Heat　conduction　in　one-dimensional　system　is　generally　found　to　be　anomalous.　However,　the　scaling　behavior　of　thermal　conductivity　with　system　length　is　still　up　for　debate　among　theoretical,　numerical　and　experimental　results.　Here　with　a　devised　adjustable　reflectivity　heat　reservoir,　we　re-address　the　anomalous　heat　conduction　in　one-dimensional　Fermi-Pasta-Ulam　beta　(FPU-beta)　lattices　under　weak　nonlinear　conditions.　Our　results　show　that,　the　boundary　thermal　resistance　has　an　important　impact　on　the　divergent　thermal　conductivity　of　one-dimensional　systems　with　weak　nonlinearity.　As　an　application　of　this　concept,　we　design　a　thermal　rectification　model　in　one　material　with　asymmetric　boundary　coupling,　which　is　in　sharp　contrast　to　the　well-known　thermal　rectifiers　composed　of　two　different　materials.　Our　findings　shed　light　on　the　intrinsic　heat　conduction　in　weak　nonlinear　lattices,　which　might　be　helpful　to　understand　the　heat　transfer　experiments　of　one-dimensional　materials,　such　as　carbon　nanotubes,　nanowires,　and　polymer　chains.