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ModifiedHankelFunctionsOfOrderOneThird

Solutions to Stokes' differential equation.

https://github.com/fgasdia/modifiedhankelfunctionsoforderonethird.jl

Science Score: 38.0%

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Keywords

differential-equations julia

Keywords from Contributors

complex-numbers complex-roots poles-finding root-finding
Last synced: 10 months ago · JSON representation ·

Repository

Solutions to Stokes' differential equation.

Basic Info
  • Host: GitHub
  • Owner: fgasdia
  • License: mit
  • Language: Julia
  • Default Branch: master
  • Size: 399 KB
Statistics
  • Stars: 0
  • Watchers: 1
  • Forks: 2
  • Open Issues: 1
  • Releases: 0
Topics
differential-equations julia
Created over 7 years ago · Last pushed over 2 years ago
Metadata Files
Readme License Citation

README.md

Modified Hankel Functions Of Order One Third and Their Derivatives

Build status DOI

Solutions to Stokes' differential equation:

From Tables of the Modified Hankel Functions of Order One-Third and of their Derivatives:

Its only singularity is an irregular singularity at infinity. The equation occurs in the description of simple cases of diffraction and of refraction of waves. The general solution of [Stokes' equation] can be written in terms of Bessel functions of order one-third. The tabulation of these Bessel functions for complex arguments would make possible the computation of solutions of [Stokes' equation] for complex arguments. The direct tabulation of solutions of [Stokes' equation] should, however, be preferred to that of Bessel functions of order one-third. Unlike Bessel's equation, [Stokes' equation] has no singularity in the finite complex plane and its solutions are single-valued, whereas the Bessel functions of order one-third are not.

Usage

```julia using ModifiedHankelFunctionsOfOrderOneThird

h1, h2, h1prime, h2prime = modifiedhankel(z) ```

The functions h₁ and h₂

An independent pair of solutions, valid for all values of , is

and

where .

The contours of integration and are

contoursofintegration

with . We take .

Solutions

Two solution approaches are used. If abs2(z) < 36, a power series solution is used. Otherwise, an asymptotic expansion is performed because of floating point limits in the power series.

Power series

Stokes' equation may be solved in a power series of , valid in the entire complex plane,

where

Asymptotic expansion

The asymptotic expansions can be used to estimate , , and their derivatives, although in general with less accuracy than the power series. Two expansions are required depending on the value of arg z. The existence of two expressions of different forms which represent asymptotically the same integral function is an example of Stokes' phenomenon.

The expansion for for is

where

See the source for the full sets of solutions.

References

The Staff of the Computation Library (1945), Tables of the modified Hankel function of order one-third and of their derivatives. Cambridge, MA: Harvard University Press.

Citing

We encourage you to cite this package if used in scientific work. See the Zenodo badge above or refer to CITATION.bib.

Owner

  • Name: Forrest Gasdia
  • Login: fgasdia
  • Kind: user

Citation (CITATION.bib) 

@misc{Gasdia19,
  author       = {Forrest Gasdia},
  title        = {ModifiedHankelFunctionsOfOrderOneThird.jl: solutions to Stokes' differential equation in the Julia programming language},
  year         = 2019,
  doi          = {10.5281/zenodo.3522565},
  url          = {https://doi.org/10.5281/zenodo.3522565}
}

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  • Total packages: 1
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  • Total dependent packages: 1
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  • Total versions: 8
juliahub.com: ModifiedHankelFunctionsOfOrderOneThird

Solutions to Stokes' differential equation.

  • Versions: 8
  • Dependent Packages: 1
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  • Downloads: 7 Total
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Dependent repos count: 9.9%
Dependent packages count: 23.0%
Forks count: 33.3%
Average: 34.9%
Stargazers count: 73.2%
Last synced: 10 months ago