  We discuss the derivation and the solutions of integro-differential equations
(variable-order time-fractional diffusion equations) following as continuous
limits for lattice continuous time random walk schemes with power-law
waiting-time probability density functions, whose parameters are
position-dependent. We concentrate on subdiffusive cases and discuss two
situations as examples: A system consisting of two parts with different
exponents of subdiffusion, and a system in which the subdiffusion exponent
changes linearly from one end of the interval to another one. In both cases we
compare the numerical solutions of generalized master equations describing the
process on the lattice with the corresponding solutions of the continuous
equations, which follow by exact solution of the corresponding equations in the
Laplace domain with subsequent numerical inversion using the Gaver-Stehfest
algorithm.
