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formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)


The general solution of impulsive systems with Riemann-Liouville fractional derivatives

Xianmin Zhang / Wenbin Ding / Hui Peng / Zuohua Liu / Tong Shu
Published Online: 2016-12-30 | DOI: https://doi.org/10.1515/math-2016-0096


In this paper, we study a kind of fractional differential system with impulsive effect and find the formula of general solution for the impulsive fractional-order system by analysis of the limit case (as impulse tends to zero). The obtained result shows that the deviation caused by impulses for fractional-order system is undetermined. An example is also provided to illustrate the result.

Keywords: Fractional differential equations; Riemann-Liouville fractional derivative; Impulse; General solution

MSC 2010: 34A08; 34A37

1 Introduction

Fractional calculus was utilized as a powerful tool to reveal the hidden aspects of the dynamics of the complex or hypercomplex systems [1-3], and the subject of fractional differential equations is gaining much attention. For details see [4-14] and the references therein.

Impulsive effects exist widely in many processes in which their states can be described by impulsive differential equations. Moreover, in case of impulsive differential equations with Caputo fractional derivative there have been numerous works about the subject [15-23], and impulsive fractional partial differential equations are widely considered in [24-29].

Motivated by the above-mentioned works, we will study the following impulsive Cauchy problem with Riemann-Liouville fractional derivative: Da+qu(t)=f(t,u(t)),t(a,T]andtti(i=1,2,,m),Δ(Ja+1qu)|t=ti=Ja+1qu(ti+)Ja+1qu(ti)=Δi(u(ti)),i=1,2,,m,Ja+1qu(a)=ua,uaC,(1) where q ∊ ℂ and ℜ(q) ∊ (0, 1), Da+q denotes left-sided Riemann-Liouville fractional derivative of order q and Ja+1q denotes left-sided Riemann-Liouville fractional integral of order 1 − q. f : J × ℂ → ℂ is an appropriate continuous function, and a = t0 < t1 < ... <tm < tm+1 = T. Here Ja+1qu(ti+)=limε0+Ja+1qu(ti+ε) and Ja+1qu(ti)=limε0Ja+1qu(ti+ε) represent the right and left limits of Ja+1qu(t)att=ti, respectively.

For impulsive system (1) we have limΔ10,Δ20,,Δm0impulsivesystem1=Da+qu(t)=f(t,u(t)),q0,1,ta,T,Ja+1qu(a)=ua,uaC,(2)

Therefore, it means that there exists a hidden condition limΔ10,Δ20,,Δm0thesolutionof impulsivesystem1=thesolutionof system2(3)

Therefore, the definition of solution for impulsive system (1) is provided as follows:

A function z(t) : [a,T] → ℂ is said to be a solution of the fractional Cauchy problem (1) if Ja+1qz(a)=ua, the equation condition Da+qz(t)=ft,zt for each t ∊ (a, T] is verified, the impulsive conditions Δ(Ja+1qz)|t=ti=Δi(z(ti)) (here i = 1,2,..., m)are satisfied, the restriction of z(t) to the interval (tk, tk+1] (here k = 0, 1, 2,...,m) is continuous, and the condition (3) holds.

Therefore, we will consider impulsive system (1) and seek some solutions of impulsive system (1) according to Definition 1.1.

The rest of this paper is organized as follows. In Section 2, some preliminaries are presented. In Section 3, we give the formula of general solution for impulsive differential equations with Riemann-Liouville fractional derivatives. In Section 4, an example is provided to expound the main result.

2 Preliminaries

Firstly, we recall some concepts of fractional calculus [2] and a property for nonlinear fractional differential equations.

The left-sided Riemann-Liouville fractional integral of order α ∊ ℂ (ℜ(α) > 0) of function x(t) is defined by Jaα+xt=1Γqattsα1xsds,t>a, where Γ is the gamma function.

The left-sided Riemann-Liouville fractional derivative of order q ∊ ℂ (ℜ(q)0) of function x(t) is defined by Daq+xt=1Γnqddtnattsnq1xsds,n=q+1,t>a.

By Lemma 2.2 in [11], the initial value problem Daq+ut=ft,ut,qCandq0,1,ta,T,Ja1q+ua=ua,uaC,(4) is equivalent to the following nonlinear Volterra integral equation of the second kind, ut=uaΓqtaq1+1Γqattsnq1fs,usds.(5)

3 Main results

Define a piecewise function u~t=1ΓqJa+1qutk+ttkq1+1Γqtkttsq1fs,usdsforttk,tk+1wherek=0,1,2,,m with Ja+1qutk+=Ja+1qutk+Δkutk. By Definition 2.2, ..., we have Daq+u~t=1ΓnqΓqddtattη1q1Ja+1qutk+ηtkq1+tknηsq1fs,usdsdη=1ΓnqΓqddttkttη1q1Ja+1qutk+ηtkq1+tknηsq1fs,usdsdη=ft,ut|ttk,tk+1 So, ũ(t) satisfies the condition of fractional derivative of (1), and it doesn’t satisfy the condition (3). Thus, we assume that ũ(t) is an approximate solution to seek the exact solution of impulsive system (1).

Let ξ be a constant. A function u(t) is a general solution of system (1) if and only if u(t) satisfies the fractional integral equation ut=uaΓqtaq1+1Γqattsq1fs,usdsforta,t1,uaΓqtaq1+1Γqattsq1fs,usds+i=1kΔiutiΓqttiq1i=1kξΔiutiΓquataq1+attsq1+fs,usdsua+atifs,usdsttiq1tittsq1+fs,usdsforttk,tk+1,(6) provided that the integral in (6) exists.

Proof. “Necessity”. First we can easily verify that Eq. (6) satisfies the hidden condition (3).

Next, Taking Riemann-Liouville fractional derivative to Eq. (6) for each t ∊ (tk, tk+1] (where k = 0, 1, 2, ...,m), we have Daq+ut=Daq+uaΓqtaq1+1Γqattsq1+fs,usds+i=1kΔiutiΓqttiq1i=1kξΔiutiΓquataq1+attsq1+fs,usdsua+atifs,usdsttiq1tittsq1+fs,usds=ft,uttaξi=1kΔiutift,uttaft,utttittk,tk+1=ft,ut|ttk,tk+1.

So, Eq. (6) satisfies Riemann-Liouville fractional derivative of system (1). Using (6) for each tk (here k = 1,2, ...,m), we get Ja+1qutk+Ja+1qutk=1Γ1qattη1q1uηdηttk+1Γ1qattη1q1uηdηt=tk=ΔkutkξΔkutkua+atfs,usdsua+atkfs,usdstktfs,usdsttk=Δkutk.

Therefore, Eq. (6) satisfies impulsive conditions of (1). Then, Eq. (6) satisfies the conditions of system (1).

“Sufficiency”. We prove that the solutions of system (1) satisfy Eq. (6) by mathematical induction. By Definition 2.1, the solution of (1) satisfies ut=uaΓqtaq1+1Γqattsq1fs,usdsforta,t1.(7)

By (7), we have Ja+1qut1+=Ja+1qut1+Δ1ut1=ua+Δ1ut1+at1fs,usds, and the approximate solution ũ(t) (for t ∊ (t1, t2]) is given by u~t=1ΓqJa+1qut1+tt1q1+1Γqt1ttsq1fs,usds=ua+Δ1ut1+at1fs,usds,Γqtt1q1+1Γqt1ttsq1fs,usdsfortt1,t2,(8) with e1(t) = u(t) – ũ(t) for t ∊ (t1, t2]. By limΔ1ut10ut=uaΓqtaq1+1Γqattsq1fs,usdsfortt1,t2, we get limΔ1ut10e1t=limΔ1ut10utu~t=uaΓqtaq1+1Γqattsq1fs,usdsua+at1fs,usdsΓqtt1q11Γqt1ttsq1fs,usds.

Then, we assume e1t=σΔ1ut1limΔ1ut10e1t=σΔ1ut1Γquataq1+attsq1fs,usdsua+at1fs,usdstt1q1t1ttsq1fs,usds. where function σ(.) is an undetermined function with σ(0) = 1. Thus, ut=u~t+e1t=1ΓqσΔ1ut1uataq1+attsq1fs,usds+Δ1ut1tt1q1+1σΔ1ut1ua+at1fs,usdstt1q1+t1ttsq1fs,usdsfortt1,t2.(9)

Using (9), we get Ja+1qut2+=Ja+1qut2+Δ2ut2=ua+Δ1ut1+Δ2ut2+at2fs,usds. Therefore, the approximate solution ũ(t) (for t ∊ (t2, t3])is given by u~t=1ΓqJa+1qut2+tt2q1+1Γqt2ttsq1fs,usds=ua+Δ2ut1+Δ2ut2at2fs,usds,Γqtt2q1+1Γqt2ttsq1fs,usdsfortt2,t3(10) with e2(t) = u(t) – ũ(t) for t ∊ (t2, t3]. Moreover, by (9), the exact solution u(t) of (1) satisfies limΔ1ut10,Δ2ut20ut=uaΓqtaq1+1Γqattsq1fs,usds,fortt2,t3,limΔ1ut10ut=1ΓqσΔ2ut2uataq1+attsq1fs,usds+Δ2ut2tt2q1+1σΔ2ut2ua+at2fs,usdstt2q1+t2ttsq1fs,usdsfortt2,t3,


Therefore, limΔ1ut10,Δ2ut20e2t=limΔ1ut10,Δ2ut20utu~t=1Γquataq1+attsq1fs,usdsua+at2fs,usdstt2q1t2ttsq1fs,usds,(11)



Then, by (11)(13), we obtain e2t=1Γq[σ(Δ1(u(t1)))+σ(Δ2(u(t2)))1]uataq1+attsq1fs,usds+Δ1(u(t1))(tt1)q1Δ1(u(t1))(tt2)q1+1σut1ua+at1fs,usdstt1q1+t1ttsq1fs,usdsσ(Δ2(u(t2)))ua+at2fs,usdstt2q1+t2ttsq1fs,usds.(14)

Thus, ut=u~t+e2t=1Γq[σ(Δ1(u(t1)))+σ(Δ2(u(t2)))1]uataq1+attsq1fs,usds+Δ1(u(t1))(tt1)q1Δ2(u(t2))(tt2)q1+1σut1ua+at1fs,usdstt1q1+t1ttsq1fs,usds+1σ(Δ2(u(t2)))ua+at2fs,usdstt2q1+t2ttsq1fs,usdsfort(t2,t3].(15)

Moreover, letting t2 → t1, we have limt2t1Daq+ut=ft,ut,qC,andq0,1,t(a,t3]andtt1andtt2,ΔJa+1qut=tk=Ja+1qutk+Ja+1qutk=Δkutk,k=1,2,Ja+1qua=ua,uaC,(16) =Daq+ut=ft,ut,qC,andq0,1,t(a,t3]andtt1,ΔJa+1qut=t1=Ja+1qut1+Ja+1qut1+Ja+1qut2+Ja+1qut2=Δ1ut1+Δ2ut2,Ja+1qua=ua,uaC,(17)

Using (9) and (15), we have 1 – σ1 + Δ2) = 1 – σ1) + 1 – σ2). Letting ρ(z) = 1 – σ(z), we get ρ(z + w) = ρ(z) + ρ(w) for ∀z, w ∊ ℂ. So, ρ(z)ξz, here ξ is a constant. Thus, ut=uaΓqtaq1+1Γqattsq1fs,usds+Δ1ut1Γqtt1q1=ξΔ1ut1Γquataq1+attsq1fs,usdsua+at1fs,usdstaq1t1ttsq1fs,usdsfortt1,t2.(18) and ut=uaΓqtaq1+1Γqattsq1fs,usds+Δ1ut1Γqtt1q1+Δ2ut2Γqtt2q1ξΔ1ut1Γquataq1+attsq1fs,usdsua+at1fs,usdstt1q1+tittsq1fs,usdsξΔ2ut2Γquataq1+attsq1fs,usdsua+at2fs,usdstt2q1+t2ttsq1fs,usdsfortt2,t3.(19) Next, for t ∊ (tn, tn+1], suppose ut=uaΓqtaq1+1Γqattsq1fs,usds+i=1nΔiutiΓqttiq1i=1nξΔiutiΓquataq1+attsq1fs,usdsua+atifs,usdsttiq1attsq1fs,usdsforttn,tn+1.(20) Using (20), we have Ja+1qutn+1+=Ja+1qutn+1+Δn+1utn+1=ua+i=1n+1Δiuti+atn+1fs,usds Thus, the approximate solution ũ(t) for t ∊ (tn+1, tn+2] is given by u~t=1ΓqJa+1qutn+1++ttn+1q1+1Γqtn+1ttsq1fs,usds=ua+i=1n+1Δiuti+atn+1fs,usds1Γqttn+1q1+1Γqtn+1ttsq1fs,usdsforttn+1,tn+2(21) with en+1(t) = u(t) – ũ(t) for t ∊ (tn+1, tn+2]. By (20), the exact solution u(t) of (1) satisfies limΔ1ut10,ut=uaΓqtaq1+1Γqattsq1fs,usdsforttn+1,tn+2,Δn+1utn+10 limΔjutj0,1jn+1ut=uaΓqtaq1+1Γqattsq1fs,usds+1in+1,andijΔiutiΓqttiq11in+1,andijξΔiutiΓquataq1+attsq1fs,usdsua+atifs,usdsttiq1tittsq1fs,usdsfortn+1,tn+2. Therefore, limΔ1ut10,en+1t=limΔ1ut10,utu~tΔn+1utn+10Δn+1utn+10=1Γquataq1+attsq1fs,usdsua+atn+1fs,usdsttn+1q1tn+1ttsq1fs,usds,(22) limΔjutj0,1jn+1en+1t=limΔjutj0,1jn+1utu~t=1Γq11jn+1andijξΔiutiuataq1+attsq1fs,usds+1jn+1andijΔiutittiq1+1jn+1andijξΔiutiua+atifs,usdsttiq1+tittsq1fs,usdsua+1jn+1andijΔiuti+atn+1fs,usdsttn+1q1tn+1ttsq1fs,usds.(23)

By (22) and (23), we obtain en+1t=1Γq11in+1ξΔiutiuataq1+attsq1fs,usds+1in+1Δiutittiq11in+1Δiutittiq1+1in+1ξΔiutiua+atifs,usdsttiq1+tittsq1fs,usdsua+atn+1fs,usdsttn+1q1+tittsq1fs,usds.(24) Thus, ut=u~t+en+1t=uaΓqtaq1+1Γqattsq1fs,usds+i=1n+1ΔiutiΓqttiq1i=1n+1ξΔiutiΓquataq1+attsq1fs,usdsua+atifs,usdsttiq1+attsq1fs,usdsforttn+1,tn+2. So, the solution of system (1) satisfies Eq. (6). So, impulsive system (1) is equivalent to the integral equation (6). The proof is now completed. ⏱

4 Example

For system (1) it is difficult to get the analytical solution when ƒ is a nonlinear function in (1). So, a linear example is given to illustrate the obtained result.

Let us consider the general solution of the impulsive fractional system D1+12ut=t,t1,3andt2,ΔJ1+112ut=2=J1+112u2+J1+112u2=δR,J1+112u1=u0R,(25)

By Theorem 3.1, the general solution of impulsive system (25) is obtained as follows: ut=u0Γ(12)t112+1Γ(12)1tts12sds,fort1,2,u0Γ(12)t112+1Γ(12)1tts12sds+δΓ(12)t212ξδΓ(12)u0t1112+1tts12sdsu0+12sdst2122tts12sdsfort2,3.(26) Next, it is verified that Eq. (26) satisfies the condition of system (25). Taking Riemann-Liouville fractional derivative to the both sides of Eq. (26), we have

(i) for t ∊ (1,2] D1+12ut=1Γ(12)ddt1ttη1121u0Γ(12)η1121+1Γ(12)1nηs121sdsdηt1,2=t|t1,2,

(ii)for t ∊ (2, 3] D1+12ut=1Γ(12)ddt1ttη121u0Γ(12)η112+1Γ(12)1nηs12sds+δΓ(12)η212ξδΓ(12)×u0η1112+1nηs12sdsu0+12sdsη2122ηηs12sdsdηt2,3={t|t1ξδ×t|t11Γ(12)Γ(12)ddt12tη121u0+12sdsη2122ηηs12sdsdηt2,3={t|t1ξδt|t1t|t2}t2,3=t|t2,3

So, Eq. (26) satisfies Riemann-Liouville fractional derivative condition of system (25). By Definition 2.1, we obtain J1+112u2+J1+112u2=1Γq1ttη121uηdηt2+1Γq1ttη121uηdηt=2=δΓ(12)Γ(12)2ntη121η2dηt2+ξδ1Γ(12)Γ(12)1ttη121u0η112+1ηηs12sdsu0+12sdsηs122nηs12sdsdηt2+=δξδu0+1tsdsu0+12sds2tsdst2+=δ. That is, Eq. (26) satisfies impulsive condition in system (25).

Finally, it is obvious that the Eq. (26) satisfies the following limit case limδ0D1+12ut=t,t1,3andt2ΔJ1+112ut=2=J1+112u2+J1+112u2+=δR,J1+112u1=u0R,=D1+12ut=t,t1,3,J1+112u1=u0R,(27) So, Eq. (26) is the general solution of system (25).


The authors are deeply grateful to the anonymous referees for their kind comments, correcting errors and improving written language, which have been very useful for improving the quality of this paper.

The work described in this paper is financially supported by the National Natural Science Foundation of China (Grant No. 21576033, 21636004, 61563023, 61261046) and State Key Development Program for Basic Research of Health and Family planning Commission of Jiangxi Province China (Grant No. 20143246), the Natural Science Foundation of Jiangxi Province (Grant No. 20151BAB207013) and the Jiujiang University Research Foundation (Grant No. 8400183).


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About the article

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Received: 2015-10-06

Accepted: 2016-11-15

Published Online: 2016-12-30

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0096.

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