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A deep learning method based on prior knowledge with dual training for solving FPK equation
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作者 彭登辉 王神龙 黄元辰 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第1期250-263,共14页
The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macrosc... The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macroscopic variables in the stochastic dynamic system. Traditional methods for solving these equations often struggle with computational efficiency and scalability, particularly in high-dimensional contexts. To address these challenges, this paper proposes a novel deep learning method based on prior knowledge with dual training to solve the stationary FPK equations. Initially, the neural network is pre-trained through the prior knowledge obtained by Monte Carlo simulation(MCS). Subsequently, the second training phase incorporates the FPK differential operator into the loss function, while a supervisory term consisting of local maximum points is specifically included to mitigate the generation of zero solutions. This dual-training strategy not only expedites convergence but also enhances computational efficiency, making the method well-suited for high-dimensional systems. Numerical examples, including two different two-dimensional(2D), six-dimensional(6D), and eight-dimensional(8D) systems, are conducted to assess the efficacy of the proposed method. The results demonstrate robust performance in terms of both computational speed and accuracy for solving FPK equations in the first three systems. While the method is also applicable to high-dimensional systems, such as 8D, it should be noted that computational efficiency may be marginally compromised due to data volume constraints. 展开更多
关键词 deep learning prior knowledge fpk equation probability density function
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A Formulation of the Porous Medium Equation with Time-Dependent Porosity: A Priori Estimates and Regularity Results
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作者 Koffi B. Fadimba 《Applied Mathematics》 2024年第10期745-763,共19页
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 de... 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. 展开更多
关键词 Porous Medium equation POROSITY Saturation equation A Priori Estimates Regularity Results
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Some Modified Equations of the Sine-Hilbert Type
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作者 闫铃娟 刘亚杰 胡星标 《Chinese Physics Letters》 SCIE EI CAS CSCD 2024年第4期1-6,共6页
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... 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. 展开更多
关键词 BILINEAR equationS equation
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An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields
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作者 Neşe İşler Acar 《Advances in Pure Mathematics》 2024年第10期785-796,共12页
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 b... 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. 展开更多
关键词 Stancu Polynomials Collocation Method Integro-Differential equations Linear equation Systems Matrix equations
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Theoretical study of particle and energy balance equations in locally bounded plasmas
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作者 Hyun-Su JUN Yat Fung TSANG +1 位作者 Jae Ok YOO Navab SINGH 《Plasma Science and Technology》 SCIE EI CAS CSCD 2024年第12期89-98,共10页
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 pl... 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). 展开更多
关键词 particle balance equation energy balance equation low temperature plasmas
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Data-Driven Ai-and Bi-Soliton of the Cylindrical Korteweg-de Vries Equation via Prior-Information Physics-Informed Neural Networks
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作者 田十方 李彪 张钊 《Chinese Physics Letters》 SCIE EI CAS CSCD 2024年第3期1-6,共6页
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 si... 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. 展开更多
关键词 equation SOLITON CYLINDRICAL
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Matrix Riccati Equations in Optimal Control
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作者 Malick Ndiaye 《Applied Mathematics》 2024年第3期199-213,共15页
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 tho... 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. 展开更多
关键词 Optimal Control Matrix Riccati equation Change of Variable
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Analytical solutions fractional order partial differential equations arising in fluid dynamics
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作者 Sidheswar Behera Jasvinder Singh Pal Virdi 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2024年第3期458-468,共11页
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 unidirectio... 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. 展开更多
关键词 the sine-cosine method He's fractional derivative analytical solution fractional Pade-Ⅱequation fractional generalized Zakharov equation
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THE STABILITY OF BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION AROUND THE HYDROSTATIC BALANCE
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作者 Saiguo XU Zhong TAN 《Acta Mathematica Scientia》 SCIE CSCD 2024年第4期1466-1486,共21页
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 Bouss... 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). 展开更多
关键词 Boussinesq equations partial dissipation stability DECAY
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The Maxwell-Heaviside Equations Explained by the Theory of Informatons
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作者 Antoine Acke 《Journal of High Energy Physics, Gravitation and Cosmology》 CAS 2024年第3期1003-1016,共14页
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 gravitatio... 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. 展开更多
关键词 GRAVITY Gravitational Field Maxwell equations Informatons
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THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARDPOTENTIAL
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作者 刘吕桥 曾娟 《Acta Mathematica Scientia》 SCIE CSCD 2024年第2期455-473,共19页
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 e... 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. 展开更多
关键词 Boltzmann equation Gevrey regularity non-cutoff hard potential
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Besov Estimates for Sub-Elliptic Equations in the Heisenberg Group
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作者 Huimin Cheng Feng Zhou 《Advances in Pure Mathematics》 2024年第9期744-758,共15页
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 Be... 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. 展开更多
关键词 Heisenberg Group Sub-Elliptic equations REGULARITY Besov Spaces
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On entire solutions of some Fermat type differential-difference equations
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作者 LONG Jian-ren QIN Da-zhuan 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2024年第1期69-88,共20页
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 ... 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]. 展开更多
关键词 entire solutions differential-difference equations EXISTENCE finite order
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Equivalence between the internal observability and equation
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作者 LIU Wen-jun TU Zhi-yu 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2024年第1期89-97,共9页
This paper is concerned with a third order in time linear Moore-Gibson-Thompson equation which describes the acoustic velocity potential in ultrasound wave program.Influenced by the work of Kaltenbacher,Lasiecka and M... This paper is concerned with a third order in time linear Moore-Gibson-Thompson equation which describes the acoustic velocity potential in ultrasound wave program.Influenced by the work of Kaltenbacher,Lasiecka and Marchand(Control Cybernet.2011,40:971-988),we establish an observability inequality of the conservative problem,and then discuss the equivalence between the exponential stabilization of a dissipative system and the internal observational inequality of the corresponding conservative system. 展开更多
关键词 Moore-Gibson-Thompson equation internal observability exponential stability
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ELLIPTIC EQUATIONS IN DIVERGENCE FORM WITH DISCONTINUOUS COEFFICIENTS IN DOMAINS WITH CORNERS
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作者 Jun CHEN Xuemei DENG 《Acta Mathematica Scientia》 SCIE CSCD 2024年第5期1903-1915,共13页
We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across th... We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners. 展开更多
关键词 elliptic equations divergence form discontinuous coefficients corner regularity
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Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg–de Vries equation in optical fibers
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作者 兰中周 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第6期119-123,共5页
Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained thro... Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced. 展开更多
关键词 complex modified KdV equation multi-soliton solutions breather-like BOUND-STATE
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Wave interaction for a generalized higher-dimensional Boussinesq equation describing the nonlinear Rossby waves
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作者 Rong SU Penghao JI Xiaojun YIN 《Journal of Oceanology and Limnology》 SCIE CAS CSCD 2024年第5期1415-1424,共10页
Based on an algebraically Rossby solitary waves evolution model,namely an extended(2+1)-dimensional Boussinesq equation,we firstly introduced a special transformation and utilized the Hirota method,which enable us to ... Based on an algebraically Rossby solitary waves evolution model,namely an extended(2+1)-dimensional Boussinesq equation,we firstly introduced a special transformation and utilized the Hirota method,which enable us to obtain multi-complexiton solutions and explore the interaction among the solutions.These wave functions are then employed to infer the influence of background flow on the propagation of Rossby waves,as well as the characteristics of propagation in multi-wave running processes.Additionally,we generated stereogram drawings and projection figures to visually represent these solutions.The dynamical behavior of these solutions is thoroughly examined through analytical and graphical analyses.Furthermore,we investigated the influence of the generalized beta effect and the Coriolis parameter on the evolution of Rossby waves. 展开更多
关键词 Rossby wave Boussinesq equation Complexiton solution Breather solution
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Implementation of the Integrated Green’s Function Method for 3D Poisson’s Equation in a Large Aspect Ratio Computational Domain
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作者 Ji Qiang Chad Mitchell +1 位作者 Remi Lehe Arianna Formenti 《Journal of Software Engineering and Applications》 2024年第9期740-749,共10页
The solution of Poisson’s Equation plays an important role in many areas, including modeling high-intensity and high-brightness beams in particle accelerators. For the computational domain with a large aspect ratio, ... The solution of Poisson’s Equation plays an important role in many areas, including modeling high-intensity and high-brightness beams in particle accelerators. For the computational domain with a large aspect ratio, the integrated Green’s function method has been adopted to solve the 3D Poisson equation subject to open boundary conditions. In this paper, we report on the efficient implementation of this method, which can save more than a factor of 50 computing time compared with the direct brute force implementation and its improvement under certain extreme conditions. 展开更多
关键词 Green’s Function Poisson equation Particle Accelerator
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Exact and heuristic formulae to compute the geodetic height from the ellipse equation
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作者 Mohamed Eleiche Ahmed Hamdi Mansi 《Geodesy and Geodynamics》 EI CSCD 2024年第2期150-155,共6页
The conversion of the cartesian coordinates of a point to its geodetic equivalent coordinates in reference to the geodetic ellipsoid is one of the main challenges in geodesy.The ellipse equation in the meridian plane ... The conversion of the cartesian coordinates of a point to its geodetic equivalent coordinates in reference to the geodetic ellipsoid is one of the main challenges in geodesy.The ellipse equation in the meridian plane significantly influences the value of the geodetic coordinates.This research analyzes this influence and how it can contribute to their solutions.The study investigates the mathematical relation between them and presents an exact formula relating to the geodetic height and the ellipse equation.In addition,a heuristic formula for the relation between the geodetic height and the ellipse equation is proposed,which is independent of the geodetic latitude and has a relative accuracy better than 99.9 %.The calculation is stable,and the cost is low. 展开更多
关键词 Ellipse equation Geodetic height Heuristic geodetic height
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ON THE STABILITY OF PERIODIC SOLUTIONS OF PIECEWISE SMOOTH PERIODIC DIFFERENTIAL EQUATIONS
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作者 Maoan HAN Yan YE 《Acta Mathematica Scientia》 SCIE CSCD 2024年第4期1524-1535,共12页
In this paper,we address the stability of periodic solutions of piecewise smooth periodic differential equations.By studying the Poincarémap,we give a sufficient condition to judge the stability of a periodic sol... In this paper,we address the stability of periodic solutions of piecewise smooth periodic differential equations.By studying the Poincarémap,we give a sufficient condition to judge the stability of a periodic solution.We also present examples of some applications. 展开更多
关键词 periodic solution Poincarémap periodic equation stability
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