*6.3. FDS Transients Expressed with the Distributed Exponential Function*

Previously, the problem of fractional transients has been focused on the pseudo-state variable dynamics, with no insight in the distributed state variable. So, we propose now to express the dynamics of *z*(ω, *t*). Let us start again with the elementary system (46):

$$D^n(\mathfrak{x}(t)) = a\mathfrak{x}(t) \quad 0 < n < 1$$

with the distributed initial condition *z*(ω, 0) ∀ω ∈ [0, ∞).

The corresponding distributed frequency model (47) leads to:

$$\frac{\partial z(\boldsymbol{\omega},t)}{\partial t} = -\omega z(\boldsymbol{\omega},t) + a \int\_0^\infty \mu\_\mathbf{n}(\boldsymbol{\xi}) z(\boldsymbol{\xi},t) d\boldsymbol{\xi}, \ \forall \boldsymbol{\omega} \in [0,\infty). \tag{59}$$

Notice that we have to separate the current frequency ω from all the other ones ξ ∈ [0, ∞). This means that the behavior of *z*(ω, *t*) depends on all the behaviors of the other state variables *z*(ξ, *t*).

Let us define δ(ξ), which is the frequency Dirac impulse verifying # ∞ 0 δ(ξ)*d*ξ = 1.

Then,

$$\int\_0^\infty \omega \delta(\xi - \omega) z(\xi, t) d\xi = \omega \int\_0^\infty \delta(\xi - \omega) z(\xi, t) d\xi = \omega z(\omega, t). \tag{60}$$

Let us define

$$
\psi(\omega,\xi) = -\omega\delta(\xi-\omega) + a\mu\_n(\xi) \tag{61}
$$

Then, (59) can be expressed as:

$$\frac{\partial z(\boldsymbol{\omega},t)}{\partial t} = -\omega \int\_0^\infty \delta(\boldsymbol{\xi},-\boldsymbol{\omega}) z(\boldsymbol{\xi},t) d\boldsymbol{\xi} + a \int\_0^\infty \mu\_n(\boldsymbol{\xi}) z(\boldsymbol{\xi},t) d\boldsymbol{\xi} = \int\_0^\infty \psi(\boldsymbol{\omega},\boldsymbol{\xi}) z(\boldsymbol{\xi},t) d\boldsymbol{\xi},\tag{62}$$

The solution *z*(ω, *t*) requires the integration of the integer-order distributed system (59) with the initial condition *z*(ω, 0) ∀ω ∈ [0, ∞). Basically, the solution verifies the integral relation:

$$\begin{aligned} z(\omega, t) &= \int\_0^t \left[ \int\_0^\infty \Psi(\omega, \xi) z(\xi, \tau) d\xi \right] d\tau + z(\omega, 0) \\ &= \,\_0I\_t^1 \left[ \int\_0^\infty \Psi(\omega, \xi) z(\xi, \tau) d\xi \right] + z(\omega, 0) \,\,\omega \in [0, \infty) \end{aligned} \tag{63}$$

This integration is performed with Picard's method, which is an iterative technique (3). At the first iteration, *z*(ω, *t*) is approximated by *z*(ω, 0).

Since *z*(ω, 0) and # ∞ 0 <sup>ψ</sup>(ω, <sup>ξ</sup>)*d*<sup>ξ</sup> are constants for <sup>0</sup> *<sup>I</sup>*<sup>1</sup> *<sup>t</sup>* , we obtain

$$z\_1(\omega, t) = z(\omega, 0) + {}\_0l\_t^1 \left[ \int\_0^\infty \psi(\omega, \xi) z(\omega, 0) d\xi \right] = z(\omega, 0) + z(\omega, 0)t \int\_0^\infty \psi(\omega, \xi) d\xi. \tag{64}$$

At the second iteration, *z*(ω, *t*) is approximated by *z*1(ω, *t*). So, we obtain

$$z\_2(\omega, t) = z(\omega, 0) + z(\omega, 0)t \int\_0^\infty \psi(\omega, \xi)d\xi + z(\omega, 0)\frac{t^2}{2} \left[\int\_0^\infty \psi(\omega, \xi)d\xi\right]^2\tag{65}$$

Additionally, at iteration *k*, we obtain

$$z\_k(\omega, t) = \sum\_{j=0}^k \frac{t^j}{j!} \left[ \int\_0^\infty \psi(\omega, \xi) d\xi \right]^j z(\omega, 0) \tag{66}$$

$$\text{Thus, } z(\omega, t) = \lim\_{k \to \infty} \sum\_{j=0}^{k} \frac{t^j}{j!} \left[ \int\_0^\infty \psi(\omega, \xi) d\xi \right]^j z(\omega, 0), \tag{67}$$

$$\text{i.e., } z(\omega, t) = \left[ \sum\_{k=0}^{\infty} \frac{t^k}{k!} \left[ \int\_0^{\infty} \psi(\omega, \xi) d\xi \right]^k \right] z(\omega, 0) \tag{68}$$

Notice that, in the integer order case, *dx*(*t*) *dt* = *ax*(*t*), so ψ(ω, ξ) ≡ *a*.

$$\text{Then, } \mathbf{x}(t) = \left[ \sum\_{k=0}^{\infty} a^k \frac{t^k}{k!} \right] \mathbf{x}(0) = e^{at} \mathbf{x}(0). \tag{69}$$

Thus, in the distributed case, we can define the distributed exponential function:

$$\Phi(t) = \exp\left(t\int\_0^\infty \psi(\omega,\xi)d\xi\right) = \sum\_{k=0}^\infty \frac{t^k}{k!} \left[\int\_0^\infty \psi(\omega,\xi)d\xi\right]^k \tag{70}$$

and

$$z(\omega, t) = \exp\left(t \int\_0^\infty \psi(\omega, \xi) d\xi\right) z(\omega, 0) = \phi(t) z(\omega, 0) \text{ } \omega \in [0, \infty), \tag{71}$$

which is the distributed generalization of Equation (69).

Then, we can generalize the distributed exponential to the linear multidimensional non-commensurate order case:

$$D^{\mathfrak{U}}(\underline{\mathfrak{x}}(t)) = A\underline{\mathfrak{x}}(t)\dim(\underline{\mathfrak{x}}(t)) = N\_\prime \tag{72}$$

where *<sup>n</sup><sup>T</sup>* <sup>=</sup> [*n*<sup>1</sup> ... *ni* ... *nN*] <sup>0</sup> <sup>&</sup>lt; *ni* <sup>≤</sup> 1.

Moreover, 0 < *ni* ≤ 1 means that the FDS system may include integer-order derivatives, since, with real systems, the dynamics are caused either by integer-order or fractionalorder derivatives.

So, (72) corresponds to the integer-order frequency-distributed differential system:

$$\begin{cases} \frac{\partial \underline{\boldsymbol{z}}(\omega, t)}{\partial t} = -\omega \underline{\boldsymbol{z}}(\omega, t) + A \int\_{0}^{\infty} [\mu\_{\underline{\Omega}}(\boldsymbol{\xi})] \underline{\boldsymbol{z}}(\boldsymbol{\xi}, t) d\boldsymbol{\xi} & \dim(\underline{\boldsymbol{z}}(t)) = N \\ \boldsymbol{0} & \infty \end{cases} \tag{73}$$
 
$$\underline{\boldsymbol{x}}(t) = \int\_{0}^{\cdot} [\mu\_{\underline{\Omega}}(\boldsymbol{\xi})] \underline{\boldsymbol{z}}(\boldsymbol{\xi}, t) d\boldsymbol{\xi}$$

with the initial condition *z*(*ω*, 0) ∀*ω* ∈ [0, ∞),

$$\text{and } \left[ \mu \underline{\mu} (\not\downarrow) \right] = \begin{bmatrix} \mu\_{\text{\tiny\$\underline{\text{\tiny\$\underline{\text{\tiny\$}}}}{}} (\not\downarrow) & & & \\ & \mu\_{\text{\tiny\$\underline{\text{\underline{\text{\underline{\text{\tiny\$}}}}}}{}} (\not\downarrow) & & \\ & 0 & \mu\_{\text{\underline{\text{\underline{\text{\underline{\text{\underline{\text{\underline{\text{\underline{\text{\underline{\text{\langle}}}}}}}}}}}} (\not\downarrow) & \\ \end{bmatrix}} \end{bmatrix} \tag{74}$$

Let us define the matrix

$$\Psi(\omega,\xi) = -\omega\delta(\xi-\omega)I + A\left[\mu\_{\mathcal{U}}(\xi)\right] \tag{75}$$

where *I* is the Identity matrix with appropriate dimension.

Then, the differential system (73) can be expressed as

$$\frac{\partial \underline{\boldsymbol{z}}(\omega, t)}{\partial t} = \int\_0^\infty \mathbb{1}(\omega, \underline{\boldsymbol{z}}) \underline{\boldsymbol{z}}(\underline{\boldsymbol{z}}, t) . \tag{76}$$

Therefore, the distributed exponential exp*t* # ∞ 0 ψ(ω, ξ)*d*ξ . function is replaced by the distributed exponential matrix:

$$\Phi(t) = \exp\left(t\int\_0^\infty \Psi(\omega,\xi)d\xi\right) = \sum\_{k=0}^\infty \frac{t^k}{k!} \left[\int\_0^\infty \Psi(\omega,\xi)d\xi\right]^k,\tag{77}$$

and the solution of (76) is:

$$\underline{\boldsymbol{\pi}}(\omega, t) = \exp\left(t \int\_0^\infty \mathbb{1}(\omega, \underline{\boldsymbol{\pi}}) d\underline{\boldsymbol{\pi}}\right) \underline{\boldsymbol{\pi}}(\omega, 0) = \Phi(t) \underline{\boldsymbol{\pi}}(\omega, 0) \,\,\boldsymbol{\omega} \in [0, \infty). \tag{78}$$

Notice that, contrary to the Mittag–Leffler matrix, the distributed exponential matrix shares the same semi-group properties (see chapter 2 of [54]) as its integer order analog, i.e.,

$$
\Phi(t, t\_0) = \Phi(t, \tau) \Phi(\tau, t\_0) \quad t\_0 < \tau < t \tag{79}
$$

Finally, the solution of the linear multidimensional non-commensurate order FDS (72) with the initial condition *z*(*ω*, 0) ∀*ω* ∈ [0, ∞) is similar to the integer-order case (5), i.e.,

$$\underline{z}(\omega, t) = \Phi(t)\underline{z}(\omega, 0) + \int\_0^t \Phi(t - \tau)\underline{B}\mu(\tau)d\tau \,\omega \in [0, \infty), \tag{80}$$

where Φ(*t*) is the distributed exponential transition matrix.
