It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment...It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular. The method consists to solve the integral equation numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.展开更多
In this paper we survey the authors' and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four s...In this paper we survey the authors' and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections: 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.展开更多
A theoretical model is developed for predicting both conduction and diffusion in thin-film ionic conductors or cables. With the linearized Poisson-Nernst-Planck(PNP)theory, the two-dimensional(2D) equations for thin i...A theoretical model is developed for predicting both conduction and diffusion in thin-film ionic conductors or cables. With the linearized Poisson-Nernst-Planck(PNP)theory, the two-dimensional(2D) equations for thin ionic conductor films are obtained from the three-dimensional(3D) equations by power series expansions in the film thickness coordinate, retaining the lower-order equations. The thin-film equations for ionic conductors are combined with similar equations for one thin dielectric film to derive the 2D equations of thin sandwich films composed of a dielectric layer and two ionic conductor layers. A sandwich film in the literature, as an ionic cable, is analyzed as an example of the equations obtained in this paper. The numerical results show the effect of diffusion in addition to the conduction treated in the literature. The obtained theoretical model including both conduction and diffusion phenomena can be used to investigate the performance of ionic-conductor devices with any frequency.展开更多
In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.T...In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.展开更多
This paper ix devoted to establishment of the Chebyshev pseudospectral domain de-composition scheme for solving two-dimensional elliptic equation. By the generalized equivalent variatiunal form, we can get the stabili...This paper ix devoted to establishment of the Chebyshev pseudospectral domain de-composition scheme for solving two-dimensional elliptic equation. By the generalized equivalent variatiunal form, we can get the stability and convergence of this new scheme.展开更多
In this paper,the approximate solutions for two different type of two-dimensional nonlinear integral equations:two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm i...In this paper,the approximate solutions for two different type of two-dimensional nonlinear integral equations:two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method.To do this,these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form.By solving these systems,unknown coefficients are obtained.Also,some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.展开更多
The Ablowitz-Ladik equation is a very important model in nonlinear mathematical physics. In this paper, the hyper- bolic function solitary wave solutions, the trigonometric function periodic wave solutions, and the ra...The Ablowitz-Ladik equation is a very important model in nonlinear mathematical physics. In this paper, the hyper- bolic function solitary wave solutions, the trigonometric function periodic wave solutions, and the rational wave solutions with more arbitrary parameters of two-dimensional Ablowitz-Ladik equation are derived by using the (GI/G)-expansion method, and the effects of the parameters (including the coupling constant and other parameters) on the linear stability of the exact solutions are analysed and numerically simulated.展开更多
Because exact analytic solution is not available, we use double expansion and boundary collocation to construct an approximate solution for a class of two-dimensional dual integral equations in mathematical physics. T...Because exact analytic solution is not available, we use double expansion and boundary collocation to construct an approximate solution for a class of two-dimensional dual integral equations in mathematical physics. The integral equations by this procedure are reduced to infinite algebraic equations. The accuracy of the solution lies in the boundary collocation technique. The application of which for some complicated initialboundary value problems in solid mechanics indicates the method is powerful.展开更多
A class of periodic initial value problems for two-dimensional Newton- Boussinesq equations are investigated in this paper. The Newton-Boussinesq equations are turned into the equivalent integral equations. With itera...A class of periodic initial value problems for two-dimensional Newton- Boussinesq equations are investigated in this paper. The Newton-Boussinesq equations are turned into the equivalent integral equations. With iteration methods, the local existence of the solutions is obtained. Using the method of a priori estimates, the global existence of the solution is proved.展开更多
Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an a...Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an arbitrary functionφ(y),a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method.By choosing the appropriate parameters,their interesting dynamic behaviors are shown in three-dimensional plots.Furthermore,novel rational solutions are generated by taking the limit of the obtained solitons.Additionally,twodimensional(2D)rogue waves(localized in both space and time)on the soliton plane are presented,we refer to them as deformed 2D rogue waves.The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane,and its evolution process is analyzed in detail.The deformed 2D rogue wave solutions are constructed successfully,which are closely related to the arbitrary functionφ(y).This new idea is also applicable to other nonlinear systems.展开更多
For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numeric...For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable.Numerical results indicate the effectiveness and accuracy of the method and con-firm the analysis.展开更多
In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial deriv...In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.展开更多
We proposed a higher-order accurate explicit finite-difference scheme for solving the two-dimensional heat equation. It has a fourth-order approximation in the space variables, and a second-order approximation in the ...We proposed a higher-order accurate explicit finite-difference scheme for solving the two-dimensional heat equation. It has a fourth-order approximation in the space variables, and a second-order approximation in the time variable. As an application, we developed the proposed numerical scheme for solving a numerical solution of the two-dimensional coupled Burgers’ equations. The main advantages of our scheme are higher accurate accuracy and facility to implement. The good accuracy of the proposed numerical scheme is tested by comparing the approximate numerical and the exact solutions for several two-dimensional coupled Burgers’ equations.展开更多
We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this work.In particular,to better capture the sharpness of the ...We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this work.In particular,to better capture the sharpness of the solution,we propose the adaptive sampling methods(ASMs)based on the residual and the gradient of the solution.We first present a residual only-based ASM denoted by ASMⅠ.In this approach,we first train the neural network using a small number of residual points and divide the computational domain into a certain number of sub-domains,then we add new residual points in the sub-domain which has the largest mean absolute value of the residual,and those points which have the largest absolute values of the residual in this sub-domain as new residual points.We further develop a second type of ASM(denoted by ASMⅡ)based on both the residual and the gradient of the solution due to the fact that only the residual may not be able to efficiently capture the sharpness of the solution.The procedure of ASMⅡis almost the same as that of ASMⅠ,and we add new residual points which have not only large residuals but also large gradients.To demonstrate the effectiveness of the present methods,we use both ASMⅠand ASMⅡto solve a number of PDEs,including the Burger equation,the compressible Euler equation,the Poisson equation over an Lshape domain as well as the high-dimensional Poisson equation.It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASMⅠor ASMⅡ,and both methods deliver much more accurate solutions than the original PINNs with the same number of residual points.Moreover,the ASMⅡalgorithm has better performance in terms of accuracy,efficiency,and stability compared with the ASMⅠalgorithm.This means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the solution.Furthermore,we also employ the similar adaptive sampling technique for the data points of boundary conditions(BCs)if the sharpness of the solution is near the boundary.The result of the L-shape Poisson problem indicates that the present method can significantly improve the efficiency,stability,and accuracy.展开更多
This paper presents a Poisson theory of the generalized Birkhoff equations, including the algebraic structure of the equations, the sufficient and necessary condition on the integral and the conditions under which a n...This paper presents a Poisson theory of the generalized Birkhoff equations, including the algebraic structure of the equations, the sufficient and necessary condition on the integral and the conditions under which a new integral can be deduced by a known integral as well as the form of the new integral.展开更多
The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the po...The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations (jumps) resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.展开更多
In this paper,we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise.The main difficulties come from the highly non-linear coefficient and the jump noise....In this paper,we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise.The main difficulties come from the highly non-linear coefficient and the jump noise.Here,we adopt a new sufficient condition for the weak convergence criterion of the large deviation principle,which was initially proposed by Matoussi,Sabbagh and Zhang(2021).展开更多
A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. T...A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena.展开更多
The motion of the self-gravitational gaseous stars can be described by the Euler-Poisson equations. The main purpose of this paper is concerned with the existence of stationary solutions of Euler-Poisson equations for...The motion of the self-gravitational gaseous stars can be described by the Euler-Poisson equations. The main purpose of this paper is concerned with the existence of stationary solutions of Euler-Poisson equations for some velocity fields and entropy functions that solve the conservation of mass and energy. Under different restriction to the strength of velocity field, we get the existence and multiplicity of the stationary solutions of Euler-Poisson system.展开更多
In this work, the three dimensional Poisson’s equation in Cartesian coordinates with the Dirichlet’s boundary conditions in a cube is solved directly, by extending the method of Hockney. The Poisson equation is appr...In this work, the three dimensional Poisson’s equation in Cartesian coordinates with the Dirichlet’s boundary conditions in a cube is solved directly, by extending the method of Hockney. The Poisson equation is approximated by 19-points and 27-points fourth order finite difference approximation schemes and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The efficiency of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results. It is shown that 19-point formula produces comparable results with 27-point formula, though computational efforts are more in 27-point formula.展开更多
文摘It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular. The method consists to solve the integral equation numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.
基金supported by 973 Key program and the Key Program from Beijing Educational Commission with No. KZ200910028002Program for New Century Excellent Talents in University (NCET)+4 种基金Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR-IHLB)The research of Sheng partially supported by NSFC (10671120)Shanghai Leading Academic Discipline Project: J50101The research of Zhang partially supported by NSFC (10671120)The research of Zheng partially supported by NSF-DMS-0603859
文摘In this paper we survey the authors' and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections: 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.
基金Project supported by the National Natural Science Foundation of China(Nos.11672265,11202182,and 11621062)the Fundamental Research Funds for the Central Universities(Nos.2016QNA4026 and2016XZZX001-05)the Open Foundation of Zhejiang Provincial Top Key Discipline of Mechanical Engineering
文摘A theoretical model is developed for predicting both conduction and diffusion in thin-film ionic conductors or cables. With the linearized Poisson-Nernst-Planck(PNP)theory, the two-dimensional(2D) equations for thin ionic conductor films are obtained from the three-dimensional(3D) equations by power series expansions in the film thickness coordinate, retaining the lower-order equations. The thin-film equations for ionic conductors are combined with similar equations for one thin dielectric film to derive the 2D equations of thin sandwich films composed of a dielectric layer and two ionic conductor layers. A sandwich film in the literature, as an ionic cable, is analyzed as an example of the equations obtained in this paper. The numerical results show the effect of diffusion in addition to the conduction treated in the literature. The obtained theoretical model including both conduction and diffusion phenomena can be used to investigate the performance of ionic-conductor devices with any frequency.
基金This work is supported by the National Natural Science Foundation of China(11661058,11761053)the Natural Science Foundation of Inner Mongolia(2017MS0107)the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region(NJYT-17-A07).
文摘In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.
文摘This paper ix devoted to establishment of the Chebyshev pseudospectral domain de-composition scheme for solving two-dimensional elliptic equation. By the generalized equivalent variatiunal form, we can get the stability and convergence of this new scheme.
文摘In this paper,the approximate solutions for two different type of two-dimensional nonlinear integral equations:two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method.To do this,these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form.By solving these systems,unknown coefficients are obtained.Also,some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.
基金Project supported by the Basic Science and the Front Technology Research Foundation of Henan Province,China(Grant Nos.092300410179 and122102210427)the Doctoral Scientific Research Foundation of Henan University of Science and Technology,China(Grant No.09001204)+1 种基金the Scientific Research Innovation Ability Cultivation Foundation of Henan University of Science and Technology,China(Grant No.011CX011)the Scientific Research Foundation of Henan University of Science and Technology(Grant No.2012QN011)
文摘The Ablowitz-Ladik equation is a very important model in nonlinear mathematical physics. In this paper, the hyper- bolic function solitary wave solutions, the trigonometric function periodic wave solutions, and the rational wave solutions with more arbitrary parameters of two-dimensional Ablowitz-Ladik equation are derived by using the (GI/G)-expansion method, and the effects of the parameters (including the coupling constant and other parameters) on the linear stability of the exact solutions are analysed and numerically simulated.
基金Project supported by the National Natural Science Foundation of China(No.K19672007)
文摘Because exact analytic solution is not available, we use double expansion and boundary collocation to construct an approximate solution for a class of two-dimensional dual integral equations in mathematical physics. The integral equations by this procedure are reduced to infinite algebraic equations. The accuracy of the solution lies in the boundary collocation technique. The application of which for some complicated initialboundary value problems in solid mechanics indicates the method is powerful.
基金Project supported by the National Natural Science Foundation of China (Nos. 10871075 and 10926101)the Natural Science Foundation of Guangdong Province of China(Nos. 9451064201003736 and 9151064201000040)
文摘A class of periodic initial value problems for two-dimensional Newton- Boussinesq equations are investigated in this paper. The Newton-Boussinesq equations are turned into the equivalent integral equations. With iteration methods, the local existence of the solutions is obtained. Using the method of a priori estimates, the global existence of the solution is proved.
基金Project supported by the National Natural Scinece Foundation of China(Grant Nos.11671219,11871446,12071304,and 12071451).
文摘Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an arbitrary functionφ(y),a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method.By choosing the appropriate parameters,their interesting dynamic behaviors are shown in three-dimensional plots.Furthermore,novel rational solutions are generated by taking the limit of the obtained solitons.Additionally,twodimensional(2D)rogue waves(localized in both space and time)on the soliton plane are presented,we refer to them as deformed 2D rogue waves.The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane,and its evolution process is analyzed in detail.The deformed 2D rogue wave solutions are constructed successfully,which are closely related to the arbitrary functionφ(y).This new idea is also applicable to other nonlinear systems.
文摘For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable.Numerical results indicate the effectiveness and accuracy of the method and con-firm the analysis.
文摘In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.
文摘We proposed a higher-order accurate explicit finite-difference scheme for solving the two-dimensional heat equation. It has a fourth-order approximation in the space variables, and a second-order approximation in the time variable. As an application, we developed the proposed numerical scheme for solving a numerical solution of the two-dimensional coupled Burgers’ equations. The main advantages of our scheme are higher accurate accuracy and facility to implement. The good accuracy of the proposed numerical scheme is tested by comparing the approximate numerical and the exact solutions for several two-dimensional coupled Burgers’ equations.
基金Project supported by the National Key R&D Program of China(No.2022YFA1004504)the National Natural Science Foundation of China(Nos.12171404 and 12201229)the Fundamental Research Funds for Central Universities of China(No.20720210037)。
文摘We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this work.In particular,to better capture the sharpness of the solution,we propose the adaptive sampling methods(ASMs)based on the residual and the gradient of the solution.We first present a residual only-based ASM denoted by ASMⅠ.In this approach,we first train the neural network using a small number of residual points and divide the computational domain into a certain number of sub-domains,then we add new residual points in the sub-domain which has the largest mean absolute value of the residual,and those points which have the largest absolute values of the residual in this sub-domain as new residual points.We further develop a second type of ASM(denoted by ASMⅡ)based on both the residual and the gradient of the solution due to the fact that only the residual may not be able to efficiently capture the sharpness of the solution.The procedure of ASMⅡis almost the same as that of ASMⅠ,and we add new residual points which have not only large residuals but also large gradients.To demonstrate the effectiveness of the present methods,we use both ASMⅠand ASMⅡto solve a number of PDEs,including the Burger equation,the compressible Euler equation,the Poisson equation over an Lshape domain as well as the high-dimensional Poisson equation.It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASMⅠor ASMⅡ,and both methods deliver much more accurate solutions than the original PINNs with the same number of residual points.Moreover,the ASMⅡalgorithm has better performance in terms of accuracy,efficiency,and stability compared with the ASMⅠalgorithm.This means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the solution.Furthermore,we also employ the similar adaptive sampling technique for the data points of boundary conditions(BCs)if the sharpness of the solution is near the boundary.The result of the L-shape Poisson problem indicates that the present method can significantly improve the efficiency,stability,and accuracy.
基金supported by the National Natural Science Foundation of China (Grant Nos 10572021 and 10772025)
文摘This paper presents a Poisson theory of the generalized Birkhoff equations, including the algebraic structure of the equations, the sufficient and necessary condition on the integral and the conditions under which a new integral can be deduced by a known integral as well as the form of the new integral.
基金supported by the National Board for Higher Mathematics,Mumbai,India under Grant No.2/48(5)/2013/NBHM(R.P.)/RD-II/688 dt 16.01.2014
文摘The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations (jumps) resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.
基金partially supported by the National Natural Science Foundation of China(11871382,12071361)partially supported by the National Natural Science Foundation of China(11971361,11731012)。
文摘In this paper,we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise.The main difficulties come from the highly non-linear coefficient and the jump noise.Here,we adopt a new sufficient condition for the weak convergence criterion of the large deviation principle,which was initially proposed by Matoussi,Sabbagh and Zhang(2021).
文摘A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena.
基金supported by NSFC (10631030, 11071094)the fund of CCNU for Ph.D students (2009021)
文摘The motion of the self-gravitational gaseous stars can be described by the Euler-Poisson equations. The main purpose of this paper is concerned with the existence of stationary solutions of Euler-Poisson equations for some velocity fields and entropy functions that solve the conservation of mass and energy. Under different restriction to the strength of velocity field, we get the existence and multiplicity of the stationary solutions of Euler-Poisson system.
文摘In this work, the three dimensional Poisson’s equation in Cartesian coordinates with the Dirichlet’s boundary conditions in a cube is solved directly, by extending the method of Hockney. The Poisson equation is approximated by 19-points and 27-points fourth order finite difference approximation schemes and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The efficiency of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results. It is shown that 19-point formula produces comparable results with 27-point formula, though computational efforts are more in 27-point formula.