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]==0.1*n[x,t],​​D[m[x,t],t]+
v
i
*D[m[x,t],x]==0.2*m[x,t],​​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]}]
NDSolve
:Warning: scaled local spatial error estimate of 383.04 at x = 6.28319 in the direction of independent variable t is much greater than the prescribed error tolerance. Grid spacing with 25 points may be too large to achieve the desired accuracy or precision. A singularity may have formed or a smaller grid spacing can be specified using the MaxStepSize or MinPoints method options.
Out[]=