In[]:=
ClearAll["Global`*"];​​
v
a
=0.1;
v
i
=0.2;β=9*10^-14;​​NDSolve[{D[n[x,t],t]+
v
a
*D[n[x,t],x]==0,​​D[m[x,t],t]+
v
i
*D[m[x,t],x]==0,​​n[0,t]==0,​​m[0,t]==0,​​n[x,0]==Sin[x],​​m[x,0]==Sin[x]},​​{n,m},{x,0,2*Pi},{t,0,4*Pi},MaxStepSizeInfinity];​​Row[{Plot3D[Evaluate[n[x,t]/.%],{x,0,2*Pi},{t,0,4*Pi},PlotRangeAll],​​Plot3D[Evaluate[m[x,t]/.%],{x,0,2*Pi},{t,0,4*Pi},PlotRangeAll]}]
Out[]=