# Solving first order partial differential equation

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.

• You are using a function that solves differential equations numerically, yet the equations have several features that are impossible to handle numerically, such as trying to solve on $(-\infty, \infty)$ and Dirac's $\delta$. If you will settle for an approximation for these, you will need to replace them with approximations by hand, i.e. use a finite range and a narrow peaked function of finite width for $\delta$. Nov 17, 2016 at 9:29

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.