  We consider the numerical approximation of different ordinary differential
equations (ODEs) and partial differential equations (PDEs) with periodic
boundary conditions involving a one-dimensional random parameter, comparing the
intrusive and non-intrusive polynomial chaos expansion (PCE) method. We
demonstrate how to modify two schemes for intrusive PCE (iPCE) which are highly
efficient in solving nonlinear reaction-diffusion equations: A second-order
exponential time differencing scheme (ETD-RDP-IF) as well as a spectral
exponential time differencing fourth-order Runge-Kutta scheme (ETDRK4). In
numerical experiments, we show that these schemes show superior accuracy to
simpler schemes such as the EE scheme for a range of model equations and we
investigate whether they are competitive with non-intrusive PCE (niPCE)
methods. We observe that the iPCE schemes are competitive with niPCE for some
model equations, but that iPCE breaks down for complex pattern formation models
such as the Gray-Scott system.
