IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v301y2017icp214-223.html
   My bibliography  Save this article

The Crane equation uuxx=−2: The general explicit solution and a case study of Chebyshev polynomial series for functions with weak endpoint singularities

Author

Listed:
  • Boyd, John P.

Abstract

The boundary value problem uuxx=−2 appears in Crane’s theory of laminar convection from a point source. We show that the solution is real only when |x|≤π/2. On this interval, denoting the constants of integration by A and s, the general solution is AV([x−s]/A) where the “Crane function” V is the parameter-free function V=exp(−{erfinv(−[2/π])x}2) and erfinv(z) is the inverse of the error function. V(x) is weakly singular at both endpoints; its Chebyshev polynomial coefficients an decrease proportionally to 1/n3. Exponential convergence can be restored by writing V(x)=∑n=0a2nT2n(z[x]) where the mapping is z=arctanh(x/℧)L2+(arctanh(x/℧))2,℧=π/2. Another option is singular basis functions. V≈(1−x2/℧2){1−0.216log(1−x2/℧2)} has a maximum pointwise error that is less 1/2000 of the maximum of the Crane function.

Suggested Citation

  • Boyd, John P., 2017. "The Crane equation uuxx=−2: The general explicit solution and a case study of Chebyshev polynomial series for functions with weak endpoint singularities," Applied Mathematics and Computation, Elsevier, vol. 301(C), pages 214-223.
    • Handle: RePEc:eee:apmaco:v:301:y:2017:i:c:p:214-223
    DOI: 10.1016/j.amc.2016.12.020
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300316307469
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2016.12.020?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    More about this item

    Keywords

    Pseudospectral; Chebyshev polynomials;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:301:y:2017:i:c:p:214-223. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.