In　this　work,　the　fractional　Laplacian　operator　is　studied　based　on　its　spectral　definition　for　a　bounded,　one-dimensional　domain.　We　first　note　that　both　the　analytical　and　numerical　solutions　of　the　fractional　diffusion　equation　subject　to　Neumann　boundary　conditions　do　not　exist　due　to　the　need　to　raise　the　zero　eigenvalue　to　a　negative　fractional　index.　To　overcome　this　issue,　we　use　the　concept　of　regularisation　and　modify　the　Neumann　boundary　conditions　to　Robin　type　by　introducing　regularisation　parameters　that　ensure　the　smallest　eigenvalue　of　both　the　Sturm–Liouville　system　and　the　matrix　representation　of　the　Laplacian　operator　is　nonzero.　We　then　observe　that　these　regularisation　parameters　can　have　a　significant　impact　on　the　solution　behaviour　of　the　fractional　diffusion　model.　This　finding　motivates　the　use　of　what　we　call　the　‘fractional　potential’　to　define　a　new　fractional　model　that　is　far　less　sensitive　to　the　regularisation　parameter　and　makes　imposing　fractional　Neumann　boundary　conditions　also　feasible.　Next,　we　compare　the　results　of　this　new　fractional　model　with　a　fractional　Riesz　problem　subject　to　Neumann　boundary　conditions　on　a　bounded　symmetric　domain.　An　interesting　finding　is　that　the　steady-state　solution　for　the　fractional　Laplacian　problem　leads　to　a　constant　function,　while　the　fractional　Riesz　operator　results　in　a　function　exhibiting　boundary　singularities.　Finally,　we　propose　a　three-layered　fractional　model　based　on　the　Laplacian　operator　to　naturally　impose　the　traditional　boundary　conditions　by　using　the　idea　of　‘geometric　reconstruction’.　We　derive　semi-analytical　and　numerical　solutions　for　this　model.　An　important　outcome　is　that　by　choosing　an　extremely　small　scale　for　the　boundary　transition　layer　on　both　sides　of　the　domain,　this　three-layered　model　can　be　used　to　approximate　the　fractional　Laplacian　problem　on　a　bounded　domain.　The　proposed　model　captures　the　fractional　diffusion　behaviour　in　the　inner　layer　well　while　describing　the　classical　diffusion　behaviour　near　the　boundary.　We　show　that　this　model　is　ideal　for　modelling　anomalous　diffusion　in　a　binary　medium　exhibiting　distinct　heterogeneity.　©　2023　Elsevier　B.V.