The　distributed-order　time　fractional　diffusion　model　with　Dirichlet　nonhomogeneous　boundary　conditions　on　a　finite　domain　is　considered.　Four　choices　of　continuous　distribution　weight　functions　with　mean　μ　and　standard　deviation　σ　are　investigated　to　study　their　impact　on　both　the　short-time　and　long-time　solution　behavior.　An　implicit　numerical　method　implemented　on　a　graded　mesh　is　proposed　to　solve　the　model　and　the　stability　and　convergence　analysis　are　presented.　Semi-analytic　solutions　are　also　derived　for　these　distributions　to　assess　the　accuracy　of　the　scheme.　Numerical　results　highlight　that　the　four　continuous　distribution　weight　functions　produce　a　short-time　solution　behavior　that　is　consistent　with　those　solutions　from　the　classical　time　fractional　partial　differential　equation　with　fractional　order　γ* = μ.　There　are　however　long-time　differences　in　the　solution　behavior　that　become　more　distinguishable　as　σ　increases.　In　particular,　we　find　a　smaller　value　of　σ　produces　more　diffuse　profiles　and　the　diffusion　rate　slows　as　σ　increases.　Furthermore,　the　asymptotic　behavior　of　the　solution　may　be　influenced　by　the　time-fractional　orders　ranging　between　the　smallest　nonzero　weight　order　and　mean　μ　for　the　continuous　uniform　and　raised　cosine　distribution　weight　functions,　respectively.　Similar　findings　are　also　observed　for　the　truncated　normal　and　beta　distributions.　©　2022　Wiley　Periodicals　LLC.