Error Estimate of the Fourier Collocation Method for the Benjamin-Ono Equation

In this paper,the Fourier collocation method for solving the generalized Benjamin-Ono equation with periodic boundary conditions is analyzed.Stability of the semi-discrete scheme is proved and error estimate in H^(1/2)-norm is obtained.

SOLUTIONS OF THE BENJAMIN-ONO EQUATION IN FRACTIONAL ORDER SOBOLEV SPACES

We show for the Benjamin-Ono equation an existence uniqeness theorem in Sobolev spaces of arbitrary fractional order s greater-than-or-equal-to 2, provided the initial data is given in the same space.

The Solitary Waves Solutions of the Internal Wave Benjamin-Ono Equation

The Benjamin-ono (BO) equation is an important nonlinear wave model which can describe the deep oceanic internal wave propagation. In this paper, the multi-algebraic solitary wave solutions for the internal wave BO equation including the linear velocity term in matrix form are given by the bilinear form. Based on the analytic solutions of the BO equation obtained in this paper and considering the hydrological parameters, the propagation of one-solitary wave and different kinds of interaction for the two-solitary waves are discussed and illustrated.

Generalized solutions to the Benjamin-Ono equations in sense of Colombeau

This paper discusses the existence and uniqueness of the generalized solution in the sense of Colombeau to the Benjamin-Ono (B-O) equation and the relationship between the new generalized solution and the classical solution.

The Cauchy Problem for the Generalized Korteweg-de Vries-Benjamin-Ono Equation with Low Regularity Data

The Cauchy problem of the generalized Korteweg-de Vries-Benjamin-Ono equation is considered, and low regularity and limit behavior of the solutions are obtained. For k = 1, local well- posedness is obtained for data in H^s（R）（s 〉 -3/4）. For k = 2, local result for data in H^S（R）（s 〉1/4） is obtained. For k = 3, local result for data in H^S（R）（s 〉 -1/6） is obtained. Moreover, the solutions of generalized Korteweg-de Vries-Benjamin-Ono equation converge to the solutions of KdV equation if the term of Benjamin-Ono equation tends to zero.

The Cauchy problem for the stochastic generalized Benjamin-Ono equation

The current paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation.By establishing the bilinear and trilinear estimates in some Bourgain spaces,we prove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data u0(x,w)∈L^(2)(Ω;H^(s)(R))which is F0-measurable with s≥1/2-α/4 andΦ∈L20,s.In particular,whenα=1,we prove that it is globally well-posed for the initial data u0(x,w)∈L2(Ω;H1(R))which is F0-measurable andΦ∈L20,1.The key ingredients that we use in this paper are trilinear estimates,the Ito formula and the Burkholder-Davis-Gundy(BDG)inequality as well as the stopping time technique.

ON THE Benjamin-Ono EQUATION AND ITS GENERALIZATION IN THE ATMOSPHERE

This paper deals with the nonlinear problems of the two-layer shallow water wave modelin the atmosphere by the multiple scale method and points out that a nonlinear Benjamin-Ono equation may be obtained if the meridional disturbance wind is weak. Furthermore, thealgebraic solitary waves and nonlinear periodic waves are also calculated and the breakup ofan algebraic solitary wave at the initial time into two is discussed. It is found that the waveform steepens in this process, which is similar to the triggering process of squall lines inthe atmosphere. On the other hand, we also point out that when the meridional disturbancewind is strong, we may obtain a modified Benjamin-Ono equation, i. e. the Benjamin-Ono-KDV equation.

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 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 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 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.

On entire solutions of some Fermat type differential-difference equations

On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].