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　mu　and　standard　deviation　sigma　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　gamma*　=　mu.　There　are　however　long-time　differences　in　the　solution　behavior　that　become　more　distinguishable　as　sigma　increases.　In　particular,　we　find　a　smaller　value　of　sigma　produces　more　diffuse　profiles　and　the　diffusion　rate　slows　as　sigma　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　mu　for　the　continuous　uniform　and　raised　cosine　distribution　weight　functions,　respectively.　Similar　findings　are　also　observed　for　the　truncated　normal　and　beta　distributions.