  In this article, we have studied the convergence behavior of the
Dirichlet-Neumann and Neumann- Neumann waveform relaxation algorithms for
time-fractional sub-diffusion and diffusion-wave equations in 1D & 2D for
regular domains, where the dimensionless diffusion coefficient takes different
constant values in different subdomains. We first observe that different
diffusion coefficients lead to different relaxation parameters for optimal
convergence. Using these optimal relaxation parameters, our analysis estimates
the slow superlinear convergence of the algorithms when the fractional order of
the time derivative is close to zero, almost finite step convergence when the
order is close to two, and in between, the superlinear convergence becomes
faster as fractional order increases. So, we have successfully caught the
transition of convergence rate with the change of fractional order of the time
derivative in estimates and verified them with numerical experiments.
