As the title says I'm trying to make DSolve solve the transport equation in 2 spatial dimensions. I tried

DSolve[{D[u[t, x, y], t] + b1 D[u[t, x, y], x] + 
b2 D[u[t, x, y], y] == 0, u[0, x, y] == g[x, y]}, u[t, x, y], {t, x, y}]

which didn't give me any output. The one-dimensional equivalent works just fine

DSolve[{D[u[t,x],t]+ b D[u[t,x],x]==0,u[0,x]==g[x]},u[t,x],{t,x}]

Do I miss something?


It's just because DSolve is relatively weak - at least as of now, so let's help it a bit by finding the general solution first:

generalsol = 
 u /. First@DSolve[D[u[t, x, y], t] + b1 D[u[t, x, y], x] + b2 D[u[t, x, y], y] == 0, 
    u, {t, x, y}]
(* Function[{t, x, y}, C[1][-b1 t + x, -b2 t + y]] *)

Then substitute it into the i.c.:

u[0, x, y] == g[x, y] /. u -> generalsol
(* C[1][x, y] == g[x, y] *)

It's clear g == C[1], so the particular solution is

generalsol[t, x, y] /. C[1] -> g
(* g[-b1 t + x, -b2 t + y] *)

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