Use findroot inside NDSolve

Question

Hi I would like to solve the following equations:

$\frac{dx(t)}{dt} = \gamma v- k x(t) \left(1-b e^{f(t)}\left(1+b e^{f(t)}\right)\right)$

where

$f(t) = k x(t) \left(1+b e^{f(t)}\right)$

As u see this the solution is self consistent, meaning that for each value of $x$ I need to determine first $f(t)$, plug it inside the equation and then solve.

Any suggestions or reference about how to use Findroot or any other method in NDSolve?

Answer 1

NDSolve can solve differential algebraic equations (DAEs) directly. Following @AccidentalFourierTransform's simplifying assumptions,

γ = 0;
b = k = 1;

sol = NDSolve[{
  x'[t] == γ v - k x[t] (1 - b Exp[f[t]] (1 + b Exp[f[t]])),
  f[t] == k x[t] (1 + b Exp[f[t]]),
  x[0] == -0.5}, {x, f}, {t, 0, 1}][[1]];

Plot[x[t] /. sol, {t, 0, 1}]

In your particular example, you can explicitly solve for f[t] using Solve and then plug that into NDSolve.

Clear[k, b]
fsol = Solve[f[t] == k x[t] (1 + b Exp[f[t]]), f[t]][[1]]
(* {f[t] -> -ProductLog[-b E^(k x[t]) k x[t]] + k x[t]} *)

γ = 0;
b = k = 1;

sol = NDSolve[{
  x'[t] == γ v - k x[t] (1 - b Exp[f[t]] (1 + b Exp[f[t]])),
  x[0] == -0.5} /. fsol, {x}, {t, 0, 1}][[1]];

gives an identical result.

Answer 2

Take $\gamma=0$ and $k=b=1$ to simplify the problem. The general case is then straightforward to implement.

Define

f[x_?NumericQ] := FindRoot[z == x (1 + Exp[z]), {z, 1}][[1, 2]]

With this, and using $x(0)=-0.5$ as the initial condition, we get

NDSolve[{x'[t] == -x[t] (1 - Exp[f[x[t]]] (1 + Exp[f[x[t]]])), x[0] == -.5}, x, {t, 0, 1}]
Plot[%[[1, 1, 2]][t], {t, 0, 1}]

Ta-da!