Generalization of Caputo-Fabrizio Fractional Derivative and Applications to Electrical Circuits

A new fractional derivative with a non-singular kernel involving exponential and trigonometric functions is proposed in this paper. The suggested fractional operator includes as a special case Caputo-Fabrizio fractional derivative. Theoretical and numerical studies of fractional differential equations involving this new concept are presented. Next, some applications to RC-electrical circuits are provided.

In this paper, a new fractional derivative with a non-singular kernel involving exponential and trigonometric functions is proposed. The introduced fractional derivative includes as a special case Caputo-Fabrizio fractional derivative. Theoretical and numerical investigations of fractional differential equations involving this new fractional operator are presented. Next, some applications to electrical circuits are provided.
In section 2, some preliminaries on harmonic analysis are presented. In section 3, we develop a general theory of fractional calculus using an arbitrary non-singular kernel. In section 4, we introduce a generalized Caputo-Fabrizio fractional derivative and study its properties. Some applications to fractional differential equations are given in section 5. A numerical method based on Picard iterations is presented in section 6 with some numerical examples. In section 7, some applications to RC-electrical circuits are provided.

SOME PRELIMINARIES ON HARMONIC ANALYSIS
We recall briefly some results on harmonic analysis that will be used later. Lemma 2.1. Folland [30]. Let ψ ∈ L 1 (R) be such that R ψ(t) dt = 1.
Hence, by Lemma 2.1, for all f ∈ L 1 (R), we have In particular, for µ ∈ L 1 (0, ∞), we have For all t > 0, we have This completes the proof of Lemma 2.2.

Definition 2.1.
We say that f is of exponential order θ , if for t large enough, one has where C > 0 and θ are constants.
We denote by L{f (t)} the Laplace transform of the function f , i.e., Recall that, if f ∈ C[0, ∞) and f is of exponential order θ , then L{f (t)}(s) exists for s > θ .
We denote by N the set of positive integers. Lemma 2.3. Schiff [31]. Let n ∈ N. If f ∈ C n [0, ∞) and for all i = 0, 1, · · · , n − 1, the function f (i) is of exponential order, then

FRACTIONAL DERIVATIVE WITH AN ARBITRARY NON-SINGULAR KERNEL
We consider the set of non-singular kernel functions Definition 3.1. Given k ∈ K, 0 < α < 1 and f ∈ C 1 [0, ∞), the fractional derivative of order α of f with respect to the non-singular kernel function k is defined by Remark 3.1. We can also define D α 0,k f for functions f ∈ AC[0, ∞) (f is an absolutely continuous function in [0, ∞)). In this case, f ′ (t) exists for almost every where t > 0 and f ′ ∈ L 1 (0, ∞).
The following properties hold.
where T α = α 1−α T. Passing to the limit as t → 0 + in the above inequality, (i) follows.
Similarly to the case n = 0, one has Using Fubini's theorem, one obtains Hence, by (3.2), one deduces that Next, using Lemma 2.3, we obtain which yields the desired result.

A GENERALIZED CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE
Consider the kernel function where a > 0 and b ≥ 0 are constants. It can be easily seen that where K is the set of kernel functions defined by (3.1). Hence, using Definition 3.2, we define the fractional derivative with respect to the kernel function k a,b as follows.
Definition 5.1. Let g ∈ C[0, T]. The fractional integral of order α of g is defined by Given f 0 ∈ R and g ∈ C 1 [0, T] with g(0) = 0, we consider the initial value problem Theorem 5.1. Problem (5.1) admits a unique solution f ∈ C 1 [0, T], which is given by Proof: Let f ∈ C 1 [0, T] be a solution of (5.1). One has By Definition 4.1, one has where On the other hand, Integrating the above equality and using that γ (0) = 0, one obtains Hence by (5.4), one deduces that Next, using (5.3), one obtains Integrating the above equality, using that f (0) = f 0 and g(0) = 0, it holds  It follows from (5.5) and (5.6) that i.e., f is a solution of (5.2).
Consider now the non-linear initial value problem where the function F :[0, T] × R → R is continuous and satisfies F(0, u 0 ) = 0.
Definition 5.2. We say that u ∈ C[0, T] is a weak solution of (5.7), if u solves the integral equation i.e., for all 0 ≤ t ≤ T.
Proof: Consider the self-mapping H : for all 0 ≤ t ≤ T. We endow C[0, T] with the norm Then (C[0, T], · ∞ ) is a Banach space. For all u, v ∈ C[0, T] and 0 ≤ t ≤ T, using (5.8), one has which yields Hence by (5.9), one deduces that H is a contraction. Therefore, the result follows from Banach fixed point theorem.

APPLICATIONS TO RC ELECTRICAL CIRCUITS
In this section, we give some applications to RC electrical circuits using the generalized Caputo-Fabrizio fractional derivative introduced in section 4. The governing ODE of an RC electrical circuit (see Figure 2) is given by where V is the voltage, R is the resistance, C is the capacitance and µ(t) is the source of volt. In this part, we consider a fractional version of (7.1) using the generalized Caputo-Fabrizio fractional derivative introduced in section 4. Namely, using the following transformation suggested in [33]: where σ is a positive parameter having dimensions of seconds, we obtain the fractional differential equation where We consider (7.3) with the source term µ(t) = sin(φt) and the initial condition In this case, (7.3) reduces to where A = − 1 κ α and B = −A. Applying the Laplace transform and using Corollary 4.3, one obtains Using (7.4), it holds (7.5) By Laplace transform inverse, one gets Examples. All simulations are obtained using MATLAB 7.5.

CONCLUSION
In this contribution, we suggested a fractional derivative involving the kernel function In the particular case (a, b) = (1, 0), the above function reduces to Caputo-Fabrizio kernel. We studied fractional differential equations via this new concept in both theoretical and numerical aspects. In the theoretical point of view, we investigated the existence and uniqueness of solutions to nonlinear fractional boundary value problems involving the new introduced fractional derivative. Namely, using Banach fixed Frontiers in Physics | www.frontiersin.org point theorem, the existence and uniqueness of weak solutions to (5.7) was established under certain conditions imposed on the non-linear term F and the parameters a, b and α. In the numerical point of view, a numerical algorithm based on Picard iterations was proposed for solving the considered problem. Numerical experiments were provided using as a model example the fractional boundary value problem (6.1).
In Figure 1, we presented the exact solution (u 1 (t)) for α = 1 and numerical solutions z 1 (t), z 3 (t), and z 10 (t) to (6.1) for α ∈ {0.95, 0.7}. One observes that for n = 10, z n (t) is close enough to u 1 (t), which confirms the convergence of the proposed algorithm. Finally, as application, we proposed a fractional model of an RC electrical circuit using the new introduced fractional derivative. One can compare the voltage V(t) obtained for different values of α in the Caputo-Fabrizio case (a, b) = (1, 0) (see Figure 3) with that obtained using different values of (a, b) (see Figures 4, 5). Namely, one can show that the voltage V(t) obtained with the use of the generalized fractional Caputo-Fabrizio derivative is more stable with respect to α than that obtained with the use of Caputo-Fabrizio fractional derivative.

DATA AVAILABILITY STATEMENT
All datasets generated for this study are included in the article/supplementary material.