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On approximation for fractional stochastic partial differential equations on the sphere |
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| Authors: | Vo?V?Anh Philip?Broadbridge Andriy?Olenko |
| Institution: | 1.School of Mathematical Sciences,Queensland University of Technology,Brisbane,Australia;2.School of Mathematics and Computational Science,Xiangtan University,Xiangtan,China;3.Department of Mathematics and Statistics,La Trobe University,Melbourne,Australia;4.School of Mathematics and Statistics,The University of New South Wales,Sydney,Australia |
| Abstract: | This paper gives the exact solution in terms of the Karhunen–Loève expansion to a fractional stochastic partial differential equation on the unit sphere \({\mathbb {S}}^{2} \subset {\mathbb {R}}^{3}\) with fractional Brownian motion as driving noise and with random initial condition given by a fractional stochastic Cauchy problem. A numerical approximation to the solution is given by truncating the Karhunen–Loève expansion. We show the convergence rates of the truncation errors in degree and the mean square approximation errors in time. Numerical examples using an isotropic Gaussian random field as initial condition and simulations of evolution of cosmic microwave background are given to illustrate the theoretical results. |
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