I’m attempting to resolve set of PDEs as proven under. As a result of `a[x_, t_] := aL*Exp[-(x - t)^2/L^2] Sin[x - t]`

when `x < -xmax`

and `t<0`

, I put `a[x,0]`

and `D/Dt(a[x,0])`

into ICs.

There seem a number of error messages, considered one of which is:

*NDSolveValue::eerr: Warning: scaled native spatial error estimate of 208.66558109964893 at t = 39.666155299563634 within the route of impartial variable x is far better than the prescribed error tolerance. Grid spacing with 135 factors could also be too massive to realize the specified accuracy or precision. A singularity could have shaped or a smaller grid spacing will be specified utilizing the MaxStepSize or MinPoints technique choices.*

I modified worth of presiciongoal and accuracygoal, however not solved. Thus, I’ve three equations:

(1) Is the ICs right for `a[x,t]`

?

(2) How one can keep away from the errors?

(3) When initional situation of n and ni are modified to `n[x,0]==0`

, `ni[x,0]==0`

, there’s one other error message, hinting stiffness at round `t=4`

. How one can remedy it?

```
Wants["DifferentialEquations`NDSolveProblems`"];
Wants["DifferentialEquations`NDSolveUtilities`"];
L = 10; b = 10; [Mu] = 1/1800; aL = 1;
[Gamma][x_, t_] := ((1 + a[x, t]^2)/(1 - u[x, t]^2))^(1/2);
[Gamma]i[x_, t_] := ((1 + [Mu]*a[x, t]^2)/(1 - ui[x, t]^2))^(1/2);
equ = {
D[a[x, t], {x, 2}] - D[a[x, t], {t, 2}] - (4/b^2) a[x, t] -
n[x, t]/[Gamma][x, t]*a[x, t] == 0,
D[[Gamma][x, t] u[x, t], t] == -eField[x, t] -
D[[Gamma][x, t], x],
D[[Gamma]i[x, t] ui[x, t], t] == -[Mu]*eField[x, t] -
D[[Gamma]i[x, t], x],
D[eField[x, t], t] == n[x, t] u[x, t] - ni[x, t] ui[x, t],
D[n[x, t], t] + D[n[x, t]*u[x, t], x] == 0 ,
D[ni[x, t], t] + D[ni[x, t]*ui[x, t], x] == 0
};
ic = {
a[x, 0] == aL*E^(-x^2/L^2) Sin[x],
By-product[0, 1][a][x,
0] == (-E^(-(x^2/L^2)) Cos[x] + 1/L E^(-(x^2/L^2)) x Sin[x])*aL,
n[x, 0] == 10, ni[x, 0] == 10,
u[x, 0] == 0, ui[x, 0] == 0, eField[x, 0] == 0
};
xmax = 40; tmax = 100;
{asol, nsol, nisol, usol, uisol, eFsol} =
NDSolveValue[{equ, ic}, {a, n, ni, u, ui, eField}, {x, -xmax,
xmax}, {t, 0, tmax}]
```

Thanks!