The simplest and probably fastest option would be to iterate your mesh and set edges vertices transparency to 0 (or any other desired value). This assumes you v got (fan shaped)adjacency information:
foreach(Triangle t in mesh)
for(int i = 0; i < 3; i++)
if(t.Adj[i] == - 1) // or ==null, either way, on the edge
t.Vert[i].a = 0.f;
t.Vert[(i + 1) % 3].a = 0.f;
In case you have opposite adjacency, it is just another mod. If you dont have neither, you can compute it. You can also distribute the alpha to adjected triangles based on distance for more in-depth smoother fadeout(just another inner loop or stack).
Other approach would be to compute extremes(x,y min/max of mesh) and map x,y on some function, which simplest would be for example
if(vert.X is less than 10% of width near to extreme) vert.A = 0.f;. Though this would work well only for convex-shaped mesh, best near-rectangle shaped.