# How to solve Partial differential equation (PDE) with delay!

I just have one question please, I searched on Mathematica documents concerning partial differential equations with delay, all I had found is ordinary differential equations(EDO), that's means equation depends on only one variable, I want to know if it is possible to also solve the EDP(equation in which state depend on two variable) with delay?

I have this example in hand if anyone can solve it please:

T = 4;
homogen =
D[f[x, t], t] - D[f[x, t], {x, 2}]  +
Integrate[
t*s*Sin[t] (f[x, s - 1])^2, {s, 0, t}] +  (3 +
Abs[f[x, t - 1]]) == 0;
(*données initiales propres *)
ic = {f[x, t /; t <= 0] == x*Sin[t], {x, 0, \[Pi]}};
(* Condition aux bord de Dirichlet*)
bc = {f[0, t] == 0, f[\[Pi], t] == 0};
(*résolution analytique de l'équation *)
sol1 = NDSolveValue[{homogen, ic, bc}, f, {x, 0, \[Pi]}, {t, -T, T}]


I can't solve it!! Help!!

• If you specify the initial conditions as "true" initial conditions, i.e. f[x, -T] = ... then you will get the following error message NDSolveValue::delpde: Delay partial differential equations are not currently supported by NDSolve. which tells you that this not possible. Commented Jul 20, 2020 at 14:43
• So, it's not possible to resolve PDE with delay using Mathematica? Commented Jul 20, 2020 at 15:32
• You may be able to convert the PDE into a system of ODEs using the Numerical Method of Lines. A similar PDE system (without delay unfortunately) was solved here by @bbgodfrey. Commented Jul 21, 2020 at 13:53
• as you said The system solved by @bbgodfrey is without delay, my main problem is the delay, without it I can solve my problem easily, actually I think Mathematica is not able to solve PDE with delay, I searched a lot for just an example, I found nothing!! Commented Jul 21, 2020 at 21:09
• You need to respond to @TimLaska so that I get notified. The links that I provided show how to convert a PDE into a system of ODEs, so there is no longer a PDE. WhenEvent also has the restriction that it cannot be applied spatially to a PDE. However, I showed a way to get around this restriction by using the Numerical Method of Lines and turning it into a system of ODEs in my answer here. DDE can be applied to systems of ODEs. Commented Jul 22, 2020 at 2:23