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The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However, discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state, and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples: The first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data. [ABSTRACT FROM AUTHOR]

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

In this paper steady‐state vibrations of the two‐degrees‐of‐freedom oscillatory systems with van der Pol coupling are investigated. The model is a system of two differential equations with weak nonlinearity. A new solving procedure based on D′Alembert's method and the method of time‐variable amplitude and phase is developed. The main advantage of the method in comparison to others is that it gives the solution of the system of two coupled weak nonlinear equations in the form that is simple to analyze, as it has the same form as the solution of the corresponding system of linear equations. In the paper two types of systems are considered: one, a two‐mass system with two degrees of freedom, and second, the one‐mass system with two degrees of freedom. The torsional vibrations of a two‐mass system and vibrations of a Jeffcott rotor with two‐degrees‐of‐freedom are analyzed. Analytically obtained results are numerically tested. It is obtained that the difference between analytic and numeric results is small and almost negligible. As the accuracy of the analytic solution is high, it is suitable for application in technics and engineering. Conclusions about steady‐state self‐sustainable oscillators, orbital, and limit cycle motions are given. [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]

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]

This paper presents an algorithm that checks if a given second‐order homogeneous linear differential equation can be reduced to the confluent Heun's equation by using the change of variables transformation and the exp‐product transformation. The main purpose of this paper is finding solutions in terms of the confluent Heun's functions for the given differential equation. [ABSTRACT FROM AUTHOR]

This paper considers the secure distributed control consensus problem of linear multi‐agent systems (MASs) under switching topologies, subject to intermittent sequential scaling attacks, which was compelled to scaling factor, attack frequency, duration and cooperative‐competitive networks. First, the scenario of a fixed topology is considered, and a novel control protocol combined with a logarithmic quantizer and relative state measurements of neighbouring agents is discussed. The sighed graph is utilized to characterize the communication topology determined by the information flow directions and captured by the graph Laplacian matrix. After that, sufficient conditions for effectiveness of the developed control methods in guiding the MASs to secure bipartite leader‐following consensus are constructed. Second, the scenario of switching topologies is considered, and it is derived that the secure bipartite consensus will be achieved if the designed state feedback control protocol with the scaling factor, the attack duration, attack frequency and switching signal are selected properly. At last, to prove the effectiveness of the designed controllers, a simulation example depended on the real‐word actual military is introduced. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new mathematical epidemic model with the effect of vaccination. We formulate a Susceptible-High risk-Infective-Recovered-Vaccinated (SHIRV) model in which the susceptible individuals with a higher probability of being infected (H) are selected as a separate class. We study the dynamical behavior of this model and define the basic reproductive number, R0. It is proved that the disease-free equilibrium is asymptotically stable if R0 < 1, and it is unstable if R0 1. Also, we investigate the existence and stability of the endemic equilibrium point analytically. For the system of differential equations of the SHIRV model, we give an approximating solution by using the Legendre-Ritz-Galerkin method. Finally, we study the influence of vaccination on measles and smallpox, two cases of the epidemic, using the proposed method in this paper. Numerical results showed that choosing high-risk people for vaccination can prevent them from getting infected and reduce mortality in the community. [ABSTRACT FROM AUTHOR]

Summary: In this paper, for the first time, a new method with closed‐form analytical equations is presented to calculate the oscillation amplitude of fourth‐order oscillators, such as quadrature oscillators. This method is actually based on the general form of the differential equations describing the structure of the fourth‐order oscillators and finding a solution for the nonlinear differential equations governing this type of oscillator. The introduced method is a general method that is valid for all fourth‐order oscillators and is also independent of the oscillation frequency. Using the proposed method, complex and time‐consuming simulation tools will no longer be needed to calculate the oscillation amplitude in a steady state. Moreover, the presented closed‐form equations help the designers to understand the design compromises and design the oscillator for their specific and desired conditions. In addition, to evaluate the correctness of the presented equations, a comprehensive analysis of the oscillation amplitude of the quadrature oscillator in the steady state is performed, and a closed‐form equation is presented for the oscillation amplitude of the oscillator in the steady state, which is proposed for the first time in this paper. A comparison between the simulation results and theoretical computations confirms the validity of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the effects of Maxwell nanoliquid transmission with MWCNT nanotube are investigated over two circular wires with opposite currents and a stretching sheet. The innovation of this paper is the examination of MWCNT nanotube temperature and nanofluid velocity and also magnetic potential and magnetic effect in the three different places of two circular wires on the sheet. The finite element method is utilized for calculating differential equations. By increasing the distance from two wires on the stretching sheet, the amount of MWCNT nanotube temperature increased, and a large vortex forms at the bottom of the sheet. Overall, the MWCNT nanotube velocity around the two circular wires at the beginning of the sheet is 46.09% higher than the middle of the sheet and 20% higher than the end of the sheet, also the MWCNT nanotube magnetic potential around the two circular wires at the beginning of the sheet is 13.40% higher than the middle of the sheet and 23% higher than the end of the sheet. The second part of the essay relates to the using response surface methodology (RSM) method in the Design‐Expert software. Based on the results obtained from the MWCNT nanotube velocity graphs and the heat transfer cantors, and magnetism effects in the Design‐Expert software, the best optimization occurred for u = 0.07, T = 1.15, and H = 4.531. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, {(G(t),F(t)),t≥0}$\lbrace (G(t), F(t)),t\ge 0\rbrace$, that is, called the resolvent family for the linear evolution process with memory, the F(t)$F(t)$ is called the memory effect family. In this paper, we prove that the families G(t)$G(t)$ and F(t)$F(t)$ are exponentially bounded, and the family (G(t),F(t))$(G(t),F(t))$ associate with an operator pair (A,L)$(A, L)$ that is called generator of the resolvent family. Using (A,L)$(A,L)$, we derive associated differential equation with memory and representation of F(t)$F(t)$ via L. These results give necessary conditions of the well‐posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on (A,L)$(A,L)$. The obtained results can be directly applied to linear delay differential equation, integro‐differential equation and functional differential equations. [ABSTRACT FROM AUTHOR]

During past decades, the study of the interaction between predator and prey species has become one of the most exciting topics in computational biology and mathematical ecology. In this paper, we aim to investigate the stability of a diseased model of susceptible, infected prey and predators around an internal steady state. To this end, the fractional derivatives based on the Mittag‐Leffler kernels in the Liouville‐Caputo concept has been taken into consideration. The existence and uniqueness of the acquired solutions to the model are also studied in this paper. In order to investigate the effects of the fractional‐order along with other existing parameters in the model, several possible scenarios have been examined. As it is seen in the proposed graphical simulations, the employed fractional operator is capable of capturing all anticipated theoretical features of the model. The numerical technique employed in this contribution is precise and efficient and can be easily adopted to investigate many fractional‐order models in biology. It is found that new proposed operators of fractional‐order can describe the real‐world phenomena even better than integer‐order differential equations because of their memory‐related properties. [ABSTRACT FROM AUTHOR]

Studying ordinary or partial differential equations or integrals using traditional asymptotic analysis, unfortunately, fails to extract the exponentially small terms and fails to derive some of their asymptotic features. In this paper, we discuss how to characterize an asymptotic behavior of a singular linear differential equation by the methods in exponential asymptotics. This paper is particularly concerned with the formulation of the series representation of a general second‐order differential equation. It provides a detailed explanation of the asymptotic behavior of the differential equation and its relation between the prefactor functions and the singulant of the expansion of the equation. Through having this relationship, one can directly uncover and investigate invisible exponentially small terms and Stokes phenomenon without doing more work for the particular type of equations. Here, we demonstrate how these terms and form of the expansion can be computed straight‐away, and, in a manner, this can be extended to the derivation of the potential Stokes and anti‐Stokes lines. [ABSTRACT FROM AUTHOR]

The paper machine cross-directional (CD) process is a large-scale spatially distributed system. It is known to be severely ill-conditioned as the gain rolls down to zero for some of the process directions. Model uncertainties in the process are inevitable resulting in a challenging robust control design problem. CD actuators are subject to min-max constraints while slice lip actuators are subject to additional bending moment limits. Because of the large number of input constraints, the industrial practice is to tune the CD controller assuming inactive constraints. The robustness of CD feedback loops to model uncertainties under constrained internal model control satisfies an integral quadratic inequality. This work develops an automatic tuning algorithm that guarantees robust stability and performance of the constrained CD feedback loop. Spatial response models are identified in a prediction error frame delivering bounds on the CD process pseudo-singular values. The CD controller is synthesized online through a linear matrix inequalities feasibility problem taking into consideration the modal space uncertainty rising from the uncertainties in the estimated parameters and the expected variations in the dynamic response. The developed tuning technique is suitable for paper machines producing different grades of paper as the CD process spatial and dynamic responses change for each grade. The performance of the tuned constrained internal model control controller is validated through comparing it to an industrial CD controller that has been implemented in paper mills as part of a commercial product. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper is dedicated to obtaining closed‐form solutions of linear difference equations which are asymptotically close to the self‐adjoint Euler‐type difference equation. In this sense, the equation is related to the Euler–Cauchy differential equation y′′+λ/t2y=0$$ {y}^{\prime \prime }+\lambda /{t}^2y=0 $$. Throughout the paper, we consider a system of sequences which behave asymptotically as an iterated logarithm. [ABSTRACT FROM AUTHOR]

The paper deals with a new type of inverse spectral problems for second‐order quadratic differential pencils when one of the boundary conditions involves arbitrary entire functions of the spectral parameter. Although various aspects of the inverse spectral theory for the pencils have been of a special interest during the last decades, such settings were considered before only in the particular case of a Sturm–Liouville equation. We develop an approach covering also the quadratic dependence on the spectral parameter in the differential equation, which is based on the completeness and basisness of certain functional systems. By this approach, we obtain a uniqueness theorem and an algorithm for solving the inverse problem along with sufficient properties of the mentioned systems. The presented results give a universal tool for studying a number of important specific situations, including various Hochstadt–Lieberman‐type inverse problems both on an interval and on geometrical graphs, which is illustrated as well. [ABSTRACT FROM AUTHOR]

Many systems in the real world are periodic due to periodic phenomena in nature. Periodic hybrid stochastic differential equations are often used to model them. In many situations, it is inappropriate to study whether the solutions of periodic hybrid stochastic differential equations will converge to an equilibrium state (say, 0 or the trivial solution) but more appropriate to discuss whether the probability distributions of the solutions will converge to a stationary distribution, known as stability in distribution. This paper aims to determine whether or not a periodic stochastic state feedback control can make a given nonlinear periodic hybrid differential equation, which is not stable in distribution, to become stable in distribution. This problem will be referred to as stabilisation in distribution by periodic noise. There is little known on this problem so far. This paper initiates the study in this direction. [ABSTRACT FROM AUTHOR]

Summary: This paper introduces an electronic circuit capable of generating a set of functions, some of them known in digital systems as the logical operators AND, OR, XOR, and so on. Using two inputs, u1 and u2, the circuit provides 16 possible output combinations. The main idea of the electronic design is based on an RC network, operational amplifiers, and voltage comparators. On the other hand, mathematically, the stable system response is used as a surface where u1 and u2 are coordinate axes forming a plane, which intersects this surface, and the output y can be seen as a circle surrounding some fixed points over this plane. The mathematical approach on this paper is intended as a groundwork for a multiple input reconfigurable logic gate that could be embedded in more complex systems. The procedure to obtain an XOR gate, represented by the f(6) function, is explained to illustrate the circuit behavior. Results of the 16 implemented functions are shown in Appendices B and C. [ABSTRACT FROM AUTHOR]

Most existing finite impulse response (FIR) filters are restricted to models without delays. This paper proposes an continuous‐time optimal unbiased FIR filter for input‐delayed systems (CTOUFFID). A new integral transformation relation was introduced to derive the FIR filter. By applying this relation, the CTOUFFID problem is represented as an optimal control problem with zero terminal state. The filter gain function is obtained by solving two coupled matrix differential equations using initial conditions. The paper also offers discussion on the horizon size and a few special cases. The main benefit of the proposed solution is that it provides the maximum likelihood estimate with no requirements for the initial values. Finally, an application with the F$$ F $$‐404 turbofan engine model is presented to demonstrate the highly robust nature of the proposed FIR filer against incomplete noise information and unspecified model uncertainties. [ABSTRACT FROM AUTHOR]

In this paper, we study the full parabolic attraction–repulsion Keller–Segel model with a fractional diffusion in ℝn$$ {\mathbb{R}}^n $$ for n=2$$ n=2 $$ or 3. We are more interested in the question that whether the solutions exist globally or blow up in finite time, which was studied in the classical attraction‐repulsion Keller‐Segel model by Jin and Wang (J. Differential Equations, 2016) through constructing a suitable energy functional. However, for the fractional attraction–repulsion Keller–Segel model, it is challenging to find a similar energy functional to study the existence of the solutions. In the present paper, under the condition of ξγ=χα$$ \xi \gamma =\chi \alpha $$, we introduce the Lp$$ {L}_p $$‐ Lq$$ {L}_q $$ estimates of the fractional derivative modulus of the solution to carry out the first step of the problem. Armed with a priori estimate, it is sufficient for us to obtain the global existence of the solutions by the Moser–Alikakos iterative method and finally arrive at the decay estimates of the solutions. [ABSTRACT FROM AUTHOR]

Aiming at such problems as poor dynamic and steady‐state performance, and non‐ideal anti‐interference performance when the traditional sliding mode control is applied in Boost converter, a non‐singular terminal sliding mode control (NTSMC) strategy based on finite time convergence is presented in this paper. Taking the inductor's current and output voltage as the state variables of the system, the average state model of Boost converter under continuous conduction mode (CCM) is established, and the differential equations of the inductor's current and output voltage are obtained. According to the above differential equation, a finite‐time convergent observer (FCO) is designed to estimate the load resistance and input voltage. By combining these estimated values with the NTSMC method, a novel non‐singular terminal sliding surface function and controller are designed, which improves the dynamic, steady‐state performance and anti‐interference performance of the system. The stability of the system is analyzed according to the Lyapunov stability theory. The simulation and experimental results show that, compared with the traditional sliding mode control, the control algorithm proposed in this paper has stronger anti‐interference ability. [ABSTRACT FROM AUTHOR]

This paper introduces a new learning algorithm for accurate, physically driven time series prediction. The fundamental assumption behind the method is that the phenomena follow Ordinary Differential Equations. We investigate the general case where the time series follows an ODE of degree m∈N $m\in \mathbb{N}$. The resulting method is a learning algorithm based on the finite differences between the values of time series. We present the application of the method in the field of geodesy for polar motion prediction, the main objective of the present paper. We show that in this application, the linear form of the method is sufficient and offers competitive predictive performance. We present a baseline solution, in which we use historical polar motion time series from 1976 to predict up to the year 2020. The prediction horizon in this case is short‐term (up to 10 days into the future). In addition, we compare the prediction accuracy in the short‐term horizon with the results of the best performing model in the first Earth Orientation Prediction Comparison Campaign. On average, a 53% improvement in prediction performance is achieved. In further analyses, we compare the prediction accuracy for both short‐term and long‐term against the results of state‐of‐the‐art methods, namely Multichannel Singular Spectrum Analysis, and a combination of Singular Spectrum Analysis and Copula sampling. We show that the proposed method in this paper can outperform the mentioned two methods in both short and long‐term horizons, with an average improvement of the prediction performance of 54% and 52%, respectively. Plain Language Summary: Physical phenomena are usually represented in mathematical terms by (ordinary) differential equations. Polar motion is an important physical parameter that describes the changes in the Earth's rotation axis with respect to the surface of the Earth. There is, however, a latency of up to several days by which the values are provided by operational data centers. Therefore, predicted values are used for applications such as spacecraft navigation that would need instantaneous information about polar motion. In order to predict polar motion time series, we introduce a new learning approach that is based on fitting ordinary differential equations to data. We specifically focus on second‐order differential equations, since they are typically more accurate and many physical phenomena follow an equation of this type. We demonstrate by extensive comparative analyses that our proposed method outperforms other methods both in short and long‐term predictions. Key Points: We introduce a new learning approach called neural ordinary differential equation differential learning for time series predictionWe present the application of the method for polar motion time series prediction, focusing on the linear form of the methodRigorous tests show that this method can achieve state‐of‐the‐art accuracy for both short and long‐term predictions [ABSTRACT FROM AUTHOR]

This paper aims to investigate the existence and uniqueness of solutions for a sixth‐order differential equation involving nonlocal and integral boundary conditions. Firstly, we obtain the properties of the relevant Green's functions. The existence result of at least one nontrivial solution is obtained by applying the Krasnoselskii–Zabreiko fixed point theorem. Moreover, we also establish the existence of unique solution to the considered problem via Hölder and Minkowski inequalities and Rus's theorem. Finally, two numerical examples are included to show the applicability of our main results. [ABSTRACT FROM AUTHOR]

In this paper, based on the differential quadrature method (DQM), matrix operators are derived for fractional integration and Caputo differentiation. These operators generalize the efficient DQM to fractional calculus. The proposed fractional differential/integral quadrature method (FDIQM) is used to solve various types of fractional ordinary and partial differential equations. FDIQM unifies the solution of multi‐integer fractional‐order differential equations leading to significant simplification in the implementation. Numerous examples are presented to demonstrate the accuracy of the operators. Other examples are presented to solve various fractional differential equations including time‐fractional sub‐diffusion equation, linear/nonlinear, and multiorder fractional differential equations. In addition, numerous boundary conditions are considered including mixed fractional derivatives. Further, a nonlinear fractional eigenvalue problem is solved efficiently, and its bifurcation diagrams are obtained. Comparisons between the proposed method and the existing ones are included, showing the ease of implementation, efficiency, and applicability of FDIQM. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the rotational periodic boundary value problem for a fractional‐order nonlinear differential equation. Applying topology‐degree theory and the Leray‐Schauder fixed‐point theorem, we prove the existence and uniqueness of solution for the fractional‐order differential system. Furthermore, the existence of solution for a nonlinear differential system with a multivalued perturbation term is investigated by using set‐valued theory and techniques of functional analysis. Two examples of applications are given at the end. [ABSTRACT FROM AUTHOR]

The maximum angular rotation attributable to the in‐plane shear deformation of flange slabs is used as generalised displacement in the conventional analysis method for I‐beams. However, the mechanical concepts are poorly understood due to the complex nature of this analysis method. Consequently, a novel strategy for analysing vertical bending in composite I‐beams is proposed in this study. This approach uses the additional deflection of composite I‐beams induced by the shear lag effect as the generalised displacement. Furthermore, this research comprehensively considers the accordion effect, shear lag and self‐equilibrium conditions for the shear lag warping stress and bending moment. Moreover, two longitudinal warping displacement difference functions are employed to accurately describe the variation of shear lag in composite I‐beams with varying flange slab widths. The differential equations of the I‐beams with corrugated steel webs in the elastic range are established based on the energy‐variation method. A complete mechanical system of a composite I‐beam is decomposed into two parts, namely, the shear lag mechanical system and the elementary beam mechanical system, which are independent of each other. The theory presented in this paper reflects the internal mechanical mechanism of composite I‐beams. The calculation accuracy is considerably improved in this study. Therefore, this method is more unambiguous and well‐defined. It enriches and advances the current analysis theory of composite structures, which can guide the design of composite I‐beams. [ABSTRACT FROM AUTHOR]

Based on the extensive application of the q$$ q $$‐series and q$$ q $$‐polynomials including (for example) the q$$ q $$‐Laguerre polynomials in several fields of the mathematical and physical sciences, we attach great importance to the equations and related application issues involving the q$$ q $$‐Laguerre polynomials. The mission of this paper is to find the general q$$ q $$‐operational equation together with the expansion issue of the bivariate q$$ q $$‐Laguerre polynomials from the perspective of q$$ q $$‐partial differential equations. We also give some applications including some q$$ q $$‐Hille‐Hardy type formulas. In addition, we present the Rogers‐type formulas and the U(n+1)$$ U\left(n&#x0002B;1\right) $$‐type generating functions for the bivariate q$$ q $$‐Laguerre polynomials by the technique based upon q$$ q $$‐operational equations. Moreover, we derive a new generalized Andrews‐Askey integral and a new transformation identity involving the bivariate q$$ q $$‐Laguerre polynomials by applying q$$ q $$‐operational equations. U(n+1)$$ U\left(n&#x0002B;1\right) $$ [ABSTRACT FROM AUTHOR]

This paper studies the existence and asymptotic properties of solutions for a semilinear measure‐driven evolution equation with nonlocal conditions on an infinite interval. The existence result of the solutions for the considered equation is established by Schauder's fixed point theorem. Then, the asymptotic stability of solutions is further proved to show that all the solutions may converge to the unique solution of the corresponding Cauchy problem. In addition, under some conditions the existence of global attracting sets and quasi‐invariant sets of mild solutions is investigated as well. Finally, an example is provided to illustrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we develop a continuous SIR epidemic model and its discrete version using the Euler method. The continuous SIR epidemic model is a widely used mathematical tool for understanding the dynamics of infectious diseases. We derive the system of differential equations that describe the rates of change in the number of individuals in each compartment and discuss the key parameters that influence the dynamics of the model. Next, we use the Euler method to discretize the continuous SIR epidemic model and obtain a discrete‐time version. We analyze the dynamics of the discrete SIR epidemic model using both analytical and numerical methods. We derive the positively invariant set of the model and discuss the initial conditions of the model and how they relate to the total population. We compare the behavior of the continuous and discrete SIR epidemic models and show that the discretized model captures the essential features of the continuous model. We also investigate the impact of key parameters such as the transmission rate, recovery rate, and death rate on the dynamics of the model. Our findings suggest that the discrete SIR epidemic model is a powerful mathematical tool for understanding the dynamics of infectious diseases and developing effective strategies for their control. By studying the behavior of the model under different scenarios, researchers can gain insights into the transmission and prevalence of infectious diseases and develop interventions that reduce their impact on public health. [ABSTRACT FROM AUTHOR]

Small noise problems are quite important for all types of stochastic differential equations. In this paper, we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter H∈(1/4,1/2]$H\in (1/4, 1/2]$. We prove a moderate deviation principle for this equation as the scale parameter tends to zero. [ABSTRACT FROM AUTHOR]

This paper is concerned with the optimal H2 filtering problem for continuous‐time systems with multiplicative noise and multiple sampled delay measurements. The reorganized observation technique is firstly applied for treating the sampled delay terms, and then an equivalent delay‐free sampled measurement is received. An optimal H2 filter is constructed by using a dynamic model with finite jumps, which can provide the state estimation of the measurement sampling interval as well as the measurement sampling moments. The filter gain is given in terms of the stabilizing solution to a set of specific Riccati differential equations. It should be pointed out that no state augmentation is required, and thus the Riccati equation developed in this paper remain the same dimension as that of the original system state. This is the clear distinction from the state augmentation method. A practical example is finally supplied to illustrate the efficiency of the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, we study asymptotic behaviors for classical thermoelastic plate equations with the Fourier law of heat conduction in the whole space ℝn$$ {\mathrm{\mathbb{R}}}^n $$, where we introduce a reduction methodology basing on third‐order (in time) differential equations and refined Fourier analysis. We derive optimal growth estimates when n⩽3$$ n\leqslant 3 $$, bounded estimates when n=4$$ n=4 $$, and decay estimates when n⩾5$$ n\geqslant 5 $$ for the vertical displacement in the L2$$ {L}^2 $$ norm. Particularly, the new critical dimension n=4$$ n=4 $$ for distinguishing the decisive role between the plate model and the Fourier law of heat conduction is discovered. Moreover, concerning the small thermal parameter in the temperature equation, we study the singular limit problem. We not only show global (in time) convergence of the vertical displacements between thermoelastic plates and structurally damped plates but also rigorously demonstrate a new second‐order profile of the solution. Our methodology can settle several closely related problems in thermoelasticity. [ABSTRACT FROM AUTHOR]

In this paper, we consider n$$ n $$‐dimensional systems of differential equations applied to model a range of physical phenomena, where n$$ n $$ represents any positive integer. The defining characteristic that we focus on are the magnitudes of the inherent physical parameters, which are frequently of different orders of magnitude for various physical systems. This property, inherent to the given physical problems, can be exploited to reduce the problems into a set of much simpler subproblems, expanding our ability to gain physical insight in terms of the underlying simpler subproblems. These orders of magnitude motivate the choice of a specific class of transformations that can be chosen to reduce the number of required dynamic iterations in a sequence‐based approach to examining such systems. This is also useful for the development of proof techniques for proving properties of mathematical models, such as their well‐posedness. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid numerical scheme based on combining exponential time differencing (ETD) and local radial basis function collocation method was constructed. Model problems with different boundary conditions were considered, and the resulting linear system was carefully analyzed. The relation between the number of points employed in the local radial basis function collocation method and the condition number of the coefficient matrix was given. For application, three typical differential equations were investigated, that is, the Allen–Cahn equation for checking the maximum‐preserving property, the combustion equation for checking the monotonicity‐preserving property, and the Gray–Scott system for checking the robustness of the proposed scheme. Numerical examples show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Mathematics and its applications try to make available a simple and accurate approach to handle nonlinear equations. The present paper investigates for the first time the application of the linearizing method to determine both the approximate frequency and displacement amplitude for the third‐order differential equations. The main merit of the linearized method is a collection of simplicity and accuracy for the solution of high‐order nonlinear conservative or non‐conservative oscillators. The current work sheds light on the approximate proposition to the damped third‐order nonlinear differential equations. The fast estimated solution is simulated to the accurate solution gained through the numerical methods. It is significant to conclude that the non‐perturbative method considered here is simpler and easier to apply than the other perturbation methods. This paper enriches some existing literature. [ABSTRACT FROM AUTHOR]

From the perspective of large‐scale system, a complex dynamical network can be regarded as the interconnected system with the node subsystem and link subsystem, which implies that the node subsystem and the link subsystem are the two main bodies of dynamical behaviors of network. Therefore, the whole stability of network is influenced by not only the dynamics of node subsystem but also the dynamics of link subsystem. According to the above view, the wholly asymptotical stability (WAS) is defined in this paper for the complex dynamical network with the model of differential equations. The WAS is used to describe the node subsystem achieves asymptotical stability when link subsystem achieves also the asymptotic stability in Lyapunov sense. For the WAS of the complex dynamical network, the corresponding criteria are derived by checking whether certain matrices are Hurwitz. The analysis results show that even if the isolated nodes are not asymptotically stable in Lyapunov sense, employing the dynamics of link can also force the node subsystem to achieve the asymptotical stability. Finally, the simulation examples show the validity of methods in this paper. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the stabilizability of nonlinear stochastic systems with no unforced dynamics. Sufficient conditions allowing to design explicitly time–varying feedback laws which render such systems asymptotically stable in probability are given. The techniques used in this work involve the stochastic Lyapunov analysis combined with the stochastic version of the La Salle invariance principle. The interest of our results is that the systems considered in the present paper cannot in general be stabilized via time‐invariant feedback laws. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we deal with a chemotaxis‐haptotaxis model with re‐establishment effect. We consider this problem in a bounded domain Ω ⊂ R N ( N = 2 , 3 ) with zero‐flux boundary conditions. Although the L ∞‐norm of the extracellular matrix density ω is easy to be obtained, the re‐establishment mechanism still cause essential difficulty due to the deficiency of regularity for ω. We use some iterative techniques to establish the W 1 , ∞ bound of uPA protease concentration v, and further obtained the L ∞ estimate of the cancer cell density u. Using these a prior estimates, we finally established the existence of global‐in‐time classical solution, which is bounded uniformly. The result of this paper fills the gap of [Pang and Wang, J. Differential Equations 263 (2017) 1269–1292; Tao and Winkler, J. Differential Equations 257 (2014) 784–815] in dimension 2 with q = 1, in [Tao and Winkler, J. Differential Equations 257 (2014) 784–815], the boundedness of the solution is left open; and in [Pang and Wang, J. Differential Equations 263 (2017) 1269–1292], the global existence and boundedness is established only for large μ. In particular, the global solvability and boundedness of smooth solutions in dimension 3 has never been touched before, this paper is the first attempt to solve this problem. [ABSTRACT FROM AUTHOR]

In terms of system modeling, random differential equations (RDEs) with colored noise are demonstrated to be more appropriate than stochastic differential equations (SDEs) relating to white noise in most practical situations. This paper discusses a category of nonlinear random systems involving time‐varying delay and exogenous disturbances. The definitions for noise‐to‐state practically stable (NSPS) and noise‐to‐state practically exponentially mean‐square stable (NSPEMS) are first given. An event‐based controller is then constructed to render the closed‐loop system NSPEMS and circumvents the Zeno behavior at the same time. Since the introduction of event‐triggered mechanism endows the closed‐loop system with hybrid behaviors, event‐based control for RDE systems still remains difficult to deal with. Within our cognitive scope, this paper serves as a pioneering attempt for event‐based control of RDE systems. And the validity of the results is corroborated by two simulation examples. [ABSTRACT FROM AUTHOR]

This paper studies optimal control and stabilization problems for continuous‐time mean‐field systems with input delay, which are the fundamental development of control and stabilization problems for mean‐field systems. There are two main contributions: (1) To the best of the authors' knowledge, the present paper is the first to establish the necessary and sufficient solvability condition for this kind of optimal control problem with input delay, and to derive the analytical form of an optimal controller through overcoming the obstacle that separation principle no longer holds for multiplicative‐noise systems. (2) For the stabilization problem, under the assumption of exact observability, it is strictly proven that the system is stabilizable if and only if the algebraic Riccati equation has a unique positive definite solution. [ABSTRACT FROM AUTHOR]

The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of harmonic analysis by means of generalized holomorphic (monogenic) functions of hypercomplex analysis. The Sturm‐Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences Vn, first encountered as a special sequence (corresponding to n=2) by Vietoris in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations, we obtain a general recurrence relation for Vn, and we derive an exponential generating function of Vn expressed by Kummer's confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

In this paper, the dynamic performance of Multi‐layered Sandwich Composite Cylindrical Open Shells (MSCCOS) with double‐layered soft cores is analyzed for the first time by adopting the First‐order Shear Deformation Theory (FSDT). For accomplishing this aim, vibration differential equations of MSCCOS are derived based on Hamilton's variance principle. Then, the closed‐form Navier solution is introduced to obtain the numerical results of differential equations. Subsequently, the effectiveness and accuracy are discussed by several comparisons. Ultimately, the dramatic effect of different structural and material parameters on natural frequencies and associated loss factors is investigated and described vividly. Because shear deformation and rotary inertia of all layers are considered in vibration differential equations, thin and moderately thick MSCCOS could be researched, and some interesting discoveries for the dynamic performance of MSCCOS are derived. Highlights: A vibration model of five‐layered composite open shell was obtained.The optimal structure parameters maximizing the damping properties are shown.Dynamics properties of five‐layered composite open shell are presented. [ABSTRACT FROM AUTHOR]

In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear χ-fractional differential equations (χ-FDEs) with variable coefficients. The fractional derivative used in this work is the χ-Caputo fractional derivative sense (χ-CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments. [ABSTRACT FROM AUTHOR]

In the realm of modeling biological systems with memory, particularly those involving intricate interactions like tumor-immune responses, the utilization of multiple time delays and Caputo-type fractional-order derivatives represents a cutting-edge approach. In this research paper, we introduce a novel fractional-order model to investigate the dynamic interplay between tumors and macrophages, a key component of the immune system, while incorporating multiple time delays into our framework. Our proposed model comprises a system of three Caputo-type fractional-order differential equations, each representing distinct cell populations: tumor cells, anti-tumor cells (specifically M1-type macrophages with pro-inflammatory properties), and pro-tumor cells (M2-type macrophages with immune-suppressive characteristics). The stability of equilibria is discussed by analyzing the characteristic equations for each case, and the existence conditions for the Hopf bifurcation are obtained according to the critical values of delay parameters. Furthermore, numerical simulations are presented in order to verify the analytical results obtained for stability and Hopf-bifurcation with respect to the two-time delay parameters τ1 and τ2. The analysis shows the rich dynamics of the model according to the fractional-order parameter and the time delay parameters. [ABSTRACT FROM AUTHOR]

Invasive tree species pose a grave global threat to biodiversity, particularly impacting vulnerable grassland ecosystems. This paper presents a novel approach to simulate the invasion of ecosystems by these aggressive species. Our innovative coupled ODE-PDE model offers explicit invasion simulations, a significant leap beyond traditional integro-difference matrix models. With robust theoretical properties, including solution positivity, and drawing upon PDE theory, our model offers deeper insights into invasion behavior. Our model reveals that invasions undergo a prolonged quiescent phase before rapidly transitioning to a consistent rate of advancement after a critical point. This study not only sheds light on invasion dynamics but also charts a course for future exploration and potential model extensions. [ABSTRACT FROM AUTHOR]

This paper presents a differential equation including both random matrices and a Liu process. Then we demonstrate that the solution to this equation exists and is unique. Under the framework of chance theory, problems of optimal control and two‐person zero‐sum game subject to differential equations are considered. An equation of optimality is provided for solving a problem of optimal control. Then equilibrium equations are proposed to identify the saddle‐point of a two‐person zero‐sum game problem. As an extension, we generalize the obtained results to the problems subject to differential equations including both random matrices and multiple Liu processes. Finally, we utilize the acquired theoretical results to analyze a portfolio selection game problem. [ABSTRACT FROM AUTHOR]

Physical laws governing population dynamics are generally expressed as differential equations. Research in recent decades has incorporated fractional‐order (non‐integer) derivatives into differential models of natural phenomena, such as reaction–diffusion systems. In this paper, we develop a method to numerically solve a multi‐component and multi‐dimensional space‐fractional system. For space discretization, we apply a Fourier spectral method that is suited for multidimensional partial differential equation systems. Efficient approximation of time‐stepping is accomplished with a locally one dimensional exponential time differencing approach. We show the effect of different fractional parameters on growth models and consider the convergence, stability, and uniqueness of solutions, as well as the biological interpretation of parameters and boundary conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the theory and application of a combined nonlinear suboptimal control and estimator based on the state‐dependent differential Riccati equation (SDDRE), aiming finite‐time tracking of manipulators that possess flexible joints. Though measuring the angular positions of links is possible for flexible joint manipulators (FJMs), it is not feasible to measure the changes in the position angles of actuators. In addition, the existence of measurement noise and the influence of unknown workspace disturbance are always unavoidable. Therefore, combining an estimator with a controller could result in full‐state feedback for the system. Designing the proposed integrated estimator algorithm includes two parts: The first part is the theoretical extension of the SDDRE approach expressed with nonlinear state variables and inputs of the system in a finite‐time horizon. In the second part, enhancing and upgrading software and hardware interface, the proposed procedure for FJMs is modeled and simulated. Then, the experimental implementation is applied in the online closed‐loop form. [ABSTRACT FROM AUTHOR]

In this paper, we analyze oscillatory properties of perturbed half‐linear differential equations (i.e., equations with one‐dimensional p‐Laplacian). The presented research covers the Euler and Riemann–Weber type equations with very general coefficients. We prove an oscillatory result and a nonoscillatory one, which show that the studied equations are conditionally oscillatory (i.e., there exists a certain threshold value that separates oscillatory and nonoscillatory equations). The obtained criteria are easy to use. Since the number of perturbations is arbitrary, we solve the oscillation behavior of the equations in the critical setting when the coefficients give exactly the threshold value. The results are new for linear equations as well. [ABSTRACT FROM AUTHOR]

In order to investigate the electromagnetic (EM) propagation characteristics and further get the channel models for indoor environments, a modified conformal finite‐difference time‐domain (CFDTD) method is introduced in this paper. Specifically, the dispersion characteristics of media are modeled by using the auxiliary differential equation (ADE) technique of the Debye model, and the EM field iterative equation is obtained by weighted‐length of various media in the conformal cells for the curved structure. The proposed method is used to simulate the channels of 2.4 GHz wireless fidelity (Wi‐Fi) network and 5G microcellular network, respectively, for the single‐ and complex multi‐room scenarios. The impulse response, power delay profile and path loss exponent are extracted from the simulation data to analyze the effective ultra‐wideband (UWB) channel characteristics. Compared with the traditional FDTD, the proposed method has the advantage of obtaining a stable and good accuracy solution and reducing the time step without the need to solve high‐order differential equations. [ABSTRACT FROM AUTHOR]