solution. By comparing the outcomes with the precise solution using graphs and tables we can verify the efficacy of the offered strategies. Also, the outcomes of the suggested methods at various fractional orders are examined, demonstrating that the findings get more accurate as the value moves from fractional order to integer order. Moreover, the offered methods are innovative, simple, and quite accurate, demonstrating that they are effective for resolving differential equations. [ABSTRACT FROM AUTHOR]

Vibration is a mechanical phenomenon that causes oscillations around an equilibrium point. Vibration is undesired in many areas, particularly engineered systems and inhabited spaces, and strategies to avoid vibration transfer to such systems have been proposed. Mechanical waves carry vibrations, and certain mechanical couplings transport vibrations more efficiently than others. By detecting equipment defects, vibration analysis (VA) in an industrial or maintenance setting seeks to save maintenance costs and downtime. Associated models from vibration settings can either be linear or nonlinear based on the modeling situation. This paper considers the Zhou Transform Method (ZTM) as a semi-analytical approach for obtaining the solutions of linear vibration mechanical models. Two illustrative examples are considered. The results obtained from the ZTM are compared with their exact forms for applicable case(s). The results show that the proposed method is very effective and reliable in obtaining approximate solutions based on the comparison. [ABSTRACT FROM AUTHOR]

Image encryption stands out as a crucial technique employed to securely transmit images across the Internet. In this paper, we introduce a novel algorithm for encrypting color images. The algorithm is built upon the principles of differential equations, XOR operations, and chaotic maps. First, the plain image is three-dimensional pixel shuffled via a logistic map. Afterward, the differential equations are used as a mathematical tool for encrypting images. The third-order ordinary differential equations are used to encrypt the shuffled images. The color values of the plain image are considered coefficients for the independent variable X. Subsequently, an alternate matrix of the same size is generated using a three-dimensional logistic map, taking into account its color values as the exponents linked to the independent variable X. A set of third-order differential equations emerged, containing an equivalent number of elements as the color values present in the plain image. This set of differential equations is addressed in the following manner: combining XOR and integration three times with respect to the independent variable X for each set of obtained differential equations while treating the integration constant as zero. Ultimately, a set of ordinary equations involving the independent variable X is derived, where the coefficients of X represent color values for the cipher image. The results from experiments and the security analysis affirm the resilience of the proposed algorithm against established security attacks. It exhibits a substantial key space, heightened key sensitivity, and a strong encryption effect. [ABSTRACT FROM AUTHOR]

In 2006, Alexander proved a result that implies for a Weingarten surface Κ1/Κ2 = c > 0, if n is the number of times a closed geodesic winds around the axis of rotation and m is the number of times the geodesic oscillates about the equator, then n/m ∈ in [1, 1/c) when 0< c <1 and n/m ∈ in (1/surd c, 1] when c >1. In this paper, we present another proof of Alexander's result for the Weingarten surfaces Κ1/Κ2 =c >0 that is simpler and more direct. Our approach uses sharp estimates of certain improper integrals to obtain the intervals for permissible ratios n/m. We numerically compute a number of closed geodesics for various combinations of (c, n,m) to illustrate the variety of patterns that are possible. [ABSTRACT FROM AUTHOR]

The present paper studies the asymptotic and oscillatory properties of solutions of odd-order differential equations with advanced arguments and in a noncanonical case. By providing new and effective relationships between the corresponding function and the solution, we present strict and new criteria for testing whether the studied equation exhibits oscillatory behavior or converges to zero. Our results contribute uniquely to oscillation theory by presenting some theorems that improve and expand upon the results found in the existing literature. We also provide an example to corroborate the validity of our proposed criteria. [ABSTRACT FROM AUTHOR]

This paper analyzes the dynamic behavior of a fishery model described by differential algebraic equations. Two patches, namely free fishing area and protected area, are included in the model. The migration of fish is symmetrical, i.e., the fish can migrate between the two patches. It is observed that a singularity-induced bifurcation occurs when the economic benefit of harvesting changes. When the economic benefit is positive, a state feedback controller is added to stabilize the system. Some examples and numerical simulations are presented to verify the theoretical results. In addition, harvesting of prey populations is used as a control measure to obtain the maximum economic benefits and ecological sustainability. The optimal solution is derived by using Pontryagin's maximum principle. Through extensive numerical simulations, it is shown that the optimal solution is capable of achieving ecosystem sustainability. [ABSTRACT FROM AUTHOR]

The performance degradation of initiating explosive devices is influenced by various internal and external factors, leading to uncertainties in their reliability and lifetime predictions. This paper proposes an uncertain degradation model based on uncertain differential equations, utilizing the Liu process to characterize the volatility in degradation rates. The ignition delay time is selected as the primary performance parameter, and the uncertain distributions, expected values and confidence intervals are derived for the model. Moment estimation techniques are employed to estimate the unknown parameters within the model. A real data analysis of ignition delay times under accelerated storage conditions demonstrates the practical applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we present a general theory for fractional-order sequential differential equations with Riemann–Liouville nabla derivatives and Caputo nabla derivatives on time scales. The explicit solution, in the case of constant coefficients, for both the homogeneous and the non-homogeneous problems, are given using the ∇-Mittag-Leffler function, Laplace transform method, operational method and operational decomposition method. In addition, we also provide some results about a solution to a new class of fractional-order sequential differential equations with convolutional-type variable coefficients using the Laplace transform method. [ABSTRACT FROM AUTHOR]

Ultra-soft soil is distributed in coastal areas around the world and has poor engineering properties. There is a significant difference in settlement between semi-rigid pile and surrounding soil under embankment load. Based on existing research results, the settlement calculation formula of ultra-soft soil composite foundation reinforced by semi-rigid pile is derived in this paper. Based on the Alamgir displacement model, assuming a three-zone model of pile skin friction with a negative skin friction plastic zone in the upper part of the pile, an elastic zone in the middle part of the pile, and a skin friction-bearing plastic zone in the lower part of the pile, the upward and downward penetrations of pile, and pile–soil slip deformation characteristics are considered. Analytical expressions for settlement calculations of semi-rigid pile composite foundations under embankments were derived based on differential equations for pile–soil load transfer in the unit cell. The influences of pile diameter and the compression modulus of the underlying layer at the pile end on the settlement characteristics of the semi-rigid pile composite foundation are discussed. The results show that the derived theoretical calculation method is in good agreement with the field measurement and laboratory model test results. Ultra-soft soil composite foundations have long settlement stabilization times and large settlement deformations. Penetration deformation occurs at the semi-rigid pile end. The relationship between pile end resistance and pile end piercing deformation is hyperbolic. The compression modulus of the underlying layer has a great influence on pile end penetration. The lower the compression modulus of the underlying layer, the larger the penetration deformation of pile end. The larger the pile diameter is, the smaller the penetration deformation is. [ABSTRACT FROM AUTHOR]

This paper addresses the active target protection problem in a three-player (Target–Attacker–Defender, TAD) conflict by proposing a cooperative integrated guidance and control (IGC) strategy. Unlike previous studies that have designed guidance and control loops separately, this work establishes an IGC model by linearizing both the translational motion and the rotational motion of the vehicles, thereby generating actuator commands directly. This model integrates the kinematics and short-period dynamics, providing a more comprehensive and accurate representation of the vehicles' characteristics. Based on the linearization and order reduction, differential game theory and the sweep method are employed to derive and analytically solve the Riccati differential equation, yielding an optimal control strategy with an explicit expression. The theoretical rigor of the proposed approach is ensured through a proof of optimality sufficiency. Furthermore, factors influencing the computational accuracy of the Riccati equation solution, including the singular values of the control matrix and condition numbers of the solution matrix, are analyzed. Taking into account the dynamic response and limitations of the actuators, numerical simulations demonstrate the effectiveness and superiority of the proposed IGC strategy in intercepting the attacker and protecting the target compared to traditional separated guidance and control designs. [ABSTRACT FROM AUTHOR]

In this paper, we present a highly efficient analytical method that combines the Laplace transform and the residual power series approach to approximate solutions of nonlinear time-fractional partial differential equations (PDEs). First, we derive the analytical method for a general form of fractional partial differential equations. Then, we apply the proposed method to find approximate solutions to the time-fractional coupled Berger equations, the time-fractional coupled Korteweg–de Vries equations and time-fractional Whitham–Broer–Kaup equations. Secondly, we extend the proposed method to solve the two-dimensional time-fractional coupled Navier–Stokes equations. The proposed method is validated through various test problems, measuring quality and efficiency using error norms E 2 and E ∞ , and compared to existing methods. [ABSTRACT FROM AUTHOR]

This paper presents π HyFlow, a modular approach to the process interaction worldview (PI). Traditionally, PI supports a set of interacting processes without enabling modular and hierarchical model definition. π HyFlow basic models define a dynamic set of processes while also enabling a modular interface for supporting model composition. π HyFlow allows an exact representation of continuous signals based on the concepts of dense outputs, and generalized sampling. π HyFlow ability to support accurate models of hybrid systems is presented through a DC-DC converter (DCC) based on a resistor–capacitor (RC) electrical circuit. The DCC output voltage can be modeled by a first-order differential equation that is solved using an exponential time differencing (ETD) integrator. The DCC network model uses a digital on-off controller and an ETD. Simulation results are generated by π HyFlow ++, a C++ implementation of the formalism. [ABSTRACT FROM AUTHOR]

In this paper, we address the effects of two fiscal policy delays on the dynamical analysis of macroeconomics. First, a time gap between the accrual of taxes and their payment is considered. Second, the time spent between the purchasing decisions and the actual expenditure is also taken into consideration. Since both these delays are significant in controlling macroeconomic conditions, this paper incorporates aforementioned delays into the IS–LM model. At first, a mathematical model is developed using delayed differential equations. Then a unique steady state solution is obtained. Around the equilibrium point, linear stability analysis is done. Also, the occurance of Hopf bifurcation is observed when delay crosses a critical point and switches in stability are also detected. Properties of Hopf bifurcation using center manifold theorem are discussed. Lastly, numerical simulations are run to verify our analysis. In this work, we considered a case study to perform simulation wherein GDP of India for last ten years is recorded for estimating some parameters. In different investment scenarios, numerical simulations corroborate the analytical findings of the model. Furthermore, rigorous analysis shows that adding the right mix of delays can help in maintaining/ regaining the stability after periods of instability, or even gaining stability in the long run. [ABSTRACT FROM AUTHOR]

Parameter identifiability methods assess whether the parameters of a model are uniquely determined by the observations. While the success of a model fit can provide some information on this, it can be valuable to determine identifiability before any fit has been attempted, or to separate identifiability from other issues. Two concepts that lean themselves well for identifiability analysis and have been underutilized are the sensitivity matrix (SM) and the Fisher information matrix (FIM). This paper presents two newly developed methods, one based on the SM and one based on the FIM. Both methods can assess local identifiability for a wide set of models, can be used with limited effort, and are freely available. The methods require the proposed model in the form of a set of differential equations, the parameter values, and the study design as input. They can be used a priori, as they do not need observed values or a successful model fit. Traditional methods provide a single categorical (yes/no) answer to the question of identifiability. In many cases, this is not very informative, and identifiability depends on study design (e.g., dose levels or observation times) and parameter values. Indicators on a continuous scale characterizing the level of identifiability would provide more detailed and relevant information, for example, to guide model development. Our two methods provide both categorical and continuous indicators. Both methods indicate which parameter combinations are difficult to identify by calculating the directions in parameter space that are least identifiable. The methods were validated with an example problem. [ABSTRACT FROM AUTHOR]

Circuit simulators are fundamentally used for solving electric circuit problems with different degrees of complexity in which node voltages and branch currents are the unknowns. This is fully understandable since they were originally created for this specific task. However, behind the curtains, powerful simulation engines based on a variety of numerical techniques operate so as to always comply with Kirchhoff's current and voltage laws. In this paper, it is shown how a simple circuital configuration, referred to as the elementary solver, consistent in two behavioral current sources in series, can be used to solve mathematical problems that go beyond electronics. Of course, the intention is not to substitute mathematical packages with well-proven calculus capacity but to increase the scope of circuit simulators for their application in other areas of research or simply for educational purposes. It is worth mentioning that no special programming skills are required (except a basic knowledge of the available tools and language) and, furthermore, that the user can operate exclusively in a graphical environment. It is shown, throughout a series of selected examples, how the method of elementary solvers (MES) works, providing a new and practical dimension to the applicability of circuit simulators. [ABSTRACT FROM AUTHOR]

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using L2 and L∞ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model. [ABSTRACT FROM AUTHOR]

This paper employed the well-known Riccati transformation method to deduce a Kneser-type oscillation criterion for second-order dynamic equations. These results are considered an extension and improvement of the known Kneser results for second-order differential equations and are new for other time scales. We have included examples to highlight the significance of the results we achieved. [ABSTRACT FROM AUTHOR]

In this paper, an SIRS epidemic model using Grunwald–Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases. The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number. The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter. Additionally, using the Grunwald–Letnikov approximation, the simulation is carried out to confirm the validity of analytic results graphically. Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021, we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one. Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new framework for deriving partitioned implicit-exponential integrators for stiff systems of ordinary differential equations and construct several time integrators of this type. The new approach is suited for solving systems of equations where the forcing term is comprised of several additive nonlinear terms. We analyze the stability, convergence, and efficiency of the new integrators and compare their performance with existing schemes for such systems using several numerical examples. We also propose a novel approach to visualizing the linear stability of the partitioned schemes, which provides a more intuitive way to understand and compare the stability properties of various schemes. Our new integrators are A-stable, second-order methods that require only one call to the linear system solver and one exponential-like matrix function evaluation per time step. [ABSTRACT FROM AUTHOR]

In this paper, the existence and number of non-contractible limit cycles of the Josephson equation $ \beta \frac{d^{2}\Phi}{dt^{2}}+(1+\gamma \cos \Phi)\frac{d\Phi}{dt}+\sin \Phi = \alpha $ are studied, where $ \phi\in \mathbb S^{1} $ and $ (\alpha,\beta,\gamma)\in \mathbb R^{3} $. Concretely, by using some appropriate transformations, we prove that such type of limit cycles are changed to limit cycles of some Abel equation. By developing the methods on limit cycles of Abel equation, we prove that there are at most two non-contractible limit cycles, and the upper bound is sharp. At last, combining with the results of the paper (Chen and Tang, J. Differential Equations, 2020), we show that the sum of the number of contractible and non-contractible limit cycles of the Josephson equation is also at most two, and give the possible configurations of limit cycles when two limit cycles appear. [ABSTRACT FROM AUTHOR]