Nesting ParametricNDSolveValue

Question

I have a problem nesting the function ParametricNDSolveValue. The documentation does not help so I hope to find some answers here.

A minimalistic example for my problem is:

PF1 = ParametricNDSolveValue[{u/uT == f'[u], f[0] == 0}, f, {u, 0, 10}, {uT}]

PF2 = ParametricNDSolveValue[{PF1[uT][u] == f'[u], f[0] == 0}, 
  f, {u, 0, 10}, {uT}]

which yields the error

ParametricNDSolveValue::dsfun: ParametricFunction[1,Internal`Bag[<1>],<<2>>,{{uT$44821},<<26>>[<2>],{},<<1>>,{1},{Automatic,0,0}},{NDSolve`base$44828,NDSolve`NDSolveParametricFunction[0,{ParametricNDSolveValue,Internal`Bag[<2>],None,ParametricNDSolveValue},<<6>>,{},All]}]<<1>><<2>>] cannot be used as a function. >>

How can I make this work?

Note:

• I'm aware that one can do this calculation in a much easier way, I am however concerned with a calculation where I have to nest the command ParametricNDSolveValue in the above way.
• I am using Mathematica 10.4.

Answer 1

Why not include both ODEs into a single ParametricNDSolveValue? For instance:

pf = ParametricNDSolveValue[
    {
    u/uT==f'[u],f[0]==0,
    f[u]==g'[u],g[0]==0
    },
    {f, g},
    {u,0,10},
    uT
];

Visualization:

Plot3D[Through[pf[uT][u]], {u, 0, 10}, {uT,0,10}]

Answer 2

Don't know why your approach doesn't work, but here is a workaround

PF1 = ParametricNDSolveValue[{u/uT == f'[u], f[0] == 0},f, {u, 0, 10}, {uT}]
sol2[uT_] :=NDSolveValue[{PF1[uT][u] == f'[u], f[0] == 0} , f, {u, 0, 10} ]
Plot[{PF1[1][u], sol2[1][u]} // Evaluate, {u, 0, 10}]