Books: 'Poisson's equation'

1

B, Palʹ͡tsev A., ed. Metody resheni͡ia uravneni͡ia Puassona v oblast͡iakh s uzkoĭ shchelʹ͡iu. Moskva: Vychislitelʹnyĭ ͡tsentr RAN, 1992.

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2

Soh, Woo Y. Direct coupling methods for time-accurate solution of incompressible Navier-Stokes equations. [Washington, DC]: National Aeronautics and Space Administration, 1992.

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3

United States. National Aeronautics and Space Administration., ed. Direct coupling methods for time-accurate solution of incompressible Navier-Stokes equations. [Washington, DC]: National Aeronautics and Space Administration, 1992.

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4

Akademi͡ia nauk SSSR. Vychislitelʹnyĭ ͡tsentr, ed. Metody resheni͡ia kraevykh zadach i asimptotiki resheniĭ pri singul͡iarnom deformirovanii oblasti. Moskva: Vychislitelʹnyĭ ͡tsentr AN SSSR, 1988.

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5

Korotkov, D. ͡IU. Metody priblizhennoĭ faktoriza͡tsii dl͡ia resheni͡ia uravneniĭ ėllipticheskogo tipa. Moskva: Vychislitelʹnyĭ ͡tsentr AN SSSR, 1989.

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6

Ponce, Augusto C. Elliptic PDEs, measures and capacities: From the Poisson equation to nonlinear Thomas-Fermi problems. Zürich: European Mathematical Society, 2016.

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7

Sorenson, Reese L. Three-dimensional zonal grids about arbitrary shapes by Poisson's equation. Moffett Field, Calif: National Aeronautics and Space Administration, Ames Research Center, 1988.

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8

Sorenson, Reese L. Three-dimensional zonal grids about arbitrary shapes by Poisson's equation. Moffett Field, Calif: National Aeronautics and Space Administration, Ames Research Center, 1988.

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9

Cooke, J. Robert. MacPoisson: Finite element analysis and Poisson's equation with the Macintosh. Ithaca, NY (P.O. Box 4448, Ithaca 14852): Cooke Publications, 1987.

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10

Center, Ames Research, ed. Three-dimensional zonal grids about arbitrary shapes by Poisson's equation. Moffett Field, Calif: National Aeronautics and Space Administration, Ames Research Center, 1988.

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11

Vorobiev, Leonid G. A symplectic Poisson solver based on fast Fourier transformation: The first trial. Tsukuba-shi, Ibaraki-ken Japan: National Laboratory for High Energy Physics, 1995.

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12

B, Volkov D., and Rachkov A. V, eds. Chislenno-analiticheskiĭ metod resheni͡ia uravneni͡ia Puassona v slozhnykh oblast͡iakh. Moskva: Vychislitelʹnyĭ ͡tsentr AN SSSR, 1990.

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13

B, Volkov D., and Rachkov A. V, eds. Reshenie zadachi o kruchenii shvellera. Moskva: Vychislitelʹnyĭ ͡tsentr AN SSSR, 1989.

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14

͡IAkovleva, E. A. Issledovanie zadachi krucheni͡ia sterzhn͡ia slozhnogo poperechnogo secheni͡ia metodom dekompozi͡tsii. Moskva: Vychislitelʹnyĭ ͡tsentr RAN, 1996.

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15

Cooke, J. Robert. Applied finite element analysis: An Apple II implementation. New York: Wiley, 1986.

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16

Blossey, Ralf. The Poisson-Boltzmann Equation. Cham: Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-24782-8.

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17

Hagley, William Andre. Self-consistent solution of Schrödinger's and Poisson's equations for arbitrary semiconductor heterostructures. Ottawa: National Library of Canada, 1993.

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18

Mayo, Anita. Fast parallel iterative solution of Poisson's and the biharmonic equations on irregular regions. New York: Courant Institute of Mathematical Sciences, New York University, 1991.

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19

United States. National Aeronautics and Space Administration. Scientific and Technical Information Branch., ed. Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. [Washington, D.C.]: National Aeronautics and Space Administration, Scientific and Technical Information Branch, 1987.

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20

United States. National Aeronautics and Space Administration. Scientific and Technical Information Branch., ed. Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. [Washington, D.C.]: National Aeronautics and Space Administration, Scientific and Technical Information Branch, 1987.

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21

Shcherbakov, Boris Alekseevich. Ustoĭchivostʹ po Puassonu dvizheniĭ dinamicheskikh sistem i resheniĭ different︠s︡ialʹnykh uravneniĭ. Kishinev: Shtiint︠s︡a, 1985.

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22

Shcherbakov, Boris Alekseevich. Ustoĭchivostʹ po Puassonu dvizheniĭ dinamicheskikh sistem i resheniĭ different͡s︡ialʹnykh uravneniĭ. Kishinev: "Shtiint͡s︡a", 1985.

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23

Chang, Sin-Chung. Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor: I, One-step method. [Washington, D.C.]: National Aeronautics and Space Administration, Scientific and Technical Information Branch, 1986.

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24

United States. National Aeronautics and Space Administration. Scientific and Technical Information Branch., ed. Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor: I, One-step method. [Washington, D.C.]: National Aeronautics and Space Administration, Scientific and Technical Information Branch, 1986.

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25

Chang, Sin-Chung. Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor: II - two-step method. Cleveland, Ohio: Lewis Research Center, 1986.

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26

Barros, Saulo R. M. The Poisson equation on the unit disk: A Multigrid solver using polar coordinates. Sankt Augustin, W.-Germany: Gesellschaft f"ur Mathematik und Datenverarbeitung, 1986.

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27

Mokhov, O. I. Symplectic and poisson geometry on loop spaces of smooth manifolds and integrable equations. [Amsterdam]: Harwood Academic Publishers, 2001.

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28

Scholma, J. K. A Lie algebraic study of some integrable systems associated with root systems. Amsterdam, the Netherlands: Centrum voor Wiskunde en Informatica, 1993.

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29

United States. National Aeronautics and Space Administration., ed. Steady and unsteady three-dimensional transonic flow computations by integral equation method: Final technical report. [Washington, DC: National Aeronautics and Space Administration, 1994.

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30

Center, Langley Research, ed. Study of Gortler vortices by compact schemes. [Hampton, Va: Langley Research Center, 1990.

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31

Colombo, Maria. Flows of Non-smooth Vector Fields and Degenerate Elliptic Equations: With Applications to the Vlasov-Poisson and Semigeostrophic Systems. Pisa: Scuola Normale Superiore, 2017.

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32

Dokshevich, A. I. Reshenii͡a︡ v konechnom vide uravneniĭ Ėĭlera-Puassona. Kiev: Nauk. dumka, 1992.

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33

O, Demuren A., and United States. National Aeronautics and Space Administration., eds. Computations of complex three-dimensional turbulent free jets. Norfolk, Va: Institute for Computational and Applied Mechanics, Old Dominion University, 1997.

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34

K, Wright, and United States. National Aeronautics and Space Administration., eds. A study entitled research on orbital plasma-electrodynamics: Progress report, period of performance, March 27, 1994 - June 29, 1994. [Washington, DC: National Aeronautics and Space Administration, 1994.

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35

J, Hüsler, and Reiss R. -D, eds. Laws of small numbers: Extremes and rare events. 2nd ed. Basel: Birkhauser Verlag, 2004.

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36

J, Hüsler, and Reiss R. -D, eds. Laws of small numbers: Extremes and rare events. Basel: Birkhäuser Verlag, 1994.

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37

Selvadurai, A. P. S. Partial Differential Equations in Mechanics 2: The Biharmonic Equation, Poisson's Equation. Springer Berlin / Heidelberg, 2010.

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38

Selvadurai, A. P. S. Partial Differential Equations in Mechanics 2: The Biharmonic Equation, Poisson's Equation. Springer London, Limited, 2013.

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39

National Aeronautics and Space Administration (NASA) Staff. Three-Dimensional Zonal Grids about Arbitrary Shapes by Poisson's Equation. Independently Published, 2019.

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40

Transition times between steady-states for heat conduction. Palmerston North, N.Z: Massey University, Dept. of Mathematics and Statistics, 1990.

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41

Sandalci, Can K. Three dimensional Monte Carlo simulator with parallel multigrid poisson solver. 1996.

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42

Deruelle, Nathalie, and Jean-Philippe Uzan. Kinetic theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0010.

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This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

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43

Deruelle, Nathalie, and Jean-Philippe Uzan. Self-gravitating fluids. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0015.

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This chapter briefly describes ‘perfect fluids’. These are characterized by their mass density ρ‎(t, xⁱ), pressure p(t, ⁱ), and velocity field v(t, ⁱ). The motion and equilibrium configurations of these fluids are determined by the equation of state, for example, p = p(ρ‎) for a barotropic fluid, and by the gravitational potential U(t, ⁱ) created at a point ⁱ by other fluid elements. The chapter shows that, given an equation of state, the equations of the problem to be solved are the continuity equation, the Euler equation, and the Poisson equation. It then considers static models with spherical symmetry, as well as polytropes and the Lane–Emden equation. Finally, the chapter studies the isothermal sphere and Maclaurin spheroids.

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44

Rajeev, S. G. Finite Difference Methods. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.003.0014.

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This chapter offers a peek at the vast literature on numerical methods for partial differential equations. The focus is on finite difference methods (FDM): approximating differential operators by functions of difference operators. Padé approximants (Fornberg) give a unifying principle for deriving the various stencils used by numericists. Boundary value problems for the Poisson equation and initial value problems for the diffusion equation are solved using FDM. Numerical instability of explicit schemes are explained physically and implicit schemes introduced. A discrete version of theClebsch formulation of incompressible Euler equations is proposed. The chapter concludes with the radial basis function method and its application to a discrete version of the Lagrangian formulation of Navier–Stokes.

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45

Poisson-Boltzmann Equation: An Introduction. Springer International Publishing AG, 2023.

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46

Mann, Peter. Hamilton’s Equations & Routhian Reduction. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0016.

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In this chapter, the Poisson bracket and angular momentum are investigated and first integrals are used to develop conservation laws as a canonical Noether’s theorem. The Poisson bracket was developed by the French mathematician Poisson in the late nineteenth century and it is a reformulation, or at least a tidying up, of Hamilton’s equations into one neat package. The Poisson bracket of a quantity with the Hamiltonian describes the time evolution of that quantity as one moves along a curve in phase space. The Lie algebra structure of symmetries in mechanics is highlighted using this formulation. The classical propagator is derived using the Poisson bracket.

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47

Deruelle, Nathalie, and Jean-Philippe Uzan. The Schwarzschild solution. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0046.

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This chapter deals with the Schwarzschild metric. To find the gravitational potential U produced by a spherically symmetric object in the Newtonian theory, it is necessary to solve the Poisson equation Δ‎U = 4π‎Gρ‎. Here, the matter density ρ‎ and U depend only on the radial coordinate r and possibly on the time t. Outside the source the solution is U = –GM/r, where M = 4π‎ ∫ ρ‎r2dr is the source mass. In general relativity the problem is to find the ‘spherically symmetric’ spacetime solutions of the Einstein equations, and the analog of the vacuum solution U = –GM/r is the Schwarzschild metric.

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48

Mann, Peter. Constrained Hamiltonian Dynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0021.

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This chapter focuses on autonomous geometrical mechanics, using the language of symplectic geometry. It discusses manifolds (including Kähler manifolds, Riemannian manifolds and Poisson manifolds), tangent bundles, cotangent bundles, vector fields, the Poincaré–Cartan 1-form and Darboux’s theorem. It covers symplectic transforms, the Marsden–Weinstein symplectic quotient, presymplectic and symplectic 2-forms, almost symplectic structures, symplectic leaves and foliation. It also discusses contact structures, musical isomorphisms and Arnold’s theorem, as well as integral invariants, Nambu structures, the Nambu bracket and the Lagrange bracket. It describes Poisson bi-vector fields, Poisson structures, the Lie–Poisson bracket and the Lie–Poisson reduction, as well as Lie algebra, the Lie bracket and Lie algebra homomorphisms. Other topics include Casimir functions, momentum maps, the Euler–Poincaré equation, fibre derivatives and the geodesic equation. The chapter concludes by looking at deformation quantisation of the Poisson algebra, using the Moyal bracket and C*-algebras to develop a quantum physics.

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49

Greenbaum, A., and Anita Mayo. Fast Parallel Iterative Solution of Poisson's and the Biharmonic Equations on Irregular Regions. Creative Media Partners, LLC, 2018.

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50

Mann, Peter. Poisson Brackets & Angular Momentum. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0017.

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This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

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Books on the topic 'Poisson's equation'

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