# 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:
•  parameter default symbol "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:
•  dimension reduction equation 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

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## 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: 20-June-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: 20-June-2024 ]}