A　new　time　and　spatial　fractional　heat　conduction　model　with　Brownian　diffusion　and　thermophoresis　is　presented　to　investigate　the　heat　and　mass　transfer　of　Maxwell　viscoelastic　nanofluid　over　a　moving　flat　plate　in　porous　medium.　New　dimensionless　variables　are　introduced　to　nondimensionalize　the　governing　equations　which　are　solved　numerically　by　L1　algorithm　and　shifted　Grunwald　formula.　The　main　observations　of　the　model　are　the　effects　of　embedded　time　and　space　fractional　parameters　on　velocity,　temperature　and　concentration　profiles　which　are　analyzed　graphically.　Numerical　computations　for　a　comparison　between　the　heat　conduction　model　with　the　time　and　spatial　fractional　derivatives　and　the　model　with　spatial　fractional　derivative　are　made.　It　is　found　that　the　heat　transfer　is　enhanced　by　the　time　and　spatial　fractional　heat　conduction　model.　Moreover,　the　larger　fractional　derivatives　refer　to　the　stronger　memory　characteristic　which　exhibits　the　physical　meanings　of　fractional　derivative.　(C)　2019　Elsevier　Ltd.　All　rights　reserved.