Title: On Approximation for Fractional Stochastic Partial Differential Equations on the Sphere,
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 (CMB) are
given to illustrate the theoretical results.