  From the smallest biological systems to the largest cosmological structures,
spatial domains undergo expansion and contraction. Within these growing
domains, diffusive transport is a common phenomenon. Mathematical models have
been widely employed to investigate diffusive processes on growing domains.
However, a standard assumption is that the domain growth is spatially uniform.
There are many relevant examples where this is not the case, such as the
colonisation of growing gut tissue by neural crest cells. As such, it is not
straightforward to disentangle the individual roles of heterogeneous growth and
diffusive transport. Here we present exact solutions to models of diffusive
transport on domains undergoing spatially non-uniform growth. The exact
solutions are obtained via a combination of transformation, convolution and
superposition techniques. We verify the accuracy of these solutions via
comparison with simulations of a corresponding lattice-based random walk. We
explore various domain growth functions, including linear growth, exponential
growth and contraction, and oscillatory growth. Provided the domain size
remains positive, we find that the derived solutions are valid. The exact
solutions reveal the relationship between model parameters, such as the
diffusivity and the type and rate of domain growth, and key statistics, such as
the survival and splitting probabilities.
