Journal articles: 'Poisson's equation Numerical solutions'

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ATKINSON, K. E. "The Numerical Evaluation of Particular Solutions for Poisson's Equation." IMA Journal of Numerical Analysis 5, no. 3 (1985): 319–38. http://dx.doi.org/10.1093/imanum/5.3.319.

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Sauer-Budge, A. M., J. Bonet, A. Huerta, and J. Peraire. "Computing Bounds for Linear Functionals of Exact Weak Solutions to Poisson's Equation." SIAM Journal on Numerical Analysis 42, no. 4 (January 2004): 1610–30. http://dx.doi.org/10.1137/s0036142903425045.

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Evans, D. J. "The numerical solution of poisson's equation in a rhombus." International Journal of Computer Mathematics 42, no. 3-4 (January 1992): 193–211. http://dx.doi.org/10.1080/00207169208804062.

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CHEN, C. S., C. S. HUANG, and K. H. LIN. "ON THE CONVERGENCE OF THE MFS–MPS SCHEME FOR 1D POISSON'S EQUATION." International Journal of Computational Methods 10, no. 02 (March 2013): 1341006. http://dx.doi.org/10.1142/s0219876213410065.

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The method of fundamental solutions (MFS) has been an effective meshless method for solving homogeneous partial differential equations. Coupled with radial basis functions (RBFs), the MFS has been extended to solve the inhomogeneous problems through the evaluation of the approximate particular solution and homogeneous solution. In this paper, we prove the the approximate solution of the above numerical process for solving 1D Poisson's equation converges in the sense of Lagrange interpolating polynomial using the result of Driscoll and Fornberg [2002].

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Ma, Zu-Hui, Weng Cho Chew, and Li Jun Jiang. "A Novel Efficient Numerical Solution of Poisson's Equation for Arbitrary Shapes in Two Dimensions." Communications in Computational Physics 20, no. 5 (November 2016): 1381–404. http://dx.doi.org/10.4208/cicp.230813.291113a.

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AbstractEven though there are various fast methods and preconditioning techniques available for the simulation of Poisson problems, little work has been done for solving Poisson's equation by using the Helmholtz decomposition scheme. To bridge this issue, we propose a novel efficient algorithm to solve Poisson's equation in irregular two dimensional domains for electrostatics through a quasi-Helmholtz decomposition technique—the loop-tree basis decomposition. It can handle Dirichlet, Neumann or mixed boundary problems in which the filling media can be homogeneous or inhomogeneous. A novel point of this method is to first find the electric flux efficiently by applying the loop-tree basis functions. Subsequently, the potential is obtained by finding the inverse of the gradient operator. Furthermore, treatments for both Dirichlet and Neumann boundary conditions are addressed. Finally, the validation and efficiency are illustrated by several numerical examples. Through these simulations, it is observed that the computational complexity of our proposed method almost scales as , where N is the triangle patch number of meshes. Consequently, this new algorithm is a feasible fast Poisson solver.

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Matsuura, T., S. Saitoh †, and D. D. Trong ‡. "Numerical solutions of the poisson equation." Applicable Analysis 83, no. 10 (October 2004): 1037–51. http://dx.doi.org/10.1080/00036810410001724616.

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Wu, C., and E. E. Kunhardt. "Numerical solution of Poisson's equation for rapidly varying driving functions." Journal of Computational Physics 84, no. 1 (September 1989): 247–54. http://dx.doi.org/10.1016/0021-9991(89)90190-3.

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Nakamura, Y., and I. Tsukabayashi. "Modified Korteweg—de Vries ion-acoustic solitons in a plasma." Journal of Plasma Physics 34, no. 3 (December 1985): 401–15. http://dx.doi.org/10.1017/s0022377800002968.

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Propagation of nonlinear ion-acoustic waves in a multi-component plasma with negative ions is investigated experimentally. At a critical concentration of negative ions, both compressive and rarefactive solitons are observed. The velocities and widths of the solitons are measured and compared with the soliton solutions of the modified Korteweg–de Vries equation and of the pseudopotential method. The modified Korteweg–de Vries equation is solved numerically to investigate overtaking collisions of a positive and a negative soliton. Fluid equations together with Poisson's equation are numerically integrated to simulate their head-on collisions.

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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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Caspi, S., M. Helm, and L. J. Laslett. "Numerical Solution of Boundary Condition to Poisson's Equation and Its Incorporation into the Program Poisson." IEEE Transactions on Nuclear Science 32, no. 5 (October 1985): 3722–24. http://dx.doi.org/10.1109/tns.1985.4334481.

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Burgos, R. B., and H. F. C. Peixoto. "SOLUTION OF 1D AND 2D POISSON'S EQUATION BY USING WAVELET SCALING FUNCTIONS." Revista de Engenharia Térmica 15, no. 2 (December 31, 2016): 68. http://dx.doi.org/10.5380/reterm.v15i2.62177.

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The use of multiresolution techniques and wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs). Therefore, the use of wavelet scaling functions as a basis in computational analysis holds some promise due to their compact support, orthogonality and localization properties. Daubechies and Deslauriers-Dubuc functions have been successfully used as basis functions in several schemes like the Wavelet- Galerkin Method (WGM) and the Wavelet Finite Element Method (WFEM). Another possible advantage of their use is the fact that the calculation of integrals of inner products of wavelet scaling functions and their derivatives can be made by solving a linear system of equations, thus avoiding the problem of using approximations by some numerical method. These inner products were defined as connection coefficients and they are employed in the calculation of stiffness matrices and load vectors. In this work, some mathematical foundations regarding wavelet scaling functions, their derivatives and connection coefficients are reviewed. A scheme based on the Galerkin Method is proposed for the direct solution of Poisson's equation (potential problems) in a meshless formulation using interpolating wavelet scaling functions (Interpolets). The applicability of the proposed method and some convergence issues are illustrated by means of a few examples.

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Mazarei, Mohammad Mehdi, and Azim Aminataei. "Numerical Solution of Poisson's Equation Using a Combination of Logarithmic and Multiquadric Radial Basis Function Networks." Journal of Applied Mathematics 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/286391.

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This paper presents numerical solution of elliptic partial differential equations (Poisson's equation) using a combination of logarithmic and multiquadric radial basis function networks. This method uses a special combination between logarithmic and multiquadric radial basis functions with a parameterr. Further, the condition number which arises in the process is discussed, and a comparison is made between them with our earlier studies and previously known ones. It is shown that the system is stable.

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Bialecki, Bernard. "Superconvergence of the orthogonal spline collocation solution of Poisson's equation." Numerical Methods for Partial Differential Equations 15, no. 3 (May 1999): 285–303. http://dx.doi.org/10.1002/(sici)1098-2426(199905)15:3<285::aid-num2>3.0.co;2-1.

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Стояновская, О. П., Н. В. Снытников, and В. Н. Снытников. "An algorithm for solving transient problems of gravitational gas dynamics: a combination of the SPH method with a grid method of gravitational potential computation." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 1 (April 2, 2015): 52–60. http://dx.doi.org/10.26089/nummet.v16r106.

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Для решения нестационарных уравнений гравитационной газовой динамики в приближении тонкого диска предложен новый численный алгоритм. Алгоритм основан на комбинации бессеточного метода сглаженных частиц (SPH, Smoothed Particle Hydrodynamics) и метода свeртки для решения уравнения Пуассона на декартовой сетке. Данный метод обладает высокой производительностью за счет того, что сеточная функция потенциала вычисляется и хранится только в плоскости диска. Работоспособность алгоритма демонстрируется в численных экспериментах по формированию структур в околозвездном диске. Сравнение результатов, полученных с использованием сеточных методов решения уравнения Пуассона в декартовой и цилиндрической геометрии, показало, что в обоих случаях удается воспроизвести решения с осевой симметрией и формирование уединенных областей повышенной плотности. A new numerical algorithm to solve the unsteady equations of gravitational gas dynamics in the thin disk approximation is proposed. This algorithm is based on a combination of the meshless SPH (Smoothed Particle Hydrodynamics) method for gas dynamics and the convolution method for solving Poisson's equation on a Cartesian grid. This convolution method is of high performance due to the fact that the grid potential function is computed and stored only in the plane of the disk. The efficiency of the algorithm is demonstrated by numerical experiments on the formation of structures in a circumstellar disk. We compare the results obtained by using the grid method for solving Poisson's equation in Cartesian and cylindrical geometry and show that in both these cases it is possible to reproduce the solutions with axial symmetry and to illustrate the formation of solitary regions of enhanced density.

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Arvas, E., R. I. Turkman, and P. S. Neelakantaswamy. "MOSFET analysis through numerical solution of Poisson's equation by the method of moments." Solid-State Electronics 30, no. 12 (December 1987): 1355–57. http://dx.doi.org/10.1016/0038-1101(87)90064-5.

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Shapiro, F. R. "The numerical solution of Poisson's equation in a pn diode using a spreadsheet." IEEE Transactions on Education 38, no. 4 (1995): 380–84. http://dx.doi.org/10.1109/13.473161.

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Jha, Navnit, Venu Gopal, and Bhagat Singh. "Geometric grid network and third-order compact scheme for solving nonlinear variable coefficients 3D elliptic PDEs." International Journal of Modeling, Simulation, and Scientific Computing 09, no. 06 (December 2018): 1850053. http://dx.doi.org/10.1142/s1793962318500538.

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By using nonuniform (geometric) grid network, a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type. Single cell discretization to the elliptic equation makes it easier to compute and exhibit stability of the numerical solutions. The monotone and irreducible property of the Jacobian matrix to the system of difference equations analyses the converging behavior of the numerical solution values. As an experiment, applications of the compact scheme to Schrödinger equations, sine-Gordon equations, elliptic Allen–Cahn equation and Poisson’s equation have been presented with root mean squared errors of exact and approximate solution values. The results corroborate the reliability and efficiency of the scheme.

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PALANICHAMY, VIMALA, and N. B. BALAMURUGAN. "ANALYTICAL MODELING OF DRAIN CURRENT, CAPACITANCE AND TRANSCONDUCTANCE IN SYMMETRIC DOUBLE-GATE MOSFETs CONSIDERING QUANTUM EFFECTS." International Journal of Nanoscience 12, no. 01 (February 2013): 1350005. http://dx.doi.org/10.1142/s0219581x13500051.

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An analytical model for double-gate (DG) MOSFETs considering quantum mechanical effects is proposed in this paper. Schrödinger and Poisson's equations are solved simultaneously using a variational approach. Solving the Poisson and Schrödinger equations simultaneously reveals quantum effects (QME) that influence the performance of DG MOSFETs. This model is developed to provide an analytical expression for inversion charge distribution function for all regions of device operation. This expression is used to calculate the other important parameters like inversion layer centroid, inversion charge, gate capacitance, drain current and transconductance. We systematically evaluate and analyze the parameters of DG MOSFETs considering QME. The analytical solutions are simple, accurate and provide good physical insight into the quantization caused by quantum confinement under various gate biases. The analytical results of this model are verified by comparing the data obtained with one-dimensional self-consistent numerical solutions of Poisson and Schrödinger equations known as SCHRED.

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Strigberger, Jack. "Numerical perturbation method for approximate solution of poisson's equation on a moderately deforming grid." International Journal for Numerical Methods in Fluids 9, no. 5 (May 1989): 599–607. http://dx.doi.org/10.1002/fld.1650090509.

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Verheest, F., T. Cattaert, E. Dubinin, K. Sauer, and J. F. McKenzie. "Whistler oscillitons revisited: the role of charge neutrality?" Nonlinear Processes in Geophysics 11, no. 4 (October 20, 2004): 447–52. http://dx.doi.org/10.5194/npg-11-447-2004.

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Abstract. When studying transverse modes propagating parallel to a static magnetic field, an apparent contradiction arises between the weakly nonlinear results obtained from the derivative nonlinear Schrödinger equation, predicting envelope solitons (where the amplitude is stationary in the wave frame, but the phase is not), and recent results for whistler oscillitons, indicating that really stationary structures of large amplitude are possible. Revisiting this problem in the fluid dynamic approach, care has been taken not to introduce charge neutrality from the outset, because this not only neglects electric stresses compared to magnetic stresses, which is reasonable, but could also imply from Poisson's equation a vanishing of the wave electric field. Nevertheless, the fixed points of the remaining equations are the same, whether charge neutrality is assumed from the outset or not, so that the solitary wave solutions at not too large amplitudes will be very similar. This is borne out by numerical simulations of the solutions under the two hypotheses, showing that the lack of correspondence with the DNLS envelope solitons indicates the limitations of the reductive perturbation approach, and is not a consequence of assuming charge neutrality.

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Yıldırım, Selçuk. "Exact and Numerical Solutions of Poisson Equation for Electrostatic Potential Problems." Mathematical Problems in Engineering 2008 (2008): 1–11. http://dx.doi.org/10.1155/2008/578723.

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Homotopy perturbation method (HPM) and boundary element method (BEM) for calculating the exact and numerical solutions of Poisson equation with appropriate boundary and initial conditions are presented. Exact solutions of electrostatic potential problems defined by Poisson equation are found using HPM given boundary and initial conditions. The same problems are also solved using the BEM. The cell integration approach is used for solving Poisson equation by BEM. The problem region containing the charge density is subdivided into triangular elements. In addition, this paper presents a numerical comparison with the HPM and BEM.

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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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Sim, Fabio M., Eka Budiarto, and Rusman Rusyadi. "Comparison and Analysis of Neural Solver Methods for Differential Equations in Physical Systems." ELKHA 13, no. 2 (October 22, 2021): 134. http://dx.doi.org/10.26418/elkha.v13i2.49097.

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Differential equations are ubiquitous in many fields of study, yet not all equations, whether ordinary or partial, can be solved analytically. Traditional numerical methods such as time-stepping schemes have been devised to approximate these solutions. With the advent of modern deep learning, neural networks have become a viable alternative to traditional numerical methods. By reformulating the problem as an optimisation task, neural networks can be trained in a semi-supervised learning fashion to approximate nonlinear solutions. In this paper, neural solvers are implemented in TensorFlow for a variety of differential equations, namely: linear and nonlinear ordinary differential equations of the first and second order; Poisson’s equation, the heat equation, and the inviscid Burgers’ equation. Different methods, such as the naive and ansatz formulations, are contrasted, and their overall performance is analysed. Experimental data is also used to validate the neural solutions on test cases, specifically: the spring-mass system and Gauss’s law for electric fields. The errors of the neural solvers against exact solutions are investigated and found to surpass traditional schemes in certain cases. Although neural solvers will not replace the computational speed offered by traditional schemes in the near future, they remain a feasible, easy-to-implement substitute when all else fails.

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CHIEN, C. S., H. T. HUANG, B. W. JENG, and Z. C. LI. "SUPERCONVERGENCE OF FEMS AND NUMERICAL CONTINUATION FOR PARAMETER-DEPENDENT PROBLEMS WITH FOLDS." International Journal of Bifurcation and Chaos 18, no. 05 (May 2008): 1321–36. http://dx.doi.org/10.1142/s0218127408021014.

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We study finite element approximations for positive solutions of semilinear elliptic eigenvalue problems with folds, and exploit the superconvergence of finite element methods (FEM). In order to apply the superconvergence of FEM for Poisson's equation in [Chen & Huang, 1995; Huang et al., 2004, 2006; Lin & Yan, 1996] to parameter-dependent problems with folds, this paper provides the framework of analysis, accompanied with the proof of the strong monotonicity of the nonlinear form. It is worthy to point out that the superconvergence of the nonlinear problem in this paper is different from that in [Chen & Huang, 1995]. A continuation algorithm is described to trace solution curves of semilinear elliptic eigenvalue problems, where the Adini elements are exploited to discretize the PDEs. Numerical results on some sample test problems with folds and bifurcations are reported.

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Kafle, J., L. P. Bagale, and D. J. K. C. "Numerical Solution of Parabolic Partial Differential Equation by Using Finite Difference Method." Journal of Nepal Physical Society 6, no. 2 (December 31, 2020): 57–65. http://dx.doi.org/10.3126/jnphyssoc.v6i2.34858.

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In the real world, many physical problems like heat equation, wave equation, Laplace equation and Poisson equation are modeled by partial differential equations (PDEs). A PDE of the form ut = α uxx, (α > 0) where x and t are independent variables and u is a dependent variable; is a one-dimensional heat equation. This is an example of a prototypical parabolic equation. The heat equation has analytic solution in regular shape domain. If the domain has irregular shape, computing analytic solution of such equations is difficult. In this case, we can use numerical methods to compute the solution of such PDEs. Finite difference method is one of the numerical methods that is used to compute the solutions of PDEs by discretizing the given domain into finite number of regions. Here, we derived the Forward Time Central Space Scheme (FTCSS) for this heat equation. We also computed its numerical solution by using FTCSS. We compared the analytic solution and numerical solution for different homogeneous materials (for different values of diffusivity α). There is instantaneous heat transfer and heat loss for the materials with higher diffusivity (α) as compared to the materials of lower diffusivity. Finally, we compared simulation results of different non-homogeneous materials.

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Mužík, Juraj, and Roman Bulko. "Free surface groundwater flow solution using boundary collocation methods." MATEC Web of Conferences 196 (2018): 03026. http://dx.doi.org/10.1051/matecconf/201819603026.

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In this paper, two meshless numerical algorithms are developed for the solution of two-dimensional steady-state diffusion equation that describes the stationary groundwater flow. The proposed numerical methods, which are truly meshless, quadrature-free and boundary only, are based on the method of fundamental solutions and singular boundary method respectively. The diffusion equation is transformed into a Poisson-type equation with a known fundamental solution. Numerical examples with moving boundary are presented and compared to the solutions obtained by the finite element method.

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Zimmermann, Tim, Massimo Pietroni, Javier Madroñero, Luca Amendola, and Sandro Wimberger. "A Quantum Model for the Dynamics of Cold Dark Matter." Condensed Matter 4, no. 4 (November 13, 2019): 89. http://dx.doi.org/10.3390/condmat4040089.

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A model for cold dark matter is given by the solution of a coupled Schrödinger–Poisson equation system. We present a numerical scheme for integrating these equations, discussing the problems arising from their nonlinear and nonlocal character. After introducing and testing our numerical approach, we illustrate key features of the system by numerical examples in 1 + 1 dimensions. In particular, we study the properties of asymptotic states to which the numerical solutions converge for artificial initial conditions.

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SHIBATA, Daisuke, and Takayuki UTSUMI. "611 Numerical solutions of Poisson equation by the CIP-BS method." Proceedings of The Computational Mechanics Conference 2008.21 (2008): 265–66. http://dx.doi.org/10.1299/jsmecmd.2008.21.265.

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Kazakova, A. O., and E. A. Mikishanina. "Numerical Modeling of Solutions of the Applied Problems for the Poisson’s Equation." IOP Conference Series: Materials Science and Engineering 1079, no. 4 (March 1, 2021): 042038. http://dx.doi.org/10.1088/1757-899x/1079/4/042038.

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Wang, Xiao, Juan Wang, Xin Wang, and Chujun Yu. "A Pseudo-Spectral Fourier Collocation Method for Inhomogeneous Elliptical Inclusions with Partial Differential Equations." Mathematics 10, no. 3 (January 19, 2022): 296. http://dx.doi.org/10.3390/math10030296.

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Inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern in many disciplines. In this paper, a pseudo-spectral collocation method, based on Fourier basis functions, is proposed for the numerical solutions of two- (2D) and three-dimensional (3D) inhomogeneous elliptic boundary value problems. We describe how one can improve the numerical accuracy by making some extra “reconstruction techniques” before applying the traditional Fourier series approximation. After the particular solutions have been obtained, the resulting homogeneous equation can then be calculated using various boundary-type methods, such as the method of fundamental solutions (MFS). Using Fourier basis functions, one does not need to use large matrices, making accrual computations relatively fast. Three benchmark numerical examples involving Poisson, Helmholtz, and modified-Helmholtz equations are presented to illustrate the applicability and accuracy of the proposed method.

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Mohanty, R. K., Rajive Kumar, and Vijay Dahiya. "Cubic Spline Iterative Method for Poisson’s Equation in Cylindrical Polar Coordinates." ISRN Mathematical Physics 2012 (February 12, 2012): 1–11. http://dx.doi.org/10.5402/2012/234516.

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Using nonpolynomial cubic spline approximation in x- and finite difference in y-direction, we discuss a numerical approximation of O(k2+h4) for the solutions of diffusion-convection equation, where k>0 and h>0 are grid sizes in y- and x-coordinates, respectively. We also extend our technique to polar coordinate system and obtain high-order numerical scheme for Poisson’s equation in cylindrical polar coordinates. Iterative method of the proposed method is discussed, and numerical examples are given in support of the theoretical results.

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Gray, J. L., and M. S. Lundstrom. "Numerical solution of Poisson's equation with application to C-V analysis of III-V heterojunction capacitors." IEEE Transactions on Electron Devices 32, no. 10 (October 1985): 2102–9. http://dx.doi.org/10.1109/t-ed.1985.22246.

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Miki, Kazuyoshi, and Toshiyuki Takagi. "Numerical solution of poisson's equation with arbitrarily shaped boundaries using a domain decomposition and overlapping technique." Journal of Computational Physics 67, no. 2 (December 1986): 263–78. http://dx.doi.org/10.1016/0021-9991(86)90262-7.

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Mohajel Sadeghi, Siamak, and Akbar Alibeigloo. "Parametric study of three-dimensional vibration of viscoelastic cylindrical shells on different boundary conditions." Journal of Vibration and Control 25, no. 19-20 (July 11, 2019): 2567–79. http://dx.doi.org/10.1177/1077546319861810.

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In this research based on theory of elasticity, free vibration behavior of a viscoelastic cylindrical shell with different boundary conditions is studied. A constitutive equation for viscoelastic material is assumed to obey the Boltzmann model and Poisson's ratio is held to be constant. Moreover, the Prony series is used to model time dependent modulus of elasticity. Governing equations of motions for simply-supported edges conditions are solved analytically using the state-space technique along the radial coordinate and the Fourier series method along the axial and circumferential directions. In the case of other edges condition a semi-analytical solution is employed by using the differential quadrature method instead of Fourier series solutions. It is worthy to note before solving the problem, that the Laplace transform is employed to convert governing differential equations from the time-domain into the Laplace domain. Then, validation of the present formulation is performed by comparing the numerical results with those published in the literature. Finally, effect of viscoelastic properties, boundary conditions, the thickness-to-radius ratio and length-to-radius ratio on the frequency behavior are studied.

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Averina, Tatyana A., and Konstantin A. Rybakov. "Using maximum cross section method for filtering jump-diffusion random processes." Russian Journal of Numerical Analysis and Mathematical Modelling 35, no. 2 (April 28, 2020): 55–67. http://dx.doi.org/10.1515/rnam-2020-0005.

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Abstract The paper is focused on problem of filtering random processes in dynamical systems whose mathematical models are described by stochastic differential equations with a Poisson component. The solution of a filtering problem supposes simulation of trajectories of solutions to a stochastic differential equation. The trajectory modelling procedure includes simulation of a Poisson flow permitting application of the maximum cross section method and its modification.

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Crouseilles, Nicolas, and Thomas Respaud. "A Charge Preserving Scheme for the Numerical Resolution of the Vlasov-Ampère Equations." Communications in Computational Physics 10, no. 4 (October 2011): 1001–26. http://dx.doi.org/10.4208/cicp.210410.211210a.

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AbstractIn this report, a charge preserving numerical resolution of the 1D Vlasov-Ampère equation is achieved, with a forward Semi-Lagrangian method introduced in [10]. The Vlasov equation belongs to the kinetic way of simulating plasmas evolution, and is coupled with the Poisson’s equation, or equivalently under charge conservation, the Ampère’s one, which self-consistently rules the electric field evolution. In order to ensure having proper physical solutions, it is necessary that the scheme preserves charge numerically. B-spline deposition will be used for the interpolation step. The solving of the characteristics will be made with a Runge-Kutta 2 method and with a Cauchy-Kovalevsky procedure.

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Zaman, Mohammad Asif. "Numerical Solution of the Poisson Equation Using Finite Difference Matrix Operators." Electronics 11, no. 15 (July 28, 2022): 2365. http://dx.doi.org/10.3390/electronics11152365.

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The Poisson equation frequently emerges in many fields of science and engineering. As exact solutions are rarely possible, numerical approaches are of great interest. Despite this, a succinct discussion of a systematic approach to constructing a flexible and general numerical Poisson solver can be difficult to find. In this introductory paper, a comprehensive discussion is presented on how to build a finite difference matrix solver that can solve the Poisson equation for arbitrary geometry and boundary conditions. The boundary conditions are implemented in a systematic way that enables easy modification of the solver for different problems. An image-based geometry-definition approach is also discussed. Python code of the numerical recipe is made publicly available. Numerical examples are presented that show how to set up the solver for different problems.

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Smith, Z. M., and S. K. Loyalka. "Numerical Solutions of the Poisson Equation: Condensation/Evaporation on Arbitrarily Shaped Aerosols." Nuclear Science and Engineering 176, no. 2 (February 2014): 154–66. http://dx.doi.org/10.13182/nse12-107.

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Yosibash, Z. "Numerical analysis on singular solutions of the Poisson equation in two-dimensions." Computational Mechanics 20, no. 4 (September 26, 1997): 320–30. http://dx.doi.org/10.1007/s004660050254.

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Gray, J. L., and M. S. Lundstrom. "A Numerical Solution of Poisson's Equation with Application to C-V Analysis of III-V Heterojunction Capacitors." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 4, no. 4 (October 1985): 546–53. http://dx.doi.org/10.1109/tcad.1985.1270156.

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Yin, Peimeng, Yunqing Huang, and Hailiang Liu. "An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation." Communications in Computational Physics 16, no. 2 (August 2014): 491–515. http://dx.doi.org/10.4208/cicp.270713.280214a.

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AbstractAn iterative discontinuous Galerkin (DG) method is proposed to solve the nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space. For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameterλ=(1), and a special initial guess forλ≫1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases ofλ=(1) andλ≪ 1. The (m+ 1)th order of accuracy forL2andmth order of accuracy forH1forPmelements are numerically obtained.

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Zhang, Xiang Dong, Lei Wang, and Da Wei Teng. "On the Treatment of Neumann Boundary Conditions in Collocation-Based Meshless Methods." Applied Mechanics and Materials 423-426 (September 2013): 1757–62. http://dx.doi.org/10.4028/www.scientific.net/amm.423-426.1757.

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The existence of Neumann boundary is a major cause of the poor accuracy and instability of collocation-based methods. Taking a Poisson equation with Neumann boundary condition as the model, the present paper studies the effects of two different radial point interpolation shape functions and their parameters on the accuracy of numerical solutions of the equation. We also study the effects of methods including fictious point method, nodes densification method and Hermite collocation method on the improvement of numerical accuracy. By comparison of analytic and numerical solutions computed using a program developed during research, we obtain parameters of shape functions and methods of treatment of Neumann boundary conditions that can be adopted to give better numerical accuracy.

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Oven, R. "Calculation of the space charge distribution in poled soda-lime glass." Journal of Physics: Condensed Matter 34, no. 5 (November 15, 2021): 055702. http://dx.doi.org/10.1088/1361-648x/ac3305.

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Abstract An analytical model of electric field assisted diffusion of ions into a multi-component glass is extended to calculate the space charge that forms between the poled layer and the potassium peak in a poled soda-lime glass. The model is compared with numerical solutions to the drift-diffusion equations and Poisson’s equation and shows good agreement. Some recent experimental results in corona poled soda-lime glass are also discussed using this model.

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Werpachowski, Roman. "On the solutions of generalized discrete Poisson equation." Linear Algebra and its Applications 430, no. 8-9 (April 2009): 1877–85. http://dx.doi.org/10.1016/j.laa.2008.09.043.

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Mirbozorgi, S. A., H. Niazmand, and M. Renksizbulut. "Electro-Osmotic Flow in Reservoir-Connected Flat Microchannels With Non-Uniform Zeta Potential." Journal of Fluids Engineering 128, no. 6 (March 24, 2006): 1133–43. http://dx.doi.org/10.1115/1.2353261.

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The effects of non-uniform zeta potentials on electro-osmotic flows in flat microchannels have been investigated with particular attention to reservoir effects. The governing equations, which consist of a Laplace equation for the distribution of external electric potential, a Poisson equation for the distribution of electric double layer potential, the Nernst-Planck equation for the distribution of charge density, and modified Navier-Stokes equations for the flow field are solved numerically for an incompressible steady flow of a Newtonian fluid using the finite-volume method. For the validation of the numerical scheme, the key features of an ideal electro-osmotic flow with uniform zeta potential have been compared with analytical solutions for the ionic concentration, electric potential, pressure, and velocity fields. When reservoirs are included in the analysis, an adverse pressure gradient is induced in the channel due to entrance and exit effects even when the reservoirs are at the same pressure. Non-uniform zeta potentials lead to complex flow fields, which are examined in detail.

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Chen, Ying-Ting, and Yang Cao. "A Coupled RBF Method for the Solution of Elastostatic Problems." Mathematical Problems in Engineering 2021 (January 22, 2021): 1–15. http://dx.doi.org/10.1155/2021/6623273.

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Radial basis function (RBF) has been widely used in many scientific computing and engineering applications, for instance, multidimensional scattered data interpolation and solving partial differential equations. However, the accuracy and stability of the RBF methods often strongly depend on the shape parameter. A coupled RBF (CRBF) method was proposed recently and successfully applied to solve the Poisson equation and the heat transfer equation (Appl. Math. Lett., 2019, 97: 93–98). Numerical results show that the CRBF method completely overcomes the troublesome issue of the optimal shape parameter that is a formidable obstacle to global schemes. In this paper, we further extend the CRBF method to solve the elastostatic problems. Discretization schemes are present in detail. With two elastostatic numerical examples, it is found that both numerical solutions of the CRBF method and the condition numbers of the discretized matrices are almost independent of the shape parameter. In addition, even if the traditional RBF methods take the optimal shape parameter, the CRBF method achieves better accuracy.

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Breden, Maxime, Claire Chainais-Hillairet, and Antoine Zurek. "Existence of traveling wave solutions for the Diffusion Poisson Coupled Model: a computer-assisted proof." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 4 (July 2021): 1669–97. http://dx.doi.org/10.1051/m2an/2021037.

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The Diffusion Poisson Coupled Model describes the evolution of a dense oxide layer appearing at the surface of carbon steel canisters in contact with a claystone formation. This model is a one dimensional free boundary problem involving drift-diffusion equations on the density of species (electrons, ferric cations and oxygen vacancies), coupled with a Poisson equation on the electrostatic potential and with moving boundary equations, which describe the evolution of the position of each unknown interfaces of the spatial domain. Numerical simulations suggest the existence of traveling wave solutions for this model. These solutions are defined by stationary profiles on a fixed size domain with interfaces moving both at the same velocity. In this paper, we present and apply a computer-assisted method in order to prove the existence of these traveling wave solutions. We also establish a precise and certified description of the solutions.

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Li, Zi-Cai, Hsin-Yun Hu, Song Wang, and Qing Fang. "Superconvergence of solution derivatives of the Shortley–Weller difference approximation to Poisson's equation with singularities on polygonal domains." Applied Numerical Mathematics 58, no. 5 (May 2008): 689–704. http://dx.doi.org/10.1016/j.apnum.2007.02.007.

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Khoie, R. "A self-consistent numerical method for simulation of quantum transport in high electron mobility transistor; part I: The Boltzmann-Poisson-Schrödinger solver." Mathematical Problems in Engineering 2, no. 3 (1996): 205–18. http://dx.doi.org/10.1155/s1024123x96000324.

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A self-consistent Boltzmann-Poisson-Schrödinger solver for High Electron Mobility Transistor is presented. The quantization of electrons in the quantum well normal to the heterojunction is taken into account by solving the two higher moments of Boltzmann equation along with the Schrödinger and Poisson equations, self-consistently. The Boltzmann transport equation in the form of a current continuity equation and an energy balance equation are solved to obtain the transient and steady-state transport behavior. The numerical instability problems associated with the simulator are presented, and the criteria for smooth convergence of the solutions are discussed. The current-voltage characteristics, transconductance, gate capacitance, and unity-gain frequency of a single quantum well HEMT is discussed. It has been found that a HEMT device with a gate length of 0.7μm, and with a gate bias voltage of 0.625 V, has a transconductance of 579.2 mS/mm, which together with the gate capacitance of 19.28 pF/cm, can operate at a maximum unity-gain frequency of 47.8 GHz.

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İlhan, Onur Alp, Hasan Bulut, Tukur Abdulkadir Sulaiman, and Haci Mehmet Baskonus. "On the new wave behavior of the Magneto-Electro-Elastic(MEE) circular rod longitudinal wave equation." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 10, no. 1 (September 4, 2019): 1–8. http://dx.doi.org/10.11121/ijocta.01.2020.00837.

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The analytical solution of the longitudinal wave equation in the MEE circular rod is analyzed by the powerful sine-Gordon expansion method. Sine - Gordon expansion is based on the well-known wave transformation and sine - Gordon equation. In the longitudinal wave equation in mathematical physics, the transverse Poisson MEE circular rod is caused by the dispersion. Some solutions with complex, hyperbolic and trigonometric functions have been successfully implemented. Numerical simulations of all solutions are given by selecting the appropriate parameter values. The physical meaning of the analytical solution explaining some practical physical problems is given.

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Journal articles on the topic 'Poisson's equation Numerical solutions'

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