PoissonPDEComponent [vars,pars]
yields a Poisson PDE term with model variables vars and model parameters pars.
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 equation1D2D
- 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 all close allBasic 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:
Related Guides
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_2025_poissonpdecomponent, author="Wolfram Research", title="{PoissonPDEComponent}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/PoissonPDEComponent.html}", note=[Accessed: 16-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_poissonpdecomponent, organization={Wolfram Research}, title={PoissonPDEComponent}, year={2022}, url={https://reference.wolfram.com/language/ref/PoissonPDEComponent.html}, note=[Accessed: 16-November-2025]}