# How to solve numerically nonlinear fractional differential equation containing at least two derivatives of non integer order?

Let's consider differential equation: $$x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))$$ with initial conditions $$x(0) = x_0, x'(0) = x_1, \ldots, x^{(n-1)}(0) = x_{n-1}$$. It is easy to solve this equation numerically by substitution $$X_1 = x, X_2 = x',\ldots, X_{n-1} = x^{(n-1)}$$ and using Euler or Runge-Kutta recurence schemes.

The analogous method cannot be applied to the equation: $$D^{\alpha} x(t) = g(t,x(t),D^{\beta}x(t))$$ where $$D^{\alpha}, D^{\beta}$$ are the Caputo derivatives, $$0<\beta<\alpha$$, and initial conditions are $$x(0) = x_0, x'(0) = x_1, \ldots, x^{(n-1)}(0) = x_{n-1}$$, where $$n = \lceil \alpha \rceil$$. How to solve such equations numerically? (if $$g$$ depends only on $$t$$ and $$x(t)$$, then I can solve this equation,for example by using Grunwald-Letnikow derivative, the problem is second derivative with non-integer order and nonlinear terms like $$x(t) \cdot D^{\beta} x(t)$$).