In　this　paper,　we　derive　the　exact　artificial　boundary　conditions　for　one-dimensional　reaction-diffusion-advection　equation　on　an　unbounded　domain.　By　employing　the　Laplace　transform,　we　reduce　the　original　unbound　domain　problem　into　a　bounded　domain　problem.　The　exact　artificial　boundary　conditions　are　given　by　Caputo-tempered　fractional　derivatives　in　the　reduced　initial-boundary　value　problem.　We　show　that　the　reduced　initial-boundary　value　problem　is　stable　with　the　exact　artificial　boundary　conditions.　We　design　a　finite　difference　scheme　for　the　reduced　finite　domain　problem.　To　save　the　computational　cost,　we　developed　a　fast　algorithm　to　solve　Caputo-tempered　derivatives　arise　in　the　boundary　conditions.　We　prove　that　the　present　difference　schemes　are　uniquely　solvable　and　unconditionally　stable　in　the　energy　norm.　Finally,　we　demonstrate　the　effectiveness　of　the　proposed　methods　by　some　numerical　examples.　©　2021　IMACS