Theory Of Equations Research Articles

Solution to the periodic problem for the impulsive hyperbolic equation with discrete memory

In this article, we consider the periodic problem for the impulsive hyperbolic equation with discrete memory. Impulsive hyperbolic equations with discrete memory arise as a mathematical model for describing physical processes in the neural networks, discontinuous dynamical systems, hybrid systems, and etc. Questions of the existence and construction of solutions to periodic problems for impulsive hyperbolic equations with discrete memory remain important issues in the theory of discontinuous partial differential equations. To find the solvability conditions of this problem we apply Dzhumabaev’s parametrization method. The coefficient conditions for the existence and uniqueness of the periodic problem for the impulsive hyperbolic equation with discrete memory are established. We offer an algorithm for determining the approximate solution to this problem and show its convergence to the exact solution of the periodic problem for the impulsive hyperbolic equation with discrete memory.

Stability and backward bifurcation in a malaria-transmission model with sterile mosquitoes

In this paper, we establish and study a malaria-transmission dynamic model of releasing sterile mosquitoes, which focuses on the mosquito populations affected by the impact of limited resources. First, we formulate a dynamic model with mosquitoes where the releasing rate of sterile mosquitoes is constant, and analyze the existence and stability of the equilibrium. Using the stability theory and method of differential equations, the threshold value of releasing sterile mosquitoes is obtained. Furthermore, we establish a malaria-transmission dynamic model with sterile release, and derive a formula for the reproductive number of infection and investigate the existence of endemic equilibria. With the symmetry of mathematical expression, it is also shown that this model may undergo backward bifurcation, where the locally stable disease-free equilibrium coexists with an endemic equilibrium. Finally, numerical simulations to illustrate our findings and brief discussions are provided.

Hammerstein equations for sparse random matrices

Abstract Finding eigenvalue distributions for a number of sparse random matrix ensembles can be reduced to solving nonlinear integral equations of the Hammerstein type. While a systematic mathematical theory of such equations exists, it has not been previously applied to sparse matrix problems. We close this gap in the literature by showing how one can employ numerical solutions of Hammerstein equations to accurately recover the spectra of adjacency matrices and Laplacians of random graphs. While our treatment focuses on random graphs for concreteness, the methodology has broad applications to more general sparse random matrices.

Fully Distributed Cooperative Output Regulation of Heterogeneous Periodic Linear Multi‐Agent Systems With Input Saturation

ABSTRACTThis article delves into the cooperative output regulation of heterogeneous continuous‐time periodic linear multi‐agent systems with input saturation under undirected graphs. Leveraging characteristics inherent in the parametric differential Lyapunov equation and cooperative output regulation theory, we formulate an observer‐based protocol. Specifically, we design distributed state observers for both the leader and followers, aiming to estimate the leader state and the follower state itself, respectively. Utilizing these estimated states, we propose a class of fully distributed observer‐based protocols to address the considered problem. In addition, we make an extension to the reduced‐order observer‐based form. The most distinguishing contributions of our proposed protocols, in contrast to existing literature, lie in the fact that the considered dynamics of each agent are time‐varying and have input saturation constraints. Notably, our protocols eliminate the need for global information of the network, operating in a fully distributed method. Finally, we apply the proposed control protocol to the multi‐spacecraft elliptical orbit rendezvous problem.

A TVD WAF scheme based on an accurate Riemann solver to simulate compressible two-phase flows

PurposeThis study aims to propose a robust total variation diminishing (TVD) weighted average flux (WAF) finite volume scheme for investigating compressible gas–liquid mixture flows.Design/methodology/approachThis study considers a two-phase flow composed of a liquid containing dispersed gas bubbles. To model this two-phase mixture, this paper uses a homogeneous equilibrium model (HEM) defined by two mass conservation laws for the two phases and a momentum conservation equation for the mixture. It is assumed that the velocity is the same for the two phases, and the density of phases is governed by barotropic laws. By applying the theory of hyperbolic equations, this study establishes an exact solution of the Riemann problem associated with the model equations, which allows to construct an exact Riemann solver within the first-order upwind Godunov scheme as well as a robust TVD WAF scheme.FindingsThe ability and robustness of the proposed TVD WAF scheme is validated by testing several two-phase flow problems involving different wave structures of the Riemann problem. Simulation results are compared against analytical solutions and other available numerical methods as well as experimental data in the literature. The proposed approach is much superior to other strategies in terms of the accuracy and ability of reconstruction.Originality/valueThe novelty of this work lies in its methodical extension of a TVD WAF scheme implementing an exact Riemann solver developed for compressible two-phase flows. Furthermore, other novelty lies on the quantitative calculation of different Riemann problem two-phase flows. Simulation results involve the verification of the constructed methods on the exact solutions of HEM without any restriction of variables.

On the existence-uniqueness and exponential estimate for solutions to stochastic functional differential equations driven by G-Lévy process

The existence-uniqueness theory for solutions to stochastic dynamic systems is always a significant theme and has received tremendous attention. This article aims to study the theory for stochastic functional differential equations (SFDEs) driven by the G-Lévy process. It derives the existence-uniqueness theorem for solutions to SFDEs driven by the G-Lévy process. Moreover, it shows the error estimation between the exact solution x(t) and Picard approximate solutions xn(t),n≥1. Ultimately, the exponential estimate has been derived.

Nonclassical dynamical behavior of solutions of partial differential-difference equations

<p>For partial differential-difference equations, a review of results regarding the relation between the type of the equation and dynamical properties of its solutions is provided. This includes the case of elliptic equations with timelike independent variables: Their solutions acquire dynamical properties (more exactly, behave as solutions of parabolic equations). The following approach to classify differential-difference equations into types, based on the property of differential-difference operators to be Fourier multipliers is applied in the following manner: An operator is treated to be elliptic if the real part of its symbol is positive, while the parabolic and hyperbolic types are defined correspondingly. It is shown that the proposed approach (being a natural extension of the classical notion of the ellipticity) is reasonable. On the other hand, fundamental novelties (compared with the classical theory of partial differential equations) occur as well. We provide conditions which guarantee the following results. For the half-space (half-plane) Dirichlet problem for elliptic equations, integral representations (of the Poisson type) of solutions are constructed, which are infinitely differentiable outside the boundary hyperplane (plane), and the asymptotic closeness of solutions (as the timelike independent variable unboundedly increases) takes place. For the Cauchy problem for parabolic equations, the same is valid, but we deal with the classical time instead of the timelike independent variable. For hyperbolic equations, multiparameter families of infinitely smooth global solutions are constructed. The said (sufficient) conditions restrict the sign of the real part of the symbol for the differential-difference operator with respect to spatial or spacelike independent variables. In a number of special cases, they might be weakened such that symbols with sign-changing real parts are admitted. The objective of the study is to observe the current stage of the classification issues for partial differential-difference equations in terms of the aforementioned approach: The ellipticity of a Fourier multiplier is defined by means of the sign (of there real part) of its symbol. Since both differential and translation operators are Fourier multipliers, methods of Fourier analysis are applicable in this study: We apply the Fourier transformation to the original partial differential-difference equation, solve the obtained ordinary differential equation, and apply the inverse Fourier transformation to the obtained solution. The main contribution obtained within this study is an efficient (workable) type concept for the fundamentally new extension of the class of partial differential equations, which is the class of partial differential-difference equations.</p>

Equivariance and partial observations in Koopman operator theory for partial differential equations

Equivariance and partial observations in Koopman operator theory for partial differential equations

Molecular integral equations theory in the near critical region of CO2

Molecular integral equations theory in the near critical region of CO2

Linking dynamics and structure in highly asymmetric ionic liquids

We explore an idealized theoretical model for ion transport within highly asymmetric ionic liquid mixtures. A primitive model-inspired system serves as a representative for asymmetric ionic materials (such as liquid crystalline salts) which quench to form disordered, partially arrested phases. Self-consistent generalized Langevin equation theory is applied to understand the connection between the size ratio of charge-matched salts and their average mobility. Within this model, we identify novel glassy states where one of the two charged species (without loss of generality, either the macro-cation or the micro-anion) is arrested, while the other retains liquid-like mobility. We discuss how this result is useful in the development of novel single-ion conducting phases in ionic liquid-based materials, for instance, conductors operating at low temperature or the technology associated with ionic liquid crystals.

Blow-up issues and peakons for a two-component high-order Fokas–Olver–Rosenau–Qiao system

In this paper, we investigate the blow-up issues and peakons for a two-component high-order Fokas–Olver–Rosenau–Qiao system, which can be view as a multi-component extension of the high-order Fokas–Olver–Rosenau–Qiao equation. We first address the local well-posedness of the Cauchy problem in the framework of the Sobolev and Besov spaces. Next, we derive the precise blow-up mechanism for strong solutions through transport equation theory, utilizing the system’s fine structure and conservation law, we present two new blow-up results about certain initial profiles in finite time. Finally, peakon solutions are discussed as well. The results are helpful to understand how two-component and high-order nonlinearities affect the blow-up solutions.

A Regularity Theory for Evolution Equations with Time-measurable Pseudo-differential Operators in Weighted Mixed-norm Sobolev-Lipschitz Spaces

AbstractThis paper investigates the existence, uniqueness, and regularity of solutions to evolution equations with time-measurable pseudo-differential operators in weighted mixed-norm Sobolev-Lipschitz spaces. We also explore trace embedding and continuity of solutions.

Modelling environmental stochasticity in a stage-structured population applying stochastic differential equations

This work presents a new approach to incorporating environmental stochasticity into stage-structured models by combining population dynamics with stochastic differential equations theory. Random fluctuations were treated as Brownian motion, where the stochastic diffusion process here was the population size. The resulting SDEs model was discussed in terms of existence, uniqueness and stability of the solution. Numerical analyses were investigated in this work to analyze and visualize our model. Our approach allowed us to project the population behavior and to calculate the stochastic growth rate. The quasi-extinction risk was also studied by setting a threshold population size under different conditions. The proposed method was applied to a real dataset of a plant species populations, and the obtained results were in excellent agreement with previous studies. Therefore, our method can be an alternative way to incorporate stochasticity in such structured population models.

Disturbance decoupling for biological fermentation systems with single input single output

The biological fermentation process has the characteristics of nonlinearity and multivariable coupling. To improve the performance of decoupling control in the fermentation process, a disturbance decoupling control based on the Lie symmetry method is proposed to obtain the analytical feedback control for a class of biological fermentation systems with single input single output (SISO). Firstly, the state-space equations and the disturbance decoupling model for a class of SISO biological fermentation systems are defined; Secondly, the key technologies and algorithm approaches of Lie symmetry theory for differential equations are introduced, and the conditions and the properties of Lie symmetry for nonlinear control systems under group action are given in detail; Finally, the derived distribution of Lie symmetric infinitesimal generators is used to prove the sufficient conditions for local disturbance decoupling in the system, and the closed-loop state feedback analytical law of the system is constructed. The proposed control method is applied to the disturbance decoupling control of mycelium concentration and substrate concentration in the biological fermentation process. Numerical simulation results show that the proposed control method can effectively improve the system decoupling control performance. Meanwhile, using Lie symmetry, the cascade decoupling standard form and the static state feedback law of biological fermentation systems can be constructed.

Poynting and polarization vectors mixed imaging condition of source time‐reversal imaging

AbstractSource time‐reversal imaging based on wave equation theory can achieve high‐precision source location in complex geological models. For the time‐reversal imaging method, the imaging condition is critical to the location accuracy and imaging resolution. The most commonly used imaging condition in time‐reversal imaging is the scalar cross correlation imaging condition. However, scalar cross‐correlation imaging condition removes the directional information of the wavefield through modulus operations to avoid the direct dot product of mutually orthogonal P‐ and S‐waves, preventing the imaging condition from leveraging the wavefield propagation direction to suppress imaging artefacts. We previously tackled this issue by substituting the imaging wavefield with the energy current density vectors of the decoupled wavefield, albeit at the cost of increased computational and storage demands. To balance artifact suppression with reduced computational and memory overhead, this work introduces the Poynting and polarization vectors mixed imaging condition. Poynting and polarization vectors mixed imaging condition utilizes the polarization and propagation direction information of the wavefield by directly dot multiplying the undecoupled velocity polarization vector with the Poynting vector, eliminating the need for P‐ and S‐wave decoupling or additional memory. Compared with scalar cross‐correlation imaging condition, this imaging condition can accurately image data with lower signal‐to‐noise ratios. Its performance is generally consistent with previous work but offers higher computational efficiency and lower memory usage. Synthetic data tests on the half‐space model and the three‐dimensional Marmousi model demonstrate the effectiveness of this method in suppressing imaging artefacts, as well as its efficiency and ease of implementation.

Radius Model for Some Cells in Human Body on Multiplicative Calculus

The cell is the basic structure and process unit that carries all the living characteristics of a living thing and has the ability to survive on its own under suitable conditions. The relationship of cell size with nutrient absorption and nutrient consumption in the cell membrane has been examined with the current model using the theory of differential equations in classical analysis. During these examinations, the cell considered was assumed to be spherical. In fact, the shapes of cells vary depending on their functional properties. Many have long appendages, cylindrical parts or branch-like structures. However, in this study, a simple global cell will be discussed, leaving all these complex situations aside. In the current model, the relationship between the change in the radius of the cell and the nutrient absorption and consumption in the cell membrane is detailed using classical differential equations. The answer to the question for which cell size is the consumption rate exactly balanced with the absorption rate was found in classical analysis. The current model consists of first-order differential equations. In this model, the dependent variables are the radius of the cell and the mass of the cell. The classical solutions of these models will be examined, the size of the cell and the cell membrane relationship will be examined, and details will be given with numerical examples. However, in order to consider this biological phenomenon from different perspectives and compare the results, the relevant event will be modeled using multiplicative analysis, one of the Non-Newtonian analyses. The new models will be solved using multiplicative analysis techniques, and the results will be compared with classical analysis. With this new model, it is planned to clarify the results obtained in the classical case, to reveal more clearly the relationship between the size of the cell and nutrient absorption and consumption in the cell membrane, and to obtain important results.

Impact of multiple time delays on bifurcation of a class of fractional nearest-neighbor coupled neural networks

In this paper, the impacts of multiple time delays on bifurcation of a class of fractional nearest-neighbor coupled neural networks are considered. Firstly, the sum of time delays is selected as a parameter, and the fractional nearest-neighbor coupled neural network model is linearized to obtain the corresponding characteristic equation. Then, utilizing stability and bifurcation theory of fractional-order delay differential equations, we investigate the effect of time delays on the system’s stability and bifurcations. The results show that when the time lag exceeds the critical value, the system will lose stability and generate Hopf bifurcation. Finally, the correctness of the conclusions in this paper is verified through numerical simulation.

Integrability and identification in multinomial choice models

McFadden's random-utility model of multinomial choice has long been the workhorse of applied research. We establish shape-restrictions with respect to price and income which are necessary and sufficient for multinomial choice-probability functions to be rationalized via random-utility models with additive but nonparametric unobserved heterogeneity and general income-effects. Our proof is constructive, and facilitates nonparametric identification of preference-distributions without requiring identification-at-infinity type arguments. A corollary shows that symmetry, a key condition for previous rationalizability results, is equivalent to absence of income-effects. Our results imply theory-consistent nonparametric bounds for choice-probabilities on counterfactual budget-sets. They also apply to widely used random-coefficient models, upon conditioning on observable choice characteristics. The theory of partial differential equations plays a key role in our analysis.

Regularity of Weak Solutions to a Class of Nonlinear Parabolic Equations in Fractional Sobolev Spaces

In this article, we study regularity of weak solutions to a class of nonlinear parabolic equations in divergence form. The main purpose is to present a regularity estimate with more general conditions on coefficients, N-functions and non-homogeneous terms in the fractional Sobolev spaces. By deriving a higher integrability estimate of weak solutions, we obtain the desired regularity estimate. In addition, the results of this article expand the regularity theory of parabolic equations in fractional Sobolev spaces and Besov spaces.

New existence results for conjoined bases of singular linear Hamiltonian systems with given Sturmian properties

In this paper we derive new existence results for conjoined bases of singular linear Hamiltonian differential systems with given qualitative (Sturmian) properties. In particular, we examine the existence of conjoined bases with invertible upper block and with prescribed number of focal points at the endpoints of the considered unbounded interval. Such results are vital for the theory of Riccati differential equations and its applications in optimal control problems. As the main tools we use a new general characterization of conjoined bases belonging to a given equivalence class (genus) and the theory of comparative index of two Lagrangian planes. We also utilize extensively the methods of matrix analysis. The results are new even for identically normal linear Hamiltonian systems. The results are also new for linear Hamiltonian systems on a compact interval, where they provide additional equivalent conditions to the classical Reid roundabout theorem about disconjugacy.