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A two-phase method for estimation of the scattering and absorption coefficients and the asymmetry factor (σs, σa and g) in the radiative transfer problem is presented. The first phase parameterizes σs and σa through g via a simplified model and performs-at a relatively low cost-a scalar optimization over g. It is shown that this gives such a good starting point that the second phase can be accurately performed by a simple Gauss-Newton method. It is also shown that as a part of the first phase can be used on its own when only σs and σa are wanted, and it is noted that this gives higher accuracy than the commonly used Kubelka-Munk method when using standardized paper industry reflectance factor measurements. The parameter estimation problem is shown to be non-trivial and ill-conditioned, and its character is analysed. It is discussed that as standard optimization methods are so sensitive to the choice of starting point for this problem that it is hard to find a starting point that gives convergence at all. The new two-phase method is illustrated by application to relevant paper industry problems, and efficiency and sensitivity measures are given. [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]

It is well-known that the Lotka–Volterra predator-prey model has a family of periodic orbits, but does not possess limit cycles and therefore the model is said to be structurally unstable. The Lotka–Volterra model is a special case of a much larger group namely the quadratic population models and it can be shown that none of them can produce limit cycles. The surprising finding is that by combining two quadratic models a quadratic population model with two limit cycles is uncovered. Although the model looks simple at first glance it provides a rich source of dynamics and deserves attention. In this paper, we revisit a model that has its origin in the work of Dubois and Closset. A set of two quadratic population models interact as piecewise defined differential equations. The model has been discussed by Ren Yongtai and Han Li, cryptically written and showing some linguistic and typographical errors, but providing an excellent vehicle for developing skills in mathematical modelling, differential equations and technology for the young researcher. We explore the model in clearer detail and supplement the theory with rich graphical illustration. The paper has the purpose of providing an example of how a young researcher, such as a postgraduate student in biomathematics, can expand on an existing model by making use of current technology. [ABSTRACT FROM AUTHOR]

The paper provides an analytical solution to the differential equation that regulates the motion of the neo-Kaleckian model in the short run. After presenting a simple open economy neo-Kaleckian model with government activity, the paper analytically derives an expression for the time of adjustment, defined as the time required for the system to make a k percent adjustment from one steady-state to another. The solution shows that there is an inverse relationship between the time of adjustment and (i) the strength of the Keynesian stability condition; (ii) the behavior of entrepreneurs underlying their decisions to more rapidly/slowly respond to changes in goods market conditions. Last, the model is calibrated for the US, showing that the vicinity of the new equilibrium is reached after a period of about 5 quarters under a baseline calibration. By formally analyzing the out-of-equilibrium trajectory of the neo-Kaleckian model, this contribution moves away from the method of comparative dynamics and provides a historical-time representation of the model's traverse. [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 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]

Recently, the author determined the higher-order differential operator having the Jacobi-Sobolev polynomials as its eigenfunctions for certain eigenvalues. These polynomials form an orthogonal system with respect to an inner product equipped with the Jacobi measure on the interval $ [-1,1] $ [ − 1 , 1 ] with parameters $ \alpha \in \mathbb {N}_{0},\beta \gt -1 $ α ∈ N 0 , β > − 1 and two point masses N, S>0 at the right end point of the interval involving functions and their first derivatives. The first purpose of the present paper is to reveal how the Jacobi-Sobolev equation reduces to the differential equation satisfied by the Laguerre-Sobolev polynomials on the positive half line via a confluent limiting process as $ \beta \rightarrow \infty $ β → ∞. Secondly, we explicitly establish the differential equation for the symmetric Gegenbauer-Sobolev polynomials by employing a quadratic transformation of the argument. Each of the three differential operators involved is of order $ 4\alpha +10 $ 4 α + 10 and symmetric with respect to the corresponding Sobolev inner product. [ABSTRACT FROM AUTHOR]

In this paper, a kind of the differential equation including a time-fractional sub-diffusion equation is considered. Through this memorandum, a well-known technique, in the time direction is adopted by the p-fractional linear multistep method (p-FLMM) according to the q-fractional backward difference formula (q-FBDF) of implicit type for q = 1, 2, 3, and the spatial direction is approximated by the second-order central difference method. The stability properties of the proposed method can be investigated in combination with the Fourier technique and $ \mathcal {Z} $ Z -transformation and also its convergence is studied by using the truncated error maximum. It is shown that the method is unconditionally stable and the orders of convergence are $ \mathcal {O}(\tau ^{p}+h^2) $ O (τ p + h 2) for $ 1 \leq p \leq ~4 $ 1 ≤ p ≤ 4 , in which p is the order of accuracy in the time direction and τ and h determine temporal and spatial stepsizes, respectively. Some numerical experiments are included to demonstrate the validity and applicability of the scheme. [ABSTRACT FROM AUTHOR]

Epidemiological models have enhanced relevance because of the COVID-19 pandemic. In this note, we emphasize visual tools that can be part of a learning module geared to teaching the SIR epidemiological model, suitable for advanced undergraduates or beginning graduate students in disciplines where the level of prior mathematical knowledge of students may not be very strong. Visual tools – phase portrait, flow field and trajectory and line plots – available in the R software are presented in a step by step manner, moving from the exponential growth model to the logistic growth model and then to the SIR model. Code for numerical simulation of differential equations and estimation of parameters is presented for the SIR model. Suggestions for students to connect the learning from these examples with research papers on COVID-19 are provided. [ABSTRACT FROM AUTHOR]

There are numerous phenomena in real world that their practical modeling in mathematics language deals with differential equations. Stochastic terms have significant roles in estimating behavior of these events such as machine tool vibrations, biology, traffic dynamics, neural networks and so on, it cannot be reasonable to consider an event without its past information, and it ends up with equations including time delays. The numerical solution of stochastic delay differential equations has always been of interest to researchers. In this paper, by using scaling function of the Daubechies wavelet, a collocation method is presented to solve a class of stochastic delay differential equations and it is tried to demonstrate the advantages of using this type of wavelet. Convergence of the method is provided and numerical experiments are reported to show application of the method in practice. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying a dynamic adhesive contact model where the body is composed of a viscoelastic material with long memory. The adhesion process is modeled by a bonding field on part of the contact surface and on which the contact and the friction are modeled by nonmonotone Clarke subdifferential conditions with adhesion. Meanwhile, on another part of the contact surface, the contact is comprised of a normal compliance contact condition while the memory effects of the obstacle are considered, the friction is modeled by a version of Coulomb's law of dry friction and the friction bound is dependent on the normal stress. The variational form of the model generates a system of a doubly-history dependent variational–hemivariational inequality as well as a differential equation. Then, on the basis of the proof of abstract variational–hemivariational inequality, the existence and uniqueness results of the contact problem are established. [ABSTRACT FROM AUTHOR]

This paper deals with integral control of a class of switched affine systems characterised by time-varying differential equations that, along with the discontinuities caused by the switching, also depend periodically on an exogenous parameter, which is an affine function of time. The goal is to design a state-dependent switching function with integral action that ensures global asymptotic tracking of a trajectory of interest. The integral action is responsible for providing robustness to the system against uncertainties and load variations, whenever the discrepancy between the model and the real system is sufficiently bounded. Due to the integrator, all the convex combinations of the dynamic matrices are non-Hurwitz. In this case, the main difficulty is to guarantee that the time derivative of the Lyapunov function is strictly negative definite, which is done by means of LMIs and conditions on the affine terms. To the best of our knowledge, this is the first Lyapunov-based switching rule with integral action able to ensure global asymptotic tracking of a pre-specified profile. When applied to AC circuits the advantage is to design the switching rule based on the original system, without resorting to averaged models. The theory is illustrated by the control of a three-phase AC-DC converter. [ABSTRACT FROM AUTHOR]

This paper is written to provide ideas and guide faculty who want to design a mathematical modeling course for undergraduate mathematics majors and minors. We discuss course goals, assignments, and projects that can be used to help students gain experience relevant for careers and mathematical modeling opportunities. The authors designed this course to build students' mathematical thought processes and toolbox, ability to analyze and evaluate mathematical models, mathematical modeling skills, and teamwork skills. The course described is intended as an upper division undergraduate course with a prerequisite of differential equations. [ABSTRACT FROM AUTHOR]

Mathematical modelling has great potential to motivate students towards studying mathematics. This article discusses several different approaches to integrating research work with a second-year undergraduate, mathematical modelling subject. I found sourcing papers from the areas of epidemiology and ecology to be a fruitful source area, particularly models involving only two or three coupled differential equations. These models were amenable to students as well as interesting and relevant to students because they came from real research papers. I will describe the use of scenarios and case studies in lectures, and group projects for assessment. The scenarios and case studies were published in a textbook that I wrote. Scenarios, case studies and projects provided an opportunity to expose students to some novel applications of differential equations. One example is developed here as a Classroom Note: modelling the dynastic cycles in Chinese history. [ABSTRACT FROM AUTHOR]

The authors present numerical analysis of singularly perturbed parabolic problems. The previous papers in this direction focussed on the convection-diffusion problems with regular type of boundary layers. It is very difficult and almost impossible as stated in the literature, e.g, in Chapter 14 of 'Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions' (1996) by Miller, John JH and O'Riordan, Eugene and Shishkin, Grigorii I, to capture singularities due to parabolic layers and develop a robust scheme for reaction-diffusion problems with general values of space shifts. The work is in progress now and this is the first paper in this direction. In this work, we are investigating such problems and develop a robust numerical scheme using Mickens's non-standard finite difference scheme and special mesh. Furthermore, numerical examples have been presented to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is to propose a new numerical scheme based on the Adams-Bashforth method for solving two component time fractional differential equations involving the Caputo-Prabhakar fractional derivative of order μ. The existence and uniqueness of the solutions based on the Adams-Bashforth method reported for two component differential equations are discussed. Also, in this paper stability and convergence are demonstrated. Moreover, two numerical example are demonstrated in order to show the validity and reliability of the proposed methods. [ABSTRACT FROM AUTHOR]

The dynamics of a simple pendulum are often presented to undergraduate engineering students in introductory courses in dynamics. It is usually the first dynamic system considered by students that is modelled by a differential equation. This paper presents the standard material given to students. It is fair to say that students are accepting this material, but many do not fully appreciate its importance as a basis for the analysis of other dynamic models. Students also believe that the behaviour of a forced simple pendulum is obvious and is not worthy of a deep analysis. This paper presents many interesting results that are available in the open literature, but is not usually presented to undergraduate students. Some of these are counter-intuitive results such as unstable points becoming stable when forced and deterministic models giving chaotic pendulum trajectories. Many results demonstrate that simple pendulums are very interesting and worthy of analysis. The extension of a forced simple pendulum to an application that all students have experience of a child playing on a swing is presented. Only a limited analysis of the swing models is presented. This leaves some open questions for students and lecturers alike to explore. [ABSTRACT FROM AUTHOR]

This paper introduces a real-time co-simulation model for the prediction of the dynamics behavior of a veneer peeling lathe machine. The real-time co-simulation model introduced is described in the framework of multi-body system dynamics. It makes use of a co-simulation procedure in which a semi-recursive multi-body approach is combined with a detailed contact description. In the employed closed-loop semi-recursive method, the constraint conditions are carried out in the system motion differential equations via the augmented Lagrangian method. A contact procedure is presented to describe the contact between the wooden log and the cutting blade and the support rollers. The results obtained by the co-simulation are compared against those experimentally measured through a peeling lathe machine in operation and an approximate finite element model of the peeling process. According to the numerical results, it is concluded that the real-time simulation model can predict dynamic performance of the complicated system of a veneer peeling lathe in real-time. Solvability of equations of motion makes the introduced model suitable to be utilized in a number of product process purposes such as marketing, training and product development projects. [ABSTRACT FROM AUTHOR]

From the perspective of large system, a directed complex dynamic network (DCDN) is regarded as being made up of the nodes subsystem (NS) and the links subsystem (LS), which are coupled to each other. Different from previous studies, which propose the dynamic model of LS with the matrix differential equations, this paper describes the dynamic behaviour of LS with the outgoing links vector at every node, by which the dynamic model of LS can be represented as the vector differential equation to form the outgoing links subsystem (OLS). Due to the fact that vectors have more flexible mathematical properties than matrices, this paper proposes the more convenient mathematic method to investigate the double tracking control problem of NS and OLS. Under the condition that the states of NS are available and the states of OLS are unavailable, the asymptotical state observer of OLS is designed, by which the tracking controllers of NS and OLS are synthesised to ensure achieving the double tracking goals. Finally, an example simulation for supporting the theoretical results is also provided. [ABSTRACT FROM AUTHOR]

Ride-share platforms are contemporary businesses that match passengers with drivers, unlike taxis that can be hailed from the street. In the literature, the problem of optimizing the operations of such companies is mostly considered in static settings. We use in this paper a dynamic model and propose differential equations to model the evolution of the system. The objective is to maximize the profit during the planning horizon. Using optimal control theory, we determine the optimal rate of change in the ride price rate. An illustrative example along with sensitivity analyses shows the effect of the system parameters on the optimal solution obtained. [ABSTRACT FROM AUTHOR]

In this paper a nonlinear string-mass structure of the vibration absorber is analyzed. This structure is convenient to be installed in vibration damping systems of high buildings for their protection in the case of earthquake. The considered string-mass structure contains a translator movable mass connected with two strings. Due to nonlinear geometric properties of the system the motion of the mass is described with a strong nonlinear second order differential equation. In the paper the approximate procedure for solving of the nonlinear equation of motion is developed. Based on the solution the influence of the string preloading force, slider mass and friction force on the vibration property of the string-mass system is investigated. It is concluded that variation of the preloading string force may be applied as a control parameter for vibration absorption and as the regulator of vibration decay time. [ABSTRACT FROM AUTHOR]

In this paper, an uncertain nonlinear switched system is actually a nonlinear switched system disturbed by subjective uncertainties, usually written as several uncertain differential equations. Stability analysis of switched systems has been discussed in depth, while few results about stability of uncertain switched systems were published before. To fill this gap, stability in measure and asymptotic stability of uncertain nonlinear switched systems with countable switches in infinite-time horizon are both considered. The essential property of the solutions to such uncertain switched systems will be illustrated and captured from different perspectives with the help of the above stability analysis. Two theorems to judge these stabilities are proposed and then verified according to uncertainty theory and Lyapunov's second method. At last, a practical model of population growth is studied to display the validness of the results derived. [ABSTRACT FROM AUTHOR]

Stiffness properties of structural members, such as beam, plate and shell, can change drastically in the presence of axial forces due to geometric effects of the nonlinear strain components. In this paper, the stability behaviour of beam-column is investigated using the governing differential equation and compared with the geometrically nonlinear finite element analysis. The lateral deflection obtained from the theoretical model matches quite accurately with the numerical values for wide range of axial to critical load ratio P/Pcr. It is shown that bending stiffness decreases linearly with the axial load. By extending the theory, an expression for the membrane stiffness of the beam-column is presented in this paper. The geometrically nonlinear finite element analysis can capture exactly the parabolic variation of the membrane stiffness as per the derived expression. It increases initially up to P/Pcr = 0.35 and decreases rapidly to negligible value near the critical load indicating buckling instability. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some finite-time control problems for stochastic linear time-varying systems, described by an Itô type differential equation. In particular, the annular stochastic finite-time stability control problem is dealt with. In this context, it is required that the norm of the state variables remains bounded between an inner and an outer ellipsoid, during a finite-interval of time. The first contribution of the paper consists of two sufficient conditions for stability, which are derived by exploiting an approach based on time-varying quadratic Lyapunov functions. The first condition requires the solution of two generalised differential Lyapunov equations; the latter the existence of a feasible solution to a pair of differential linear matrix inequalities. As particular cases, we obtain sufficient conditions for stochastic finite-time stability (in this last case, the lower ellipsoid collapses to the origin). From the analysis conditions, a sufficient condition is derived for the annular stochastic finite-time stabilisation via both state and output feedback. Some numerical examples illustrate the application of the proposed methodology and show that our approach attains less conservative results than those obtainable with the existing literature. Moreover, an application of the annular stochastic finite-time stability approach to the stabilisation problem of a satellite with respect to the geomagnetic field is presented. [ABSTRACT FROM AUTHOR]

AbstractThis paper aims to investigate a class of boundary value problems generated by a third-order differential equation with distributional potentials and some dissipative boundary conditions. We first construct an operator related to the problem, then prove that the operator is dissipative and show some eigenvalue properties of this operator. Further using Krein’s theorem we prove the completeness theorems of the operator. [ABSTRACT FROM AUTHOR]

AbstractMultistate Markov models are a canonical parametric approach for data modeling of observed or latent stochastic processes supported on a finite state space. Continuous-time Markov processes describe data that are observed irregularly over time, as is often the case in longitudinal medical data, for example. Assuming that a continuous-time Markov process is time-homogeneous, a closed-form likelihood function can be derived from the Kolmogorov forward equations – a system of differential equations with a well-known matrix-exponential solution. Unfortunately, however, the forward equations do not admit an analytical solution for continuous-time, time-inhomogeneous Markov processes, and so researchers and practitioners often make the simplifying assumption that the process is piecewise time-homogeneous. In this paper, we provide intuitions and illustrations of the potential biases for parameter estimation that may ensue in the more realistic scenario that the piecewise-homogeneous assumption is violated, and we advocate for a solution for likelihood computation in a truly time-inhomogeneous fashion. Particular focus is afforded to the context of multistate Markov models that allow for state label misclassifications, which applies more broadly to hidden Markov models (HMMs), and Bayesian computations bypass the necessity for computationally demanding numerical gradient approximations for obtaining maximum likelihood estimates (MLEs). Supplemental materials are available online. [ABSTRACT FROM AUTHOR]

In this paper the forced axial vibration of a nanobeam by deploying the nonlocal strain gradient theory exposed to a distributed moving force, is investigated. The assumed nanobeam in this paper is under the axial constant distributed load. The governing equations of the vibration of the nanobeam are obtained via the classic beam theory and the Hamilton principle. The determined partial differential equations are converted to differential equations by applying the assumed modes method. The dynamic axial displacement of the nanobeam along its length, is obtained by solving the differential equations via the convolution integral. The effect of length scale parameter, nonlocal parameter, velocity parameter, the constant distributed load and the excitation frequency on the maximum nondimensional dynamic axial displacement of the nanobeam are analyzed. Accordingly, the effect of these two parameters on the maximum nondimensional axial displacement of the nanobeam under the harmonic moving load is of significance importance. [ABSTRACT FROM AUTHOR]

The GM (1, N) model is an important prediction model among gray models, but the conventional GM (1, N) model shows big errors sometimes mainly due to the structural inconsistency between the gray differential equation for parameter estimation and the whitening equation for prediction. The paper derives the modified model of conventional gray differential equation based on the whitening equation, and the gray differential equation obtained in this way has the structure basically consistent with the structure of whitening equation. The paper estimates parameters with the gray differential equation of modified GM (1, N) model and gives three parameter estimation methods with different background values. The time response equation for conventional GM (1, N) model generally has no analytical solution, but the paper gives a specific solving method for the time response equation of modified model. The example proves that the modified GM (1, N) model given and the models built with the methods proposed improve the simulation and prediction accuracy significantly compared with the conventional model. [ABSTRACT FROM AUTHOR]

Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. Therefore, exponential Euler differences for fractional-order differential models are novel and significant. In allusion to fuzzy Cohen–Grossberg neural networks of fractional order, this paper establishes a Mittag–Leffler Euler difference model on the basis of the constant variation formula in fractional calculus. In the second place, the existence of a unique global bounded solution and equilibrium point is studied by using the contractive mapping principle. Thirdly, global exponential stability and synchronisation of the derived difference models are investigated by employing some inequality techniques and reductio. At last, some illustrative examples and numerical simulations are employed to demonstrate the main results of this paper. [ABSTRACT FROM AUTHOR]

Free transverse vibration of variable cross-section cantilever FG beams with nonlinear profiles is investigated in this paper. Four thickness functions, namely, linear, parabolic, sinusoidal, and exponential functions are assumed for variation of the cross-section of the beam. Linear and exponential grading rules for axially varying material properties are covered in this paper. The governing differential equation is solved using the weighted residual collocation method with the exact solution shape functions for the uniform beam as the trial functions. This choice for the trial functions showed an increase in the convergence rate. The Gauss–Legendre points are used as the collocation points to reduce the fluctuations in the convergence curve as usually encountered in the point collocation method. The effects of the taper parameter for all kinds of thickness functions on the natural frequencies are studied. The effects of various parameters, including taper parameter, profile, and grading function on the natural frequencies are investigated. Also, a series of finite element simulations are performed for comparison purposes. It is observed that the obtained results are in good agreement with the numerical simulations and available published data with vastly reduced computational cost as a result of using the collocation method. [ABSTRACT FROM AUTHOR]

This paper studies adaptive finite-time control problem of a class of nonlinear systems with dynamic and parametric uncertainties. The power of systems in this paper is any positive odd rational number but not necessary equal to or greater than one. To solve this problem, finite-time input-to-state stability (FTISS) is used to characterise the unmeasured dynamic uncertainty. By skillfully combining Lyapunov function, sign function, adding a power integrator, adaptive control and FTISS methods, an adaptive state feedback controller is designed to drive the states to the origin in finite time while maintaining all the closed-loop signals bounded. [ABSTRACT FROM AUTHOR]

This paper considers fuzzy boundary value problems associated with second-order fuzzy differential equations under granular differentiability. Using a horizontal membership function, we present the notion of second-order granular differentiability for fuzzy functions. Using the granular differentiability concept, we interpret two kinds of two-point boundary value problems for second-order fuzzy differential equations. Sufficient conditions are established for the existence and uniqueness of solutions to these fuzzy boundary value problems. An algorithm is presented for solving non-linear fuzzy boundary value problems under granular differentiability. We provide one example and two engineering applications to demonstrate the algorithm's effectiveness and results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the new concept of non-local heat conduction equation to generalized magneto-thermoelastic problem of two dimensional isotropic and homogeneous half-space in presence of heat-flux at the boundary surface. By using the harmonic plane waves, the governing equations are transformed to the vector matrix differential equation which is then solved by eigenvalue method. The analytical closed form solutions for displacement component, temperature distribution and stress components have been made and comparisons are also illustrated graphically with the theory of non-local dual-phase-lag (NLDPL) and non-local Lord-Shulman (NLLS) theory for different values of physical parameters. The significant effects of non-local variables as well as phase lagging parameters on displacements, temperature distribution and stress components are studied by means of graphically and concluding remarks are drawn. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of multi-term time-fractional measure differential equations with initial conditions in Banach spaces. Firstly, we look for a suitable concept of S-asymptotically ω-periodic mild solution to our concerned problem, by means of the Laplace transform and (β , γ k) -resolvent family { S β , γ k (t) } t ≥ 0 . Secondly, the existence of S-asymptotically ω-periodic mild solutions for the mentioned system is obtained by utilizing regulated functions and fixed point theorem. Finally, as the application of abstract results, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic singular value and spectral distributions of matrices A n arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices A n give rise to a sequence { A n } n , which often turns out to be a GLT sequence. In this paper, we provide an extension of the theory of GLT sequences: we show that any sequence of diagonal sampling matrices constructed from asymptotically uniform samples of an almost everywhere continuous function falls in the class of GLT sequences. We also detail a few representative applications of this result in the context of finite difference discretizations of DEs with discontinuous coefficients. [ABSTRACT FROM AUTHOR]

Ayala-Gilpin ecosystem is one of the most famous differential equation models based on experimental and theoretical analysis. Most of the previous works on the almost periodic solution and its stability are mainly about the existence of an almost-periodic solution and its global stability. In fact, the real ecosystem presents multiple stable states due to the influence of various external factors. However, there are few researches on multiple almost-periodic solutions and local stability of Ayala-Gilpin ecosystem. Therefore, we focus on the existence and local exponential stability of multiple almost-periodic solutions for a classical nonlinear competitive Ayala-Gilpin ecosystem provided with varying-lags and control terms in this paper. Firstly, a class of functions having only two zeros in (− ∞ , + ∞) is investigated. Next, based on the existence of zeros of these functions, we obtain some sufficient conditions to ensure the existence of at least four almost-periodic positive solutions by utilising theorem of coincidence degree and some inequality techniques. Furthermore, we also conclude that this ecosystem exists at least two almost-periodic positive solutions under relatively weak conditions. Finally, we build some Lyapunov functionals to discuss the local exponential stability of each almost-periodic positive solution in its own region. As an application, a numerical simulation inspect the effectiveness of our major results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss mathematical modeling opportunities that can be included in an introductory Differential Equations course. In particular, we focus on the development of and extensions to the single salty tank model. Typically, salty tank models are included in course materials with matter-of-fact explanations. These explanations miss the opportunity for students to develop rate of change equations for themselves. In this paper, we highlight an open-ended single salty tank task that provides the foundation for more complicated salty tank models that we also detail herein. Furthermore, we discuss instructional implementation techniques that instructors can use in their classes to meet their specific learning goals. [ABSTRACT FROM AUTHOR]

In this paper, we establish the asymptotic boundary of traveling waves (first-component) in a discrete diffusive epidemic model, whose existence is well-known. Meanwhile, we derive the nonexistence of traveling wave solutions with non-positive wave speed. We solve two open problems left in two papers [Fu et al., Nonlinear Convex A. 17 (2016), pp. 1739–1751; Wu, J. Differential Equations 262 (2017), pp. 272–282]. [ABSTRACT FROM AUTHOR]

In this paper we propose a solution to the regulation problem of holonomic mechanical systems without velocity measurements by constructing nonlinear PI controllers. An approach based on the Lyapunov functional method is used to analyse the stability of the closed-loop system which has the form of Volterra integro-differential equation. The paper presents the quasi-invariance principle for Volterra nonlinear integral differential equations. Furthermore, we present sufficient conditions that one can use to conclude the asymptotic stability of the closed-loop system via using a Lyapunov functional with semi-definite time derivative. The simulation results of the two-link robot manipulator demonstrate the effectiveness of the proposed PI controller. [ABSTRACT FROM AUTHOR]

This paper is devoted to one open extremal problem of Dubinin in geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider Dubinin’s problem of the maximum of product of inner radii ofnnon-overlapping domains containing points of the unit circle and the powerof the inner radius of a domain containing the origin. The problem was formulated in 1994 in the work of Dubinin and then repeated in his monograph in 2014. Currently it is not solved in general. In this paper, we obtained a solution of this problem for some concrete values ofnand. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sign-changing points of nontrivial real-valued solutions of homogeneous Sturm–Liouville differential equations of the form − d (d u / d α) + u d β = 0 , where d α is a positive Borel measure supported everywhere on (a , b) and d β is a locally finite real Borel measure on (a , b). Since solutions for such equations are functions of locally bounded variation, sign-changing points are the natural generalization of zeros. We prove that sign-changing points for each nontrivial real-valued solution are isolated in (a , b). We also prove a Sturm-type separation theorem for two nontrivial linearly independent solutions and conclude the paper by proving a Sturm-type comparison theorem for two differential equations with distinct potentials d β. [ABSTRACT FROM AUTHOR]

In this paper, we prove new Liouville-type properties of fully nonlinear subelliptic equations modelled on the Heisenberg group. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential equations. [ABSTRACT FROM AUTHOR]

The development by Hewlett-Packard of memristor, a fourth fundamental electronic component, has gained attention in the fields of memory technology and neuromorphic. As memristor components are not currently available commercially, modelling of memristors with associated circuit elements is required to build application circuits. In this work, based on the Hewlett-Packard memristor, we implement a new Chua's chaotic circuit, which is a simple combination of electronic components which exhibit chaos. The nonlinear element in Chua's circuit is substituted with an active memristor-based circuit consisting of a combination of negative resistor and TiO2-based memristor, which allows the circuit to exhibit chaos without scaling the differential equations and produces realistic voltages and currents across the circuit components. This paper indicates for the first time, chaotic oscillations are produced using memristor-based Chua's circuit where actual physical equations, describing the memristor, are applied. The analysis of the proposed single memristor-based chaotic is discussed. Such a circuit is of interest in chaos-based systems for applications such as wireless communication. [ABSTRACT FROM AUTHOR]

The parameter values of complex media can be uncertain in reality and uncertainty quantification (UQ) in the simulation of wave propagation is essential. In this paper, an efficient surrogate model based on the artificial neural network (ANN) is developed to imitate the concerned ground penetrating radar (GPR) calculation. A two-dimensional GPR system is simulated with the auxiliary differential equation (ADE) finite-difference time-domain (FDTD), and the soil is considered as a nonmagnetic medium with the frequency-dependent dielectric permittivity represented by a two-term Debye model. A solid metallic object and a piece of dry granite are buried in the dispersive soil whose parameter values are considered uncertain. The framework of the surrogate model is presented for GPR system modeling. Different popular activation functions are applied to the proposed surrogate model and their results are analyzed. Compared with other activation functions, the activation function of exponential linear unit has a better performance in the training process. Moreover, the dropout method is used to reduce overfitting to a large extent. At last, the well trained surrogate model is applied for UQ to GPR calculation instead of running one thousand full-wave GPR simulations, and the results show its accuracy and efficiency. [ABSTRACT FROM AUTHOR]

This article studies a new class of nonautonomous second-order nonlocal measure driven control systems involving state-dependent delay (SDD). The system presented in this paper can deal with a large class of hybrid systems with their Zeno behaviour having no restrictions. We use the theory of measure of noncompactness to develop our main results. The existence of a solution to the considered problem is obtained via the Mönch fixed point theorem. The approximate controllability of the aforementioned system is investigated under the assumption that the associated linear system is approximately controllable. To investigate our main results, the integral is taken in the Bochner–Stieltjes sense that makes the obtained results more general. In the end, an example is formulated for the practical usefulness of the established theory. [ABSTRACT FROM AUTHOR]

We consider a doubly stochastic Markov chain, where the transition intensities are modelled as diffusion processes. Here we present a forward partial integro-differential equation (PIDE) for the transition probabilities. This is a generalisation of Kolmogorov’s forward differential equation. In this set-up, we define forward transition rates, generalising the concept of forward rates, e.g. the forward mortality rate. These models are applicable in e.g. life insurance mathematics, which is treated in the paper. The results presented follow from the general forward PIDE for stochastic processes, of which the Fokker–Planck differential equation and Kolmogorov’s forward differential equation are the two most known special cases. We end the paper by considering the semi-Markov case, which can also be considered a special case of a general forward partial integro-differential equation. [ABSTRACT FROM AUTHOR]

The decision-making of cash holdings is very important for the daily operation of enterprises. This present paper tries to establish uncertain optimal cash holding models with the constraint of safe cash holding area, and discusses the solutions of the models by establishing Lagrange function under KKT condition. On the one hand, this paper enriches the existing cash holding models, on the other hand, it is also a comprehensive discussion on the application of uncertainty theory in cash holding management. [ABSTRACT FROM AUTHOR]

In this paper, the Chebyshev collocation numerical method is developed for solving generalized thermoelasticity problems of the isotropic homogeneous two dimensional media. The coupled thermoelastic equations are derived based on Lord-Shulman (LS) and Green-Lindsay (GL) theories. The temperature and displacement fields are approximated in space domain by linear combinations of Chebyshev polynomials. Also, the direct collocation method is applied to governing differential equations to generate the system of differential equations with respect to time. The resulted set of differential equations are solved in time domain by Wilson method. Both temperature and traction loadings are considered to be applied at the left side of the media. The obtained results from the present paper for the classical coupled thermoelasticity of two dimensional finite domains are compared with the same results extracted analytically in the literature and a very close agreement is observed. [ABSTRACT FROM AUTHOR]

In this paper, a recurrent neural network with mixed delays which plays an important role is considered. We are concerned with the existence, uniqueness and global exponential stability of the doubly measure pseudo almost automorphic solutions. First, we establish results that are interesting on the functional space of such functions like composition theorem. Second, by employing the fixed-point theorem and some properties of the measure pseudo almost automorphic functions, some sufficient conditions for the existence, uniqueness and global exponential stability of the doubly measure pseudo almost automorphic solutions have been established. Our results obtained in this paper are new. Finally, two commonly used numerical examples are given to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]