SOlvaBILITY OF HIGHER INDEX TIME-VARYING LINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS

Linear differential-algebraic equations (DAEs) with time-varying coefficients A(t)x(1)(t) + B(t)x(t) = q(t), which are tractable with a higher index. are discussed. Their essential properties are investigated. Some equivalent system,,; are given. Using them the paper shows how to state properly initial and boundary conditions for these DAEs. The existence and uniqueness theory of the solution of the initial and boundary value problems for higher index DAEs are proposed.

ON A REGULARIZATION OF INDEX 2 DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH PROPERLY STATED LEADING TERM

In this article, linear regular index 2 DAEs A（t）[D（t）x（t）]＇ ＋ B（t）x（t） = q（t） are considered. Using a decoupling technique, initial condition and boundary condition are properly formulated. Regular index 1 DAEs are obtained by a regularization method. We study the behavior of the solution of the regularization system via asymptotic expansions. The error analysis between the solutions of the DAEs and its regularization system is given.

NONNEGATIVITY OF SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL-ALGEBRAIC EQUATIONS

Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.

LYAPUNOV-LIKE EXPONENTIAL STABILITY AND UNSTABILITY OF DIFFERENTIAL-ALGEBRAIC EQUATION

In this paper, Lyapunov-like exponential stability and unstability of differentialalgebraic equation are considered from the viewpoint of stability of system motion, and the criteria of exponential stability and unstability of nonlinear nonautonomous differential-algebraic equation are given by using Lyapunov-like function similar to ordinary differential equation.

SOLVING NONLINEAR DELAY-DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH SINGULAR PERTURBATION VIA BLOCK BOUNDARY VALUE METHODS

Block boundary value methods(BBVMs)are extended in this paper to obtain the numerical solutions of nonlinear delay-differential-algebraic equations with singular perturbation(DDAESP).It is proved that the extended BBVMs in some suitable conditions are globally stable and can obtain a unique exact solution of the DDAESP.Besides,whenever the classic Lipschitz conditions are satisfied,the extended BBVMs are preconsistent and pth order consistent.Moreover,through some numerical examples,the correctness of the theoretical results and computational validity of the extended BBVMs is further confirmed.

Legendre Neural Network for Solving Linear Variable Coefficients Delay Differential-Algebraic Equations with Weak Discontinuities

In this paper,we propose a novel Legendre neural network combined with the extreme learning machine algorithm to solve variable coefficients linear delay differential-algebraic equations with weak discontinuities.First,the solution interval is divided into multiple subintervals by weak discontinuity points.Then,Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials on each subinterval.Finally,the parameters of the neural network are obtained by training with the extreme learning machine.The numerical examples show that the proposed method can effectively deal with the difficulty of numerical simulation caused by the discontinuities.

Convergence of Linear Multistep Methods and One-Leg Methods for Index-2 Differential-Algebraic Equations with a Variable Delay

Linear multistep methods and one-leg methods are applied to a class of index-2 nonlinear differential-algebraic equations with a variable delay.The corresponding convergence results are obtained and successfully confirmed by some numerical examples.The results obtained in this work extend the corresponding ones in literature.

ON SOLVABILITY AND WAVEFORM RELAXATION METHODS FOR LINEAR VARIABLE-COEFFICIENT DIFFERENTIAL-ALGEBRAIC EQUATIONS

This paper is concerned with the solvability and waveform relaxation methods of linear variable-coefficient differential-algebraic equations （DAEs）. Most of the previous works have been focused on linear variable-coefficient DAEs with smooth coefficients and data, yet no results related to the convergence rate of the corresponding waveform relaxation methods has been obtained. In this paper, we develope the solvability theory for the linear variable-coefficient DAEs on Legesgue square-integrable function space in both traditional and least squares senses, and determine the convergence rate of the waveform relaxation methods for solving linear variable-coefficient DAEs.

A Formulation of the Porous Medium Equation with Time-Dependent Porosity: A Priori Estimates and Regularity Results

We consider a generalized form of the porous medium equation where the porosity ϕis a function of time t: ϕ=ϕ(x,t): ∂(ϕS)∂t−∇⋅(k(S)∇S)=Q(S).In many works, the porosity ϕis either assumed to be independent of (or to depend very little of) the time variable t. In this work, we want to study the case where it does depend on t(and xas well). For this purpose, we make a change of unknown function V=ϕSin order to obtain a saturation-like (advection-diffusion) equation. A priori estimates and regularity results are established for the new equation based in part on what is known from the saturation equation, when ϕis independent of the time t. These results are then extended to the full saturation equation with time-dependent porosity ϕ=ϕ(x,t). In this analysis, we make explicitly the dependence of the various constants in the estimates on the porosity ϕby the introduced transport vector w, through the change of unknown function. Also we do not assume zero-flux boundary, but we carry the analysis for the case Q≡0.

Some Modified Equations of the Sine-Hilbert Type

Three modified sine-Hilbert(sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided.Based on these bilinear equations, some exact solutions to the three modified equations are derived.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

Theoretical study of particle and energy balance equations in locally bounded plasmas

In this study,new particle and energy balance equations have been developed to predict the electron temperature and density in locally bounded plasmas.Classical particle and energy balance equations assume that all plasma within a reactor is completely confined only by the reactor walls.However,in industrial plasma reactors for semiconductor manufacturing,the plasma is partially confined by internal reactor structures.We predict the effect of the open boundary area(A′_(L,eff))and ion escape velocity(u_(i))on electron temperature and density by developing new particle and energy balance equations.Theoretically,we found a low ion escape velocity(u_(i)/u_(B)≈0.2)and high open boundary area(A′_(L,eff)/A_(T,eff)≈0.6)to result in an approximately 38%increase in electron density and an 8%decrease in electron temperature compared to values in a fully bounded reactor.Additionally,we suggest that the velocity of ions passing through the open boundary should exceedω_(pi)λ_(De)under the condition E^(2)_(0)?(Φ/λ_(De))^(2).

Data-Driven Ai-and Bi-Soliton of the Cylindrical Korteweg-de Vries Equation via Prior-Information Physics-Informed Neural Networks

By the modifying loss function MSE and training area of physics-informed neural networks(PINNs),we propose a neural networks model,namely prior-information PINNs(PIPINNs).We demonstrate the advantages of PIPINNs by simulating Ai-and Bi-soliton solutions of the cylindrical Korteweg-de Vries(cKdV)equation.

Matrix Riccati Equations in Optimal Control

In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied thoroughly, matrix Riccati equation of which scalar Riccati equations is a particular case, is much less investigated. This article proposes a change of variable that allows to find explicit solution of the Matrix Riccati equation. We then apply this solution to Optimal Control.

Analytical solutions fractional order partial differential equations arising in fluid dynamics

This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.

The Maxwell-Heaviside Equations Explained by the Theory of Informatons

In the articles “Newtons Law of Universal Gravitation Explained by the Theory of Informatons” and “The Gravitational Interaction between Moving Mass Particles Explained by the Theory of Informatons” the gravitational interaction has been explained by the hypothesis that information carried by informatons is the substance of gravitational fields, i.e. the medium that the interaction in question makes possible. From the idea that “information carried by informatons” is its substance, it has been deduced that—on the macroscopic level—a gravitational field manifests itself as a dual entity, always having a field- and an induction component (Egand Bg) simultaneously created by their common sources. In this article we will mathematically deduce the Maxwell-Heaviside equations from the kinematics of the informatons. These relations describe on the macroscopic level how a gravitational field (Eg, Bg) is generated by whether or not moving masses and how spatial and temporal changes of Egand Bgare related. We show that there is no causal link between Egand Bg.

THE STABILITY OF BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION AROUND THE HYDROSTATIC BALANCE

This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmospheric and oceanographic phenomena.The Boussinesq system focused on here is anisotropic,and involves only horizontal dissipation and thermal damping.In the 2D case R^(2),due to the lack of vertical dissipation,the stability and large-time behavior problems have remained open in a Sobolev setting.For the spatial domain T×R,this paper solves the stability problem and gives the precise large-time behavior of the perturbation.By decomposing the velocity u and temperatureθinto the horizontal average(ū,θ)and the corresponding oscillation(ū,θ),we can derive the global stability in H~2 and the exponential decay of(ū,θ)to zero in H^(1).Moreover,we also obtain that(ū_(2),θ)decays exponentially to zero in H^(1),and thatū_(1)decays exponentially toū_(1)(∞)in H^(1)as well;this reflects a strongly stratified phenomenon of buoyancy-driven fluids.In addition,we establish the global stability in H^(3)for the 3D case R^(3).

THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARDPOTENTIAL

In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary L^(2) weighted estimates.

Prediction of ILI following the COVID-19 pandemic in China by using a partial differential equation

The COVID-19 outbreak has significantly disrupted the lives of individuals worldwide.Following the lifting of COVID-19 interventions,there is a heightened risk of future outbreaks from other circulating respiratory infections,such as influenza-like illness(ILI).Accurate prediction models for ILI cases are crucial in enabling governments to implement necessary measures and persuade individuals to adopt personal precautions against the disease.This paper aims to provide a forecasting model for ILI cases with actual cases.We propose a specific model utilizing the partial differential equation(PDE)that will be developed and validated using real-world data obtained from the Chinese National Influenza Center.Our model combines the effects of transboundary spread among regions in China mainland and human activities’impact on ILI transmission dynamics.The simulated results demonstrate that our model achieves excellent predictive performance.Additionally,relevant factors influencing the dissemination are further examined in our analysis.Furthermore,we investigate the effectiveness of travel restrictions on ILI cases.Results can be used to utilize to mitigate the spread of disease.

Besov Estimates for Sub-Elliptic Equations in the Heisenberg Group

In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.