PoissonPDEComponent

PoissonPDEComponent[vars,pars]

yields a Poisson PDE term with model variables vars and model parameters pars.

Details

  • PoissonPDEComponent returns a sum of differential operators to be used as a part of partial differential equations:
  • PoissonPDEComponent can be used to model Poisson equations with dependent variable , independent variables and time variable .
  • Stationary model variables vars are vars={u[x1,…,xn],{x1,…,xn}}.
  • Time-dependent model variables vars are vars={u[t,x1,…,xn],t,{x1,…,xn}}.
  • The PoissonPDEComponent is based on a diffusion and source term:
  • The Poisson PDE term is realized as a DiffusionPDETerm with –1 as a diffusion coefficient and a SourcePDETerm with coefficient resulting in .
  • The following model parameters pars can be given:
  • parameterdefaultsymbol
    "PoissonSourceTerm"1
    "RegionSymmetry"None
  • The source term coefficient is a scalar.
  • The source term coefficient can depend on time, space, parameters and the dependent variables.
  • A possible choice for the parameter "RegionSymmetry" is "Axisymmetric".
  • "Axisymmetric" region symmetry represents a truncated cylindrical coordinate system where the cylindrical coordinates are reduced by removing the angle variable as follows:
  • dimensionreductionequation
    1D
    2D
  • The diffusion coefficient 1 affects the meaning of NeumannValue.
  • If the PoissonPDEComponent depends on parameters that are specified in the association pars as …,keypi…,pivi,…, the parameters are replaced with .

Examples

open allclose all

Basic Examples  (4)

Define a Poisson PDE component:

Define a Poisson PDE component with a symbolic coefficient:

Find the eigenvalues of a Poisson PDE component:

Solve a Poisson equation:

Visualize the result:

Scope  (1)

Define a 2D axisymmetric Poisson equation:

Activate the equation:

Applications  (1)

Solve an axisymmetric Poisson problem in a solid cylinder. Define the variables and parameters:

The solid cylinder can be approximated by a 2D rectangle that represents a cross section of the solid. Create the 2D rectangle using Polygon:

Set up the boundary conditions:

Set up the PDE:

Solve the PDE:

Visualize the solution with DensityPlot:

The exact solution is given by . Visualize the error between the exact solution and the 2D axisymmetric solution:

Wolfram Research (2020), PoissonPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/PoissonPDEComponent.html (updated 2022).

Text

Wolfram Research (2020), PoissonPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/PoissonPDEComponent.html (updated 2022).

CMS

Wolfram Language. 2020. "PoissonPDEComponent." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/PoissonPDEComponent.html.

APA

Wolfram Language. (2020). PoissonPDEComponent. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PoissonPDEComponent.html

BibTeX

@misc{reference.wolfram_2024_poissonpdecomponent, author="Wolfram Research", title="{PoissonPDEComponent}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/PoissonPDEComponent.html}", note=[Accessed: 05-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_poissonpdecomponent, organization={Wolfram Research}, title={PoissonPDEComponent}, year={2022}, url={https://reference.wolfram.com/language/ref/PoissonPDEComponent.html}, note=[Accessed: 05-November-2024 ]}