  Anomalous diffusion phenomena have been observed in many complex physical and
biological systems. One significant advance recently is the physical extension
of particle's motion in static medium to uniformly (and even nonuniformly)
expanding medium. The dynamic mechanism of particle's motion in the
nonuniformly expanding medium has only been investigated in the framework of
continuous-time random walk. To study more physical observables and supplement
the theory of the expanding medium problems, we characterize the nonuniformly
expanding medium with a spatial-temporal dependent scale factor $a(x,t)$, and
build the Langevin picture describing the particle's motion in the nonuniformly
expanding medium. By introducing a new coordinate, besides of the existing
comoving and physical coordinates, we build the relation between the
nonuniformly expanding medium and the uniformly expanding one, and further
obtain the moments of the comoving and physical coordinates. Both exponential
and power-law formed scale factor are considered to uncover the combined
effects of the particle's intrinsic diffusion and the nonuniform expansion of
medium. Our detailed theoretical analyses and simulations provide the
foundation for studying more expanding medium problems.
