Theory Of Differential Equations Research Articles

Study on the Solutions of Impulsive Integrodifferential Equations of Mixed Type Based on Infectious Disease Dynamical Systems

Since ancient times, infectious diseases have been a major source of harm to human health. Therefore, scientists have established many mathematical models in the history of fighting infectious diseases to study the law of infection and then analyzed the practicability and effectiveness of various prevention and control measures, providing a scientific basis for human prevention and research of infectious diseases. However, due to the great differences in the transmission mechanisms and modes of many diseases, there are many kinds of infectious disease dynamic models, which make the research more and more difficult. With the continuous progress of infectious disease research technology, people have adopted more ways to prevent and interfere with the derivation and spread of infectious disease, which will make the state of infectious disease system change in an instant. The mutation of this state can be described more scientifically and reasonably by the mathematical impulse dynamic system, which makes the research more practical. Based on this, a time-delay differential system model of infectious disease under impulse effect was established by means of impulse differential equation theory. A class of periodic boundary value problems for impulsive integrodifferential equations of mixed type with integral boundary conditions was studied. The existence of periodic solutions of these equations was obtained by using the comparison theorem, upper and lower solution methods, and the monotone iteration technique. Finally, combined with the practical application, the established time-delay differential system model was applied to the prediction of the stability and persistence of the infectious disease dynamic system, and the correctness of the conclusion was further verified. This study provides some reference for the prevention and treatment of infectious diseases.

Review of Research on the Qualitative Theory of Differential Equations at St. Petersburg University. I. Stable Periodic Points of Diffeomorphisms with Homoclinic Points and Systems with Weakly Hyperbolic Invariant Sets

Review of Research on the Qualitative Theory of Differential Equations at St. Petersburg University. I. Stable Periodic Points of Diffeomorphisms with Homoclinic Points and Systems with Weakly Hyperbolic Invariant Sets

Stability of the isotropic conductivity reconstruction using magnetic resonance electrical impedance tomography (MREIT)

Abstract Magnetic resonance electrical impedance tomography (MREIT) is a high-resolution imaging modality that aims to reconstruct the objects' conductivity distributions at low frequency using the measurable $z$-th component of the magnetic flux density obtained from an MRI scanner. Traditional reconstruction algorithms in MREIT use two data subject to two linearly independent current densities. However, the temporal resolution of such a MREIT image is relatively low. Recently, a single current MREIT has been proposed to improve the temporal resolution. Even though a series of reconstruction algorithms have been proposed in the last two decades, the theoretical studies of MREIT are still quite limited. This paper presents the stability theorems for two datum and a single data-based isotropic conductivity reconstruction using MREIT. Using the regularity theory of elliptic partial differential equations, we prove that the only instability in the inverse problem of MREIT comes from taking the second-order derivative of the measured data $B_z$, the $z$-th component of the magnetic flux density. To get a stable reconstruction from the noisy $B_z$ data, we note that the edge structure of $\nabla B_z$ reveals the edge features in the unknown conductivity and provides an edge-preserving denoising approach for the $\nabla B_z$ data. We use a modified Shepp-Logan phantom model to validate the proposed theory and the denoising approach.

The soliton and periodic solutions for the perturbed sine–cosine–Gordon equation

In the paper, the existence of soliton and periodic solutions is studied for the perturbed sine–cosine–Gordon equation. The corresponding traveling wave system is converted to a regularly Hamilton system via using bifurcation theory of differential equations and the geometric singular perturbation theory. Then by using Melnikov's method and symbolic computation, periodic wave solutions, solitary solutions, and kink and antikink solutions are obtained. The wave speed selection principles are given, respectively. Numerical simulations are performed to illustrate our theoretical analysis.

Mathematical Modeling of the Transmission Dynamics of Gumboro Disease

Gumboro disease is a viral poultry disease that causes immune suppression on the infected birds leading to poor production, mortality, and exposure to secondary infections, hence a major threat in the poultry industry worldwide. A mathematical model of the transmission dynamics of Gumboro disease is developed in this paper having four compartments of chicken population and one compartment of Gumboro pathogen population. The basic reproduction number Rog is derived, and the dynamical behaviors of both the disease-free equilibrium (DFE) and endemic equilibrium are analyzed using the ordinary differential equation theory. From the analysis, we found that the system exhibits an asymptotic stable DFE whenever Rog<1 and an asymptotic stable EE whenever Rog>1. The numerical simulation to verify the theoretical results was carried out using MATLAB ode45 solver, and the results were found to be consistent with the theoretical findings.

The Maximal and Minimal Distributions of Wealth Processes in Black–Scholes Markets

The Black–Scholes formula is an important formula for pricing a contingent claim in complete financial markets. This formula can be obtained under the assumption that the investor’s strategy is carried out according to a self-financing criterion; hence, there arise a set of self-financing portfolios corresponding to different contingent claims. The natural questions are: If an investor invests according to self-financing portfolios in the financial market, what are the maximal and minimal distributions of the investor’s wealth on some specific interval at the terminal time? Furthermore, if such distributions exist, how can the corresponding optimal portfolios be constructed? The present study applies the theory of backward stochastic differential equations in order to obtain an affirmative answer to the above questions. That is, the explicit formulations for the maximal and minimal distributions of wealth when adopting self-financing strategies would be derived, and the corresponding optimal (self-financing) portfolios would be constructed. Furthermore, this would verify the benefits of diversified portfolios in financial markets: that is, do not put all your eggs in the same basket.

Stability analysis of a delayed SEIRQ epidemic model with diffusion

In this paper, we investigate the effect of spatial diffusion and delay on the dynamical behavior of the SEIRQ epidemic model. The introduction of the delay in this model makes it more realistic and modelizes the latency period. In addition, the consideration of an SEIRQ model with diffusion aims to better understand the impact of the spatial heterogeneity of the environment and the movement of individuals on the persistence and extinction of disease. First, we determined a threshold value $R_0$ of the delayed SEIRQ model with diffusion. Next, By using the theory of partial functional differential equations, we have shown that the unique disease-free equilibrium is asymptotically stable , what is proven by the numericals scchema. Moreover,we search under their condition the endemic equilibrium is asymptotically stable .

Gradient Flow of the Sinai–Ruelle–Bowen Entropy

Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow’s ordinary differential equation representation. This gradient flow has close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation.

On infinite spectra of oscillation exponents of third order linear differential equations

The research topic of this work is at the junction of the theory of Lyapunov exponents and oscillation theory. In this paper, we study the spectra (i.e., the sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential equations with coefficients continuous on the positive semi-axis. In the first part of the paper, we build a third order linear differential equation with the following property: the spectra of all upper and lower strong and weak exponents of oscillation of strict and non-strict signs, zeros, roots and hyper roots contain a countable set of different essential values, both metrically and topologically. Moreover, all these values are implemented on the same sequence of solutions of the constructed equation, that is, for each solution from this sequence, all of the oscillation exponents coincide with each other. In the construction of the indicated equation and in the proof of the required results, we used analytical methods of the qualitative theory of differential equations and methods from the theory of perturbations of solutions of linear differential equations, in particular, the author’s technique for controlling the fundamental system of solutions of such equations in one special case. In the second part of the paper, the existence of a third order linear differential equation with continuum spectra of the oscillation exponents is established, wherein the spectra of all oscillation exponents fill the same segment of the number axis with predetermined arbitrary positive incommensurable ends. It turned out that for each solution of the constructed differential equation, all of the oscillation exponents coincide with each other. The obtained results are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations.

Coexistence and locally exponential stability of multiple equilibrium points for fractional-order impulsive control Cohen–Grossberg neural networks

Different from the existing multiple Mittag-Leffler stability or multiple asymptotic stability, the multiple exponential stability, which has explicit and faster convergence rate, is investigated in this paper for fractional-order impulsive control Cohen–Grossberg neural networks. First, by using the definition of Dirac delta function, the fractional-order control Cohen–Grossberg neural networks are translated into fractional-order impulsive neural networks, in which pulse effect relies on the fractional order of the addressed system. Then, based on maximum norm, 1−norm and general q−norm (q=2n), a series of novel criteria are obtained respectively to ensure that such n−neuron neural networks can have ∏i=1n(2Li+1) total equilibrium points and ∏i=1n(Li+1) locally exponentially stable equilibrium points, by utilizing the known fixed point theorem, the method of average impulsive interval, the theory of fractional-order differential equations, and the method of Lyapunov function. This paper’s investigation reveals the effects of impulsive function, impulsive interval, and fractional order on the dynamic behaviors. Finally, theoretical results are shown to be effective by four illustrative examples.

Control theory for fractional differential Sylvester matrix equations with Caputo fractional derivative

In this manuscript, we introduce the concept of first-time controllability and observability concerning fractional differential Sylvester matrix equations employing the Caputo fractional derivative. Our work establishes necessary and sufficient conditions for controllability and observability, wherein controllability equates to having a controllability matrix with full rank and observability aligns with an observability Gramian matrix that is nonsingular. Furthermore, we provide several theorems addressing the observability of fractional differential Sylvester matrix equations. Finally, we offer two illustrative examples to demonstrate the efficacy and application of our established results.

The Impact of the Monetary Policy of the US Federal Reserve on the Ratio of the US Dollar To the Japanese Yen in the Foreign Exchange Market during 2015-2023

The information base of the study is the time series of the daily exchange rate of the currency pair USD/JPY, the US 10-year Treasury bond (TB 10), the Japanese government bond 10-year (JGB 10), the volatility index (VIX) of the Chicago Board Options Exchange (CBOE), which calculated by Bloomberg. Study period: December 1, 2015 (the date the Federal Reserve raised interest rates, which initiated a new regime – began easing its ZIRP zero interest rate policy) to December 29, 2023. The relationship between monetary policy and the currency pair USD/JPY is evaluated using 3 approaches: 1) a rally dynamic system based on the standard theory of nonlinear differential equations aimed at eliminating psychological noise (1/f noise, pink noise, trend noise with spectrum capacity); 2) regression of instrumental variables with heteroskedasticity aimed at managing unobservable sources of variability; 3) an event tracking method to measure anomalous asset price response, analyzing the daily price action of USD/JPY to detect market-related fluctuations in the exchange rate and separate the impact of general market trends from related differences in interest rates. The study found that the US Federal Reserve’s (Fed’s) monetary policy only occasionally affects the USD/JPY exchange rate, and it changes mainly when monetary policy deviates from expected future trajectories. Within the studied period, this happened only 3 times: growth on December 16, 2015; emergency rate reduction due to COVID-19 on March 15, 2020; and the first increase of 75 basis points on June 15, 2022 (the article estimates the impact of each of these episodes on the exchange rate). Statements by the Fed on monetary policy affect the price dynamics of the USD/JPY pair in the short term (since short-term psychological expectations play a decisive role in shaping the dynamics of the currency market), but not often (if the statement on monetary policy is consistent with market forecasts, it is practically will not affect exchange rates). The Fed’s adoption of tools such as “forward guidance” and official statements to inform the future course of monetary policy indirectly affects the price dynamics of USD/JPY. The price action of USD/JPY was driven by the fundamentals of the difference in interest rates between the US and Japanese bonds, which are influenced by the monetary policy of the US Federal Reserve and the Bank of Japan (BoJ). In 2023, it could be hard to attempt to challenge FX intervention except to curb “excess volatility and disorderly movements in exchange rates” like the Minister of Finance’s (MoF) 2022 intervention.

Limit Cycles of the Discontinuous Piecewise Differential Systems Separated by a Nonregular Line and Formed by a Linear Center and a Quadratic One

During the last decades, the study of discontinuous piecewise differential systems has become an interesting subject of research due to the important applications of this kind of systems to model natural phenomena. In the qualitative theory of differential equations, one of the interesting problems is the detection of the number of limit cycles and their configurations which remains open to date, except for very particular families of differential equations. Here, we are inspired to study the maximum number of limit cycles of the discontinuous piecewise differential systems separated by a nonregular line and formed by a linear center and one of the four classes of quadratic centers. The main tool used to prove our main results is based on the first integrals of such systems. All the computations of this paper are verified using the algebraic manipulator Mathematica.

Data-driven discovery of stochastic dynamical systems with [formula omitted]-stable Lévy noise based on residual networks

Identifying evolution laws of complex systems from massive data has been a challenging issue in the cross field of stochastic dynamics and data science with greater and greater concern in recent years. So far, significant progress has been made in dealing with cases where differential equations are driven by Gaussian noise, yet work for non-Gaussian cases remains great room for growth. In this paper, a data-driven approach based on deep residual networks (ResNets) is proposed to extract governing laws of stochastic dynamical systems perturbed by non-Gaussian α-stable Lévy noise. Specifically, we firstly separate the non-Gaussian noise from drift term for subsequent extraction according to the fundamental theory of differential equations. For the noise part, the Lévy jump measure and noise intensity are estimated via some properties of α-stable distribution. Next the residual network is employed to identify drift. Numerical results of systems of 1 to 3 dimensions are given to demonstrate the feasibility of this method. It is necessary to emphasize here that the contribution of this method is mainly manifested in its effective resolution of the identification for complex non-Gaussian stochastic dynamical systems via cutting-edge machine learning techniques.

A boundary value problem for the fourth-order degenerate equation of the mixed type

Many problems in mechanics, physics, and geophysics lead to solving partial differential equations that are not included in the known classes of elliptic, parabolic or hyperbolic equations. Such equations, as a rule, began to be called non-classical equations of mathematical physics. The theory of degenerate equations is one of the central branches of the modern theory of partial differential equations. This is primarily due to the identification of a variety of applied problems, the mathematical modeling of which serves the study of various types of degenerate equations. The study of boundary value problems for mixed type’s equations of the fourth-order with power-law degeneration remains relevant. In this work, a boundary value problem in a rectangular domain for a degenerate equation of the fourth-order mixed-type is posed and investigated. Well-posedness of the boundary value problem for a fourth-order partial differential equation is established by proving the existence and uniqueness of the solution. Under sufficient conditions, a solution to the problem under consideration was explicitly found by the variable separation method.

An Approximation to the Invariant Measure of the Limiting Diffusion of G/Ph/n + GI Queues in the Halfin–Whitt Regime and Related Asymptotics

In this paper, we develop a stochastic algorithm based on the Euler–Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of [Formula: see text] queues in the Halfin–Whitt regime. Specifically, we prove a nonasymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion. To establish the error bound, we employ the recently developed Stein’s method for multidimensional diffusions, in which the regularity of Stein’s equation obtained by the partial differential equation (PDE) theory plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of [Formula: see text] queues and its Euler–Maruyama scheme. In particular, the variances in the CLT and MDP associated with the limiting diffusion are determined by Stein’s equation and Malliavin calculus, in which properties of a mollified diffusion and an associated weighted occupation time play a crucial role. Funding: X. Jin is supported in part by the Fundamental Research Funds for the Central Universities [Grants JZ2022HGQA0148 and JZ2023HGTA0170]. G. Pang is supported in part by the U.S. National Science Foundation [Grants DMS-1715875 and DMS-2216765]. L. Xu is supported in part by the National Nature Science Foundation of China [Grant 12071499], Macao Special Administrative Region [Grant FDCT 0090/2019/A2], and the University of Macau [Grant MYRG2018-00133-FST]. This work was supported by U.S. National Science Foundation [Grant DMS-2108683].

A rumor propagation model in multilingual environment with time and state dependent impulsive control

In this paper we are interested in the problem of modeling, analysis and control of rumor spreading on online social networks. As the diversity of languages and the difference of user nodes in the network, a heterogeneous rumor spreading model with two language spreaders is proposed, and the impulsive isolation control strategy is considered. Firstly, the model with periodic impulsive isolation control and time-delay recovery rate is discussed. Based on the stability theory of impulsive differential equations, some conditions regarding the impulsive period and control gains are given for the rumor asymptotically disappearance and persistent existence. Secondly, a new type of event-triggered impulsive isolation control method is developed to address the tradeoff between restraining rumor and minimizing the cost of control, and some conditions are given to make the rumor disappear asymptotically. Furthermore, an improved event-triggered mechanism is established based on the fixed-time stability theory, which provides the conditions for the rumor to disappear within a fixed-time. Finally, some numerical simulations and an actual example are given to verify the rationality of the model and the effectiveness of the control methods.

A brief survey on the development and applications of Goebel’s coincidence point theorem in differential and integral equations

Abstract Goebel’s coincidence theorem, a remarkably simple extension of Banach’s contraction principle widely applied in analysis, has found significant utility in the theory of differential and integral equations. Over the past fifty years, researchers have endeavored to generalize the definition of a metric space, thereby extending the scope of Goebel’s coincidence theorem to diverse settings. This survey paper overviews valuable insights into Goebel’s coincidence theorem’s historical background, its relevance in the field of fixed point theory, and its practical implications in solving problems related to differential and integral equations.

Improving Transferability for Cross-Domain Trajectory Prediction via Neural Stochastic Differential Equation

Multi-agent trajectory prediction is crucial for various practical applications, spurring the construction of many large-scale trajectory datasets, including vehicles and pedestrians. However, discrepancies exist among datasets due to external factors and data acquisition strategies. External factors include geographical differences and driving styles, while data acquisition strategies include data acquisition rate, history/prediction length, and detector/tracker error. Consequently, the proficient performance of models trained on large-scale datasets has limited transferability on other small-size datasets, bounding the utilization of existing large-scale datasets. To address this limitation, we propose a method based on continuous and stochastic representations of Neural Stochastic Differential Equations (NSDE) for alleviating discrepancies due to data acquisition strategy. We utilize the benefits of continuous representation for handling arbitrary time steps and the use of stochastic representation for handling detector/tracker errors. Additionally, we propose a dataset-specific diffusion network and its training framework to handle dataset-specific detection/tracking errors. The effectiveness of our method is validated against state-of-the-art trajectory prediction models on the popular benchmark datasets: nuScenes, Argoverse, Lyft, INTERACTION, and Waymo Open Motion Dataset (WOMD). Improvement in performance gain on various source and target dataset configurations shows the generalized competence of our approach in addressing cross-dataset discrepancies.

Mathematical modelling of COVID-19 outbreak using caputo fractional derivative: stability analysis

The novel coronavirus SARS-Cov-2 is a pandemic condition and poses a massive menace to health. The governments of different countries and their various prohibitory steps to restrict the virus's expanse have changed individuals' communication processes. Due to physical and financial factors, the population's density is more likely to interact and spread the virus. We establish a mathematical model to present the spread of the COVID-19 in worldwide. In this article, we propose a novel mathematical model (‘ S L I I q I h R P ’) to assess the impact of using hospitalization, quarantine measures, and pathogen quantity in controlling the COVID-19 pandemic. We analyse the boundedness of the model's solution by employing the Laplace transform approach to solve the fractional Gronwall's inequality. To ensure the uniqueness and existence of the solution, we rely on the Picard-Lindelof theorem. The model's basic reproduction number, a crucial indicator of epidemic potential, is determined based on the greatest eigenvalue of the next-generation matrix. We then employ stability theory of fractional differential equations to qualitatively examine the model. Our findings reveal that both locally and globally, the endemic equilibrium and disease-free solutions demonstrate symptomatic stability. These results shed light on the effectiveness of the proposed interventions in managing and containing the COVID-19 outbreak.