HelmholtzPDEComponent

HelmholtzPDEComponent[vars,pars]

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

Details

• HelmholtzPDEComponent returns a sum of differential operators to be used as a part of partial differential equations:
• HelmholtzPDEComponent can be used to model Helmholtz 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 HelmholtzPDEComponent is based on a diffusion and reaction term:
• The Helmholtz PDE term is realized as a DiffusionPDETerm with 1 as diffusion coefficient and a ReactionPDETerm with coefficient , resulting in .
• The following model parameters pars can be given:
•  parameter default symbol "HelmholtzEigenvalue" 1 "RegionSymmetry" None
• The reaction term coefficient is a scalar.
• The reaction term coefficient can depend on time, space, parameters and the dependent variables.
• If the HelmholtzPDEComponent depends on parameters that are specified in the association pars as ,keypi,pivi,], the parameters are replaced with .
• 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 HelmholtzPDEComponent 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 Helmholtz equation:

Activate the equation:

Define a Helmholtz equation with a symbolic coefficient:

Define a Helmholtz equation with an eigenvalue of 2:

Confirm the first eigenvalue of a Helmholtz PDE term specified to have an eigenvalue of 2:

Scope(1)

Define a 2D axisymmetric Helmholtz equation:

Activate the equation:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_helmholtzpdecomponent, author="Wolfram Research", title="{HelmholtzPDEComponent}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/HelmholtzPDEComponent.html}", note=[Accessed: 20-June-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_helmholtzpdecomponent, organization={Wolfram Research}, title={HelmholtzPDEComponent}, year={2022}, url={https://reference.wolfram.com/language/ref/HelmholtzPDEComponent.html}, note=[Accessed: 20-June-2024 ]}