How to pass arguments in NIntegrate and NDSolve

Question

I am solving PDEs with NDSolve and NIntegrate, but I do not know how to pass arguments correctly. My orignial code is very complicated, so I simplfied it to clarify the problems I met.

1. Firstly, I solved the simplified code without passing the arguments:

   ClearAll[x, y,r, t]; ss = NDSolve[{ x'[t] == (NIntegrate[x[t]rr, {r, 0, 5}] + y[t] + t)*y[t], y'[t] == -x[t], x[0] == 1, y[0] == 1}, {x, y}, {t, 10}]; Plot[Evaluate[{x[t], y[t]} /. ss], {t, 0, 10}]

2. Then I used the skil of passing arguments to solve the problems, and the code becomes:

   ClearAll[x, y, z, zz, t]; u[x_, r_] := xr; v[u_?NumericQ] := NIntegrate[ur, {r, 0, 5}]; z[v_, y_, t_] := v + y + t;

   s = NDSolve[{ x'[t] == z[t, t, t]*y[t], y'[t] == -x[t], x[0] == 1, y[0] == 1}, {x, y}, {t, 10}]; Plot[Evaluate[{x[t], y[t]} /. s], {t, 0, 10}] But I got different and wrong results. Why?

3. I also tried to pass the arguments in another way, but I just got the error messages. Why?

   ClearAll[x, y, u, v, z, zz, t, f]; u[x_, r_] := xr; v[x_?NumberQ] := NIntegrate[ur, {r, 0, 5}]; z[v_, y_, t_] := v + y + t;

   s = NDSolve[{ x'[t] == z[v[u[x[t]]], y[t], t]*y[t], y'[t] == -x[t], x[0] == 1, y[0] == 1}, {x, y}, {t, 10}];

4. I also tried another way to pass the arguments, and the result is correct and same to 1). But I prefer not to use it due to my complicated original code.

   ClearAll[x, y, u, v, z, zz, t, f]; u[x_, r_] := xr; v[x_?NumberQ] := NIntegrate[ur, {r, 0, 5}]; f = NIntegrate[u[x[t], r]*r, {r, 0, 5}]; z[v_, y_, t_] := v + y + t;

   s = NDSolve[{x'[t] == z[f, y[t], t]*y[t], y'[t] == -x[t], x[0] == 1, y[0] == 1}, {x, y}, {t, 10}];

   Plot[Evaluate[{x[t], y[t]} /. s], {t, 0, 10}]

How to pass arguments in complicated codes correctly?

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The original equations look like this. They are integration-differential equations.

The only independent variables is z, the aim is to solve $\nu$[z], qr[z] and qi[z] from z=0 to z. The parameters are: A[z], B[z], D[z], d[z], and ne[$\rho$,z], $\gamma$[$\rho$,z], $\epsilon$r[$\rho$,z] and $\epsilon$i[$\rho$,z]. The integration of $\rho$ ranges from d to infinite.

Equations:

I wrote the code with error messages: NDSolve::ndnum: Encountered non-numerical value for a derivative at z == 0.1.

From equations to code, different names are used: $\rho$ -->r, D-->D0, d-->dmin. $\gamma$ -->$\gamma_0$

ClearAll[ν, qr, qi, r, γ, ne, ϵr, ϵi, dmin, A, B, D0, aL, Zni, νei];
aL = 10;
Zni = 0.001;
νei = 0.001;
γ= (1 + aL^2*ν^2*Exp[-2*qr*r^2])^0.5;
ne= Zni - 4*qr*γ*(1 -γ^(-2))*(1 - qr*r^2*(1 + γ^(-2)));
ϵr= 1 - ne/γ;
ϵi= 4*Pi*νei*ne;
dmin= Re[qr^(-0.5)*(-0.5*Log[Zni/(8*aL^2*ν^2*qr) (1+sqrt[4+(Zni/(4*qr))^2])])^0.5];
D0[ne_?NumericQ, qr_?NumericQ] := NIntegrate[16*Pi*νei*ne*Exp[-2*qr*r^2]*qr*r, {r, dmin, Infinity}];
A[ν_?NumericQ, qr_?NumericQ, ϵr_?NumericQ, qi_?NumericQ,ϵi_?NumericQ, dmin_?NumericQ] := 
  ν*NIntegrate[r*Exp[-qr*r^2]*((1-ϵr)*Sin[qi*r^2]-ϵi*Cos[qi*r^2]), {r, dmin, Infinity}];
B[ν_?NumericQ, qr_?NumericQ, ϵr_?NumericQ, qi_?NumericQ,ϵi_?NumericQ, dmin_?NumericQ] := 
  ν*NIntegrate[r*Exp[-qr*r^2]*((1-ϵr)*Cos[qi*r^2]+ϵi*Sin[qi*r^2]), {r, dmin, Infinity}];
s = NDSolve[{
  ν'[z] == -A[ν[z], qr[z], ϵr,qi[z], ϵi, dmin]qr[z] -  D0[ne, qr[z]]ν[z],
  qr'[z] == -2 A[ν[z], qr[z], ϵr,qi[z], ϵi, dmin]qr[z]^2/ν[z]-D0[ne,qr[z]]qr[z], 
  qi'[z] == -3 A[ν[z], qr[z], ϵr,qi[z], ϵi, dmin]qr[z]qi[z]/ν[z] +

               B[ν[z], qr[z], ϵr,qi[z], ϵi, dmin]qr[z]^2/ν[z] -

               D0[ne, qr[z]]qi[z],
  ν[0.1] == 1., qr[0.1] == 10^(-4), qi[0.1] == 0.}, {ν, qr, qi}, {z, 0.1, 10000.}]

Answer 1

As mentioned in the comments above, your understanding for function is wrong. You'd better put more effort in learning the core language of Mathematica, or even consider revisiting the definition of function in math. The following is a not-the-simplest but correct way to code your equation set in Mathematica, do read it carefully:

Clear["`*"]
vei = 1/1000;
r0 = Sqrt[1 + v^2 Exp[-2 qr rho^2]];
ne = 1/1000 - 4 qr r0 (1 - 1/r0^2) (1 - qr rho^2 (1 + 1/r0^2));
er = 1 - ne/r0; ei = 4 Pi vei ne;

d[v_, qr_] = Sqrt[(-Log@(1/1000 1/(v^2 qr) (1 + Sqrt[4 + (1/(1000 4 qr))^2])))]/Sqrt[qr];

{coreA[v_, qr_, qi_, rho_], coreB[v_, qr_, qi_, rho_]} = 
  Exp[-qr rho^2] rho RotationMatrix[qi rho^2].{ei, 1 - er};
int[core_, v_, qr_, qi_?NumericQ] := 
 v NIntegrate[core[v, qr, qi, rho] , {rho, d[v, qr], Infinity}, MaxRecursion -> 40]

coreD[v_, qr_, rho_] = 16 Pi vei ne Exp[-2 qr rho^2] qr rho;
intD[v_, qr_?NumericQ] := 
 NIntegrate[coreD[v, qr, rho], {rho, 0, Infinity}, MaxRecursion -> 40]

(* Notice I've actually modified the sign of A! *)    
eqn = With[{v = v@z, qr = qr@z, qi = qi@z}, 
   With[{intA = int[coreA, v, qr, qi], intB = int[coreB, v, qr, qi], 
     intD = intD[v, qr]}, {D[v, z] == intA qr - intD v, 
     D[qr, z] == 2 intA qr^2/v - intD qr, 
     D[qi, z] == 3 intA qi qr/v + intB qr^2/v - intD qi}]];

bc = {v[0] == 1, qr[0] == 10^(-4), qi[0] == 0};

{solv, solqr, solqi} = NDSolveValue[{eqn, bc}, {v, qr, qi}, {z, 0, 1}]
(* Some warning generated, but a result is still given *)
Plot[{Re@#[z], Im@#@z} &@solv // Evaluate, {z, 0, 1}]

When solving the equation set, NIntegrate spits out a lot of warning, I'm not sure about the reason. Perhaps it's a clue of divergence and the model should be checked, perhaps a higher WorkingPrecision or MaxRecursion is needed. Anyway I'd like to stop here and leave the remaining work to you.