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Journal articles on the topic 'Poisson equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 10 May 2025

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1

Daribayev, Beimbet, Aksultan Mukhanbet, and Timur Imankulov. "Implementation of the HHL Algorithm for Solving the Poisson Equation on Quantum Simulators." Applied Sciences 13, no. 20 (October 20, 2023): 11491. http://dx.doi.org/10.3390/app132011491.

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The Poisson equation is a fundamental equation of mathematical physics that describes the potential distribution in static fields. Solving the Poisson equation on a grid is computationally intensive and can be challenging for large grids. In recent years, quantum computing has emerged as a potential approach to solving the Poisson equation more efficiently. This article uses quantum algorithms, particularly the Harrow–Hassidim–Lloyd (HHL) algorithm, to solve the 2D Poisson equation. This algorithm can solve systems of equations faster than classical algorithms when the matrix A is sparse. The main idea is to use a quantum algorithm to transform the state vector encoding the solution of a system of equations into a superposition of states corresponding to the significant components of this solution. This superposition is measured to obtain the solution of the system of equations. The article also presents the materials and methods used to solve the Poisson equation using the HHL algorithm and provides a quantum circuit diagram. The results demonstrate the low error rate of the quantum algorithm when solving the Poisson equation.

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2

Alqhtani, Manal, Khaled M. Saad, Rasool Shah, Wajaree Weera, and Waleed M. Hamanah. "Analysis of the Fractional-Order Local Poisson Equation in Fractal Porous Media." Symmetry 14, no. 7 (June 27, 2022): 1323. http://dx.doi.org/10.3390/sym14071323.

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This paper investigates the fractional local Poisson equation using the homotopy perturbation transformation method. The Poisson equation discusses the potential area due to a provided charge with the possibility of area identified, and one can then determine the electrostatic or gravitational area in the fractal domain. Elliptic partial differential equations are frequently used in the modeling of electromagnetic mechanisms. The Poisson equation is investigated in this work in the context of a fractional local derivative. To deal with the fractional local Poisson equation, some illustrative problems are discussed. The solution shows the well-organized and straightforward nature of the homotopy perturbation transformation method to handle partial differential equations having fractional derivatives in the presence of a fractional local derivative. The solutions obtained by the defined methods reveal that the proposed system is simple to apply, and the computational cost is very reliable. The result of the fractional local Poisson equation yields attractive outcomes, and the Poisson equation with a fractional local derivative yields improved physical consequences.

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3

Fortunato, Daniel, and Alex Townsend. "Fast Poisson solvers for spectral methods." IMA Journal of Numerical Analysis 40, no. 3 (November 29, 2019): 1994–2018. http://dx.doi.org/10.1093/imanum/drz034.

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Abstract Poisson’s equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference (FD) and finite element methods, fast Poisson solvers for spectral methods have remained elusive. Here we derive spectral methods for solving Poisson’s equation on a square, cylinder, solid sphere and cube that have optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom used to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of FD matrices, our solver exploits a separated spectra property that holds for our carefully designed spectral discretizations. Without parallelization we can solve Poisson’s equation on a square with 100 million degrees of freedom in under 2 min on a standard laptop.

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4

EL-HANBALY, A. M., and A. ELGARAYHI. "Exact solutions of the collisional Vlasov equation." Journal of Plasma Physics 59, no. 1 (January 1998): 169–77. http://dx.doi.org/10.1017/s0022377897006132.

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The symmetry group of the Vlasov–Fokker–Planck equation (VFPE) is constructed. The effects of the Poisson equation on this group is studied, and different types of similarity solutions of the whole system of equations (VFPE+Poisson equation) are obtained.

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5

Yuan, Hong Fen, and Valery V. Karachik. "DUNKL-POISSON EQUATION AND RELATED EQUATIONS IN SUPERSPACE." Mathematical Modelling and Analysis 20, no. 6 (November 23, 2015): 768–81. http://dx.doi.org/10.3846/13926292.2015.1112856.

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Abstract In this paper, we investigate the Almansi expansion for solutions of Dunkl-polyharmonic equations by the 0-normalized system for the Dunkl-Laplace operator in superspace. Moreover, applying the 0-normalized system, we construct solutions to the Dunkl-Helmholtz equation, the Dunkl-Poisson equation, and the inhomogeneous Dunkl-polyharmonic equation in superspace.

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6

Schmidt, Gunther, and Hartmut Strese. "BEM for poisson equation." Engineering Analysis with Boundary Elements 10, no. 2 (1992): 119–23. http://dx.doi.org/10.1016/0955-7997(92)90040-e.

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7

Vedenyapin, V. V., T. V. Salnikova, and S. Ya Stepanov. "Vlasov-Poisson-Poisson equations, critical mass and kordylewski clouds." Доклады Академии наук 485, no. 3 (May 21, 2019): 276–80. http://dx.doi.org/10.31857/s0869-56524853276-280.

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A derivation of the Vlasov-Poisson-Poisson equation is proposed for studying stationary solutions of a system of gravitating charged particles in vicinity of triangular libration points (Kordylevsky cloud). Stationary solutions are sought as functions of integrals, which leads to elliptic nonlinear equations for the potentials of the gravitational and electrostatic fields. This gives a critical mass: for bodies with large masses dominates gravitation forces, and for bodies with smaller masses - electrostatic forces.

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8

Gharib, Gharib M., and Rania Saadeh. "Reduction of the Self-dual Yang-Mills Equations to Sinh-Poisson Equation and Exact Solutions." WSEAS TRANSACTIONS ON MATHEMATICS 20 (October 26, 2021): 540–46. http://dx.doi.org/10.37394/23206.2021.20.57.

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The geometric properties of differential systems are used to demonstrate how the sinh-poisson equation describes a surface with a constant negative curvature in this paper. The canonical reduction of 4-dimensional self dual Yang Mills theorem is the sinh-poisson equation, which explains pseudo spherical surfaces. We derive the B¨acklund transformations and the travelling wave solution for the sinh-poisson equation in specific. As a result, we discover exact solutions to the self-dual Yang-Mills equations.

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9

Labovskiy, S. M., and M. F. Getimane. "POISSON PROBLEM FOR A LINEAR FUNCTIONAL DIFFERENTIAL EQUATION." Tambov University Reports. Series: Natural and Technical Sciences 21, no. 1 (2016): 76–81. http://dx.doi.org/10.20310/1810-0198-2016-21-1-76-81.

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10

Zhang, Zhehao. "A New Fractional Poisson Process Governed by a Recursive Fractional Differential Equation." Fractal and Fractional 6, no. 8 (July 29, 2022): 418. http://dx.doi.org/10.3390/fractalfract6080418.

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This paper proposes a new fractional Poisson process through a recursive fractional differential governing equation. Unlike the homogeneous Poison process, the Caputo derivative on the probability distribution of k jumps with respect to time is linked to all probability distribution functions of j jumps, where j is a non-negative integer less than or equal to k. The distribution functions of arrival times are derived, while the inter-arrival times are no longer independent and identically distributed. Further, this new fractional Poisson process can be interpreted as a homogeneous Poisson process whose natural time flow has been randomized, and the underlying time randomizing process has been studied. Finally, the conditional distribution of the kth order statistic from random number samples, counted by this fractional Poisson process, is also discussed.

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11

JIN, SHI, XIAOMEI LIAO, and XU YANG. "THE VLASOV–POISSON EQUATIONS AS THE SEMICLASSICAL LIMIT OF THE SCHRÖDINGER–POISSON EQUATIONS: A NUMERICAL STUDY." Journal of Hyperbolic Differential Equations 05, no. 03 (September 2008): 569–87. http://dx.doi.org/10.1142/s021989160800160x.

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In this paper, we numerically study the semiclassical limit of the Schrödinger–Poisson equations as a selection principle for the weak solution of the Vlasov–Poisson in one space dimension. Our numerical results show that this limit gives the weak solution that agrees with the zero diffusion limit of the Fokker–Planck equation. We also numerically justify the multivalued solution given by a moment system of the Vlasov–Poisson equations as the semiclassical limit of the Schrödinger–Poisson equations.

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12

Kremer, Gilberto Medeiros. "Post-Newtonian Jeans Equation for Stationary and Spherically Symmetrical Self-Gravitating Systems." Universe 8, no. 3 (March 13, 2022): 179. http://dx.doi.org/10.3390/universe8030179.

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The post-Newtonian Jeans equation for stationary self-gravitating systems is derived from the post-Newtonian Boltzmann equation in spherical coordinates. The Jeans equation is coupled with the three Poisson equations from the post-Newtonian theory. The Poisson equations are functions of the energy-momentum tensor components which are determined from the post-Newtonian Maxwell–Jüttner distribution function. As an application, the effect of a central massive black hole on the velocity dispersion profile of the host galaxy is investigated and the influence of the post-Newtonian corrections are determined.

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13

Li, Zhen, Zuoqiang Shi, and Jian Sun. "Point Integral Method for Solving Poisson-Type Equations on Manifolds from Point Clouds with Convergence Guarantees." Communications in Computational Physics 22, no. 1 (May 3, 2017): 228–58. http://dx.doi.org/10.4208/cicp.111015.250716a.

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AbstractPartial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Due to the complicated geometrical structure of the manifold, it is difficult to get efficient numerical method to solve PDE on manifold. In the paper, we propose a method called point integral method (PIM) to solve the Poisson-type equations from point clouds. Among different kinds of PDEs, the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most important. In PIM, the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown function. This feature makes the integral equation easy to be discretized from point cloud. In the paper, we explain the derivation of the integral equations, describe the point integral method and its implementation, and present the numerical experiments to demonstrate the convergence of PIM.

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14

Lu, Lin, Xiaokai He, and Xing Zhou. "The Period Function of the Generalized Sine-Gordon Equation and the Sinh-Poisson Equation." Mathematics 12, no. 16 (August 10, 2024): 2474. http://dx.doi.org/10.3390/math12162474.

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In this paper, we consider the generalized sine-Gordon equation ψtx=(1+a∂x2)sinψ and the sinh-Poisson equation uxx+uyy+σsinhu=0, where a is a real parameter, and σ is a positive parameter. Under different conditions, e.g., a=0, a≠0, and σ>0, the periods of the periodic wave solutions for the above two equations are discussed. By the transformation of variables, the generalized sine-Gordon equation and sinh-Poisson equations are reduced to planar dynamical systems whose first integral includes trigonometric terms and exponential terms, respectively. We successfully handle the trigonometric terms and exponential terms in the study of the monotonicity of the period function of periodic solutions.

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15

Marian, Daniela, Sorina Anamaria Ciplea, and Nicolaie Lungu. "Hyers–Ulam Stability of Order k for Euler Equation and Euler–Poisson Equation in the Calculus of Variations." Mathematics 10, no. 15 (July 22, 2022): 2556. http://dx.doi.org/10.3390/math10152556.

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In this paper, we define and study Hyers–Ulam stability of order 1 for Euler’s equation and Hyers–Ulam stability of order m−1 for the Euler–Poisson equation in the calculus of variations in two special cases, when these equations have the form y″(x)=f(x) and y(m)(x)=f(x), respectively. We prove some estimations for Jyx−Jy0x, where y is an approximate solution and y0 is an exact solution of the corresponding Euler and Euler-Poisson equations, respectively. We also give two examples.

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16

Krzywicki, A., and T. Nadzieja. "Poisson-Boltzmann equation in ℝ³." Annales Polonici Mathematici 54, no. 2 (1991): 125–34. http://dx.doi.org/10.4064/ap-54-2-125-134.

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17

LAMPERT, MURRAY. "The nonlinear Poisson–Boltzmann equation." Nature 315, no. 6015 (May 1985): 159. http://dx.doi.org/10.1038/315159a0.

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18

Krowne, Clifford M. "Semiconductor heterostructure nonlinear Poisson equation." Journal of Applied Physics 65, no. 4 (February 15, 1989): 1602–14. http://dx.doi.org/10.1063/1.342952.

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19

Woelki, Stefan, and Hans-Helmut Kohler. "A modified Poisson–Boltzmann equation." Chemical Physics 261, no. 3 (November 2000): 411–19. http://dx.doi.org/10.1016/s0301-0104(00)00277-9.

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20

Woelki, Stefan, and Hans-Helmut Kohler. "A modified Poisson–Boltzmann equation." Chemical Physics 261, no. 3 (November 2000): 421–38. http://dx.doi.org/10.1016/s0301-0104(00)00278-0.

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21

Mehrazin, Hashem. "Laplace equation and Poisson integral." Applied Mathematics and Computation 145, no. 2-3 (December 2003): 451–63. http://dx.doi.org/10.1016/s0096-3003(02)00499-x.

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22

Mauser, N. J. "The Schrödinger-Poisson-Xα equation." Applied Mathematics Letters 14, no. 6 (August 2001): 759–63. http://dx.doi.org/10.1016/s0893-9659(01)80038-0.

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23

Munteanu, Ovidiu, Chiung-Jue Anna Sung, and Jiaping Wang. "Poisson equation on complete manifolds." Advances in Mathematics 348 (May 2019): 81–145. http://dx.doi.org/10.1016/j.aim.2019.03.019.

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24

Ghasemi Moghaddam, Razieh, and Tofigh Allahviranloo. "On the fuzzy Poisson equation." Fuzzy Sets and Systems 347 (September 2018): 105–28. http://dx.doi.org/10.1016/j.fss.2017.12.013.

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25

Grunert, K., and Khai T. Nguyen. "On the Burgers–Poisson equation." Journal of Differential Equations 261, no. 6 (September 2016): 3220–46. http://dx.doi.org/10.1016/j.jde.2016.05.028.

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26

Parida, Mamata, and Sudarsan Padhy. "Electro-osmotic flow of a third-grade fluid past a channel having stretching walls." Nonlinear Engineering 8, no. 1 (January 28, 2019): 56–64. http://dx.doi.org/10.1515/nleng-2017-0112.

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Abstract The electro-osmotic flow of a third grade fluid past a channel having stretching walls has been studied in this paper. The channel height is taken much greater than the thickness of the electric double layer comprising of the Stern and diffuse layers. The equations governing the flow are obtained from continuity equation, the Cauchy’s momentum equation and the Poisson-Boltzmann equation. The Debye-Hückel approximation is adopted to linearize the Poisson-Boltzmann equation. Suitable similarity transformations are used to reduce the resulting non-linear partial differential equation to ordinary differential equation. The reduced equation is solved numerically using damped Newton’s method. The results computed are presented in form of graphs.

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27

KC, Gokul, and Ram Prasad Dulal. "Adaptive Finite Element Method for Solving Poisson Partial Differential Equation." Journal of Nepal Mathematical Society 4, no. 1 (May 14, 2021): 1–18. http://dx.doi.org/10.3126/jnms.v4i1.37107.

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Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adaptive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically.

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28

Nathiya, N., and C. Amulya Smyrna. "Infinite Schrödinger networks." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 4 (December 2021): 640–50. http://dx.doi.org/10.35634/vm210408.

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Finite-difference models of partial differential equations such as Laplace or Poisson equations lead to a finite network. A discretized equation on an unbounded plane or space results in an infinite network. In an infinite network, Schrödinger operator (perturbed Laplace operator, $q$-Laplace) is defined to develop a discrete potential theory which has a model in the Schrödinger equation in the Euclidean spaces. The relation between Laplace operator $\Delta$-theory and the $\Delta_q$-theory is investigated. In the $\Delta_q$-theory the Poisson equation is solved if the network is a tree and a canonical representation for non-negative $q$-superharmonic functions is obtained in general case.

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29

BONILLA, LUIS L., and JUAN S. SOLER. "HIGH-FIELD LIMIT OF THE VLASOV–POISSON–FOKKER–PLANCK SYSTEM: A COMPARISON OF DIFFERENT PERTURBATION METHODS." Mathematical Models and Methods in Applied Sciences 11, no. 08 (November 2001): 1457–68. http://dx.doi.org/10.1142/s0218202501001410.

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A reduced drift-diffusion (Smoluchowski–Poisson) equation is found for the electric charge in the high-field limit of the Vlasov–Poisson–Fokker–Planck system, both in one and three dimensions. The corresponding electric field satisfies a Burgers equation. Three methods are compared in the one-dimensional case: Hilbert expansion, Chapman–Enskog procedure and closure of the hierarchy of equations for the moments of the probability density. Of these methods, only the Chapman–Enskog method is able to systematically yield reduced equations containing terms of different order.

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30

Wu, Zhen. "Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration." Journal of the Australian Mathematical Society 74, no. 2 (April 2003): 249–66. http://dx.doi.org/10.1017/s1446788700003281.

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AbstractWe first give the existence and uniqueness result and a comparison theorem for backward stochastic differential equations with Brownian motion and Poisson process as the noise source in stopping time (unbounded) duration. Then we obtain the existence and uniqueness result for fully coupled forward-backward stochastic differential equation with Brownian motion and Poisson process in stopping time (unbounded) duration. We also proved a comparison theorem for this kind of equation.

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31

Abdallah, S., and C. F. Smith. "Three-Dimensional Solutions for Inviscid Incompressible Flow in Turbomachines." Journal of Turbomachinery 112, no. 3 (July 1, 1990): 391–98. http://dx.doi.org/10.1115/1.2927672.

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A primitive variable formulation is used for the solution of the incompressible Euler equation. In particular, the pressure Poisson equation approach using a nonstaggered grid is considered. In this approach, the velocity field is calculated from the unsteady momentum equation by marching in time. The continuity equation is replaced by a Poisson-type equation for the pressure with Neumann boundary conditions. A consistent finite-difference method, which insures the satisfaction of a compatibility condition necessary for convergence, is used in the solution of the pressure equation on a nonstaggered grid. Numerical solutions of the momentum equations are obtained using the second-order upwind differencing scheme, while the pressure Poisson equation is solved using the line successive overrelaxation method. Three turbo-machinery rotors are tested to validate the numerical procedure. The three rotor blades have been designed to have similar loading distributions but different amounts of dihedral. Numerical solutions are obtained and compared with experimental data in terms of the velocity components and exit swirl angles. The computed results are in good agreement with the experimental data.

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32

van Driel, M., J. Kemper, and C. Boehm. "On the modelling of self-gravitation for full 3-D global seismic wave propagation." Geophysical Journal International 227, no. 1 (June 21, 2021): 632–43. http://dx.doi.org/10.1093/gji/ggab237.

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SUMMARY We present a new approach to the solution of the Poisson equation present in the coupled gravito-elastic equations of motion for global seismic wave propagation in time domain aiming at the inclusion of the full gravitational response into spectral element solvers. We leverage the Salvus meshing software to include the external domain using adaptive mesh refinement and high order shape mapping. Together with Neumann boundary conditions based on a multipole expansion of the right-hand side this minimizes the number of additional elements needed. Initial conditions for the iterative solution of the Poisson equation based on temporal extrapolation from previous time steps together with a polynomial multigrid method reduce the number of iterations needed for convergence. In summary, this approach reduces the extra cost for simulating full gravity to a similar order as the elastic forces. We demonstrate the efficacy of the proposed method using the displacement from an elastic global wave propagation simulation (decoupled from the Poisson equation) at $200\, \mbox{s}$ dominant period to compute a realistic right-hand side for the Poisson equation.

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33

Khairi, Fathul, and Malahayati. "Penerapan Fungsi Green dari Persamaan Poisson pada Elektrostatika." Quadratic: Journal of Innovation and Technology in Mathematics and Mathematics Education 1, no. 1 (April 30, 2021): 56–79. http://dx.doi.org/10.14421/quadratic.2021.011-08.

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The Dirac delta function is a function that mathematically does not meet the criteria as a function, this is because the function has an infinite value at a point. However, in physics the Dirac Delta function is an important construction, one of which is in constructing the Green function. This research constructs the Green function by utilizing the Dirac Delta function and Green identity. Furthermore, the construction is directed at the Green function of the Poisson's equation which is equipped with the Dirichlet boundary condition. After the form of the Green function solution from the Poisson's equation is obtained, the Green function is determined by means of the expansion of the eigen functions in the Poisson's equation. These results are used to analyze the application of the Poisson equation in electrostatic.

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34

Johannessen, Kim. "A Nonlinear Differential Equation Related to the Jacobi Elliptic Functions." International Journal of Differential Equations 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/412569.

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A nonlinear differential equation for the polar angle of a point of an ellipse is derived. The solution of this differential equation can be expressed in terms of the Jacobi elliptic function dn(u,k). If the polar angle is extended to the complex plane, the Jacobi imaginary transformation properties and the dependence on the real and complex quarter periods can be described. From the differential equation of the polar angle, exact solutions of the Poisson Boltzmann and the sinh-Poisson equations are found in terms of the Jacobi elliptic functions.

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35

HEJAZI, S. REZA, AZADEH NADERIFARD, SOLEIMAN HOSSEINPOUR, and ELHAM DASTRANJ. "Symmetries, Noether’s Theorem, Conservation Laws and Numerical Simulation for Space-Space-Fractional Generalized Poisson Equation." Kragujevac Journal of Mathematics 47, no. 5 (2023): 713–25. http://dx.doi.org/10.46793/kgjmat2305.713h.

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In the present paper Lie theory of diﬀerential equations is expanded for ﬁnding symmetry geometric vector ﬁelds of Poisson equation. Similarity variables extracted from symmetries are applied in order to ﬁnd reduced forms of the considered equation by using Erdélyi-Kober operator. Conservation laws of the space-space-fractional generalized Poisson equation with the Riemann-Liouville derivative are investigated via Noether’s method. By means of the concept of non-linear self-adjointness, Noether’s operators, formal Lagrangians and conserved vectors are computed. A collocation technique is also applied to give a numerical simulation of the problem.

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36

Kuzmych, V. A., M. A. Novotarskyi, and O. B. Nesterenko. "SOLVING POISSON EQUATION WITH CONVOLUTIONAL NEURAL NETWORKS." Radio Electronics, Computer Science, Control, no. 1 (April 4, 2022): 48. http://dx.doi.org/10.15588/1607-3274-2022-1-6.

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Context. The Poisson equation is the one of fundamental differential equations, which used to simulate complex physical processes, such as fluid motion, heat transfer problems, electrodynamics, etc. Existing methods for solving boundary value problems based on the Poisson equation require an increase in computational time to achieve high accuracy. The proposed method allows solving the boundary value problem with significant acceleration under the condition of acceptable loss of accuracy. Objective. The aim of our work is to develop artificial neural network architecture for solving a boundary value problem based on the Poisson equation with arbitrary Dirichlet and Neumann boundary conditions. Method. The method of solving boundary value problems based on the Poisson equation using convolutional neural network is proposed. The network architecture, structure of input and output data are developed. In addition, the method of training dataset generation is described. Results. The performance of the developed artificial neural network is compared with the performance of the numerical finite difference method for solving the boundary value problem. The results showed an acceleration of the computational speed in x10–700 times depending on the number of sampling nodes. Conclusions. The proposed method significantly accelerated speed of solving a boundary value problem based on the Poisson equation in comparison with the numerical method. In addition, the developed approach to the design of neural network architecture allows to improve the proposed method to achieve higher accuracy in modeling the process of pressure distribution in areas of arbitrary size.

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37

SALORT, DELPHINE. "TRANSPORT EQUATIONS WITH UNBOUNDED FORCE FIELDS AND APPLICATION TO THE VLASOV–POISSON EQUATION." Mathematical Models and Methods in Applied Sciences 19, no. 02 (February 2009): 199–228. http://dx.doi.org/10.1142/s0218202509003401.

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The aim of this paper is to give new dispersive tools for certain kinetic equations. As an application, we study the three-dimensional Vlasov–Poisson equation for initial data having strictly less than six moments in [Formula: see text] where the nonlinear term E is a priori unbounded. We prove via new dispersive effects that in fact the force field E is smooth in space at the cost of a localization in a ball and an averaging in time. We deduce new conditions to bound the density ρ in L∞ and to have existence and uniqueness of global weak solutions of the Vlasov–Poisson equation with bounded density for initial data strictly less than six moments in [Formula: see text]. The proof is based on a new approach which consists in establishing a priori dispersion estimates (moment effects) on the one hand for linear transport equations with unbounded force fields and on the other hand along the trajectories of the Vlasov–Poisson equation.

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38

Sundaram, Arunachalam. "Numerical Approximation of Poisson Equation Using the Finite Difference Method." International Journal for Research in Applied Science and Engineering Technology 10, no. 10 (October 31, 2022): 1513–18. http://dx.doi.org/10.22214/ijraset.2022.47247.

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Abstract: In this paper, we implement the Finite Difference Method to approximate the homogeneous form of the Poisson equation. The Poisson equation is discretized using the central difference approximation of the second derivative and the grid function is determined by the five point method approximates the exact solution of the Poisson equation. The finite difference approximation is consistent and convergent. The method of solving the numerical approximation of Poisson equation is implemented using Python Programming

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39

Odesskii, Alexander, Vladimir Rubtsov, and Vladimir Sokolov. "Parameter-dependent associative Yang–Baxter equations and Poisson brackets." International Journal of Geometric Methods in Modern Physics 11, no. 09 (October 2014): 1460036. http://dx.doi.org/10.1142/s0219887814600366.

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We discuss associative analogues of classical Yang–Baxter equation (CYBE) meromorphically dependent on parameters. We discover that such equations enter in a description of a general class of parameter-dependent Poisson structures and double Lie and Poisson structures in sense of Van den Bergh. We propose a classification of all solutions for one-dimensional associative Yang–Baxter equations (AYBE).

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40

Abdallah, S., and H. G. Smith. "Computation of Inviscid Incompressible Flow Using the Primitive Variable Formulation." Journal of Turbomachinery 108, no. 1 (July 1, 1986): 68–75. http://dx.doi.org/10.1115/1.3262026.

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The primitive variable formulation originally developed for the incompressible Navier–Stokes equations is applied for the solution of the incompressible Euler equations. The unsteady momentum equation is solved for the velocity field and the continuity equation is satisfied indirectly in a Poisson-type equation for the pressure (divergence of the momentum equation). Solutions for the pressure Poisson equation with derivative boundary conditions exist only if a compatibility condition is satisfied (Green’s theorem). This condition is not automatically satisfied on nonstaggered grids. A new method for the solution of the pressure equation with derivative boundary conditions on a nonstaggered grid [25] is used here for the calculation of the pressure. Three-dimensional solutions for the inviscid rotational flow in a 90 deg curved duct are obtained on a very fine mesh (17 × 17 × 29). The use of a fine grid mesh allows for the accurate prediction of the development of the secondary flow. The computed results are in good agreement with the experimental data of Joy [15].

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Xu, Qiuyan, and Zhiyong Liu. "Alternating Asymmetric Iterative Algorithm Based on Domain Decomposition for 3D Poisson Problem." Mathematics 8, no. 2 (February 19, 2020): 281. http://dx.doi.org/10.3390/math8020281.

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Poisson equation is a widely used partial differential equation. It is very important to study its numerical solution. Based on the strategy of domain decomposition, the alternating asymmetric iterative algorithm for 3D Poisson equation is provided. The solution domain is divided into several sub-domains, and eight asymmetric iterative schemes with the relaxation factor for 3D Poisson equation are constructed. When the numbers of iteration are odd or even, the computational process of the presented iterative algorithm are proposed respectively. In the calculation of the inner interfaces, the group explicit method is used, which makes the algorithm to be performed fast and in parallel, and avoids the difficulty of solving large-scale linear equations. Furthermore, the convergence of the algorithm is analyzed theoretically. Finally, by comparing with the numerical experimental results of Jacobi and Gauss Seidel iterative algorithms, it is shown that the alternating asymmetric iterative algorithm based on domain decomposition has shorter computation time, fewer iteration numbers and good parallelism.

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Ndayisenga, Serge, Leonid A. Sevastianov, and Konstantin P. Lovetskiy. "Finite-difference methods for solving 1D Poisson problem." Discrete and Continuous Models and Applied Computational Science 30, no. 1 (February 25, 2022): 62–78. http://dx.doi.org/10.22363/2658-4670-2022-30-1-62-78.

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The paper discusses the formulation and analysis of methods for solving the one-dimensional Poisson equation based on finite-difference approximations - an important and very useful tool for the numerical study of differential equations. In fact, this is a classical approximation method based on the expansion of the solution in a Taylor series, based on which the recent progress of theoretical and practical studies allowed increasing the accuracy, stability, and convergence of methods for solving differential equations. Some of the features of this analysis include interesting extensions to classical numerical analysis of initial and boundary value problems. In the first part, a numerical method for solving the one-dimensional Poisson equation is presented, which reduces to solving a system of linear algebraic equations (SLAE) with a banded symmetric positive definite matrix. The well-known tridiagonal matrix algorithm, also known as the Thomas algorithm, is used to solve the SLAEs. The second part presents a solution method based on an analytical representation of the exact inverse matrix of a discretized version of the Poisson equation. Expressions for inverse matrices essentially depend on the types of boundary conditions in the original setting. Variants of inverse matrices for the Poisson equation with different boundary conditions at the ends of the interval under study are presented - the Dirichlet conditions at both ends of the interval, the Dirichlet conditions at one of the ends and Neumann conditions at the other. In all three cases, the coefficients of the inverse matrices are easily found and the algorithm for solving the problem is practically reduced to multiplying the matrix by the vector of the right-hand side.

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Riazi, N., and M. Mohammadi. "Composite Lane-Emden Equation as a Nonlinear Poisson Equation." International Journal of Theoretical Physics 51, no. 4 (November 9, 2011): 1276–83. http://dx.doi.org/10.1007/s10773-011-1003-8.

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Tsai, Miao-Yu, Chia-Ni Sun, and Chao-Chun Lin. "Concordance correlation coefficients estimated by modified variance components and generalized estimating equations for longitudinal overdispersed Poisson data." Statistical Methods in Medical Research 31, no. 2 (December 20, 2021): 267–86. http://dx.doi.org/10.1177/09622802211065156.

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For longitudinal overdispersed Poisson data sets, estimators of the intra-, inter-, and total concordance correlation coefficient through variance components have been proposed. However, biased estimators of quadratic forms are used in concordance correlation coefficient estimation. In addition, the generalized estimating equations approach has been used in estimating agreement for longitudinal normal data and not for longitudinal overdispersed Poisson data. Therefore, this paper proposes a modified variance component approach to develop the unbiased estimators of the concordance correlation coefficient for longitudinal overdispersed Poisson data. Further, the indices of intra-, inter-, and total agreement through generalized estimating equations are also developed considering the correlation structure of longitudinal count repeated measurements. Simulation studies are conducted to compare the performance of the modified variance component and generalized estimating equation approaches for longitudinal Poisson and overdispersed Poisson data sets. An application of corticospinal diffusion tensor tractography study is used for illustration. In conclusion, the modified variance component approach performs outstandingly well with small mean square errors and nominal 95% coverage rates. The generalized estimating equation approach provides in model assumption flexibility of correlation structures for repeated measurements to produce satisfactory concordance correlation coefficient estimation results.

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Mendoza, M., S. Succi, and H. J. Herrmann. "High-order kinetic relaxation schemes as high-accuracy Poisson solvers." International Journal of Modern Physics C 26, no. 05 (March 25, 2015): 1550055. http://dx.doi.org/10.1142/s0129183115500552.

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We present a new approach to find accurate solutions to the Poisson equation, as obtained from the steady-state limit of a diffusion equation with strong source terms. For this purpose, we start from Boltzmann's kinetic theory and investigate the influence of higher-order terms on the resulting macroscopic equations. By performing an appropriate expansion of the equilibrium distribution, we provide a method to remove the unnecessary terms up to a desired order and show that it is possible to find, with high level of accuracy, the steady-state solution of the diffusion equation for sizeable Knudsen numbers. In order to test our kinetic approach, we discretize the Boltzmann equation and solve the Poisson equation, spending up to six order of magnitude less computational time for a given precision than standard lattice Boltzmann methods (LBMs).

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ZHONG, YONGMIN, BIJAN SHIRINZADEH, GURSEL ALICI, and JULIAN SMITH. "HAPTIC DEFORMATION SIMULATION WITH POISSON EQUATION." International Journal of Image and Graphics 06, no. 03 (July 2006): 445–73. http://dx.doi.org/10.1142/s0219467806002355.

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This paper presents a new methodology for the deformation of soft objects by drawing an analogy between Poisson equation and elastic deformation. The potential energy stored in an elastic body as a result of a deformation caused by an external force is propagated among mass points by Poisson equation. An improved Poisson model is developed for propagating the energy generated by the external force in a natural manner. A cellular neural network (CNN) model is established to solve the Poisson model for the real-time requirement of soft object deformation. A haptic virtual reality system has been established for deformation simulation with force feedback. This proposed methodology not only deals with local and large-range deformations, but also accommodates isotropic, anisotropic and inhomogeneous materials by simply modifying constitutive coefficients, as well as predicts the mechanical behaviors of living tissues.

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Nós, Rudimar Luiz, and João Pedro Santos Brito Micheletti. "Solução numérica da equação de poisson 2d e 3d em malhas estruturadas." ForScience 9, no. 2 (February 21, 2022): e01091. http://dx.doi.org/10.29069/forscience.2021v9n2.e1091.

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Resumo Apresentamos neste trabalho a solução numérica de algumas equações de Poisson, uma equação diferencial parcial elíptica de segunda ordem, em malhas estruturadas bidimensionais e tridimensionais. Na determinação da solução numérica, empregamos o método iterativo SOR para solucionar o sistema de equações lineares proveniente da discretização da equação de Poisson por intermédio do método de diferenças finitas. Além disso, construímos algumas soluções manufaturadas 2D e 3D para a equação de Poisson, testamos valores ótimos para o parâmetro de sobrerrelaxação no método SOR e analisamos o comportamento dos métodos empregados na solução numérica de problemas 2D com singularidades. Na visualização das soluções manufaturadas e numéricas 2D e 3D, utilizamos, respectivamente, o Matlab e o Tecplot 360. Concluímos que a convergência do método SOR é lenta em problemas com condições de contorno de Neumann e em problemas com singularidades fortes. Palavras-chave: Método de diferenças finitas. Método SOR. Soluções manufaturadas. Abstract Numerical solution of 2d and 3d poisson equation in structured meshes We present in this work the numerical solution of some Poisson equations, an elliptic partial differential equation of second order, in two-dimensional and three-dimensional structured meshes. In determining the numerical solution, we used the iterative SOR method to solve the system of linear equations arising from the discretization of the Poisson equation using the finite difference method. Furthermore, we build some 2D and 3D manufactured solutions for the Poisson equation, and test optimal values for the over-relaxation parameter in the SOR method and analyze the behavior of the methods used in the numerical solution of 2D problems with singularities. In the visualization of the 2D and 3D manufactured and numerical solutions, we used, respectively, Matlab and Tecplot 360. We concluded that the convergence of the SOR method is slow in problems with Neumann boundary conditions and in problems with strong singularities. Keywords: Finite difference method. SOR method. Manufactured solutions.

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Gilmore, Steven, and Khai T. Nguyen. "SBV regularity for Burgers-Poisson equation." Journal of Mathematical Analysis and Applications 500, no. 1 (August 2021): 125095. http://dx.doi.org/10.1016/j.jmaa.2021.125095.

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Jebari, Khalid, Mohammed Madiafi, and Abdelaziz El Moujahid. "Solving Poisson Equation by Genetic Algorithms." International Journal of Computer Applications 83, no. 5 (December 18, 2013): 1–6. http://dx.doi.org/10.5120/14441-2597.

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Shatnawi, Rouba. "Atomic Solution of Poisson Type Equation." European Journal of Pure and Applied Mathematics 15, no. 2 (April 30, 2022): 397–402. http://dx.doi.org/10.29020/nybg.ejpam.v15i2.4318.

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