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

Xianmin Zhang; Wenbin Ding; Hui Peng; Zuohua Liu; Tong Shu

doi:10.1515/math-2016-0096

Article Open Access

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

Xianmin Zhang , Wenbin Ding , Hui Peng , Zuohua Liu and Tong Shu

Published/Copyright: December 30, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

Abstract

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]andt≠ti(i=1,2,…,m),Δ(Ja+1−qu)|t=ti=Ja+1−qu(ti+)−Ja+1−qu(ti−)=Δi(u(ti−)),i=1,2,…,m,Ja+1−qu(a)=ua,ua∈C,(1)

where q ∊ ℂ and ℜ(q) ∊ (0, 1), Da+q denotes left-sided Riemann-Liouville fractional derivative of order q and Ja+1−q denotes left-sided Riemann-Liouville fractional integral of order 1 − q. f : J × ℂ → ℂ is an appropriate continuous function, and a = t₀ < t₁ < ... <t_m < t_m+1 = T. Here Ja+1−qu(ti+)=limε→0+Ja+1−qu(ti+ε) and Ja+1−qu(ti−)=limε→0−Ja+1−qu(ti+ε) represent the right and left limits of Ja+1−qu(t)att=ti, respectively.

For impulsive system (1) we have

limΔ1→0,Δ2→0,⋯,Δm→0impulsivesystem1=Da+qu(t)=f(t,u(t)),q∈0,1,t∈a,T,Ja+1−qu(a)=ua,ua∈C,(2)

Therefore, it means that there exists a hidden condition

limΔ1→0,Δ2→0,⋯,Δm→0thesolutionof impulsivesystem1=thesolutionof system2(3)

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

Definition 1.1

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

Definition 2.1

The left-sided Riemann-Liouville fractional integral of order α ∊ ℂ (ℜ(α) > 0) of function x(t) is defined by

Jaα+xt=1Γq∫att−sα−1xsds,t>a,

where Γ is the gamma function.

Definition 2.2

The left-sided Riemann-Liouville fractional derivative of order q ∊ ℂ (ℜ(q) ≥ 0) of function x(t) is defined by

Daq+xt=1Γn−qddtn∫att−sn−q−1xsds,n=ℜq+1,t>a.

By Lemma 2.2 in [11], the initial value problem

Daq+ut=ft,ut,q∈Candℜq∈0,1,t∈a,T,Ja1−q+ua=ua,ua∈C,(4)

is equivalent to the following nonlinear Volterra integral equation of the second kind,

ut=uaΓqt−aq−1+1Γq∫att−sn−q−1fs,usds.(5)

3 Main results

Define a piecewise function

u~t=1ΓqJa+1−qutk+t−tkq−1+1Γq∫tktt−sq−1fs,usdsfort∈tk,tk+1wherek=0,1,2,…,m

with Ja+1−qutk+=Ja+1−qutk−+Δkutk−. By Definition 2.2, ..., we have

Daq+u~t=1Γn−qΓqddt∫att−η1−q−1Ja+1−qutk+η−tkq−1+∫tknη−sq−1fs,usdsdη=1Γn−qΓqddt∫tktt−η1−q−1Ja+1−qutk+η−tkq−1+∫tknη−sq−1fs,usdsdη=ft,ut|t∈tk,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).

Theorem 3.1

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=uaΓqt−aq−1+1Γq∫att−sq−1fs,usdsfort∈a,t1,uaΓqt−aq−1+1Γq∫att−sq−1fs,usds+∑i=1kΔiuti−Γqt−tiq−1−∑i=1kξΔiuti−Γquat−aq−1+∫att−sq−1+fs,usds−ua+∫atifs,usdst−tiq−1−∫titt−sq−1+fs,usdsfort∈tk,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 ∊ (t_k, t_k+1] (where k = 0, 1, 2, ...,m), we have

Daq+ut=Daq+uaΓqt−aq−1+1Γq∫att−sq−1+fs,usds+∑i=1kΔiuti−Γqt−tiq−1−∑i=1kξΔiuti−Γquat−aq−1+∫att−sq−1+fs,usds−ua+∫atifs,usdst−tiq−1−∫titt−sq−1+fs,usds=ft,utt≥a−ξ∑i=1kΔiuti−ft,utt≥a−ft,utt≥tit∈tk,tk+1=ft,ut|t∈tk,tk+1.

So, Eq. (6) satisfies Riemann-Liouville fractional derivative of system (1). Using (6) for each t_k (here k = 1,2, ...,m), we get

Ja+1−qutk+−Ja+1−qutk−=1Γ1−q∫att−η1−q−1uηdηt→tk+−1Γ1−q∫att−η1−q−1uηdηt=tk=Δkutk−−ξΔkutk−ua+∫atfs,usds−ua+∫atkfs,usds−∫tktfs,usdst→tk=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=uaΓqt−aq−1+1Γq∫att−sq−1fs,usdsfort∈a,t1.(7)

By (7), we have Ja+1−qut1+=Ja+1−qut1−+Δ1ut1−=ua+Δ1ut1−+∫at1fs,usds, and the approximate solution ũ(t) (for t ∊ (t₁, t₂]) is given by

u~t=1ΓqJa+1−qut1+t−t1q−1+1Γq∫t1tt−sq−1fs,usds=ua+Δ1ut1−+∫at1fs,usds,Γqt−t1q−1+1Γq∫t1tt−sq−1fs,usdsfort∈t1,t2,(8)

with e₁(t) = u(t) – ũ(t) for t ∊ (t₁, t₂]. By

limΔ1ut1−→0⁡ut=uaΓqt−aq−1+1Γq∫att−sq−1fs,usdsfort∈t1,t2,

we get

limΔ1ut1−→0e1t=limΔ1ut1−→0ut−u~t=uaΓqt−aq−1+1Γq∫att−sq−1fs,usds−ua+∫at1fs,usdsΓqt−t1q−1−1Γq∫t1tt−sq−1fs,usds.

Then, we assume

e1t=σΔ1ut1−limΔ1ut1−→0e1t=σΔ1ut1−Γquat−aq−1+∫att−sq−1fs,usds−ua+∫at1fs,usdst−t1q−1−∫t1tt−sq−1fs,usds.

where function σ(.) is an undetermined function with σ(0) = 1. Thus,

ut=u~t+e1t=1ΓqσΔ1ut1−uat−aq−1+∫att−sq−1fs,usds+Δ1ut1−t−t1q−1+1−σΔ1ut1−ua+∫at1fs,usdst−t1q−1+∫t1tt−sq−1fs,usdsfort∈t1,t2.(9)

Using (9), we get Ja+1−qut2+=Ja+1−qut2−+Δ2ut2−=ua+Δ1ut1−+Δ2ut2−+∫at2fs,usds. Therefore, the approximate solution ũ(t) (for t ∊ (t₂, t₃])is given by

u~t=1ΓqJa+1−qut2+t−t2q−1+1Γq∫t2tt−sq−1fs,usds=ua+Δ2ut1−+Δ2ut2−∫at2fs,usds,Γqt−t2q−1+1Γq∫t2tt−sq−1fs,usdsfort∈t2,t3(10)

with e2(t) = u(t) – ũ(t) for t ∊ (t₂, t₃]. Moreover, by (9), the exact solution u(t) of (1) satisfies

limΔ1ut1−→0,Δ2ut2−→0⁡ut=uaΓqt−aq−1+1Γq∫att−sq−1fs,usds,fort∈t2,t3,limΔ1ut1−→0⁡ut=1ΓqσΔ2ut2−uat−aq−1+∫att−sq−1fs,usds+Δ2ut2−t−t2q−1+1−σΔ2ut2−ua+∫at2fs,usdst−t2q−1+∫t2tt−sq−1fs,usdsfort∈t2,t3,

limΔ2ut2−→0⁡ut=1ΓqσΔ1ut1−uat−aq−1+∫att−sq−1fs,usds+Δ1ut1−t−t1q−1+1−σΔ1ut1−ua+∫at1fs,usdst−t1q−1+∫t1tt−sq−1fs,usdsfort∈t2,t3.

Therefore,

limΔ1ut1−→0,Δ2ut2−→0e2t=limΔ1ut1−→0,Δ2ut2−→0ut−u~t=1Γquat−aq−1+∫att−sq−1fs,usds−ua+∫at2fs,usdst−t2q−1−∫t2tt−sq−1fs,usds,(11)

limΔ1ut1−→0e2t=limΔ1ut1−→0ut−u~t=σΔ2ut2−Γquat−aq−1+∫att−sq−1fs,usds−ua+∫at2fs,usdst−t2q−1−∫t2tt−sq−1fs,usds,(12)

limΔ2ut2−→0e2t=limΔ2ut2−→0ut−u~t=1ΓqσΔ1ut1−uat−aq−1+∫att−sq−1fs,usds+Δ1ut1−t−t1q−1−Δ1ut1−t−t2q−1+1−σΔ1ut1−ua+∫at1fs,usdst−t1q−1−∫t1tt−sq−1fs,usds−ua+Δ1ut1−+∫at2fs,usdst−t2q−1−∫t2tt−sq−1fs,usds.(13)

Then, by (11) – (13), we obtain

e2t=1Γq[σ(Δ1(u(t1−)))+σ(Δ2(u(t2−)))−1]uat−aq−1+∫att−sq−1fs,usds+Δ1(u(t1−))(t−t1)q−1−Δ1(u(t1−))(t−t2)q−1+1−σut1−ua+∫at1fs,usdst−t1q−1+∫t1tt−sq−1fs,usds−σ(Δ2(u(t2−)))ua+∫at2fs,usdst−t2q−1+∫t2tt−sq−1fs,usds.(14)

Thus,

ut=u~t+e2t=1Γq[σ(Δ1(u(t1−)))+σ(Δ2(u(t2−)))−1]uat−aq−1+∫att−sq−1fs,usds+Δ1(u(t1−))(t−t1)q−1−Δ2(u(t2−))(t−t2)q−1+1−σut1−ua+∫at1fs,usdst−t1q−1+∫t1tt−sq−1fs,usds+1−σ(Δ2(u(t2−)))ua+∫at2fs,usdst−t2q−1+∫t2tt−sq−1fs,usdsfort∈(t2,t3].(15)

Moreover, letting t₂→ t₁, we have

limt2→t1Daq+ut=ft,ut,q∈C,andℜq∈0,1,t∈(a,t3]andt≠t1andt≠t2,ΔJa+1−qut=tk=Ja+1−qutk+−Ja+1−qutk−=Δkutk−,k=1,2,Ja+1−qua=ua,ua∈C,(16)

=Daq+ut=ft,ut,q∈C,andℜq∈0,1,t∈(a,t3]andt≠t1,ΔJa+1−qut=t1=Ja+1−qut1+−Ja+1−qut1−+Ja+1−qut2+−Ja+1−qut2−=Δ1ut1−+Δ2ut2−,Ja+1−qua=ua,ua∈C,(17)

Using (9) and (15), we have 1 – σ(Δ₁ + Δ₂) = 1 – σ(Δ₁) + 1 – σ(Δ₂). 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=uaΓqt−aq−1+1Γq∫att−sq−1fs,usds+Δ1ut1−Γqt−t1q−1=ξΔ1ut1−Γquat−aq−1+∫att−sq−1fs,usds−ua+∫at1fs,usdst−aq−1−∫t1tt−sq−1fs,usdsfort∈t1,t2.(18)

and

ut=uaΓqt−aq−1+1Γq∫att−sq−1fs,usds+Δ1ut1−Γqt−t1q−1+Δ2ut2−Γqt−t2q−1−ξΔ1ut1−Γquat−aq−1+∫att−sq−1fs,usds−ua+∫at1fs,usdst−t1q−1+∫titt−sq−1fs,usds−ξΔ2ut2−Γquat−aq−1+∫att−sq−1fs,usds−ua+∫at2fs,usdst−t2q−1+∫t2tt−sq−1fs,usdsfort∈t2,t3.(19)

Next, for t ∊ (t_n, t_n+1], suppose

ut=uaΓqt−aq−1+1Γq∫att−sq−1fs,usds+∑i=1nΔiuti−Γqt−tiq−1−∑i=1nξΔiuti−Γquat−aq−1+∫att−sq−1fs,usds−ua+∫atifs,usdst−tiq−1−∫att−sq−1fs,usdsfort∈tn,tn+1.(20)

Using (20), we have

Ja+1−qutn+1+=Ja+1−qutn+1−+Δn+1utn+1−=ua+∑i=1n+1Δiuti−+∫atn+1fs,usds

Thus, the approximate solution ũ(t) for t ∊ (t_n+1, t_n+2] is given by

u~t=1ΓqJa+1−qutn+1++t−tn+1q−1+1Γq∫tn+1tt−sq−1fs,usds=ua+∑i=1n+1Δiuti−+∫atn+1fs,usds1Γqt−tn+1q−1+1Γq∫tn+1tt−sq−1fs,usdsfort∈tn+1,tn+2(21)

with e_n+1(t) = u(t) – ũ(t) for t ∊ (t_n+1, t_n+2]. By (20), the exact solution u(t) of (1) satisfies

limΔ1ut1−→0,ut=uaΓqt−aq−1+1Γq∫att−sq−1fs,usdsfort∈tn+1,tn+2,⋮Δn+1utn+1−→0

limΔjutj−→0,1≤j≤n+1⁡ut=uaΓqt−aq−1+1Γq∫att−sq−1fs,usds+∑1≤i≤n+1,andi≠jΔiuti−Γqt−tiq−1−∑1≤i≤n+1,andi≠jξΔiuti−Γquat−aq−1+∫att−sq−1fs,usds−ua+∫atifs,usdst−tiq−1−∫titt−sq−1fs,usdsfor∈tn+1,tn+2.

Therefore,

limΔ1ut1−→0,en+1t=limΔ1ut1−→0,ut−u~t⋮⋮Δn+1utn+1−→0Δn+1utn+1−→0=1Γquat−aq−1+∫att−sq−1fs,usds−ua+∫atn+1fs,usdst−tn+1q−1−∫tn+1tt−sq−1fs,usds,(22)

limΔjutj−→0,1≤j≤n+1en+1t=limΔjutj−→0,1≤j≤n+1ut−u~t=1Γq1−∑1≤j≤n+1andi≠jξΔiuti−uat−aq−1+∫att−sq−1fs,usds+∑1≤j≤n+1andi≠jΔiuti−t−tiq−1+∑1≤j≤n+1andi≠jξΔiuti−ua+∫atifs,usdst−tiq−1+∫titt−sq−1fs,usds−ua+∑1≤j≤n+1andi≠jΔiuti−+∫atn+1fs,usdst−tn+1q−1−∫tn+1tt−sq−1fs,usds.(23)

By (22) and (23), we obtain

en+1t=1Γq1−∑1≤i≤n+1ξΔiuti−uat−aq−1+∫att−sq−1fs,usds+∑1≤i≤n+1Δiuti−t−tiq−1−∑1≤i≤n+1Δiuti−t−tiq−1+∑1≤i≤n+1ξΔiuti−ua+∫atifs,usdst−tiq−1+∫titt−sq−1fs,usds−ua+∫atn+1fs,usdst−tn+1q−1+∫titt−sq−1fs,usds.(24)

Thus,

ut=u~t+en+1t=uaΓqt−aq−1+1Γq∫att−sq−1fs,usds+∑i=1n+1Δiuti−Γqt−tiq−1−∑i=1n+1ξΔiuti−Γquat−aq−1+∫att−sq−1fs,usds−ua+∫atifs,usdst−tiq−1+∫att−sq−1fs,usdsfort∈tn+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.

Example 4.1

Let us consider the general solution of the impulsive fractional system

D1+12ut=t,t∈1,3andt≠2,ΔJ1+1−12ut=2=J1+1−12u2+−J1+1−12u2−=δ∈R,J1+1−12u1=u0∈R,(25)

By Theorem 3.1, the general solution of impulsive system(25)is obtained as follows:

ut=u0Γ(12)t−1−12+1Γ(12)∫1tt−s−12sds,fort∈1,2,u0Γ(12)t−1−12+1Γ(12)∫1tt−s−12sds+δΓ(12)t−2−12−ξδΓ(12)u0t−1112+∫1tt−s−12sds−u0+∫12sdst−2−12∫2tt−s−12sdsfort∈2,3.(26)

Next, it is verified thatEq. (26)satisfies the condition of system(25). Taking Riemann-Liouville fractional derivative to the both sides ofEq. (26), we have

(i) for t ∊ (1,2]

D1+12ut=1Γ(12)ddt∫1tt−η1−12−1u0Γ(12)η−112−1+1Γ(12)∫1nη−s12−1sdsdηt∈1,2=t|t∈1,2,

(ii)for t ∊ (2, 3]

D1+12ut=1Γ(12)ddt∫1tt−η12−1u0Γ(12)η−1−12+1Γ(12)∫1nη−s−12sds+δΓ(12)η−2−12−ξδΓ(12)×u0η−1112+∫1nη−s−12sds−u0+∫12sdsη−2−12−∫2ηη−s−12sdsdηt∈2,3={t|t⩾1−ξδ×t|t⩾1−1Γ(12)Γ(12)ddt∫12t−η12−1u0+∫12sdsη−2−12−∫2ηη−s−12sdsdηt∈2,3={t|t⩾1−ξδt|t⩾1−t|t⩾2}t∈2,3=t|t∈2,3

So, Eq. (26)satisfies Riemann-Liouville fractional derivative condition of system(25). By Definition 2.1, we obtain

J1+1−12u2+−J1+1−12u2−=1Γq∫1tt−η12−1uηdηt→2+−1Γq∫1tt−η12−1uηdηt=2−=δΓ(12)Γ(12)∫2nt−η12−1η−2dηt→2+−ξδ1Γ(12)Γ(12)∫1tt−η12−1u0η−1−12+∫1ηη−s−12sds−u0+∫12sdsη−s−12−∫2nη−s−12sdsdηt→2+=δ−ξδu0+∫1tsds−u0+∫12sds−∫2tsdst→2+=δ.

That is, Eq. (26)satisfies impulsive condition in system(25).

Finally, it is obvious that the Eq. (26) satisfies the following limit case

limδ→0D1+12ut=t,t∈1,3andt≠2ΔJ1+1−12ut=2=J1+1−12u2+−J1+1−12u2+=δ∈R,J1+1−12u1=u0∈R,=D1+12ut=t,t∈1,3,J1+1−12u1=u0∈R,(27)

So, Eq. (26) is the general solution of system(25).

E-mail: z6x2m@sohu.com, z6x2m@qq.com

Acknowledgement

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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Received: 2015-10-6

Accepted: 2016-11-15

Published Online: 2016-12-30

Published in Print: 2016-1-1

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