Recent　studies　highlight　that　diffusion　processes　in　highly　heterogeneous,　fractal-like　me-dia　can　exhibit　anomalous　transport　phenomena,　which　motivates　us　to　consider　the　use　of　generalised　transport　models　based　on　fractional　operators.　In　this　work,　we　harness　the　properties　of　the　distributed-order　space　fractional　potential　to　provide　a　new　perspective　on　dealing　with　boundary　conditions　for　nonlocal　operators　on　finite　domains.　Firstly,　we　consider　a　homogeneous　space　distributed-order　model　with　Beta　distribution　weight.　An　a　priori　estimate　based　on　the　L-2　norm　is　presented.　Secondly,　utilising　finite　Fourier　and　Laplace　transform　techniques,　the　analytical　solution　to　the　model　is　derived　in　terms　of　Kummer＇s　confluent　hypergeometric　function.　Moreover,　the　finite　volume　method　com-bined　with　Jacobi-Gauss　quadrature　is　applied　to　derive　the　numerical　solution,　which　demonstrates　high　accuracy　even　when　the　weight　function　shows　near　singularity.　Finally,　a　one-dimensional　two-layered　problem　involving　the　use　of　the　fractional　Laplacian　oper-ator　and　space　distributed-order　operator　is　developed　and　the　correct　form　of　boundary　conditions　to　impose　is　analysed.　Utilising　the　idea　of　＇geometric　reconstruction＇,　we　intro-duce　a　＇transition　layer＇　whereby　the　fractional　operator　index　varies　from　fractional　order　to　integer　order　across　a　fine　layer　at　the　boundary　of　the　domain　when　transitioning　from　the　complex　internal　structure　to　the　external　conditions　exposed　to　the　medium.　An　im-portant　observation　is　that　for　a　fractional　dominated　case,　the　diffusion　behaviour　in　the　main　layer　is　similar　to　fractional　diffusion,　while　near　the　boundary　the　behaviour　transi-tions　to　the　case　of　classical　diffusion.　(C)　2022　Elsevier　Inc.　All　rights　reserved.