Theory Of Differential Equations Research Articles

Fractional-Order Sequential Linear Differential Equations with Nabla Derivatives on Time Scales

In this paper, we present a general theory for fractional-order sequential differential equations with Riemann–Liouville nabla derivatives and Caputo nabla derivatives on time scales. The explicit solution, in the case of constant coefficients, for both the homogeneous and the non-homogeneous problems, are given using the ∇-Mittag-Leffler function, Laplace transform method, operational method and operational decomposition method. In addition, we also provide some results about a solution to a new class of fractional-order sequential differential equations with convolutional-type variable coefficients using the Laplace transform method.

A MATHEMATICAL MODELING ON THE ENVIRONMENTAL IMPACT BY TRAFFIC JAM: A CASE STUDY IN DHAKA, BANGLADESH

Traffic jams are a common incident in Bangladesh, particularly in major cities like Dhaka. Traffic jams not only cause significant delays and economic losses but also contribute to substantial environmental degradation. This study object to quantify the environmental impact of traffic jam in Bangladesh using mathematical modeling, a case study of Dhaka city is presented, focusing on air pollution and greenhouse gas emissions. The model analyzed by using stability theory of differential equations with equilibrium analysis by convincing stable or unstable equilibrium. Moreover, the analytical results are validated by numerical simulations for describing the complex behaviors of the model. Overall, the model represents an important tool for investigating environmental impact of traffic jam and this model can help inform policy decisions and promote more sustainable practices.

A discontinuous Galerkin method for a coupled Stokes–Biot problem

The interaction between fluid and poroelastic structure is a complex problem that couples the Stokes equations with the Biot system. In this work, we rewrite the poroelastic equations using three fields (displacement, fluid pressure, and total pressure) to account for locking phenomenon in fluid–structure interaction and the oscillation of numerical solutions. Then, in order to solve the coupled system, we use the weighted discontinuous Galerkin finite element method for spatial discretization and propose a semi-discrete scheme. In addition, we propose a fully discrete scheme using the backward Euler’s method for temporal discretization. In the framework of the Galerkin approximation, the stability and errors of semi- and fully discrete schemes are analyzed using the theory of differential algebraic equations and weak compactness arguments. Through numerical experiments, we validate the theoretical rate of convergence, explain the behavior of the method in modeling the interaction between surface and groundwater hydrological systems, investigate the robustness of the method with respect to physical parameters, and simulate channel filtering. In particular, the numerical results are free of locking features, reducing non-physical numerical oscillations.

Group Classification of Second-Order Linear Neutral Differential Equations

In this paper, we shall extend the method of obtaining symmetries of ordinary differential equations to second-order non-homogeneous functional differential equations with variable coefficients. The existing research for delay differential equations defines a Lie-Bäcklund operator and uses the invariant manifold theorem to obtain the infinitesimal generators of the Lie group. However, we shall use a different approach that requires Taylor’s theorem for a function of several variables to obtain a Lie invariance condition and the determining equations for second-order functional differential equations. Certain standard results from the theory of ordinary differential equations have been employed to simplify the equation under study. The symmetry analysis of this equation was found to be non-trivial for arbitrary variable coefficients. In such cases, by selecting certain specific functions, arising in most practical models, we find the symmetries that are seen to be in terms of Bessel’s functions, Mathieu functions, etc. We then make a complete group classification of the second-order linear neutral differential equation, for which there is no existing literature.

Non-chattering finite-time and fixed-time flocking of Cucker–Smale systems

In this paper, the non-chattering finite/fixed-time flocking problems of Cucker–Smale (C–S) systems are investigated. To address the chattering issue, two novel chattering-free protocols are proposed to achieve the flocking of C–S systems within a finite/fixed time. Using the stability theory of differential equations, sufficient conditions for achieving the flocking are established, and the upper bounds of convergence time are given. The connection between the chattering-free flocking protocols and the classic finite/fixed-time control schemes involving the signum function is shown for the first time. Moreover, we discover a correlation between the settling time and the group size. Finally, the effectiveness of the presented protocols is validated via several simulations.

Fixed Time Synchronization of Stochastic Takagi–Sugeno Fuzzy Recurrent Neural Networks with Distributed Delay under Feedback and Adaptive Controls

In this paper, the stochastic Takagi–Sugeno fuzzy recurrent neural networks (STSFRNNS) with distributed delay is established based on the Takagi–Sugeno (TS) model and the fixed time synchronization problem is investigated. In order to synchronize the networks, we design two kinds of controllers: a feedback controller and an adaptive controller. Then, we obtain the synchronization criteria in a fixed time by combining the Lyapunov method and the related inequality theory of the stochastic differential equation and calculate the stabilization time for the STSFRNNS. In addition, to verify the authenticity of the theoretical results, we use MATLABR2023A to carry out numerical simulation.

APPLICATION OF DIFFERENTIAL EQUATIONS IN VARIOUS FIELDS OF SCIENCE

The article, "Application of Differential Equations in Various Fields of Science," explores the use of differential equations for modeling economic and natural phenomena. It examines two main models of economic dynamics: the Evans model for the market of a single product, and the Solow model for economic growth.The author emphasizes the importance of proving the existence of solutions to differential equations in order to verify the accuracy of mathematical models. They also discuss the role of electronic computers in developing the theory of differential equations and its connection with other branches of mathematics such as functional analysis, algebra, and probability theory.Furthermore, the article highlights the significance of various solution methods for differential equations, including the Fourier method, Ritz method, Galerkin method, and perturbation theory.Special attention is paid to the theory of partial differential equations, the theory of differential operators, and problems arising in physics, mechanics, and technology. Differential equations are the theoretical foundation of almost all scientific and technological models and a key tool for understanding various processes in science, such as in physics, chemistry, and biology.Examples of processes described by differential equations include normalreproduction, explosive growth, and the logistic curve. Cases of using differential equations to model deterministic, finite-dimensional, and differentiable phenomena, as well as the impact of catch quotas on population dynamics, are discussed.In conclusion, the significance of differential equations for research and their role in stimulating the development of new mathematical areas is emphasized.

Dynamical behaviors of a network-based SIR epidemic model with saturated incidence and pulse vaccination

Pulse vaccination is an effective strategy to restrain the spread of infectious diseases. This paper proposes a network-based SIR epidemic model incorporating a saturated force of infection and vertical transmission along with pulse vaccination strategy and continuous treatment plan. Dynamical behaviors of the model are analyzed in virtue of the theory for impulsive differential equations. We give the boundedness of solutions and derive the expression of basic reproduction number R0. By the Floquet theorem and comparison principle, we show that the disease-free periodic solution is globally asymptotically stable when R0<1. Moreover, a sufficient condition for the uniform permanence of the system is also obtained. Finally, numerical analyses are given to substantiate the theoretical results and a case for Rubella is studied.

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

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-component of the magnetic flux density obtained from an MRI scanner. Traditional reconstruction algorithms in MREIT use two datum subject to two linearly independent current densities. However, the temporal resolution of such a MREIT image is relatively low. Recently, a single current harmonic Bz algorithm 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 Bz , the z-component of the magnetic flux density. To get a stable reconstruction from the noisy Bz data, we note that the edge structure of ∇Bz reveals the edge features in the unknown conductivity and provides an edge-preserving denoising approach for the ∇Bz data. We use a modified Shepp–Logan phantom model to validate the proposed theory and the denoising approach.

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

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.

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.