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An Efficient Accurate Direct Solution of Poisson's Equation for Computation of Meteorological Parameters
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作者 D.R.Chakraborty P.S.Salvekar 《Advances in Atmospheric Sciences》 SCIE CAS CSCD 1989年第4期501-508,共8页
Poisson's equation is solved numerically by two direct methods, viz. Block Cyclic Reduction (BCR) method and Fourier Method. Qualitative and quantitative comparison between the numerical solutions obtained by two ... Poisson's equation is solved numerically by two direct methods, viz. Block Cyclic Reduction (BCR) method and Fourier Method. Qualitative and quantitative comparison between the numerical solutions obtained by two methods indicates that BCR method is superior to Fourier method in terms of speed and accuracy. Therefore. BCR method is applied to solve (?)2(?)= ζ and (?)2X= D from observed vorticity and divergent values. Thereafter the rotational and divergent components of the horizontal monsoon wind in the lower troposphere are reconstructed and are com pared with the results obtained by Successive Over-Relaxation (SOR) method as this indirect method is generally in more use for obtaining the streamfunction ((?)) and velocity potential (X) fields in NWP models. It is found that the results of BCR method are more reliable than SOR method. 展开更多
关键词 An Efficient Accurate Direct Solution of Poisson’s equation for Computation of Meteorological Parameters
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BIOLOGICAL INVASION PROBLEM WITH FREE BOUNDARY NONLOCAL DIFFUSION EQUATION
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作者 HE Yu-rong ZHANG Ya-rong 《数学杂志》 2025年第1期48-56,共9页
In order to better describe the phenomenon of biological invasion,this paper introduces a free boundary model of biological invasion.Firstly,the right free boundary is added to the equation with logistic terms.Secondl... In order to better describe the phenomenon of biological invasion,this paper introduces a free boundary model of biological invasion.Firstly,the right free boundary is added to the equation with logistic terms.Secondly,the existence and uniqueness of local solutions are proved by the Sobolev embedding theorem and the comparison principle.Finally,according to the relevant research data and contents of red fire ants,the diffusion area and nest number of red fire ants were simulated without external disturbance.This paper mainly simulates the early diffusion process of red fire ants.In the early diffusion stage,red fire ants grow slowly and then spread over a large area after reaching a certain number. 展开更多
关键词 Free boundary biological invasion differential equation
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Normalized Solutions of Nonlinear Choquard Equations with Nonconstant Potential
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作者 LI Nan XU Liping 《应用数学》 北大核心 2025年第1期14-29,共16页
In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with ... In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with prescribed 2-norm has some normalized solutions by introducing variational methods. 展开更多
关键词 Nonlinear Choquard equation Potential function Variational method Normalized solution
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Two Second-Order Ecient Numerical Schemes for the Boussinesq Equations
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作者 LIU Fang WANG Danxia ZHANG Jianwen 《应用数学》 北大核心 2025年第1期114-129,共16页
In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,t... In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,the original Boussinesq system is transformed into an equivalent one.Then we discretize it using the second-order backward di erentiation formula(BDF2)and Crank-Nicolson(CN)to obtain two second-order time-advanced schemes.In both numerical schemes,a pressure-correction method is employed to decouple the velocity and pressure.These two schemes possess the desired property that they can be fully decoupled with satisfying unconditional stability.We rigorously prove both the unconditional stability and unique solvability of the discrete schemes.Furthermore,we provide detailed implementations of the decoupling procedures.Finally,various 2D numerical simulations are performed to verify the accuracy and energy stability of the proposed schemes. 展开更多
关键词 Scalar auxiliary variable approach Pressure-correction method Fully decoupled Unconditional stability Boussinesq equations
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WELL-POSEDNESS AND PEAKON SOLUTIONS FOR A HIGHER ORDER CAMASSA-HOLM TYPE EQUATION
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作者 CHEN shuang 《数学杂志》 2025年第1期57-71,共15页
In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belon... In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belongs to the Sobolev space H'(R)for some s>2.In addition,we obtain the weak formulation of this equation and prove the existence of both single peakon solution and a multi-peakon dynamic system. 展开更多
关键词 Generalized higher order Camassa-Holm type equation Local well-posedness PEAKON
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DEEP NEURAL NETWORKS COMBINING MULTI-TASK LEARNING FOR SOLVING DELAY INTEGRO-DIFFERENTIAL EQUATIONS
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作者 WANG Chen-yao SHI Feng 《数学杂志》 2025年第1期13-38,共26页
Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay di... Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay differential equations(DDEs)and delay integrodifferential equations(DIDEs)with constant delays,primarily due to their low regularity at delayinduced breaking points.In this paper,a DNN method that combines multi-task learning(MTL)which is proposed to solve both the forward and inverse problems of DIDEs.The core idea of this approach is to divide the original equation into multiple tasks based on the delay,using auxiliary outputs to represent the integral terms,followed by the use of MTL to seamlessly incorporate the properties at the breaking points into the loss function.Furthermore,given the increased training dificulty associated with multiple tasks and outputs,we employ a sequential training scheme to reduce training complexity and provide reference solutions for subsequent tasks.This approach significantly enhances the approximation accuracy of solving DIDEs with DNNs,as demonstrated by comparisons with traditional DNN methods.We validate the effectiveness of this method through several numerical experiments,test various parameter sharing structures in MTL and compare the testing results of these structures.Finally,this method is implemented to solve the inverse problem of nonlinear DIDE and the results show that the unknown parameters of DIDE can be discovered with sparse or noisy data. 展开更多
关键词 Delay integro-differential equation Multi-task learning parameter sharing structure deep neural network sequential training scheme
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Hoeffding's inequality for Markov processes via solution of Poisson's equation
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作者 Yuanyuan LIU Jinpeng LIU 《Frontiers of Mathematics in China》 SCIE CSCD 2021年第2期543-558,共16页
We investigate Hoeffding's inequality for both discrete-time Markov chains and continuous-time Markov processes on a general state space. Our results relax the usual aperiodicity restriction in the literature, and... We investigate Hoeffding's inequality for both discrete-time Markov chains and continuous-time Markov processes on a general state space. Our results relax the usual aperiodicity restriction in the literature, and the explicit upper bounds in the inequalities are obtained via the solution of Poisson's equation. The results are further illustrated with applications to queueing theory and reective diffusion processes. 展开更多
关键词 Hoeffding's inequality Markov process poisson's equation
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Zero extension for Poisson's equation 被引量:1
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作者 Yongyong Cai Shulin Zhou 《Science China Mathematics》 SCIE CSCD 2020年第4期721-732,共12页
In this paper,we present a necessary and sufficient condition to guarantee that the extended function of the solution for Poisson's equation in a smaller domain by zero extension is still the solution of the corre... In this paper,we present a necessary and sufficient condition to guarantee that the extended function of the solution for Poisson's equation in a smaller domain by zero extension is still the solution of the corresponding extension problem in a larger domain.We prove the results under the frameworks of classical solutions,strong solutions and weak solutions.Furthermore,we give some observations for the nonlinear pLaplace equation. 展开更多
关键词 poisson's equation ZERO EXTENSION a necessary and SUFFICIENT CONDITION
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Uniqueness and stability in an inverse problem for a Poisson's equation
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作者 Li-xin FENG~(1+) Yan-cheng QU~2 1 School of Mathematical Sciences,Heilongjiang University,Harbin 150080,China 2 Research Center of Control and Inertial Technology,Harbin Institute of Technology,Harbin 150001,China 《Science China Mathematics》 SCIE 2007年第7期967-976,共10页
Consider the Poisson’s equation Ψ″ (x) = ?e v?Ψ + e Ψ?v ? N(x) with the Dirichlet boundary data, and we mainly investigate the inverse problem of determining the unknown function N(x) from a parameter function fa... Consider the Poisson’s equation Ψ″ (x) = ?e v?Ψ + e Ψ?v ? N(x) with the Dirichlet boundary data, and we mainly investigate the inverse problem of determining the unknown function N(x) from a parameter function family. Some uniqueness and stability results in the inverse problem are obtained. 展开更多
关键词 Poisson’s equation WELL-POSEDNESS inverse problem 65N30 35R30
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Noether's and Poisson's methods for solving differential equation x_s^((m))=F_s(t,x_k^((m-2)) ,x_k^((m-1)))
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作者 何光 梅凤翔 《Chinese Physics B》 SCIE EI CAS CSCD 2008年第3期822-824,共3页
This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether me... This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether method and the Poisson method. Then the solution of the higher-order equation can be obtained by integrating the solution of the second-order equation. 展开更多
关键词 Noether's method Poisson's method higher order ordinary differential equation integration
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SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION 被引量:1
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作者 向长林 郑高峰 《Acta Mathematica Scientia》 SCIE CSCD 2024年第2期420-430,共11页
This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽... This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems. 展开更多
关键词 fourth order elliptic equation regularity theory Morrey space decay estimates Riesz potential
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Measured boundary parameterization based on Poisson's equation
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作者 Jun-jie CAO Zhi-xun SU Xiu-ping LIU Hai-chuan BI 《Journal of Zhejiang University-Science C(Computers and Electronics)》 SCIE EI 2010年第3期187-198,共12页
One major goal of mesh parameterization is to minimize the conformal distortion. Measured boundary parameteri-zations focus on lowering the distortion by setting the boundary free with the help of distance from a cent... One major goal of mesh parameterization is to minimize the conformal distortion. Measured boundary parameteri-zations focus on lowering the distortion by setting the boundary free with the help of distance from a center vertex to all the boundary vertices. Hence these parameterizations strongly depend on the determination of the center vertex. In this paper,we introduce two methods to determine the center vertex automatically. Both of them can be used as necessary supplements to the existing measured boundary methods to minimize the common artifacts as a result of the obscure choice of the center vertex. In addition,we propose a simple and fast measured boundary parameterization method based on the Poisson's equation. Our new approach generates less conformal distortion than the fixed boundary methods. It also generates more regular domain boundaries than other measured boundary methods. Moreover,it offers a good tradeoff between computation costs and conformal distortion compared with the fast and robust angle based flattening (ABF++). 展开更多
关键词 Mesh parameterization Poisson's equation Measured boundary
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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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