NDSolve and assuming

Question

I am trying to solve the system of PDE with NDSolve and I got an error message like this

NDSolve::deqn: "Equation or list of equations expected instead of

ConditionalExpression[[Eta]r[[Xi]r]==Ir[[Xi]r] (26/125 (1+Times[<<2>>])+2 Re[Times[<<2>>]])+Il[[Xi]l] (Im[Times[<<2>>]+Times[<<4>>]+Times[<<2>>]]-13/125 Re[Times[<<2>>]])+Sqrt[Il[[Xi]l]Ir[[Xi]r]] (Cos[[Phi][<<3>>]] (Times[<<2>>]+Times[<<2>>])-(Times[<<2>>]+Times[<<2>>]) Sin[[Phi][<<3>>]]),<<1>>] in the first argument {{ConditionalExpression[[Eta]r[[Xi]r]==Ir[[Xi]r] (Times[<<2>>]+Times[<<2>>])+Il[[Xi]l] (Im[<<1>>]+Times[<<2>>])+Sqrt[Times[<<2>>]] (Times[<<2>>]+Times[<<3>>]),<<1>>],<<4>>,<<1>>},{[CapitalPhi][0,0,0]==[Pi],[Eta]r[0]==1.57389,[Eta]l[0]==1.57389}}.

I have 2 questions related to this problem:

• Can I somehow see what is <<1>>? As I undertand from here http://reference.wolfram.com/language/ref/Skeleton.html <<1>> is a short form of some expression, when mathematica think that it is too long. I just want to see full text of this error.

• Can I tell to mathematica, that some of my unknown variables are real and more than 0. I am afraid, that I got this error message, because I have something like Sqrt[Ir*Il] in my system of PDE. and if mathematica think, that Ir and Il can be negative or complex, it will be very difficult to solve this system. I just want to tell mathematica try to find soliution only for positive Ir and Il. I tried to use Assuming like this

Assuming[Ir is Reals && Ir > 0, NDSolve[...]]

but it does not help anyhow.

Update with full example. I have many Greek symbols, when I copied it, it became chnaged a little bit

ϵr:=Rationalize[0.001];
ϵl:=Rationalize[0.002];
KH:=Rationalize[0.12];
L := 1;
S := Sqrt[3]/4*L^2;
μ:=4*S/(WL*L);
ϕ0 :=Pi;



systemtosolve={
ηr[ξr]==Im[β[ξr]]*Ir[ξr]+Im[θl[ξl,ξr]]*Il[ξl] + (Im[σr[ξl, ξr]]*Cos[ϕ[t,ξl,ξr]]-Re[σr[ξl, ξr]]*Sin[ϕ[t,ξl,ξr]])*Sqrt[Ir[ξr]*Il[ξl]],

ηl[ξl]==Im[β[ξl]]*Il[ξl]+Im[θr[ξl,ξr]]*Ir[ξr] + (Im[σl[ξl, ξr]]*Cos[ϕ[t,ξl,ξr]]+Re[σl[ξl, ξr]]*Sin[ϕ[t,ξl,ξr]])*Sqrt[Ir[ξr]*Il[ξl]],

μ *D[Φ[t,ξl,ξr],t]==(ωj[ξr,ωab,ku]-ωj[ξl,ωab,ku])-c/L*KH*(
Re[Z[ξr]-Z[ξl]]
-Ir[ξr]*(Re[β[ξr]]-Re[θr[ξl,ξr]])
+Il[ξl]*(Re[β[ξl]]-Re[θl[ξl,ξr]])
-2*(Re[(σr[ξl, ξr]-σl[ξl, ξr])/2]*Cos[ϕ[t,ξl,ξr]]-Im[(σr[ξl, ξr]+σl[ξl, ξr])/2]*Sin[ϕ[t,ξl,ξr]])*Sqrt[Ir[ξ]*Il[ξ]]),

ηr[ξr]==Im[Z[ξr]]-ϵr/KH,

ηl[ξl]==Im[Z[ξl]]-ϵl/KH,

ϕ[t,ξl,ξr]==μ *Φ[t,ξl,ξr]+KH*Re[Z[ξr]-Z[ξl]]-ϕ0
};



ic = {
   Φ[0, 0, 0] == Pi,
   ηr[0] == Im[Z[0]] - ϵr/KH,
   ηl[0] == Im[Z[0]] - ϵr/KH
   };



vars = {ηr, ηl, Φ, Ir, Il};


syssolution = Assuming[ηr∈Reals&&ηr>0&&ηl∈Reals&&ηl>0Ir[ξr]∈Reals&&Ir[ξr]>0&&Il[ξl]∈Reals&&Il[ξl]>0&&Φ∈Reals&&Φ>0ξr∈Reals&& ξl∈Reals && ξr < 0.5 && ξr > -0.5&& ξl < 0.5 && ξl > -0.5, NDSolve[{systemtosolve,ic},vars,{t,0,10^-6},{ξr,-0.1,0.1},{ξl,-0.1,0.1}]];


NDSolve::deqn: "Equation or list of equations expected instead of ConditionalExpression[ηr[ξr]==Ir[ξr]\ (26/125\ (1+Times[<<2>>])+2\ Re[Times[<<2>>]])+Il[ξl]\ (Im[Times[<<2>>]+Times[<<4>>]+Times[<<2>>]]-13/125\ Re[Times[<<2>>]])+Sqrt[Il[ξl]\Ir[ξr]]\ (Cos[ϕ[<<3>>]]\ (Times[<<2>>]+Times[<<2>>])-(Times[<<2>>]+Times[<<2>>])\ Sin[ϕ[<<3>>]]),<<1>>] in the first argument {{ConditionalExpression[ηr[ξr]==Ir[ξr]\ (Times[<<2>>]+Times[<<2>>])+Il[ξl]\ (Im[<<1>>]+Times[<<2>>])+Sqrt[Times[<<2>>]]\ (Times[<<2>>]+Times[<<3>>]),<<1>>],<<4>>,<<1>>},{Φ[0,0,0]==π,ηr[0]==1.57389,ηl[0]==1.57389}}."

There are functions β, θl, σr, Z which are normal complex functions and they are working good, because before I have a graphs for all of them.

I made a screenshot for better understanding (Image is a small in preview. Please save it and open separately to have a good quality.)

Update2

I tried to minimize my system and I found the main reason, but I still do not know what I can do.

My code looks like this

KH := Rationalize[0.12]; 
L := 1; 
S := Sqrt[3]/4*L^2;
μ := 4*S/(WL*L); ϕ0 := Pi;

systemtosolve = {    D[Φ[t, ξl, ξr], t] == 1 +
     Cos[μ *Φ[t, ξl, ξr] + 
       KH*Re[Z[ξr] - Z[ξl]] - ϕ0]    };

ic = {    Φ[0, ξl, ξr] == Pi    };
vars = {Φ};

syssolution =    NDSolve[{systemtosolve, ic},     vars, {t, 0, 1}, {ξr, -0.1, 0.1}, {ξl, -0.1, 0.1}];

NDSolve::deqn: Equation or list of equations expected instead of ConditionalExpression[(Φ^(1,0,0))[t,ξl,ξr]==1-Cos[3/25 Plus[<<2>>]+1250000000/791 Power[<<2>>] Φ[<<3>>]],(((0.
-2. I) ξl∈Reals&&Re[(0. +2. I) ξl]<0.208)||Re[(0. -1. I) ξl]>-0.104)&&(((0. -2. I) ξr∈Reals&&Re[(0. +2. I) ξr]<0.208)||Re[(0. -1. I) ξr]>-0.104)] in the first argument {{ConditionalExpression[(Φ^(1,0,0))[t,ξl,ξr]==1-Cos[Plus[<<2>>]],((Times[<<2>>]∈Reals&&Re[<<1>>]<0.208)||Re[Times[<<2>>]]>-0.104)&&((Times[<<2>>]∈Reals&&Re[<<1>>]<0.208)||Re[Times[<<2>>]]>-0.104)]},{Φ[0,ξl,ξr]==π}}.
>>

The problem dissappeared if I remove function Z[] This function is defined like this

Z[x_] := 2*I*Integrate[Exp[-y^2.0 + 2.0*I*y*x], {y, 0, Infinity}];

When I just calculate Z function or draw it's graph everything is fine, but NDSolve do not like this function somehow.

Can you help me to understand, what is wrong with Z function?

Answer 1

The OP still left out a definition (WL), but the problem is that Integrate generates a ConditionalExpression, on which NDSolve chokes.

2*I*Integrate[Exp[-y^2.0 + 2.0*I*y*x], {y, 0, Infinity}]
(*
  ConditionalExpression[
   2 I E^(-1. x^2) (0.886227 + (0. + 
         0.886227 I) Erfi[(1. + 0. I) x]), ((0. - 2. I) x ∈ 
       Reals && Re[(0. + 2. I) x] < 0) || Re[(0. - 1. I) x] > 0]
*)

One can specify the option GenerateConditions -> False, but perhaps the results may not be correct? So let's try using assumptions, since x is real. It's also risky to use approximate numbers when using Integrate, so let's change 2.0 to 2.

2*I*Integrate[Exp[-y^2 + 2*I*y*x], {y, 0, Infinity}, Assumptions -> x ∈ Reals]
(*
  2 I (1/2 E^-x^2 Sqrt[π] + I DawsonF[x])
*)

I tested the OP's code (from "Update2") with the definition WL = 1, and a solution is produced without error.

Note: Given the problems with Integrate and complex-transcendental integrals, one should check this integral before proceeding.