Research article

Estimation of generalized fractional integral operators with nonsingular function as a kernel

  • 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

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  • 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)2Nl=1ailsin(θlθi),i=1,2,,N, (1)

    where Pisource>0 is power source (the energy feeding rate) of the ith node, I0 is the moment of inertia, KD>0 is the ratio between the dissipated energy by the turbine and the square of the angular velocity, and ail>0 is the maximum transmitted power between the ith and lth nodes. The phase angle θi of ith node is given by θi(t)=Ωt+˜θi(t) with a standard frequency Ω (50 or 60 Hz) and small deviations ˜θi(t). Power plants in a big connected network should be synchronized to the same frequency. If loads are too strong and unevenly distributed or if some major fault or a lightening occurs, an oscillator (power plant) may lose synchronization. In that situation the synchronization landscape may change drastically and a blackout may occur. The transient stability of power grids can be regarded as a synchronization problem for nonstationary generator rotor angles aiming to restore synchronism subject to local excitations. Therefore, the region of attraction of synchronized states is a central problem for the transient stability.

    By denoting ωi=PisourceKD and assuming ailKD=KN(i,l{1,2,,N}) and I=0 in (1), Choi et. al derived a Kuramoto-like model for the power grid as follows [5]:

    (˙θi)2=ωi+KNNl=1sin(θlθi),˙θi>0,i=1,2,,N. (2)

    Here, the setting ˙θi>0 was made in accordance to the observation in system (1) that the grid is operating at a frequency close to the standard frequency with small deviations. In order to ensure that the right-hand side of (2) is positive, it is reasonable to assume that ωi>K,i=1,2,,N. This model can be used to understand the emergence of synchronization in networks of oscillators. As far as the authors know, there are few analytical results. In [5], the authors considered this model with identical natural frequencies and prove that complete phase synchronization occurs if the initial phases are distributed inside an arc with geodesic length less than π/2.

    If (˙θi)2 in (2) is replaced by ˙θi, the model is the famous Kuramoto model [13]; for its synchronization analysis we refer to [3,6,10], etc. Sakaguchi and Kuramoto [16] proposed a variant of the Kuramoto model to describe richer dynamical phenomena by introducing a phase shift (frustration) in the coupling function, i.e., by replacing sin(θlθi) with sin(θlθi+α). They inferred the need of α from the empirical fact that a pair of oscillators coupled strongly begin to oscillate with a common frequency deviating from the simple average of their natural frequencies. Some physicists also realized via many experiments that the emergence of the phase shift term requires larger coupling strength K and longer relaxation time to exhibit mutual synchronization compared to the zero phase shift case. This is why we call the phase shift a frustration. The effect of frustration has been intensively studied, for example, [11,12,14]. In power grid model using Kuramoto oscillators, people use the phase shift to depict the energy loss due to the transfer conductance [7]. In this paper, we will incorporate the phase shift term in (2) and study the Kuramoto-like model with frustration α(π4,π4):

    (˙θi)2=ωi+KNNl=1sin(θlθi+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:=max1iNωi,ω_:=min1iNωi,D(ω):=ˉωω_,θM:=max1iNθi,θm:=min1iNθi,D(θ):=θMθm,νM:=max1iNνi,νm:=min1iNνi,D(ν):=νMνm,θνM{θj|νj=νM},θνm{θj|νj=νm}.

    In this paper, we consider the system

    (˙θi)2=ωi+KNNl=1sin(θlθi+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 θ:=(θ1,θ2,,θN) be a dynamical solution to the system (4). We say

    1. it exhibits asymptotically complete phase synchronization if

    limt(θi(t)θj(t))=0,ij.

    2. it exhibits asymptotically complete frequency synchronization if

    limt(˙θi(t)˙θj(t))=0,ij.

    For identical oscillators without frustration, we have the following result.

    Theorem 2.2. Let θ:=(θ1,θ2,,θN) be a solution to the coupled system (4) with α=0,ωi=ω0,i=1,2,,N and ω0>K. If the initial configuration

    θ0A:={θ[0,2π)N:D(θ)<π},

    then there exits λ1,λ2>0 and t0>0 such that

    D(θ(t))D(θ0)eλ1t,t0. (5)

    and

    D(ν(t))D(ν(t0))eλ2(tt0),tt0. (6)

    Next we introduce the main result for nonidentical oscillators with frustration. For ˉω<12sin2|α|, we set

    Kc:=D(ω)2ˉω12ˉωsin|α|>0.

    For suitable parameters, we denote by D1 and D the two angles as follows:

    sinD1=sinD:=ˉω+K(D(ω)+Ksin|α|)Kω_K,0<D1<π2<D<π.

    Theorem 2.3. Let θ:=(θ1,θ2,,θN) be a solution to the coupled system (4) with K>Kc and [ω_,ˉω][K+1,12sin2|α|). If

    θ0B:={θ[0,2π)N|D(θ)<D|α|},

    then for any small ε>0 with D1+ε<π2, there exists λ3>0 and T>0 such that

    D(ν(t))D(ν(T))eλ3(tT),tT. (7)

    Remark 1. If the parametric conditions in Theorem 2.3 are fulfilled, the reference angles D1 and D are well-defined. Indeed, because K>Kc and ˉω<12sin2|α|, we have

    D(ω)2ˉω12ˉωsin|α|<K,12ˉωsin|α|>0.

    This implies

    2ˉω(D(ω)+Ksin|α|)K<1.

    Then, by ω_K+1 and Kˉω we obtain

    sinD1=sinD:=ˉω+K(D(ω)+Ksin|α|)Kω_K2ˉω(D(ω)+Ksin|α|)K<1.

    Remark 2. In order to make 1<K+1<12sin2|α|, it is necessary to assume α(π4,π4). This is the reason for the setting α(π4,π4).

    In this subsection we consider the system (4) with identical natural frequencies and zero frustration:

    (˙θi)2=ω0+KNNl=1sin(θlθi),˙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 θ:=(θ1,θ2,,θN) be a solution to the coupled system (8), then for any i,j{1,2,,N}, we have

    (˙θi˙θj)(˙θi+˙θj)=2KNNl=1cos(θlθi+θj2)sinθjθi2.

    Proof. It is immediately obtained by (8).

    Lemma 3.2. Suppose θ:=(θ1,θ2,,θN) be a solution to the coupled system (8) and Ω>K, then we have

    ˙θiω0+K.

    Proof. It follows from (8) and ˙θi>0 that we have

    (˙θi)2=ω0+KNNl=1sin(θlθi)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 D(θ(t)) and D(ν(t)). Note that D(θ(t)) is Lipschitz continuous and differentiable except at times of collision between the extremal phases and their neighboring phases. Therefore, for the collision time t at which D(θ(t)) is not differentiable, we can use the so-called Dini derivative to replace the classic derivative. In this manner, we can proceed the analysis with differential inequality for D(θ(t)). For D(ν(t)) we can proceed in the similar way. In the following context, we will always use the notation of classic derivative for D(θ(t)) and D(ν(t)).

    Lemma 3.3. Let θ:=(θ1,θ2,,θN) be a solution to the coupled system (8) and Ω>K. If the initial configuration θ(0)A, then θ(t)A,t0.

    Proof. For any θ0A, there exists D(0,π) such that D(θ0)<D. Let

    T:={T[0,+)|D(θ(t))<D,t[0,T)}.

    Since D(θ0)<D, and D(θ(t)) is a continuous function of t, there exists η>0 such that

    D(θ(t))<D,t[0,η).

    Therefore, the set T is not empty. Let T:=supT. We claim that

    T=. (9)

    Suppose to the contrary that T<. Then from the continuity of D(θ(t)), we have

    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))=(θMθm)(˙θM˙θm)=(θMθm)1˙θM+˙θm2KNNl=1cos(θlθM+θm2)sin(θmθM2)(θMθm)1˙θM+˙θm2KNNl=1cosD2sin(θmθM2)(θMθm)1ω0+KKNNl=1cosD2sin(θmθM2)=2KcosD2ω0+KD(θ)2sinD(θ)2KcosD2πω0+KD(θ)2,t[0,T).

    Here we used the relations

    D2<D(θ)2θlθM20θlθm2D(θ)2<D2

    and

    xsinx2πx2,x[π2,π2].

    Therefore, we have

    ddtD(θ)KcosD2πω0+KD(θ),t[0,T), (10)

    which implies that

    D(θ(T))D(θ0)eKcosD2πω0+KT<D(θ0)<D.

    This is contradictory to D(θ(T))=D. The claim (9) is proved, which yields the desired result.

    Now we can give a proof for Theorem 2.2.

    Proof of Theorem 2.2.. According to Lemma 3.3, we substitute T= into (10), then (5) is proved with λ1=KcosD2πω0+K.

    On the other hand, by (5) there exist t0 and δ(0<δ<π2) such that D(θ(t))δ for tt0. Now we differentiate (8) to find

    ˙νi=K2NνiNl=1cos(θlθi)(νlνi).

    Using Lemma 3.2, we now consider the temporal evolution of D(ν(t)):

    ddtD(ν)=˙νM˙νm=K2NνMNl=1cos(θlθνM)(νlνM)K2NνmNl=1cos(θlθνm)(νlνm)Kcosδ2NνMNl=1(νlνM)Kcosδ2NνmNl=1(νlνm)K2Ncosδω0+KNl=1(νlνM)K2Ncosδω0+KNl=1(νlνm)=Kcosδ2Nω0+KNl=1(νlνMνl+νm)=Kcosδ2ω0+KD(ν),tt0.

    This implies that

    D(ν(t))D(ν(t0))eKcosδ2ω0+K(tt0),tt0,

    and proves (6) with λ2=Kcosδ2ω0+K.

    Remark 3. Theorem 2.2 shows, as long as the initial phases are confined inside an arc with geodesic length strictly less than π, complete phase synchronization occurs exponentially fast. This extends the main result in [5] where the initial phases for synchronization need to be confined in an arc with geodesic length less than π/2.

    In this subsection, we prove the main result for nonidentical oscillators with frustration.

    Lemma 3.4. Let θ:=(θ1,θ2,,θN) be a solution to the coupled system (4), then for any i,j{1,2,,N}, we have

    (˙θi˙θj)(˙θi+˙θj)D(ω)+KNNl=1[sin(θlθi+α)sin(θlθj+α)].

    Proof. By (4) and for any i,j{1,2,,N}

    (˙θi˙θj)(˙θi+˙θj)=(˙θi)2(˙θj)2,

    the result is immediately obtained.

    Lemma 3.5. Let θ:=(θ1,θ2,,θN) be a solution to the coupled system (4) and ω_K+1, then we have

    ˙θi[ω_K,ˉω+K],i=1,2,,N.

    Proof. From (4), we have

    ω_K(˙θi)2ˉω+K,i=1,2,,N,

    and also because ˙θi>0. The following lemma gives a trapping region for nonidentical oscillators.

    Lemma 3.6. Let θ:=(θ1,θ2,,θN) be a solution to the coupled system (4) with K>Kc and [ω_,ˉω][K+1,12sin2|α|). If the initial configuration θ0B, then θ(t)B for t0.

    Proof. We define the set T and its supremum:

    T:={T[0,+)|D(θ(t))<D|α|,t[0,T)},T:=supT.

    Since θ0B, and note that D(θ(t)) is a continuous function of t, we see that the set T is not empty and T is well-defined. We now claim that

    T=.

    Suppose to the contrary that T<. Then from the continuity of D(θ(t)), we have

    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)+KNNl=1(sin(θlθM+α)sin(θlθm+α))]I.

    For I, we have

    I=D(ω)+KcosαNNl=1[sin(θlθM)sin(θlθm)]+KsinαNNl=1[cos(θlθM)cos(θlθm)].

    We now consider two cases according to the sign of α.

    (1) α[0,π4). In this case, we have

    ID(ω)+KcosαsinD(θ)ND(θ)Nl=1[(θlθM)(θlθm)]+KsinαNNl=1[1cosD(θ)]=D(ω)K[sin(D(θ)+α)sinα]=D(ω)K[sin(D(θ)+|α|)sin|α|].

    (2) α(π4,0). In this case, we have

    ID(ω)+KcosαsinD(θ)ND(θ)Nl=1[(θlθM)(θlθm)]+KsinαNNl=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(θlθM)θlθM,sin(θlθm)θlθmsinD(θ)D(θ),

    and

    cosD(θ)cos(θlθM),cos(θlθm)1,l=1,2,,N.

    Since D(θ(t))+|α|<D<π for t[0,T), we obtain

    ID(ω)K[sin(D(θ)+|α|)sin|α|] (11)
    D(ω)+Ksin|α|KsinDD(D(θ)+|α|). (12)

    By (12) and Lemma 3.5 we have

    12ddtD(θ)2D(θ)1˙θM+˙θm(D(ω)+Ksin|α|KsinDD(D(θ)+|α|))=D(ω)+Ksin|α|˙θM+˙θmD(θ)KsinDD(˙θM+˙θm)D(θ)(D(θ)+|α|)D(ω)+Ksin|α|2ω_KD(θ)KsinDD2ˉω+KD(θ)(D(θ)+|α|),t[0,T).

    Then we obtain

    ddtD(θ)D(ω)+Ksin|α|2ω_KKsinD2Dˉω+K(D(θ)+|α|),t[0,T),

    i.e.,

    ddt(D(θ)+|α|)D(ω)+Ksin|α|2ω_KKsinD2Dˉω+K(D(θ)+|α|)=KsinD2ˉω+KKsinD2Dˉω+K(D(θ)+|α|),t[0,T).

    Here we used the definition of sinD. By Gronwall's inequality, we obtain

    D(θ(t))+|α|D+(D(θ0)+|α|D)eKsinD2Dˉω+Kt,t[0,T),

    Thus

    D(θ(t))(D(θ0)+|α|D)eKsinD2Dˉω+Kt+D|α|,t[0,T).

    Let tT and we have

    D(θ(T))(D(θ0)+|α|D)eKsinD2Dˉω+KT+D|α|<D|α|,

    which is contradictory to D(θ(T))=D|α|. Therefore, we have

    T=.

    That is,

    D(θ(t))D|α|,t0.

    Lemma 3.7. Let θ:=(θ1,θ2,,θN) be a solution to the coupled system (4) with K>Kc and [ω_,ˉω][K+1,12sin2|α|). If the initial configuration θ(0)B, then

    ddtD(θ(t))D(ω)+Ksin|α|2ω_KK2ˉω+Ksin(D(θ)+|α|),t0.

    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|2ω_KD(θ)Ksin(D(θ)+|α|)2ˉω+KD(θ),t0.

    The proof is completed.

    Lemma 3.8. Let θ:=(θ1,θ2,,θN) be a solution to the coupled system (4) with K>Kc and [ω_,ˉω][K+1,12sin2|α|). If the initial configuration θ(0)B, then for any small ε>0 with D1+ε<π2, there exists some time T>0 such that

    D(θ(t))<D1|α|+ε,tT.

    Proof. Consider the ordinary differential equation:

    ˙y=D(ω)+Ksin|α|2ω_KK2ˉω+Ksiny,y(0)=y0[0,D). (13)

    It is easy to find that y=D1 is a locally stable equilibrium of (13), while y=D is unstable. Therefore, for any initial data y0 with 0<y0<D, the trajectory y(t) monotonically approaches y. Then, for any ε>0 with D1+ε<π2, there exists some time T>0 such that

    |y(t)y|<ε,tT.

    In particular, y(t)<y+ε for tT. By Lemma 3.7 and the comparison principle, we have

    D(θ(t))+|α|<D1+ε,tT,

    which is the desired result.

    Remark 4. Since

    sinD1D(ω)K+sin|α|>sin|α|,

    we have D1>|α|.

    Proof of Theorem 2.3. It follows from Lemma 3.8 that for any small ε>0, there exists some time T>0 such that

    suptTD(θ(t))<D1|α|+ε<π2.

    We differentiate the equation (4) to find

    ˙νi=K2NνiNl=1cos(θlθi+α)(νlνi),νi>0.

    We now consider the temporal evolution of D(ν(t)):

    ddtD(ν)=˙νM˙νm=K2NνMNl=1cos(θlθνM+α)(νlνM)K2NνmNl=1cos(θlθνm+α)(νlνm)K2NνMNl=1cos(D1+ε)(νlνM)K2NνmNl=1cos(D1+ε)(νlνm)Kcos(D1+ε)2Nˉω+KNl=1(νlνMνl+νm)=Kcos(D1+ε)2ˉω+KD(ν),tT,

    where we used

    cos(θlθνM+α),cos(θlθνm+α)cos(D1+ε),andνM,νmˉω+K.

    Thus we obtain

    D(ν(t))D(ν(T))eKcos(D1+ε)2ˉω+K(tT),tT,

    and proves (7) with λ3=Kcos(D1+ε)2ˉω+K.

    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 π. For nonidentical oscillators with frustration, we present a framework to guarantee the emergence of frequency synchronization.

    We would like to thank the anonymous referee for his/her comments which helped us to improve this paper.



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