Bessel function has a significant role in fractional calculus having immense applications in physical and theoretical approach. Present work aims to introduce fractional integral operators in which generalized multi-index Bessel function as a kernel, and develop some important special cases which are connected with fractional operators in fractional calculus. Here, we construct important links to familiar findings from some individual occurrence with our key outcomes.
Citation: Iqra Nayab, Shahid Mubeen, Rana Safdar Ali, Gauhar Rahman, Abdel-Haleem Abdel-Aty, Emad E. Mahmoud, Kottakkaran Sooppy Nisar. Estimation of generalized fractional integral operators with nonsingular function as a kernel[J]. AIMS Mathematics, 2021, 6(5): 4492-4506. doi: 10.3934/math.2021266
[1] | Xiaoxue Zhao, Zhuchun Li . Synchronization of a Kuramoto-like model for power grids with frustration. Networks and Heterogeneous Media, 2020, 15(3): 543-553. doi: 10.3934/nhm.2020030 |
[2] | Tingting Zhu . Synchronization of the generalized Kuramoto model with time delay and frustration. Networks and Heterogeneous Media, 2023, 18(4): 1772-1798. doi: 10.3934/nhm.2023077 |
[3] | Seung-Yeal Ha, Yongduck Kim, Zhuchun Li . Asymptotic synchronous behavior of Kuramoto type models with frustrations. Networks and Heterogeneous Media, 2014, 9(1): 33-64. doi: 10.3934/nhm.2014.9.33 |
[4] | Tingting Zhu . Emergence of synchronization in Kuramoto model with frustration under general network topology. Networks and Heterogeneous Media, 2022, 17(2): 255-291. doi: 10.3934/nhm.2022005 |
[5] | Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li . Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Networks and Heterogeneous Media, 2017, 12(1): 1-24. doi: 10.3934/nhm.2017001 |
[6] | Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang . Uniform stability and mean-field limit for the augmented Kuramoto model. Networks and Heterogeneous Media, 2018, 13(2): 297-322. doi: 10.3934/nhm.2018013 |
[7] | Seung-Yeal Ha, Hansol Park, Yinglong Zhang . Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration. Networks and Heterogeneous Media, 2020, 15(3): 427-461. doi: 10.3934/nhm.2020026 |
[8] | Seung-Yeal Ha, Se Eun Noh, Jinyeong Park . Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Networks and Heterogeneous Media, 2015, 10(4): 787-807. doi: 10.3934/nhm.2015.10.787 |
[9] | Vladimir Jaćimović, Aladin Crnkić . The General Non-Abelian Kuramoto Model on the 3-sphere. Networks and Heterogeneous Media, 2020, 15(1): 111-124. doi: 10.3934/nhm.2020005 |
[10] | Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun . Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks and Heterogeneous Media, 2013, 8(4): 943-968. doi: 10.3934/nhm.2013.8.943 |
Bessel function has a significant role in fractional calculus having immense applications in physical and theoretical approach. Present work aims to introduce fractional integral operators in which generalized multi-index Bessel function as a kernel, and develop some important special cases which are connected with fractional operators in fractional calculus. Here, we construct important links to familiar findings from some individual occurrence with our key outcomes.
Synchronization in complex networks has been a focus of interest for researchers from different disciplines[1,2,4,8,15]. In this paper, we investigate synchronous phenomena in an ensemble of Kuramoto-like oscillators which is regarded as a model for power grid. In [9], a mathematical model for power grid is given by
Pisource=I¨θi˙θi+KD(˙θi)2−N∑l=1ailsin(θl−θi),i=1,2,…,N, | (1) |
where
By denoting
(˙θi)2=ωi+KNN∑l=1sin(θl−θi),˙θi>0,i=1,2,…,N. | (2) |
Here, the setting
If
(˙θi)2=ωi+KNN∑l=1sin(θl−θi+α),˙θi>0,i=1,2,…,N. | (3) |
We will find a trapping region such that any nonstationary state located in this region will evolve to a synchronous state.
The contributions of this paper are twofold: First, for identical oscillators without frustration, we show that the initial phase configurations located in the half circle will converge to complete phase and frequency synchronization. This extends the analytical results in [5] in which the initial phase configuration for synchronization needs to be confined in a quarter of circle. Second, we consider the nonidentical oscillators with frustration and present a framework leading to the boundness of the phase diameter and complete frequency synchronization. To the best of our knowledge, this is the first result for the synchronization of (3) with nonidentical oscillators and frustration.
The rest of this paper is organized as follows. In Section 2, we recall the definitions for synchronization and summarize our main results. In Section 3, we give synchronization analysis and prove the main results. Finally, Section 4 is devoted to a concluding summary.
Notations. We use the following simplified notations throughout this paper:
νi:=˙θi,i=1,2,…,N,ω:=(ω1,ω2,…,ωN),ˉω:=max1≤i≤Nωi,ω_:=min1≤i≤Nωi,D(ω):=ˉω−ω_,θM:=max1≤i≤Nθi,θm:=min1≤i≤Nθi,D(θ):=θM−θm,νM:=max1≤i≤Nνi,νm:=min1≤i≤Nνi,D(ν):=νM−νm,θνM∈{θj|νj=νM},θνm∈{θj|νj=νm}. |
In this paper, we consider the system
(˙θi)2=ωi+KNN∑l=1sin(θl−θi+α),˙θi>0,α∈(−π4,π4),θi(0)=θ0i,i=1,2,…,N. | (4) |
Next we introduce the concepts of complete synchronization and conclude this introductory section with the main result of this paper.
Definition 2.1. Let
1. it exhibits asymptotically complete phase synchronization if
limt→∞(θi(t)−θj(t))=0,∀i≠j. |
2. it exhibits asymptotically complete frequency synchronization if
limt→∞(˙θi(t)−˙θj(t))=0,∀i≠j. |
For identical oscillators without frustration, we have the following result.
Theorem 2.2. Let
θ0∈A:={θ∈[0,2π)N:D(θ)<π}, |
then there exits
D(θ(t))≤D(θ0)e−λ1t,t≥0. | (5) |
and
D(ν(t))≤D(ν(t0))e−λ2(t−t0),t≥t0. | (6) |
Next we introduce the main result for nonidentical oscillators with frustration. For
Kc:=D(ω)√2ˉω1−√2ˉωsin|α|>0. |
For suitable parameters, we denote by
sinD∞1=sinD∞∗:=√ˉω+K(D(ω)+Ksin|α|)K√ω_−K,0<D∞1<π2<D∞∗<π. |
Theorem 2.3. Let
θ0∈B:={θ∈[0,2π)N|D(θ)<D∞∗−|α|}, |
then for any small
D(ν(t))≤D(ν(T))e−λ3(t−T),t≥T. | (7) |
Remark 1. If the parametric conditions in Theorem 2.3 are fulfilled, the reference angles
D(ω)√2ˉω1−√2ˉωsin|α|<K,1−√2ˉωsin|α|>0. |
This implies
√2ˉω(D(ω)+Ksin|α|)K<1. |
Then, by
sinD∞1=sinD∞∗:=√ˉω+K(D(ω)+Ksin|α|)K√ω_−K≤√2ˉω(D(ω)+Ksin|α|)K<1. |
Remark 2. In order to make
In this subsection we consider the system (4) with identical natural frequencies and zero frustration:
(˙θi)2=ω0+KNN∑l=1sin(θl−θi),˙θi>0,i=1,2,…,N. | (8) |
To obtain the complete synchronization, we need to derive a trapping region. We start with two elementary estimates for the transient frequencies.
Lemma 3.1. Suppose
(˙θi−˙θj)(˙θi+˙θj)=2KNN∑l=1cos(θl−θi+θj2)sinθj−θi2. |
Proof. It is immediately obtained by (8).
Lemma 3.2. Suppose
˙θi≤√ω0+K. |
Proof. It follows from (8) and
(˙θi)2=ω0+KNN∑l=1sin(θl−θi)≤ω0+K. |
Next we give an estimate for trapping region and prove Theorem 2.2. For this aim, we will use the time derivative of
Lemma 3.3. Let
Proof. For any
T:={T∈[0,+∞)|D(θ(t))<D∞,∀t∈[0,T)}. |
Since
D(θ(t))<D∞,t∈[0,η). |
Therefore, the set
T∗=∞. | (9) |
Suppose to the contrary that
D(θ(t))<D∞,t∈[0,T∗),D(θ(T∗))=D∞. |
We use Lemma 3.1 and Lemma 3.2 to obtain
12ddtD(θ(t))2=D(θ(t))ddtD(θ(t))=(θM−θm)(˙θM−˙θm)=(θM−θm)1˙θM+˙θm2KNN∑l=1cos(θl−θM+θm2)sin(θm−θM2)≤(θM−θm)1˙θM+˙θm2KNN∑l=1cosD∞2sin(θm−θM2)≤(θM−θm)1√ω0+KKNN∑l=1cosD∞2sin(θm−θM2)=−2KcosD∞2√ω0+KD(θ)2sinD(θ)2≤−KcosD∞2π√ω0+KD(θ)2,t∈[0,T∗). |
Here we used the relations
−D∞2<−D(θ)2≤θl−θM2≤0≤θl−θm2≤D(θ)2<D∞2 |
and
xsinx≥2πx2,x∈[−π2,π2]. |
Therefore, we have
ddtD(θ)≤−KcosD∞2π√ω0+KD(θ),t∈[0,T∗), | (10) |
which implies that
D(θ(T∗))≤D(θ0)e−KcosD∞2π√ω0+KT∗<D(θ0)<D∞. |
This is contradictory to
Now we can give a proof for Theorem 2.2.
Proof of Theorem 2.2.. According to Lemma 3.3, we substitute
On the other hand, by (5) there exist
˙νi=K2NνiN∑l=1cos(θl−θi)(νl−νi). |
Using Lemma 3.2, we now consider the temporal evolution of
ddtD(ν)=˙νM−˙νm=K2NνMN∑l=1cos(θl−θνM)(νl−νM)−K2NνmN∑l=1cos(θl−θνm)(νl−νm)≤Kcosδ2NνMN∑l=1(νl−νM)−Kcosδ2NνmN∑l=1(νl−νm)≤K2Ncosδ√ω0+KN∑l=1(νl−νM)−K2Ncosδ√ω0+KN∑l=1(νl−νm)=Kcosδ2N√ω0+KN∑l=1(νl−νM−νl+νm)=−Kcosδ2√ω0+KD(ν),t≥t0. |
This implies that
D(ν(t))≤D(ν(t0))e−Kcosδ2√ω0+K(t−t0),t≥t0, |
and proves (6) with
Remark 3. Theorem 2.2 shows, as long as the initial phases are confined inside an arc with geodesic length strictly less than
In this subsection, we prove the main result for nonidentical oscillators with frustration.
Lemma 3.4. Let
(˙θi−˙θj)(˙θi+˙θj)≤D(ω)+KNN∑l=1[sin(θl−θi+α)−sin(θl−θj+α)]. |
Proof. By (4) and for any
(˙θi−˙θj)(˙θi+˙θj)=(˙θi)2−(˙θj)2, |
the result is immediately obtained.
Lemma 3.5. Let
˙θi∈[√ω_−K,√ˉω+K],∀i=1,2,…,N. |
Proof. From (4), we have
ω_−K≤(˙θi)2≤ˉω+K,∀i=1,2,…,N, |
and also because
Lemma 3.6. Let
Proof. We define the set
T:={T∈[0,+∞)|D(θ(t))<D∞∗−|α|,∀t∈[0,T)},T∗:=supT. |
Since
T∗=∞. |
Suppose to the contrary that
D(θ(t))<D∞∗−|α|,t∈[0,T∗),D(θ(T∗))=D∞∗−|α|. |
We use Lemma 3.4 to obtain
12ddtD(θ)2=D(θ)ddtD(θ)=D(θ)(˙θM−˙θm)≤D(θ)1˙θM+˙θm[D(ω)+KNN∑l=1(sin(θl−θM+α)−sin(θl−θm+α))]⏟I. |
For
I=D(ω)+KcosαNN∑l=1[sin(θl−θM)−sin(θl−θm)]+KsinαNN∑l=1[cos(θl−θM)−cos(θl−θm)]. |
We now consider two cases according to the sign of
(1)
I≤D(ω)+KcosαsinD(θ)ND(θ)N∑l=1[(θl−θM)−(θl−θm)]+KsinαNN∑l=1[1−cosD(θ)]=D(ω)−K[sin(D(θ)+α)−sinα]=D(ω)−K[sin(D(θ)+|α|)−sin|α|]. |
(2)
I≤D(ω)+KcosαsinD(θ)ND(θ)N∑l=1[(θl−θM)−(θl−θm)]+KsinαNN∑l=1[cosD(θ)−1]=D(ω)−K[sin(D(θ)−α)+sinα]=D(ω)−K[sin(D(θ)+|α|)−sin|α|]. |
Here we used the relations
sin(θl−θM)θl−θM,sin(θl−θm)θl−θm≥sinD(θ)D(θ), |
and
cosD(θ)≤cos(θl−θM),cos(θl−θm)≤1,l=1,2,…,N. |
Since
I≤D(ω)−K[sin(D(θ)+|α|)−sin|α|] | (11) |
≤D(ω)+Ksin|α|−KsinD∞∗D∞∗(D(θ)+|α|). | (12) |
By (12) and Lemma 3.5 we have
12ddtD(θ)2≤D(θ)1˙θM+˙θm(D(ω)+Ksin|α|−KsinD∞∗D∞∗(D(θ)+|α|))=D(ω)+Ksin|α|˙θM+˙θmD(θ)−KsinD∞∗D∞∗(˙θM+˙θm)D(θ)(D(θ)+|α|)≤D(ω)+Ksin|α|2√ω_−KD(θ)−KsinD∞∗D∞∗2√ˉω+KD(θ)(D(θ)+|α|),t∈[0,T∗). |
Then we obtain
ddtD(θ)≤D(ω)+Ksin|α|2√ω_−K−KsinD∞∗2D∞∗√ˉω+K(D(θ)+|α|),t∈[0,T∗), |
i.e.,
ddt(D(θ)+|α|)≤D(ω)+Ksin|α|2√ω_−K−KsinD∞∗2D∞∗√ˉω+K(D(θ)+|α|)=KsinD∞∗2√ˉω+K−KsinD∞∗2D∞∗√ˉω+K(D(θ)+|α|),t∈[0,T∗). |
Here we used the definition of
D(θ(t))+|α|≤D∞∗+(D(θ0)+|α|−D∞∗)e−KsinD∞∗2D∞∗√ˉω+Kt,t∈[0,T∗), |
Thus
D(θ(t))≤(D(θ0)+|α|−D∞∗)e−KsinD∞∗2D∞∗√ˉω+Kt+D∞∗−|α|,t∈[0,T∗). |
Let
D(θ(T∗))≤(D(θ0)+|α|−D∞∗)e−KsinD∞∗2D∞∗√ˉω+KT∗+D∞∗−|α|<D∞∗−|α|, |
which is contradictory to
T∗=∞. |
That is,
D(θ(t))≤D∞∗−|α|,∀t≥0. |
Lemma 3.7. Let
ddtD(θ(t))≤D(ω)+Ksin|α|2√ω_−K−K2√ˉω+Ksin(D(θ)+|α|),t≥0. |
Proof. It follows from (11) and Lemma 3.5, Lemma 3.6 and that we have
12ddtD(θ)2=D(θ)ddtD(θ)≤D(θ)1˙θM+˙θm[D(ω)−K(sin(D(θ)+|α|)−sin|α|)]=D(ω)+Ksin|α|˙θM+˙θmD(θ)−Ksin(D(θ)+|α|)˙θM+˙θmD(θ)≤D(ω)+Ksin|α|2√ω_−KD(θ)−Ksin(D(θ)+|α|)2√ˉω+KD(θ),t≥0. |
The proof is completed.
Lemma 3.8. Let
D(θ(t))<D∞1−|α|+ε,t≥T. |
Proof. Consider the ordinary differential equation:
˙y=D(ω)+Ksin|α|2√ω_−K−K2√ˉω+Ksiny,y(0)=y0∈[0,D∞∗). | (13) |
It is easy to find that
|y(t)−y∗|<ε,t≥T. |
In particular,
D(θ(t))+|α|<D∞1+ε,t≥T, |
which is the desired result.
Remark 4. Since
sinD∞1≥D(ω)K+sin|α|>sin|α|, |
we have
Proof of Theorem 2.3. It follows from Lemma 3.8 that for any small
supt≥TD(θ(t))<D∞1−|α|+ε<π2. |
We differentiate the equation (4) to find
˙νi=K2NνiN∑l=1cos(θl−θi+α)(νl−νi),νi>0. |
We now consider the temporal evolution of
ddtD(ν)=˙νM−˙νm=K2NνMN∑l=1cos(θl−θνM+α)(νl−νM)−K2NνmN∑l=1cos(θl−θνm+α)(νl−νm)≤K2NνMN∑l=1cos(D∞1+ε)(νl−νM)−K2NνmN∑l=1cos(D∞1+ε)(νl−νm)≤Kcos(D∞1+ε)2N√ˉω+KN∑l=1(νl−νM−νl+νm)=−Kcos(D∞1+ε)2√ˉω+KD(ν),t≥T, |
where we used
cos(θl−θνM+α),cos(θl−θνm+α)≥cos(D∞1+ε),andνM,νm≤√ˉω+K. |
Thus we obtain
D(ν(t))≤D(ν(T))e−Kcos(D∞1+ε)2√ˉω+K(t−T),t≥T, |
and proves (7) with
In this paper, we presented synchronization estimates for the Kuramoto-like model. We show that for identical oscillators with zero frustration, complete phase synchronization occurs exponentially fast if the initial phases are confined inside an arc with geodesic length strictly less than
We would like to thank the anonymous referee for his/her comments which helped us to improve this paper.
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