Numerical solution of a parabolic equation on the sphere using Laplace transforms and radial basis functions


  • Quoc Thong Le Gia
  • William McLean



parabolic equation, radial basis function, unit sphere


We propose a method to construct numerical solutions of a parabolic equation on the unit sphere. The time discretisation uses Laplace transformation and quadrature. The spatial approximation employs radial basis functions restricted to the sphere. The method provides a parallel algorithm to construct high accuracy numerical solutions. References
  • Debao Chen, Valdir A. Menegatto, and Xingping Sun. A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 131:2733--2740, 2003. doi:10.1090/S0002-9939-03-06730-3
  • Q. T. Le Gia. Approximation of parabolic PDEs on spheres using spherical basis functions. Adv. Comput. Math., 22:377--397, 2005. doi:10.1007/s10444-003-3960-9
  • William McLean and Vidar Thomee. Numerical solution via Laplace transforms of a fractional order evolution equation. J. Integral Equations Appl., 22:57--94, 2010. doi:10.1216/JIE-2010-22-1-57
  • M. N. Ozisik. Heat conduction. John Wiley and Sons, Inc., New York, 1993.
  • E. B. Saff and A. B. J. Kuijlaars. Distributing many points on a sphere. Math. Intelligencer, 19:5--11, 1997. doi:10.1007/BF03024331
  • I. J. Schoenberg. Positive definite functions on spheres. Duke Math. J., 9:96--108, 1942. doi:10.1215/S0012-7094-42-00908-6
  • Dongwoo Sheen, Ian H .Sloan, and Vidar Thomee. A parallel method for time-discretization of parabolic equations based on contour integral representation and quadrature. Math. Comp., 69:177--195, 1999. doi:10.1090/S0025-5718-99-01098-4
  • Holger Wendland. Scattered Data Approximation. Cambridge University Press, Cambridge, 2005. doi:10.2277/0521843359
  • Yuan Xu and E. W. Cheney. Strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 116:977--981, 1992. doi:10.1090/S0002-9939-1992-1096214-6





Proceedings Computational Techniques and Applications Conference