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:
  •  del .(1 del u(x))^(︷^(  diffusion term   )) +k^2 u(x)^(︷^( 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:
  • parameterdefaultsymbol
    "HelmholtzEigenvalue"1
  • 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 .

Examples

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:

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

Text

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

BibTeX

@misc{reference.wolfram_2020_helmholtzpdecomponent, author="Wolfram Research", title="{HelmholtzPDEComponent}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HelmholtzPDEComponent.html}", note=[Accessed: 20-January-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_helmholtzpdecomponent, organization={Wolfram Research}, title={HelmholtzPDEComponent}, year={2020}, url={https://reference.wolfram.com/language/ref/HelmholtzPDEComponent.html}, note=[Accessed: 20-January-2021 ]}

CMS

Wolfram Language. 2020. "HelmholtzPDEComponent." Wolfram Language & System Documentation Center. Wolfram Research. 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