Theory Of Differential Equations Research Articles

Exact solutions for analog Hawking effect in dielectric media

In the framework of the analog Hawking radiation for dielectric media, we analyze a toy model and also the 2D reduction of the Hopfield model for a specific monotone and realistic profile for the refractive index. We are able to provide exact solutions, which do not require any weak dispersion approximation. The theory of Fuchsian ordinary differential equations is the basic tool for recovering exact solutions, which are rigorously identified and involve the so-called generalized hypergeometric functions F34(α1,α2,α3,α4;β1,β2,β3;z). A complete set of connection formulas are available, both for the subcritical case and for the transcritical one, and also the Stokes phenomenon occurring in the problem is fully discussed. From the physical point of view, we focus on the problem of thermality. Under suitable conditions, the Hawking temperature is deduced, and we show that it is in full agreement with the expression deduced in other frameworks under various approximations. Published by the American Physical Society 2024

Bounded Kukles Systems of Degree Three

One of the most important tasks in the qualitative theory of differential equations in the plane is the study of global asymptotic stability: an equilibrium point that is globally attractive. It is known that if an equilibrium point of a planar vector field is globally asymptotically stable, then the vector field is bounded. A planar vector field is said to be bounded if the forward orbit of every point enters and remains in a compact set. In this paper, we study cubic Kukles systems that are bounded which is the first step toward the characterization of cubic Kukles systems that are globally asymptotically stable. We emphasize that these systems form a seven-parameter family of polynomial differential systems. We obtain a total of 25 cubic Kukles sub-systems that are bounded and we provide 11 phase portraits of such sub-systems in a neighborhood of infinity.

The non-exponential decay characteristics of HTS NI-class coils in sudden discharge experiments

Abstract The decay of the voltage and magnetic field of the high-temperature superconductivity (HTS) no-insulation-class (NI-class) coil in the sudden discharge experiment is usually considered to be exponential. However, our experiment found that it is not accurate to describe the decay characteristics of the NI-class coil with a simple exponential function. Combined with equivalent circuit models of the NI-class coil, we use the theory of ordinary differential equations to explain the non-exponential decay characteristics and point out that it is closer to the matrix exponential characteristics. Based on this finding, we believe that the non-uniform turn-to-turn contact resistivity (TTCR) is an important cause of the non-exponential decay characteristics of the NI-class coil. Reverse thinking leads us to realize that the voltage decay curve of the NI-class coil in the sudden discharge experiment contains the distribution information of its TTCR. Therefore, we propose a method for measuring the TTCR of NI-class coils based on the matrix exponential model. The co-simulation combining the finite element model and the equivalent circuit model shows that the non-uniformly distributed TTCR has a great influence on the transient process of the NI-class coil, which makes the current and current density in the transient process vary greatly in different regions of the NI-class coil. Those findings mean that the simulation study of the NI-class coils and magnets with uniform TTCR usually used in the past may be far from the actual situation.

Sensitivity analysis and global stability of epidemic between Thais and tourists for Covid -19

This study employs a mathematical model to analyze and forecast the severe outbreak of SARS-CoV-2 (Severe Acute Respiratory Syndrome Coronavirus 2), focusing on the socio-economic ramifications within the Thai population and among foreign tourists. Specifically, the model examines the impact of the disease on various population groups, including susceptible (S), exposed (E), infected (I), quarantined (Q), and recovered (R) individuals among tourists visiting the country. The stability theory of differential equations is utilized to validate the mathematical model. This involves assessing the stability of both the disease-free equilibrium and the endemic equilibrium using the basic reproduction number. Emphasis is placed on local stability, the positivity of solutions, and the invariant regions of solutions. Additionally, a sensitivity analysis of the model is conducted. The computation of the basic reproduction number (R0) reveals that the disease-free equilibrium is locally asymptotically stable when R0 is less than 1, whereas the endemic equilibrium is locally asymptotically stable when R0 exceeds 1. Notably, both equilibriums are globally asymptotically stable under the same conditions. Through numerical simulations, the study concludes that the outcome of COVID-19 is most sensitive to reductions in transmission rates. Furthermore, the sensitivity of the model to all parameters is thoroughly considered, informing strategies for disease control through various intervention measures.

Exponential stability of a non-uniform Euler-Bernoulli beam with axial force

The aim of this paper is to investigate the boundary stabilization of a non-uniform Euler-Bernoulli beam with axial force generated by the equation ρ(x)utt+(σ(x)uxx)xx−(q(x)ux)x=0 defined in (0,1)×(0,∞), where the coefficients ρ(x)>0, σ(x)>0, and q(x)≥0. By adopting a new technique based on a qualitative theory of fourth-order linear differential equations, we prove that the spectrum of the system operator is confined to the open left-half complex plane and that all the associated eigenvalues are geometrically simple. Using this result and the Riesz basis approach, we obtain the exponential stability and the optimal decay rate of the system. This study extends the earlier paper by B.-Z. Guo [SIAM J. Control Optim., 40 (2002), pp. 1905–1923], where no axial force acting on the system (i.e., q≡0).

Stability of open-loop Nash equilibria for [formula omitted]-person nonzero-sum bounded rationality generalized differential games

The problem of determining the existence of Nash equilibria in n-person nonzero-sum generalized differential games is highly intricate and constrained by the advancement of partial differential equations theory. There is limited existing research literature on this subject. This paper presents an existence theorem for open-loop Nash equilibria employing the Fan-Glicksberg fixed point theorem. The n-person nonzero-sum bounded rationality generalized differential game model is formulated by introducing a bounded rationality function, and its structural stability and robustness are studied. The conclusions indicate that in the sense of Baire classification, most n-person nonzero-sum bounded rationality generalized differential games are structurally stable and robust in the set of ɛ-open-loop Nash equilibria, and we can approximate the equilibrium set obtained with full rationality generalized differential games by the ɛk-open-loop Nash equilibria set obtained with bounded rationality generalized differential games.

Controllability results of discrete boundary control system in Banach spaces

In this paper, we study the boundary controllability of the discrete-time control system in Banach spaces. The aim is to establish sufficient conditions for the existence and uniqueness of solutions, as well as for the approximate controllability of a discrete-time boundary control system. The form of the solution to the considered system and its existence and uniqueness are obtained using non-linear functional analysis, theory of difference equations, semigroup theory and contraction mapping principle. Furthermore, we achieve the approximate controllability results through a sequential approach. In the end, we present some examples to illustrate the application of these analytical results.

Ulam–Hyers–Rassias Mittag-Leffler stability of ϖ–fractional partial differential equations

This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results.

A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks

Kolmogorov–Arnold Networks (KANs) were recently introduced as an alternative representation model to MLP. Herein, we employ KANs to construct physics-informed machine learning models (PIKANs) and deep operator models (DeepOKANs) for solving differential equations for forward and inverse problems. In particular, we compare them with physics-informed neural networks (PINNs) and deep operator networks (DeepONets), which are based on the standard MLP representation. We find that although the original KANs based on the B-splines parameterization lack accuracy and efficiency, modified versions based on low-order orthogonal polynomials have comparable performance to PINNs and DeepONet, although they still lack robustness as they may diverge for different random seeds or higher order orthogonal polynomials. We visualize their corresponding loss landscapes and analyze their learning dynamics using information bottleneck theory. Our study follows the FAIR principles so that other researchers can use our benchmarks to further advance this emerging topic.

Lie group analysis, solitary wave solutions and conservation laws of Schamel Burger’s equation

This paper presents a Lie group analysis of the Schamel Burger’s equation, notable for producing shock-type traveling waves in distinctive physical contexts. We determine the infinitesimal generators for this equation using the Lie group theory of differential equations. By applying Lie point symmetries, we establish commutation relations, the adjoint representation, and identify the optimal system of sub-algebras. Using elements from this optimal system, we perform symmetry reductions, resulting in various nonlinear ordinary differential equations (ODEs). Some of these reductions yield exact explicit solutions, while others necessitate the use of the new auxiliary equation method to obtain optical soliton solutions. We illustrate the dynamics of these soliton solutions graphically through both two and three-dimensional representations of wave structures. Additionally, we compute the conservation laws for the Schamel Burger’s equation by applying Ibragimov’s theorem, deriving conserved quantities corresponding to its point Lie symmetries. This analysis underscores our novel contribution, offering insights not previously explored in the literature.

ЭКИНЧИ ТАРТИПТЕГИ ТУУНДУСУНА КАРАТА ЧЕЧИЛБЕГЕН ДИФФЕРЕНЦИАЛДЫК ТЕҢДЕМЕ ҮЧҮН ЧЕКТИК МАСЕЛЕНИН ЧЫГАРЫЛЫШЫН КОЛЛОКАЦИЯ-АСИМПТОТИКАЛЫК МЕТОД МЕНЕН ТАБУУ

Boundary value problems for ordinary differential equations is one of the sections of the general theory of differential equations. The difference between the boundary value problem and the Cauchy problem is that the solution of the differential equation must satisfy boundary conditions connecting the values of the desired function at more than one point. The most common boundary value problem refers to two-point boundary value problems for which boundary conditions are set at two points at the ends of the interval on which solutions are being sought. In addition, the special case of two-point boundary value problems also covers such an important section of the theory of differential equations as the theory of oscillations. In this article, the problem of constructing an asymptotic expansion in a small parameter for solving a boundary value problem for a second-order differential equation of an unresolved relative to the highest derivative of the desired solution is considered. The solution of the boundary value problem is found according to the algorithm of the collocation-asymptotic method. A numerical example of a boundary value problem is considered and an asymptotic decomposition is constructed for solving problems with a relatively small parameter, including the first two terms of the expansions.

Stability analysis of a nonlinear malaria transmission epidemic model using an effective numerical scheme

Malaria is a fever condition that results from Plasmodium parasites, which are transferred to humans by the attacks of infected female Anopheles mosquitos. The deterministic compartmental model was examined using stability theory of differential equations. The reproduction number was obtained to be asymptotically stable conditions for the disease-free, and the endemic equilibria were determined. More so, the qualitatively evaluated model incorporates time-dependent variable controls which was aimed at reducing the proliferation of malaria disease. The reproduction number R o\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left(o\\right)$$\\end{document} was determined to be an asymptotically stable condition for disease free and endemic equilibria. In this paper, we used various schemes such as Runge–Kutta order 4 (RK-4) and non-standard finite difference (NSFD). All of the schemes produce different results, but the most appropriate scheme is NSFD. This is true for all step sizes. Various criteria are used in the NSFD scheme to assess the local and global stability of disease-free and endemic equilibrium points. The Routh–Hurwitz condition is used to validate the local stability and Lyapunov stability theorem is used to prove the global asymptotic stability. Global asymptotic stability is proven for the disease-free equilibrium when R0≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}\\le 1$$\\end{document}. The endemic equilibrium is investigated for stability when R0≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}\\ge 1$$\\end{document}. All of the aforementioned schemes and their effects are also numerically demonstrated. The comparative analysis demonstrates that NSFD is superior in every way for the analysis of deterministic epidemic models. The theoretical effects and numerical simulations provided in this text may be used to predict the spread of infectious diseases.

Theory on Linear L-Fractional Differential Equations and a New Mittag–Leffler-Type Function

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two.

State estimation based on measurement from a part of nodes for a delayed Itô type of complex network with Markovian mode-dependent parameters

This article is devoted to coping with the state estimate problem for a class of Itô stochastic complex network. The target network under consideration is a hybrid system involving Markovian jumping parameters and mode-dependent distributed time-delays, and the system noise is characterized by the Brownian Motion. The problem addressed is to design a state estimator to track the states of target network under the assumption that the network outputs are from only a portion of network nodes. By utilizing the Lyapunov stability method and the Itô stochastic differential equation theory, sufficient conditions are educed so that the state estimation error of the estimator is mean-square exponentially ultimately bounded. In particular, in the case that the system is noise-free, the derived conditions can ensure the state estimation error is mean-square exponentially stable. Furthermore, the estimator design can be made by solving a set of matrix inequalities. Lastly, a numerical example is exploited to indicates the validity of the theoretical results.

New fixed-time/preassigned-time stability results of impulsive systems and its application to synchronization of delayed octonion-valued neural networks

In this paper, in order to obtain more accurate settling time estimation and overcome the complexity of the theoretical analysis caused by using decomposition method, the problem of fixed-time (F-T) and preassigned-time (P-T) synchronization of delayed octonion-valued neural networks with impulsive effects is explored by utilizing direct octonion approach. First, a new fixed-time stability theorem for impulsive system is presented by using the comparison principle, average impulsive interval and Bernoulli differential equation theory. Second, to solve the problems of non associative and non commutative laws of octonion, some novel lemmas are introduced, which are the basis of the direct octonion approach. Third, under the improved F-T stability result, a new P-T stability theorem is derived and the settling time is preassigned according to realistic need and is irrelevant with any initial values and any parameters. Then, based on newly established theorems and developed controller strategies, some sufficient conditions are deduced to guarantee F-T and P-T synchronization of the discussed systems. Finally, a numerical simulation is provided to verify the effectiveness of the theoretical results.

Dynamical analysis of the pest control model with multiple delays and impulsive effects

In recent years, the suppression of disease population in pest control has gained popularity and remains a pressing issue in agricultural production. Additionally, the mathematical modeling method is often exploited to study the dynamics of different infectious diseases. A new pest control model with multiple delays and impulsive effects is developed and investigated. Firstly, the global attractivity of infection-free periodic solution of the system is proved by applying the theory of impulsive differential equation and comparative theorem. Secondly, the persistence of the infection-free periodic solution of the proposed model is also proved. Finally, the theoretical analysis results are validated by numerical simulations and some suggestions on how to better control the pests are presented.

Analyzing crop production: Unraveling the impact of pests and pesticides through a fractional model

The continuous growth of the human population raises concerns about food, fiber, and agricultural insecurity. Meeting the escalating demand for agricultural products due to this population surge makes protecting crops from pests becomes imperative. While farmers use chemical pesticides as crop protectors, the extensive use of these chemicals adversely affects both human health and the environment. In this research work, we formulate a nonlinear mathematical model using the Caputo fractional (CF) operator to investigate the effects of pesticides on crop yield dynamics. We assume that pesticides are sprayed proportional to the density of pest density and pests not entirely reliant on crops. The feasibility of every possible nonnegative equilibrium and its stability characteristics are explored utilizing the stability theory of fractional differential equations. Our model analysis reveals that in a continuous spray approach, the roles of pesticide abatement rate and pesticide uptake rate can be interchanged. Furthermore, we have identified the optimal time profile for pesticide spraying rate. This profile proves effective in minimizing both the pest population and the associated costs. To provide a practical illustration of our analytical findings and to showcase the impact of key parameters on the system’s dynamics, we conducted numerical simulations. These simulations are conducted employing the generalized Adams–Bashforth–Moulton method, which allowed us to vividly demonstrate the real-world implications of our research.

Analyzing crop production: Unraveling the impact of pests and pesticides through a fractional model

The continuous growth of the human population raises concerns about food, fiber, and agricultural insecurity. Meeting the escalating demand for agricultural products due to this population surge makes protecting crops from pests becomes imperative. While farmers use chemical pesticides as crop protectors, the extensive use of these chemicals adversely affects both human health and the environment. In this research work, we formulate a nonlinear mathematical model using the Caputo fractional (CF) operator to investigate the effects of pesticides on crop yield dynamics. We assume that pesticides are sprayed proportional to the density of pest density and pests not entirely reliant on crops. The feasibility of every possible nonnegative equilibrium and its stability characteristics are explored utilizing the stability theory of fractional differential equations. Our model analysis reveals that in a continuous spray approach, the roles of pesticide abatement rate and pesticide uptake rate can be interchanged. Furthermore, we have identified the optimal time profile for pesticide spraying rate. This profile proves effective in minimizing both the pest population and the associated costs. To provide a practical illustration of our analytical findings and to showcase the impact of key parameters on the system’s dynamics, we conducted numerical simulations. These simulations are conducted employing the generalized Adams–Bashforth–Moulton method, which allowed us to vividly demonstrate the real-world implications of our research.

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.