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Advances in Computational Mathematics

, Volume 40, Issue 2, pp 353–375 | Cite as

Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions

  • Quoc Thong Le Gia
  • William McLean
Article

Abstract

We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions restricted to the sphere. The method allows us to construct high accuracy numerical solutions in parallel. We establish L 2 error estimates for smooth and nonsmooth initial data, and describe some numerical experiments.

Keywords

Parabolic equations Laplace transforms Unit sphere Radial basis functions 

Mathematics Subject Classifications (2010)

35R01 65N30 

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References

  1. 1.
    Bailey, W.E.: An Elementary Treatise on Fourier’s Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. Dover, New York (1959)Google Scholar
  2. 2.
    Butcher, J.C.: Numerical Methods for Differential Equations. Wiley, New York (2010)Google Scholar
  3. 3.
    Chen, D., Menegatto, V.A., Sun, X.: A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Am. Math. Soc. 131, 2733–2740 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Le Gia, Q.T., McLean, W.: Numerical solution of a parabolic equation on the sphere using Laplace transforms and radial basis functions. In: McLean, W., Roberts, A.J. (eds.) Proceedings of the 15th Biennial Computational Techniques and Applications Conference, CTAC-2010, vol. 52 of ANZIAM J., pp. C89—C102 (2011)Google Scholar
  5. 5.
    Le Gia, Q.T.: Approximation of parabolic PDEs on spheres using spherical basis functions. Adv. Comput. Math. 22, 377–397 (2005)CrossRefMathSciNetGoogle Scholar
  6. 6.
    McLean, W., Thomée, V.: Numerical solution via Laplace transforms of a fractional order evolution equation. J. Integr. Equ. Appl. 22, 57–94 (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Müller, C.: Spherical Harmonics, volume 17 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1966)Google Scholar
  8. 8.
    National Institute of Standards and Technology: Digital Library of Mathematical Functions. Release date 2011-07-01, http://dlmf.nist.gov/
  9. 9.
    Suleiman,M., Jaafar, A., Jawias, N.I.C., Ismail, F.: Fourth order four-stage diagonally implicit Runge-Kutta method for linear ordinary differential equations. Malays. J. Math. Sci. 4, 95–105 (2010)Google Scholar
  10. 10.
    Nitsche, J.A., Schatz, A.H.: Interior estimates for Ritz–Galerkin methods. Math. Comput. 28, 937–958 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Saff, E.B., Kuijlaars, A.B.J.: Distributing many points on a sphere. Math. Intell. 19, 5–11 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Schoenberg, I.J.: Positive definite function on spheres. Duke Math. J. 9, 96–108 (1942)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Sheen, D., Sloan, I.H., Thomée, V.: A parallel method for time-discretization of parabolic equations based on contour integral representation and quadrature. Math. Comput. 69, 177–195 (1999)CrossRefGoogle Scholar
  14. 14.
    Szegö, G.: Orthogonal Polynomials. American Mathematical Society, New York (1959)zbMATHGoogle Scholar
  15. 15.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  16. 16.
    Tran, T., Le Gia, Q.T., Sloan, I.H., Stephan, E.P.: Boundary integral equations on the sphere with radial basis functions: error analysis. Appl. Numer. Math. 59, 2857–2871 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Tran, T., Pham, T.D.: Pseudodifferential Equations on the Sphere with Spherical Radial Basis Functions: Error Analysis. Applied Mathematics Report 2008/11, The University of New South Wales (2008)Google Scholar
  18. 18.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  19. 19.
    Xu, Y., Cheney, E.W.: Strictly positive definite functions on spheres. Proc. Am. Math. Soc. 116, 977–981 (1992)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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