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

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

In this paper on a basis of functional analysis methods we justified theoretically the possibility of different orders root-polynomial functions application for interpolation and approximation of the boundary trajectory of an electron beam in case of its propagation in ionized gas with compensation of the space charge of the beam electrons. It is shown, that the root-polynomial functions satisfy to the second-order differential equation, describing the boundary trajectory of the beam electrons under such physical conditions. The results of interpolation and approximation of the boundary trajectory of the electron beam by root-polynomial functions from the second to the fifth order under the following physical conditions are presented. The interpolation results are compared with the corresponded results of the differential equation solution for the boundary trajectory of the electron beam using Runge-Kutta numerical method of the fourth order. These results are considered as reference ones for the interpolation task. To solve the approximation problem, in this paper an iterative algorithm based on the calculation of both values of the function and its derivatives at reference points is proposed. The approximation task is solved for a sample of numerical data obtained by experimental electron-beam equipment for real processes of current electron-beam technologies, which led to a rather large value of the experimental measurement error due to the effect of random factors associated with thermal treatment of products with electron beam. Test calculations show that the error of interpolation and approximation of numerical data, describing the boundary trajectory of electron beam in case of its propagation in ionized gas, does not exceed a few percent. The theoretical and practical results obtained in this paper are interesting for a wide range of specialists who are engaged in the physics of electron beams, the development of electron-beam technological equipment and implementation of current electron-beam technologies into industry. [ABSTRACT FROM AUTHOR]

Calculus equation is an important tool for mathematical research and plays an important role in most natural science research. Since the beginning of the eighteenth century, people have gradually used differential and integral equations to solve physical problems. In general, several different aspects of differential equations in the field of mathematics are concerned and studied by most scholars. However, this paper studies and establishes the optimal model for numerical solution of differential equations through deep learning and genetic algorithm. In this paper, the solution of ordinary differential equations is solved through the use of polynomial function space, while the linear combination of simple function x and its product nx can obtain multinomial function space. The space function form of polynomial is very simple, and the operation ability is very strong. Almost all functions can be approximated, and the function space can be transformed by a simple function. Through data simulation test results, it can be found that the oscillation of neural network output is stronger and stronger with the increase in depth, that is to say, the deeper depth endows the neural network with stronger oscillation properties, so for the oscillation function, the depth neural network fitting effect is better than the shallow neural network. Therefore, by combining deep learning and genetic algorithm, this paper studies and establishes the optimal model for numerical solution of differential equations, and finds that the deep neural network can largely complete data simulation. [ABSTRACT FROM AUTHOR]

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

This paper develops a novel set of displacement temperature potential functions to solve the thermoelastodynamic problems in functionally graded transversely isotropic media subjected to thermal source. For this purpose, three-dimensional heat and wave equations are considered to obtain the displacement temperature equations of motion for functionally graded materials. In the present study, a systematic method is used to decouple the elasticity and heat equations. Hence one sixth-order differential equation and two second-order differential equations are obtained. Completeness of the solution is proved using a retarded logarithmic Newtonian potential function for functionally graded transversely isotropic domain. To verify the obtained solution, in a simpler case, potential functions are generated for homogeneous transversely isotropic media that coincide with respective equations. Presented potential functions can be used to solve the problems in various media like infinite and semi-infinite space, beams and columns, plates, shells, etc., with arbitrary boundary conditions and subjected to arbitrary mechanical and thermal loads. [ABSTRACT FROM AUTHOR]

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

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

This paper is devoted to studying a class of fuzzy fractional switched implicit differential equations (FFSIDEs) with non-instantaneous impulses that there are few papers considering this issue. Considering switching law and the memory property of fractional calculus, we first present a formula of solution for FFSIDEs with non-instantaneous impulses. Subsequently, based on a sequence of Picard functions, we explore the existence of solutions for the addressed equations by successive approximation. Furthermore, Ulam–Hyers (U–H) stability for this considered equations is derived. The main results are obtained using fuzzy-valued fractional calculus and nonlinear analysis. Finally, two numerical examples illustrating the theoretical result are given. [ABSTRACT FROM AUTHOR]

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

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

This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order 3 < u ˆ ≤ 4 and includes a term Θ (β , Z (β)) alongside boundary conditions. Through the application of a fixed point theorem in appropriate function spaces, we consider nonlocal conditions along with necessary assumptions under which solutions to the given FDE exist. Furthermore, we offer an example to illustrate the results. [ABSTRACT FROM AUTHOR]

Hypergeometric functions of one and many variables play an important role in various branches of modern physics and mathematics. We often encounter hypergeometric functions with indices linearly dependent on a small parameter with respect to which we need to perform Laurent expansions. Moreover, it is desirable that such expansions be expressed in terms of well-known functions that can be evaluated with arbitrary precision. To solve this problem, we use the method of differential equations and the reduction of corresponding differential systems to a canonical basis. In this paper, we are interested in the generalized hypergeometric functions of one variable and in the Appell and Lauricella functions and their expansions in terms of the Goncharov polylogarithms. Particular attention is paid to the case of rational indices of the considered hypergeometric functions when the reduction to the canonical basis involves a nontrivial variable change. The paper comes with a Mathematica package Diogenes, which provides an algorithmic implementation of the required steps. [ABSTRACT FROM AUTHOR]

Mathematical models play a crucial role in controlling and preventing the spread of diseases. Based on the communication characteristics of diseases, it is necessary to take into account some essential epidemiological factors such as the time delay that takes an individual to progress from being latent to become infectious, the infectious age which refers to the duration since the initial infection and the occurrence of reinfection after a period of improvement known as relapse, etc. Moreover, age-structured models serve as a powerful tool that allows us to incorporate age variables into the modeling process to better understand the effect of these factors on the transmission mechanism of diseases. In this paper, motivated by the above fact, we reformulate an SEIR model with relapse and age structure in both latent and infected classes. Then, we investigate the asymptotic behavior of the model by using the stability theory of differential equations. For this purpose, we introduce the basic reproduction number R 0 of the model and show that this threshold parameter completely governs the stability of each equilibrium of the model. Our approach to show global attractivity is based on the fluctuation lemma and Lyapunov functionals method with some results on the persistence theory. The conclusion is that the system has a disease-free equilibrium which is globally asymptotically stable if R 0 < 1 , while it has only a unique positive endemic equilibrium which is globally asymptotically stable whenever R 0 > 1 . Our results imply that early diagnosis of latent infection with decrease in both transmission and relapse rates may lead to control and restrict the spread of disease. The theoretical results are illustrated with numerical simulations, which indicate that the age variable is an essential factor affecting the spread of the epidemic. [ABSTRACT FROM AUTHOR]

This paper initializes higher-order uncertain calculus that deals with higher-order differentiation and multiple integration of uncertain process based on uncertainty theory. Fubini theorem and fundamental theorem of higher-order uncertain calculus are derived. Finally, this paper rigorously defines higher-order uncertain differential equations and introduces some analytic methods for solving these equations. [ABSTRACT FROM AUTHOR]

In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation driven by a highly irregular vector field and study the effect of this noise on such dynamical systems. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the "roughness" and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds, in a certain sense, to the Nash–Moser iterative scheme in combination with a new concept of "higher order averaging operators along highly fractal stochastic curves". This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations. [ABSTRACT FROM AUTHOR]

Uncertainty theory is an area in axiomatic mathematics recently proposed by Professor Baoding Liu and aiming to deal with belief degrees. Retrieving 1004 journal articles from the Web of Science database between 2008 and 2019, and utilizing CiteSpace and Pajek software, we analyze the publications per year and by geographical distribution, productive scholars and their cooperation, key journals, highly cited articles and main paths of the field. In this way, seven key sub-fields of uncertainty theory and their research potential are derived. The results show the following: (1) The literature on uncertainty theory follows a linear growth trend, involves an extensive network of 1000 scholars worldwide and is published in 300 journals, indicating thus that uncertainty theory has become increasingly attractive, and its academic influence is gradually expanding. (2) Seven key sub-fields of uncertainty theory have clearly been identified, including the axiomatic system, uncertain programming, uncertain sets, uncertain logic, uncertain differential equations, uncertain risk analysis, and uncertain processes. Among them, uncertain differential equations and programming are the two main sub-fields with the largest numbers of published papers. Furthermore, for evaluating the research potential of sub-fields, maturity and recent attention indicators are calculated using the citations, total number of publications, quantity of most cited literature and half-life. Based on these indicators, uncertain processes shows the greatest development potential, and has remained a hot topic in recent years, being mainly concentrated on the uncertain renewal reward process, optimal control of discrete-time uncertain systems, and uncertain linear quadratic optimal control. Additionally, uncertain risk analysis is ranked second, and focuses on the analysis of expected losses, investment risk, and structural reliability of uncertain systems. [ABSTRACT FROM AUTHOR]

In the framework of uncertainty theory, this paper investigates the pricing problem of American swaption. By assuming that the floating interest rate obeys an uncertain differential equation, the pricing formula of American swaption is derived. Furthermore, parameter estimation of the uncertain interest rate model is given, and the uncertain hypothesis test shows that the uncertain interest rate model fits the Shanghai interbank offered rate well. Finally, as a byproduct, this paper also indicates that stochastic differential equations cannot model real-world interest rates. [ABSTRACT FROM AUTHOR]

Without taking into account external friction, small transverse vibrations of a viscoelastic pipeline of unit length are described for nonnegative values of time in dimensionless variables by an integro-differential equation with hinged conditions at the ends and with initial conditions. The solution of this equation can be written in terms of an operator semigroup. In the present paper, we establish that this equation generates a semigroup that is analytic in some sector of the right half-plane. [ABSTRACT FROM AUTHOR]

This paper presents a mathematical demonstration that particular observed energetic particle distributions of space physics in the particle velocity magnitude, v ∈ [ 0 , ∞) , are not rigorously Kappa distributions. The method is based on the distribution function of a test particle of mass m in a heat bath of particles of mass M . The distribution function is given by a Fokker-Planck equation. The particles interact via Coulomb collisions and a second diffusion coefficient that represents the effects of wave-particle interactions. For the particular wave-particle diffusion coefficient that varies inversely with the particle velocity, the steady distribution for m / M → 0 is a Kappa distribution which is the solution of a Pearson ordinary differential equation. The analysis of the observed distributions versus v ∈ (0 , ∞) employed in this paper is based on the Pearson differential equation and applied to several published distributions. The chosen distributions are representative and shown not to be Kappa distributions. Thus, the Tsallis nonextensive entropy which yields uniquely the Kappa distribution does not explain the occurrence of the myriad of nonequilibrium distributions in space physics. [ABSTRACT FROM AUTHOR]

A hybrid system interacts with the discrete and continuous dynamics of a physical dynamical system. The notion of a hybrid system gives embedded control systems a great advantage. The Langevin differential equation can accurately depict many physical phenomena and help researchers effectively represent anomalous diffusion. This paper considers a fractional hybrid Langevin differential equation, including the ψ-Caputo fractional operator. Furthermore, some novel boundaries selected are considered to be a problem. We used the Schauder and Banach fixed-point theorems to prove the existence and uniqueness of solutions to the considered problem. Additionally, the Ulam-Hyer stability is evaluated. Finally, we present a representative example to verify the theoretical outcomes of our findings. [ABSTRACT FROM AUTHOR]

This paper is mainly focused on the solution of Sylvester matrix differential equations with time-independent coefficients. We propose a new approach based on the construction of a particular constant solution which allows to construct an approximate solution of the differential equation from that of the corresponding algebraic equation. Moreover, when the matrix coefficients of the differential equation are large, we combine the constant solution approach with Krylov subspace methods for obtaining an approximate solution of the Sylvester algebraic equation, and thus form an approximate solution of the large-scale Sylvester matrix differential equation. We establish some theoretical results including error estimates and convergence as well as relations between the residuals of the differential and its corresponding algebraic Sylvester matrix equation. We also give explicit benchmark formulas for the solution of the differential equation. To illustrate the efficiency of the proposed approach, we perform numerous numerical tests and make various comparisons with other methods for solving Sylvester matrix differential equations. [ABSTRACT FROM AUTHOR]

Brück’s conjecture asserts that if a non-constant entire function f(z) with hyper-order ρ 2 (f) ∉ N ∪ { ∞ } shares one finite value a CM (counting multiplicities) with its derivative, then f ′ - a = c (f - a) , for some non-zero constant c. This conjecture has been affirmed for entire functions with finite order and hyper-order less than one. In this paper, we show that Brück’s conjecture is true for entire functions that satisfy second order differential equations with meromorphic coefficients of finite order. [ABSTRACT FROM AUTHOR]

In this paper, the authors studied sufficient conditions to understand the oscillatory behavior of solutions of nonlinear neutral delay (and advanced) differential equation (E) with deviating arguments. Employing Banach fixed point theorem, the authors established the existence of bounded positive solutions of (E). The authors solved various examples using MATLAB software to understand the applications of the main theorems. Moreover, the authors analyzed the effect of delay terms on the behavior of solutions of (E). This paper has improved the results obtained in Basu [2] and Chatzarakis [11]. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach to analyzing the dynamics of COVID-19 using nonstandard finite difference (NSFD) schemes. Our model incorporates both asymptomatic and symptomatic infected individuals, allowing for a more comprehensive understanding of the epidemic's spread. We introduce an unconditionally stable NSFD system that eliminates the need for traditional Runge–Kutta methods, ensuring dynamical consistency and numerical accuracy. Through rigorous numerical analysis, we evaluate the performance of different NSFD strategies and validate our analytical findings. Our work demonstrates the benefits of using NSFD schemes for modeling infectious diseases, offering advantages in terms of stability and efficiency. We further illustrate the dynamic behavior of COVID-19 under various conditions using numerical simulations. The results from these simulations demonstrate the effectiveness of the proposed approach in capturing the epidemic's complex dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we answer the questions posed by Gundersen and Yang about the entire solutions of a class of nonlinear homogeneous binomial differential equations and obtain explicit forms of all the entire solutions of this type of differential equations. Moreover, we provide some examples to demonstrate that the equation solutions we obtained are accurate. [ABSTRACT FROM AUTHOR]

Although many researchers have examined the synchronization characteristics of numerous dynamic models, equipment that utilizes synchronization phenomena with high yield, optimal isolation effect, and robust stability performance has not yet been completely into reality. In this paper, a novel model scheme with three bodies driven by four vibrators is presented, and the aim is to clarify the mechanism of synchronization and stability and find the self-balancing behavior of the system where the dynamic load transmitted to the foundation is zero. The motion differential equations of the system and the theoretical criteria for achieving stable synchronization behavior are provided. The kinetic and coupling dynamic characteristics of the system are discussed in detail through numerical analyses, including the stable states of four vibrators, phase relationships among three bodies, synchronization and stability ability coefficients, and a maximum of the coupling torque, etc. It shows that the self-balancing behavior of the system is occurring in Region 2, where the reverse relative motion of two rigid bodies exhibits stronger harmonic vibration, and the isolation body embodies no vibration, which probably leads to the minimum of transmitting the dynamic loads to the foundation. Additionally, it found the diversity of the nonlinear system, which is in all the other Regions except for Region 2. The characteristics analysis and simulation results verify the validity and feasibility of the theoretical investigation. [ABSTRACT FROM AUTHOR]

Background: Numerical methods are used to solve differential equations, but few are effective for nonlinear ordinary differential equations (ODEs) of order higher than one. This paper proposes a new method for such ODEs, based on Taylor series expansion. The new method is a second-order method for second-order ODEs, and it is equivalent to the central difference method, a well-known method for solving differential equations. The new method is also simple to implement for higher-order differential equations. The proposed technique was applied to solve the Van der Pol and Van der Pol–Duffing equations. It is stable over a wide range of nonlinearity and produces accurate and reliable results. For the self-excitation Van der Pol equation, the proposed technique was applied with different values of nonlinear damping. Results: The results were compared with those obtained using the ODE15s solver in MATLAB. The two sets of results showed excellent agreement. For the forced Van der Pol–Duffing equation, the proposed technique was applied with different values of exciting force amplitude and frequency. It was found that for certain conditions, the solution obtained using the proposed technique differed from that obtained using ODE15s. Conclusions: The solution obtained using the proposed technique showed good agreement with the solutions obtained using ODE45 and Runge–Kutta fourth order. The results show that the proposed approach is very simple to apply and produces acceptable error. It is a powerful and versatile tool for solving of high-order nonlinear differential equations accurately. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a mixed boundary value problem for a complete Sturm–Liouville equation, where the coefficients can also be negative. In particular, the existence of infinitely many distinct positive solutions to the given problem is obtained by using critical point theory. [ABSTRACT FROM AUTHOR]

Recent research indicates the need for improved models of physical phenomena with multiple shocks. One of the newest methods is to use differential inclusions instead of differential equations. In this work, we intend to investigate the existence of solutions for an m-dimensional system of quantum differential inclusions. To ensure the existence of the solution of inclusions, researchers typically rely on the Arzela–Ascoli and Nadler's fixed point theorems. However, we have taken a different approach and utilized the endpoint technique of the fixed point theory to guarantee the solution's existence. This sets us apart from other researchers who have used different methods. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables, and some figures. The paper ends with an example. [ABSTRACT FROM AUTHOR]

In this paper, we study a new splitting method for the semi-linear system of ordinary differential equation, where the linear part is stiff. Firstly, the stiff part is split into two parts. The first stiff part, that is called the stiff-cutter and expected to be easily inverted, is implicitly treated. The second stiff part and the remaining nonlinear part are explicitly treated. Therefore, such stiff-cut method can be fast implemented and save the CPU time. Theoretically, we rigorously prove that the proposed method is unconditionally stable and convergent, if the stiff-cutter is chosen to be well-matched in the stiff part. As an application, we apply our method to solve a spatial-fractional reaction-diffusion equation and give a way for how to choose a suitable stiff-cutter. Finally, numerical experiments are carried out to illustrate the accuracy and efficiency of the proposed stiff-cut method. [ABSTRACT FROM AUTHOR]

We derive Melnikov type conditions for the persistence of periodic solutions in perturbed slowly varying discontinuous differential equations. In contrast to Battelli and Fečkan (Mathematics 9(19):2449, 2021) we assume that the unperturbed (frozen) equation has a family of periodic solutions depending on some parameters. The result of this paper are motivated by and extend a result in Wiggins and Holmes (SIAM J Math Anal 18:592–611, 1987) where the authors considered a two dimensional hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation. [ABSTRACT FROM AUTHOR]

Any organism in nature will inevitably be affected by uncertain factors. The deterministic model and stochastic model are no longer suitable for population dynamics analysis under uncertain noise environment. In order to simulate these problems more reasonably, we propose an uncertain logistic population model with Allee effect, which describes the population dynamic behavior through uncertain differential equation. In this paper, the solution and α -path of the uncertain Logistic population model with Allee effect are given, and the behavior analysis of the solution is also discussed. Besides, some numerical examples are put forward to illustrate the conclusions obtained in the paper. [ABSTRACT FROM AUTHOR]

We study the question of the algebraic difference independence of the Euler gamma function Γ and the functions in a certain class F , which contains those Dirichlet series as L-functions in the extended Selberg class S ♯ and some periodic functions. The main results in this paper are the difference analogues of the corresponding results from Lü (J Math Anal Appl 462(2):1195–1204, 2018) that showed that the Euler gamma function Γ and the functions in F can not satisfy a class of algebraic differential equations with meromorphic coefficients ϕ of Nevanlinna's characteristics satisfying T (r , ϕ) = o (r) , as r → ∞. Examples are provided to show that the main results in this paper, in a sense, are best possible. [ABSTRACT FROM AUTHOR]

Abstract—Elastic bending stresses and deformations in the lithosphere are usually calculated based on the Kirchhoff–Love theory for thin plates. The criterion for its applicability is the smallness of the ratio of plate thickness to plate length. In oceanic plates, due to the buoyancy force of the mantle, the main deformations are not uniformly distributed along the plate but concentrate in the vicinity of the subduction zone. Therefore, the effective length of the bending part of the plate is a few fractions of its actual length, and the plate thinness criterion is partially violated. In this paper, we analyze the possibility of applying thick plate bending equations. The existing variational theories of 3D bending of thick plates are substantially more complicated than the Kirchhoff–Love theory, as they involve solving three differential equations instead of one, and have limited application due to their complexity. Since geophysical applications frequently use 2D models, in this paper we analyze in detail the potential and accuracy of the thick plate bending theory for 2D models. After the conversion to the 2D plane strain and plane stress approximation, the original 3D Reissner thick plate bending equations are written out in the form similar to the Kirchhoff equations with additive corrections and are supplemented with the explicit formulas for longitudinal displacement. The comparison of the analytical solutions of the 2D Reissner equations with the exact solutions shows that the 2D approximation only provides a correction for the plate deflection function. However, this correction refines the Kirchhoff–Love theory by almost an order of magnitude. At the same time, the solution of the equations in this case turns out to be almost as simple as the solution of the thin plate equations. [ABSTRACT FROM AUTHOR]

This paper is devoted to describing the asymptotic behavior of solutions to linear differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. In this paper, to construct a uniform asymptotics, we use resurgent analysis methods based on the Laplace–Borel integral transformation. [ABSTRACT FROM AUTHOR]

In this paper, we suggest a new type of tasks for control theory for stochastic dynamical systems — programmed control with Probability 1 (PCP1). PCP1 is an application of an invariant theory. We use the PCP1 concept for dynamical processes described by a system of Itô differential equations with jump-diffusion (GSDES). The considered equations include the drift, the diffusion, and the jumps, together or not. Features of our approach are both a wide set of dynamical systems and investigation of such systems for their unique trajectories. Our method is based on the concept of a stochastic first integral (SFI) for GSDES and its equations which author studied before. The purpose of the present paper is to construct a differential equation system (both stochastic and deterministic) using a known set of FIs for the investigating process. Several examples are given. [ABSTRACT FROM AUTHOR]

The geometry description plays a central role in many engineering applications and directly influences the quality of the computer simulation results. The geometry of a space curve can be completely defined in terms of two parameters: the horizontal and vertical curvatures, or equivalently, the curve curvature and torsion. In this paper, distinction is made between the track angle and space-curve bank angle, referred to in this paper as the Frenet bank angle. In railroad vehicle systems, the track bank angle measures the track super-elevation required to define a balance speed and achieve a safe vehicle operation. The formulation of the track space-curve differential equations in terms of Euler angles, however, shows the dependence of the Frenet bank angle on two independent parameters, often used as inputs in the definition of the track geometry. This paper develops the general differential equations that govern the track geometry using the Euler angle sequence adopted in practice. It is shown by an example that a curve can be twisted and vertically elevated but not super-elevated while maintaining a constant vertical-development angle. The continuity conditions at the track segment transitions are also examined. As discussed in the paper, imposing curvature continuity does not ensure continuity of the tangent vectors at the curve/spiral intersection. Several curve geometries that include planar and helix curves are used to explain some of the fundamental issues addressed in this study. [ABSTRACT FROM AUTHOR]

This paper provides a class of upper and lower solution definitions for second-order coupled systems by transforming the fourth-order differential equation into a second-order differential system. Then, by constructing a homotopy parameter and utilizing the maximum principle, we propose an upper and lower solutions method for studying a class of second-order coupled systems with Dirichlet boundary conditions and obtain an existence result. [ABSTRACT FROM AUTHOR]

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid. [ABSTRACT FROM AUTHOR]

Mathematical modelling has become a key tool in pharmacological analysis, towards understanding dynamics of cell signalling and quantifying ligand-receptor interactions. Ordinary differential equation (ODE) models in receptor theory may be used to parameterise such interactions using timecourse data, but attention needs to be paid to the theoretical identifiability of the parameters of interest. Identifiability analysis is an often overlooked step in many bio-modelling works. In this paper we introduce structural identifiability analysis (SIA) to the field of receptor theory by applying three classical SIA methods (transfer function, Taylor Series and similarity transformation) to ligand-receptor binding models of biological importance (single ligand and Motulsky-Mahan competition binding at monomers, and a recently presented model of a single ligand binding at receptor dimers). New results are obtained which indicate the identifiable parameters for a single timecourse for Motulsky-Mahan binding and dimerised receptor binding. Importantly, we further consider combinations of experiments which may be performed to overcome issues of non-identifiability, to ensure the practical applicability of the work. The three SIA methods are demonstrated through a tutorial-style approach, using detailed calculations, which show the methods to be tractable for the low-dimensional ODE models. [ABSTRACT FROM AUTHOR]

Purpose: The deep or hard formation is an important development direction of oil and gas energy resource exploration, and new equipment for accelerating rock breaking is urgently needed. Therefore, a new composite impact acceleration tool is proposed in this paper. Methods: Based on the tool, the dynamic theoretical model and differential equation of the drill string system are established. And simulate the rock breaking characteristics of PDC (Polycrystalline Diamond Compact) cutters under composite impact. Results: The new tool can change these results from periodic fluctuation to be stable such as the angular velocity of the drill bit, driving and friction torque. Simultaneously, the angular displacement changes from discontinuous to continuous increase. The increase of WOB (Weight on Bit) could enhance PDC invasion depth and improve rock breaking efficiency, but it may cause bouncing of drill bit. The rotational speed of 60 rpm (revolution per minute) is the critical point in the cases. When the rotational speed increases above 60 rpm, the specific energy of rock breaking reduces and the efficiency improves. However, it shows the opposite characteristics when the rotational speed increases below 60 rpm. Conclusions: The application of the composite impact acceleration tool can significantly suppress the stick–slip vibration and improves dynamics stability of drill string system. Also, appropriate selection of key parameters such as WOB or rotational speed can improve drilling efficiency. [ABSTRACT FROM AUTHOR]

Purpose: The real-time dynamics of the single-link flexible manipulator is a challenging problem due to its inherent instability and in situ vibration. In order to add the criticalities to these real-time dynamics, run-time vibration does play a pivotal role in designing a robust control system for the flexible robotic manipulator. Methodology: Governing differential equations and the boundary conditions are usually considered exact of the single-link manipulator and the formulation leads to an Eigenvalue problem where the elements of the matrices are in exact form. Double parametric form has been used to solve the fuzzy differential equation. Results and Conclusions: In this paper, a new idea has been introduced in the above problem for a representative single-link robotic manipulator considering the uncertainty in the associated parameter(s) in the governing differential equation, which may mimic the actual scenario of the real environment. The uncertainty has been considered in terms of a novel fuzzy model which agrees with the crisp case model too. [ABSTRACT FROM AUTHOR]

This paper presents a bilinear Chebyshev pseudo-spectral method to compute European and American option prices under the two-asset Black–Scholes and Heston models. We expand a function and its derivatives into their Chebyshev series, so the differentiation matrices that act on the Chebyshev coefficients are sparse and better conditioned. First, the equation is spatially discretized using a bilinear pseudo-spectral method created by Chebyshev polynomials and a first–order matrix differential equation (MDE) is obtained. Then, using the Kronecker product, this equation is converted to a system of first–order ODEs. For solving the European options, by using an eigenvalue decomposition method, the arising system will be analytically solved. Therefore, the arising errors are because of spatial discretization and quadrature errors. For solving the American options, the approach is combined with the operator splitting method or penalty method to obtain the temporal discretization. By avoiding some transforms to convert the equation to a constant coefficient equation without mixed derivatives, we will obtain more accurate solutions. Also, transforming the European option into a set of separated ODEs by an eigenvalue decomposition causes to reduce the computational complexity. We also consider the hedge ratios which show the sensitivity of an option to the stock prices. Several numerical examples are included to show the accuracy and efficiency of the proposed approach. The results show that the spectral convergence can be achieved for models with smooth functions. [ABSTRACT FROM AUTHOR]

In this paper, we extend the classical theory of normal forms for continuous and difference dynamical systems to dynamic equations on discrete time scales. As consequences of the well known results from the theory of analytic differential equations, we obtain some versions of the Poincaré and Siegel theorems for dynamic equations on discrete time scales. Using these results and known results on the stability of dynamic equations on time scales, we obtain some stability results for the nonlinear dynamic equations. We also prove some results on generic properties of bifurcation curves and the saddle-node bifurcation for one-parameter families of dynamic equations on arbitrary time scales. [ABSTRACT FROM AUTHOR]

In this paper, by applying fractional quantum calculus, we study a nonlinear Duffing-type equation with three sequential fractional q-derivatives. We prove the existence and uniqueness results by using standard fixed-point theorems (Banach and Schaefer fixed-point theorems). We also discuss the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of the mentioned Duffing problem. Finally, we present an illustrative example and nice application; a Duffing-type oscillator equation with regard to our outcomes. [ABSTRACT FROM AUTHOR]

The adaptive predefined settling time performance tracking control for a class of uncertain nonlinear network systems with fast time-varying or even abrupt reference signals is studied. In this paper, an improved performance function with time-varying boundary (following the change of reference signal) is designed to overcome the problem that the traditional prescribed performance control (PPC) can only converge to a constant in steady state. Compared with the existing adaptive finite time prescribed performance congestion control for tracking fixed value reference signals, an adaptive predefined settling time prescribed performance congestion control strategy is proposed by means of some new recursive construction (introducing nonautonomous differential equations (NDE)) and analysis innovation. In addition, the network system considered in this paper is more general. Under the framework of backstepping, NDE is used iteratively to control the approximation error with boundary inequality. It is proved theoretically that all signals of the closed-loop system are predefined settling time bounded. Finally, the effectiveness of the proposed control strategy is verified by simulations. [ABSTRACT FROM AUTHOR]

Parameter estimation is one of key issues in the study of uncertain differential equations. In this paper, one novel estimation method named the estimating function technique of uncertain differential equations based on uncertain integrals is proposed, which is the first method based on uncertain integrals appeared in the literature. Compared to existing methods, our approach fully utilizes information from both the parameters and observed data. In certain settings, previous methods can be viewed as special cases of our technique. One numerical example with different transform functions is provided to illustrate our method and the efficiency is shown by residual analysis. Additionally, a case study on Alibaba stock prices is included to demonstrate the application of our method. The paper concludes with final remarks and recommendations for future research. [ABSTRACT FROM AUTHOR]

Industrial facilities, as an essential part of socio-economic systems, are susceptible to disruptions caused by earthquakes. Such disruptions may result from direct structural damage to facilities or their loss of functionality due to impacts on their support facilities and infrastructure systems. Decisions to improve the seismic performance of industrial facilities should ideally be informed by risk (and resilience) analysis, taking into account their loss of functionality and the following recovery under the influence of various sources of uncertainty. Rather than targeting specific individual facilities like a hazardous chemical plant, our objective is to quantify the resilience of interacting industrial facilities (i.e., networked industrial facilities) in the face of uncertain seismic events while accounting for their functional dependencies on infrastructure systems. A specific facility, such as a hazardous chemical plant, can be a compound node in the network representation, interacting with other facilities and their supporting infrastructure components. In this context, a compound node is a complex system in its own right. To this end, this paper proposes a formulation to model the functionality of interacting industrial facilities and infrastructure using a system of coupled differential equations, representing dynamic processes on interdependent networked systems. The equations are subject to uncertain initial conditions and have uncertain coefficients, capturing the effects of uncertainties in earthquake intensity measures, structural damage, and post-disaster recovery process. The paper presents a computationally tractable approach to quantify and propagate various sources of uncertainty through the formulated equations. It also discusses the recovery of damaged industrial facilities and infrastructure components under resource and implementation constraints. The effects of changes in structural properties and networks' connectivity are incorporated into the governing equations to model networks' functionality recovery and quantify their resilience. The paper illustrates the proposed approach for the seismic resilience analysis of a hypothetical but realistic shipping company in the city of Memphis in Tennessee, United States. The example models the effects of dependent water and power infrastructure systems on the functionality disruption and recovery of networked industrial facilities subject to seismic hazards. [ABSTRACT FROM AUTHOR]

Background: The modeling and identification of contact interface restoring forces of different parts of mechanical systems has received increasing attention in the past years. Micro-vibro-impacts and Micro/macro-slip damping mechanisms as the initial source of nonlinear behavior in structural contact interfaces. The above-mentioned mechanisms could appear whilst the amplitude of the vibration is increased in frictional contact mechanisms. Purpose: The aim of this paper is to examine the nonlinear behavior of the frictional contact interfaces in the presence of variable contact preload induced by micro-vibro-impacts. The Bouc-Wen model is widely used in various engineering fields such as electromagnetics, mechanical, and civil engineering; and can be used to represent the hysteric behavior of the system and/or mechanisms. However, it is presumed that the contact preload is constant and uniform on the entire rough interface among the modeling of the frictional contact interface and its properties. Methods: Meanwhile, the micro-vibro-impacts is negligible on the contact interface. Moreover, this study is aimed to investigate the effects of variable preload mechanisms in the contact region and to propose a modified Bouc-Wen model that takes the effects of micro-vibro-impacts on the contact interface preload into account, as well as the resulting shear stiffness on the connection interface during macro-slip. Results: The unknown parameters of the modified Bouc-Wen model are estimated/identified by matching the experimentally obtained hysteresis loops. Additionally, the Hill stability criterion is used to investigate the stability of the resulting differential equations with time-varying parameters in the proposed modified Bouc-Wen hysteretic model. Conclusions: According to the simulation results, the modified Bouc-Wen model captures the friction forces on the contact interfaces, accurately. [ABSTRACT FROM AUTHOR]