3.2.1. Caputo Derivative Definition

This definition corresponds to first differentiate *x*(*t*) and then calculates a fractional integral with order (1−*n*). Since 0 < *n* < 1, then 0 < 1−*n* < 1.

$${}^{C}D\_{t}^{n}(x(t)) = I\_{t}^{1-n} \left(\frac{d\mathbf{x}}{dt}\right) = h\_{1-n}(t) \* \frac{d\mathbf{x}}{dt} \tag{12}$$

Definition (12) clearly shows that the Caputo derivative corresponds to the Riemann– Liouville integral of the derivative of *x*(*t*).

In the Laplace domain, the Caputo derivative definition leads to

$$L\left\{ \left( {}^{\mathbb{C}}D\_{t}^{\mathbb{H}}(\mathbf{x}) \right) \right\} = \frac{1}{\mathbf{s}^{\frac{1}{1-n}}} L\left\{ \frac{d\mathbf{x}(t)}{dt} \right\} = \frac{1}{\mathbf{s}^{\frac{1}{1-n}}} [\mathbf{s}X(\mathbf{s}) - \mathbf{x}(0)].\tag{13}$$

$$\text{So, } X(s) = \frac{1}{s^{\eta}} L\left\{ ^C D\_t^{\eta}(x) \right\} + \frac{x(0)}{s},\tag{14}$$

Or in the time domain,

$$\mathbf{x}(t) = l\_t^\mathbf{n} \left( ^C D\_t \mathbf{n} \right) + \mathbf{x}(0). \tag{15}$$

Thus, the solution of FDE/FDS (6) according to the Caputo approach is

$$\mathbf{x}(t) = {}\_{0}l\_{t}^{n}(f(\mathbf{x}(t), u(t))) + \mathbf{x}(0) \quad \text{for } \mathbf{t} \ge 0 \quad 0 < n < 1,\tag{16}$$

where *x*(0) is interpreted as the initial condition of the FDE/FDS and also as the initial value of the Riemann–Liouville integral.

This simple integral Equation (16), apparently equivalent to the integer order case (2), has made the success of the Caputo derivative approach.
