Solving first order partial differential equation

Question

I used below code to solve a first order partial equation, but there is no response and I can not understand it. The code is as below:

l = 100; a = 35;  d = 1;

ph = NDSolveValue[{D[Exp[Abs[kk] t] u[t, kk], 
t] == -
I Pi a t kk (Exp[t (Abs[kk] - Abs[kk + 2 Pi/l])] +
Exp[t (Abs[kk] - Abs[kk - 2 Pi/l])] +
d/2 Exp[t (Abs[kk] - Abs[kk - 4 Pi/l])] +
d/2 Exp[t (Abs[kk] - Abs[kk + 4 Pi/(3 l)])])/(2 l),
u[0, kk] == DiracDelta[kk - 4 Pi /l], u[t, Pi/l] == u[t, 2 Pi/l]} , u, {t, 0, 100}, {kk,  -Infinity, Infinity}]

Could anyone understand what is wrong? Note that u is a periodic function of x (position) with period = l. kk is showing a Fourier transform of x.

Answer 1

Using the bounds of the periodic condition as the region bounds:

l = 100; a = 35; d = 1;
ph = NDSolveValue[{D[Exp[Abs[kk] t] u[t, kk], 
     t] == -I Pi a t kk (Exp[t (Abs[kk] - Abs[kk + 2 Pi/l])] + 
        Exp[t (Abs[kk] - Abs[kk - 2 Pi/l])] + 
        d/2 Exp[t (Abs[kk] - Abs[kk - 4 Pi/l])] + 
        d/2 Exp[t (Abs[kk] - Abs[kk + 4 Pi/(3 l)])])/(2 l), 
   u[0, kk] == DiracDelta[kk - 4 Pi/l], u[t, Pi/l] == u[t, 2 Pi/l]}, 
  u, {t, 0, 100}, {kk, Pi/l, 2 Pi/l}]

Plot3D[Evaluate[ReIm[ph[t, kk]]], {t, 0, 100}, {kk, Pi/l, 2 Pi/l}, 
 PlotRange -> All]

You may want to approximate the DiractDelta with a smoothed function and it should be inside the region.