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ORIGINAL RESEARCH article

Front. Phys., 20 March 2020
Sec. Statistical and Computational Physics
This article is part of the Research Topic New Trends in Fractional Differential Equations with Real-World Applications in Physics View all 16 articles

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

  • 1Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia
  • 2Department of Mathematics, National Institute of Technology, Jamshedpur, India

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.

1. Introduction

In the recent decades, the theory of fractional calculus has brought the attention of a great number of researchers in various disciplines. Indeed, it was observed that the use of fractional derivatives is very useful for modeling many problems in engineering sciences (see e.g., [110]). Various notions of fractional derivatives exist in the literature. The basic notions are those introduced by Riemann-Liouville and Caputo (see e.g., [11]), which involve the singular kernel k(t,s)=(t-s)-αΓ(1-α), 0 < α < 1. These fractional derivatives play an important role for modeling many phenomena in physics. However, as it was mentioned in Caputo and Fabrizio [12], certain phenomena related to material heterogeneities cannot be well-modeled using Riemann-Liouville or Caputo fractional derivatives. Due to this fact, Caputo and Fabrizio [12] suggested a new fractional derivative involving the non-singular kernel k(t,s)=e-α(t-s)1-α, 0 < α < 1. Later, Caputo-Fabrizio fractional derivative was used by many authors for modeling various problems in engineering sciences (see e.g., [1324]). Furthermore, other fractional derivatives with non-singular kernels were introduced by some authors (see e.g., [10, 2529]).

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.

2. 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 ∈ L1(ℝ) be such that

ψ(t)dt=1.

Consider the sequence of functions {ψε}ε>0 defined by

ψε(t)=1εψ(tε),  t.

If μ ∈ L1(ℝ), then

ψε*μL1(),  ε>0

and

limε0+ψε*μ-μL1()=0,

where * denotes the convolution product.

Lemma 2.2. Let ψ ∈ L1(0, ∞) be such that

0ψ(t)dt=1.    (2.1)

Consider the sequence of functions {ψε}ε>0 defined by

ψε(t)=1εψ(tε),  t>0.

If μ ∈ L1(0, ∞), then the sequence of functions {Iεμ}ε>0 defined by

Iεμ(t)=0tψε(t-s)μ(s)ds,  t>0

satisfies the following properties:

IεμL1(0,),  ε>0

and

limε0+Iεμ-μL1(0,)=0.

Proof: For any function f defined almost every where in (0, ∞), let

f~(t)={f(t)a.e.t>0,0ift0.

From (2.1), one has ψ~L1() and

ψ~(t)dt=1.

Hence, by Lemma 2.1, for all fL1(ℝ), we have

ψε~*fL1(),  ε>0

and

limε0+ψε~*f-fL1()=0,

where

ψε~(t)=1εψ~(tε),  t.

In particular, for μ ∈ L1(0, ∞), we have

ψε~*μ~L1(),  ε>0    (2.2)

and

limε0+ψε~*μ~-μ~L1()=0.    (2.3)

For all t > 0, we have

ψε~*μ~(t)=ψε~(t-s)μ~(s)ds                 =0tψε(t-s)μ(s)ds                 =Iεμ(t).

Hence, using (2.2) and (2.3), one obtains

0|Iεμ(t)|dt=0|ψε~*μ~(t)|dtψε~*μ~L1()<

and

Iεμ-μL1(0,)=0|ψε~*μ~(t)-μ~(t)|dt                                   ψε~*μ~-μ~L1()0as  ε0+.

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

|f(t)|Ceθt,

where C > 0 and θ are constants.

We denote by L{f(t)} the Laplace transform of the function f, i.e.,

L{f(t)}(s)=0e-stf(t)dt.

Recall that, if fC[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 fCn[0, ∞) and for all i = 0, 1, ⋯ , n − 1, the function f(i) is of exponential order, then

L{f(n)(t)}(s)=snL{f(t)}(s)-i=1nsi-1f(n-i)(0).

3. Fractional Derivative With an Arbitrary Non-singular Kernel

We consider the set of non-singular kernel functions

K={kC[0,)L1(0,):0k(σ)dσ=1}.    (3.1)

Definition 3.1. Given kK, 0 < α < 1 and fC1[0, ∞), the fractional derivative of order α of f with respect to the non-singular kernel function k is defined by

(D0,kαf)(t)=11-α0tk(α(t-s)1-α)f(s)ds,  t>0.

Remark 3.1. We can also define D0,kαf for functions fAC[0, ∞) (f is an absolutely continuous function in [0, ∞)). In this case, f′(t) exists for almost every where t > 0 and f′ ∈ L1(0, ∞).

The following properties hold.

Theorem 3.1. Let kK and fC1[0, ∞). Then

(i) For all 0 < α < 1,

limt0+(D0,kαf)(t)=0.

(ii) If f′ ∈ L1(0, ∞), one has

D0,kαfL1(0,),  0<α<1

and

limα1-D0,kαf-fL1(0,)=0.

Proof: (i) Let 0 < α < 1. For 0 < t<T < ∞, one has

|(D0,kαf)(t)|kL(0,Tα)fL(0,T)1-αt,

where Tα=α1-αT. Passing to the limit as t → 0+ in the above inequality, (i) follows.(ii) Suppose that f′ ∈ L1(0, ∞). For 0 < α < 1, let ε=1-αα. One has

(D0,kαf)(t)=ε+1ε0tk(1ε(t-s))f(s)ds                    =(ε+1)0t1εk(1ε(t-s))f(s)ds                    =(ε+1)0tkε(t-s)f(s)ds,  t>0,

where

kε(x)=1εk(xε),  x>0.

Hence, using Lemma 2.2, (ii) follows.

Definition 3.2. Given kK, 0 < α < 1, n ∈ ℕ∪{0} and fCn + 1[0, ∞), the fractional derivative of order α+n of f with respect to the non-singular kernel k is defined by

(D0,kα+nf)(t)=11-α0tk(α(t-s)1-α)f(n+1)(s)ds,  t>0.

Remark 3.2. We can also define D0,kα+nf for functions fACn + 1[0, ∞). In this case, fn + 1(t) exists for almost every where t > 0 and f(n + 1)L1(0, ∞).

Similarly to the case n = 0, one has

Theorem 3.2. Let kK, n ∈ ℕ ∪{0} and fCn + 1[0, ∞). Then

(i) For all 0 < α < 1,

limt0+(D0,kα+nf)(t)=0.

(ii) If f(n + 1)L1(0, ∞), then

D0,kα+nfL1(0,),  0<α<1

and

limα1-D0,kα+nf-f(n+1)L1(0,)=0.

Remark 3.3. From the assertion (ii) of Theorem 3.2, if f(n+1) ∈ L1(0, ∞), one has

limα1-(D0,kα+nf)(t)=f(n+1)(t),  a.e.t>0.

Theorem 3.3. Given kK, 0 < α < 1, n ∈ ℕ ∪{0} and fCn + 1[0, ∞) with f(i), i = 0, 1, ⋯  , n, are of exponential order, one has

L{(D0,kα+nf)(t)}(s)=11-α(sn+1L{f(t)}(s)-i=1n+1si-1f(n+1-i)(0))L{kα(t)}(s),

where

kα(t)=k(αt1-α),  t>0.

Proof: One has

L{(D0,kα+nf)(t)}(s)     =0e-ts(D0,kα+nf)(t)dt     =0e-ts(11-α0tk(α(t-σ)1-α)f(n+1)(σ)dσ)dt.

Using Fubini's theorem, one obtains

L{(D0,kα+nf)(t)}(s)  =11-α0f(n+1)(σ)(σe-tsk(α(t-σ)1-α)dt)dσ.    (3.2)

Using the change of variable τ = t − σ, it holds

σe-tsk(α(t-σ)1-α)dt=e-σs0e-τsk(ατ1-α)dτ=e-σsL{kα(t)}(s).

Hence, by (3.2), one deduces that

L{(D0,kα+nf)(t)}(s)=11-αL{f(n+1)(t)}(s)L{kα(t)}(s).

Next, using Lemma 2.3, we obtain

L{(D0,kα+nf)(t)}(s)=11-α(sn+1L{f(t)}(s)-i=1n+1si-1f(n+1-i)(0))L{kα(t)}(s),

which yields the desired result.

4. A Generalized Caputo-Fabrizio Fractional Derivative

Consider the kernel function

ka,b(t)=(a2+b2a)e-atcos(bt),  t0,

where a > 0 and b ≥ 0 are constants. It can be easily seen that

ka,bK,    (4.1)

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 ka, b as follows.

Definition 4.1. Given a > 0, b ≥ 0, 0 < α < 1, n ∈ ℕ∪{0} and fCn + 1[0, ∞), the fractional derivative of order α+n of f with respect to the kernel function ka, b is defined by

(D0,a,bα+nf)(t)=(11-α)(a2+b2a)                              0te-aα(t-s)1-αcos(bα(t-s)1-α)f(n+1)(s)ds,  t>0.

Remark 4.1. Taking a = 1 and b = 0 in the above definition, one obtains

(D0,1,0α+nf)(t)=( CFD0α+nf)(t),  t>0, 

where  CFD0α+n is the Caputo–Fabrizio fractional derivative operator of order α + n (see [12]).

Remark 4.2. Definition 4.1 can be extended to the case of functions fCn + 1[0, T], where 0 < T < ∞.

From (4.1) and Theorem 3.2, one deduces that

Corollary 4.1. Let a > 0, b ≥ 0, n ∈ ℕ∪{0} and fCn + 1[0, ∞). Then

(i) For all 0 < α < 1,

limt0+(D0,a,bα+nf)(t)=0.

(ii) If f(n + 1)L1(0, ∞), then

D0,a,bα+nfL1(0,),  0<α<1

and

limα1-D0,a,bα+nf-f(n+1)L1(0,)=0.

Let

ka,b,α(t)=ka,b(αt1-α),  t>0,

that is,

ka,b,α(t)=(a2+b2a)e-aαt1-αcos(bαt1-α),  t>0.

Lemma 4.1. Abramowitz and Stegun [32]. Let a > 0, b ≥ 0 and 0 < α < 1. Then

L{ka,b,α(t)}(s)=(1-α)(a2+b2)a[(1-α)s+αa((1-α)s+αa)2+b2α2],                                                  s>0.

Using Theorem 3.3 and Lemma 4.1, one deduces that

Corollary 4.2. Let a > 0, b ≥ 0, 0 < α < 1, n ∈ ℕ∪{0} and fCn + 1[0, ∞) with f(i), i = 0, 1, ⋯ , n, are of exponential order. Then

L{(D0,a,bα+nf)(t)}(s)        =(a2+b2)a(sn+1L{f(t)}(s)-i=1n+1si-1f(n+1-i)(0))          [(1-α)s+αa((1-α)s+αa)2+b2α2],  s>0.

For n = 0, one obtains

Corollary 4.3. Let a > 0, b ≥ 0, 0 < α < 1 and fC1[0, ∞) with f is of exponential order. Then

L{(D0,a,bαf)(t)}(s)=(a2+b2)a(sL{f(t)}(s)-f(0))                [(1-α)s+αa((1-α)s+αa)2+b2α2].

5. Applications to Fractional Differential Equations

Let a > 0, b ≥ 0, 0 < T < ∞ and 0 < α < 1.

Definition 5.1. Let gC[0, T]. The fractional integral of order α of g is defined by

(I0,a,bαg)(t)=a(1-α)a2+b2g(t)          + α(0tg(σ)dσ-b2a2+b20te-aα(t-σ)1-αg(σ)dσ),                                                                                0tT,

with (I0,a,bαg)(0)=0.

Given f0 ∈ ℝ and gC1[0, T] with g(0) = 0, we consider the initial value problem

{(D0,a,bαf)(t)=g(t),  0<t<T,f(0)                 =f0.    (5.1)

Theorem 5.1. Problem (5.1) admits a unique solution fC1[0, T], which is given by

f(t)=f0+(I0,a,bαg)(t),  0tT.    (5.2)

Proof: Let fC1[0, T] be a solution of (5.1). One has

(D0,a,bαf)(t)=g(t),  0<t<T.    (5.3)

By Definition 4.1, one has

(D0,a,bαf)(t)=(11-α)(a2+b2a)                   {f(t)+0tddt(e-aα(t-s)1-αcos(bα(t-s)1-α))f(s)ds}                   =(11-α)(a2+b2a)f(t)                   -(aα1-α)(11-α)(a2+b2a)                   0te-aα(t-s)1-αcos(bα(t-s)1-α)f(s)ds                   -(bα1-α)(11-α)(a2+b2a)                   0te-aα(t-s)1-αsin(bα(t-s)1-α)f(s)ds                   =(11-α)(a2+b2a)f(t)-(aα1-α)g(t)                   -(bα1-α)(11-α)(a2+b2a)γ(t),    (5.4)

where

γ(t)=0te-aα(t-s)1-αsin(bα(t-s)1-α)f(s)ds.

On the other hand,

γ(t)=0tddt(e-aα(t-s)1-αsin(bα(t-s)1-α))f(s)ds=-(aα1-α)γ(t)+(bα1-α)0te-aα(t-s)1-αcos(bα(t-s)1-α)f(s)ds=-(aα1-α)γ(t)+(abαa2+b2)g(t).

Integrating the above equality and using that γ(0) = 0, one obtains

γ(t)=abαa2+b20te-aα(t-s)1-αg(s)ds.

Hence by (5.4), one deduces that

(D0,a,bαf)(t)=(11-α)(a2+b2a)f(t)-(aα1-α)g(t)                             -(bα1-α)20te-aα(t-s)1-αg(s)ds.

Next, using (5.3), one obtains

f(t)=a2αa2+b2g(t)+(ab2α2(1-α)(a2+b2))0te-aα(t-s)1-αg(s)ds               +a(1-α)a2+b2g(t).

Integrating the above equality, using that f(0) = f0 and g(0) = 0, it holds

f(t)-f0=(a2αa2+b2)0tg(σ)dσ+a(1-α)a2+b2g(t)               +(ab2α2(1-α)(a2+b2))0t0σe-aα(σ-s)1-αg(s)dsdσ    (5.5)

On the other hand, using Fubini's theorem, one gets

0t0σe-aα(σ-s)1-αg(s)dsdσ=0tg(s)eaαs1-α(ste-aασ1-αdσ)ds=(1-αaα)0tg(s)ds-(1-αaα)0te-aα(t-s)1-αg(s)ds.    (5.6)

It follows from (5.5) and (5.6) that

f(t)=f0+(I0,a,bαg)(t),

i.e., f is a solution of (5.2).

Suppose now that f satisfies (5.2). Clearly, one has fC1[0, T]. Since g(0) = 0, one has f(0) = f0. On the other hand, an elementary calculation shows that (D0,a,bαf)(t)=g(t) for all 0 < t < T. Therefore, f is a solution of (5.2).

Consider now the non-linear initial value problem

{(D0,a,bαu)(t)=F(t,u(t)),  0<t<T,u(0)                  =u0,    (5.7)

where the function F:[0, T] × ℝ → ℝ is continuous and satisfies F(0, u0) = 0.

Definition 5.2. We say that uC[0, T] is a weak solution of (5.7), if u solves the integral equation

u(t)=u0+(I0,a,bαF(·,u(·))(t),  0tT,

i.e.,

u(t)=u0+a(1-α)a2+b2F(t,u(t))+α(0tF(σ,u(σ))dσ-b2a2+b20te-aα(t-σ)1-αF(σ,u(σ))dσ),

for all 0 ≤ tT.

Remark 5.1. Observe that, if FC1([0, T] × ℝ), and uC1[0, T] is a solution of (5.7), then uC[0, T] is a weak solution of (5.7).

Theorem 5.2. Suppose that

|F(t,η)-F(t,ξ)||η-ξ|,  (η,ξ)2,    (5.8)

where ℓ > 0 is a constant. If

(Aα+(α+Bα)T)<1,    (5.9)

where Aα=a(1-α)a2+b2 and Bα=αb2a2+b2, then (5.7) admits a unique weak solution u*C[0, T]. Moreover, for any z0C[0, T], the Picard sequence {zn} defined by

zn+1(t)=u0+a(1-α)a2+b2F(t,zn(t))+α(0tF(σ,zn(σ))dσ-b2a2+b20te-aα(t-σ)1-αF(σ,zn(σ))dσ),

for all 0 ≤ tT, converges uniformly to u*.

Proof: Consider the self-mapping H : C[0, T] → C[0, T] defined by

(Hu)(t)=u0+a(1-α)a2+b2F(t,u(t))+α(0tF(σ,u(σ))dσ-b2a2+b20te-aα(t-σ)1-αF(σ,u(σ))dσ),

for all 0 ≤ tT. We endow C[0, T] with the norm

u=max{|u(t)|:0tT}.

Then (C[0, T], ||·||) is a Banach space. For all u, vC[0, T] and 0 ≤ tT, using (5.8), one has

|(Hu)(t)(Hv)(t)|Aα|F(t,u(t))F(t,v(t)|+α0t|F(σ,u(σ))F(σ,v(σ))|dσ+Bα0teaα(tσ)1α|F(σ,u(σ))F(σ,v(σ))|dσAαuv+αTuv+BαTuv=(Aα+(α+Bα)T)uv,

which yields

Hu-Hv(Aα+(α+Bα)T)u-v.

Hence by (5.9), one deduces that H is a contraction. Therefore, the result follows from Banach fixed point theorem.

6. Numerical Solution via Picard Iteration

Consider the initial value problem

{(D0,1,1αu)(t)=u(t)3+et,  0<t<1,u(0)                 =-3,    (6.1)

where 0 < α < 1. For α = 1, (6.1) reduces to

{u(t)=u(t)3+et,  0<t<1,u(0)=-3.    (6.2)

The exact solution of (6.2) is given by

u1(t)=32et-92et3,  0t1.

(6.1) is a special case of (5.7) with T = 1, a = b = 1, u0 = −3 and F(t,x)=x3+et. One can check easily that F satisfies (5.8) with =13. Moreover, one has

(Aα+(α+Bα)T)=13(12+α)<1.

Hence by Theorem 5.2, (6.1) has a unique weak solution u*C[0, 1]. Consider now the Picard sequence {zn} ⊂ C[0, 1] given by z0(t) = −3 and

zn+1(t)=-3+(1-α)2F(t,zn(t))  +α(0tF(σ,zn(σ))dσ-120te-α(t-σ)1-αF(σ,zn(σ))dσ),    (6.3)

for all n = 0, 1, , 2, ⋯  By Theorem 5.2, the sequence {zn} converges uniformly to u*. In Figure 1A, for α = 0.95, we plot u1(t) [the exact solution of (6.2)], z1(t), z3(t), and z10(t). In Figure 1B, for α = 0.7, we plot z1(t), z3(t), and z10(t).

FIGURE 1
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Figure 1. Picard iterations for different values of α. (A) α = 0.95. (B) α = 0.7.

7. 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

dV(t)dt+V(t)RC=μ(t)RC,    (7.1)

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]:

ddt1σ1-αD0,a,bα,  a>0,b0,0<α<1,    (7.2)

where σ is a positive parameter having dimensions of seconds, we obtain the fractional differential equation

(D0,a,bαV)(t)+1καV(t)=1καμ(t),    (7.3)

where

κα=RCσ1-α.
FIGURE 2
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Figure 2. RC circuit.

We consider (7.3) with the source term

μ(t)=sin(ϕt)

and the initial condition

V(0)=0.    (7.4)

In this case, (7.3) reduces to

(D0,a,bαV)(t)=AV(t)+Bsin(ϕt),

where A=-1κα and B = −A. Applying the Laplace transform and using Corollary 4.3, one obtains

(a2+b2)a(sL{V(t)}(s)-V(0))[(1-α)s+αa((1-α)s+αa)2+b2α2]=AL{V(t)}(s)+Bϕs2+ϕ2.

Using (7.4), it holds

L{V(t)}(s)=Bϕs2+ϕ2(sFα,a,b(s)-A)-1,

where

Fα,a,b(s)=(a2+b2)a[(1-α)s+αa((1-α)s+αa)2+b2α2].    (7.5)

By Laplace transform inverse, one gets

V(t)=L-1{Bϕs2+ϕ2(sFα,a,b(s)-A)-1}(t).

Examples. All simulations are obtained using MATLAB 7.5. Consider an RC circuit with R = 10Ω, C = 0.1F, ϕ = 15 and σ = RCα. In this case, we have κα=αα-1(RC)α, A = −α1−α(RC)−α and B = α1−α(RC)−α. Figure 3 shows the voltage V(t) for different values of α in the case (a, b) = (1, 0) (Caputo-Fabrizio case). Figure 4 shows the voltage V(t) for different values of α in the case (a,b)=(2,2). Figure 5 shows the voltage V(t) for different values of α in the case (a, b) = (10, 3).

FIGURE 3
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Figure 3. Graph of the voltage in the RC circuit for different values of α with μ(t) = sin(15t) and (a, b) = (1, 0).

FIGURE 4
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Figure 4. Graph of the voltage in the RC circuit for different values of α with μ(t) = sin(15t) and (a,b)=(2,2).

FIGURE 5
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Figure 5. Graph of the voltage in the RC circuit for different values of α with μ(t) = sin(15t) and (a, b) = (10, 3).

8. Conclusion

In this contribution, we suggested a fractional derivative involving the kernel function

ka,b(t,s)=(11-α)(a2+b2a)e-aα(t-s)1-αcos(bα(t-s)1-α),                                                  a>0,b0,0<α<1.

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 non-linear fractional boundary value problems involving the new introduced fractional derivative. Namely, using Banach fixed 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 (u1(t)) for α = 1 and numerical solutions z1(t), z3(t), and z10(t) to (6.1) for α ∈ {0.95, 0.7}. One observes that for n = 10, zn(t) is close enough to u1(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.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Funding

BS was supported by Researchers Supporting Project RSP-2019/4, King Saud University, Riyadh, Saudi Arabia.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: fractional derivative, non-singular kernel, Picard iteration, RC-electrical circuit, convergence

Citation: Alshabanat A, Jleli M, Kumar S and Samet B (2020) Generalization of Caputo-Fabrizio Fractional Derivative and Applications to Electrical Circuits. Front. Phys. 8:64. doi: 10.3389/fphy.2020.00064

Received: 25 December 2019; Accepted: 28 February 2020;
Published: 20 March 2020.

Edited by:

Jagdev Singh, JECRC University, India

Reviewed by:

Haci Mehmet Baskonus, Harran University, Turkey
Devendra Kumar, University of Rajasthan, India

Copyright © 2020 Alshabanat, Jleli, Kumar and Samet. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Sunil Kumar, skumar.math@nitjsr.ac.in

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