Research Article The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

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1 Hindwi Advnce in Mhemicl Phyic Volume 207, Aricle ID , pge hp://doi.org/0.55/207/ Reerch Aricle The Generl Soluion of Differenil Equion wih Cpuo-Hdmrd Frcionl Derivive nd Noninnneou Impule Xinzhen Zhng, Zuohu Liu, 2 Hui Peng, 3 Xinmin Zhng, 3 nd Shiyong Yng 3 School of Chemicl nd Environmenl Engineering, Jiujing Univeriy, Jiujing, Jingxi , Chin 2 School of Chemiry nd Chemicl Engineering, Chongqing Univeriy, Chongqing , Chin 3 School of Elecronic Engineering, Jiujing Univeriy, Jiujing, Jingxi , Chin Correpondence hould be ddreed o Xinmin Zhng; z6x2m@26.com Received 25 Sepember 206; Acceped 22 November 206; Publihed 9 Februry 207 Acdemic Edior: Kliyperuml Nkkeern Copyrigh 207 Xinzhen Zhng e l. Thi i n open cce ricle diribued under he Creive Common Aribuion Licene, which permi unrericed ue, diribuion, nd reproducion in ny medium, provided he originl work i properly cied. Bed on ome recen work bou he generl oluion of frcionl differenil equion wih innneou impule, Cpuo- Hdmrd frcionl differenil equion wih noninnneou impule i udied in hi pper. An equivlen inegrl equion wih ome undeermined conn i obined for hi frcionl order yem wih noninnneou impule, which men h here i generl oluion for he impulive yem. Nex, n exmple i given o illure he obined reul.. Inroducion Impulive differenil equion re ued in modeling of biology nd phyic nd engineering nd o forh o decribe brup chnge of he e cerin inn. However, he clicl impulive model in which innneou impule were minly conidered in mo of he exiing pper cn no decribe ome procee uch evoluion procee in phrmcoherpy. Hence, Hernández nd O Regn [] nd Pierri e l. [2] preened nd udied kind of differenil equion wih noninnneou impule. Nex, he frcionl differenil equion wih noninnneou impule were conidered in [3, 4]. Recenly, Hdmrd frcionl clculu i geing enion which i n imporn pr of heory of frcionl clculu [5]. The work in [6 ] mde developmen in fundmenl heorem of Hdmrd frcionl clculu. A Cpuoype modificion of Hdmrd frcionl derivive which i clled Cpuo-Hdmrd frcionl derivive w given in [2], nd i fundmenl heorem were proved in [3, 4]. Furhermore, ome work in [5 2] uncover h here i generl oluion for everl frcionl differenil equion wih innneou impule. Therefore, we will ry o conider he generl oluion for differenil equion wih Cpuo-Hdmrd frcionl derivive nd noninnneou impule: C-HD q + u () =f(, u ()), u () =g k (, u ()), u () =u C, (, k+ ], k=0,,...,n, ( k, ], k=,2,...,n, where q C, R(q) (0, ), nd>0nd C-H D q i he + lef-ide Cpuo-Hdmrd frcionl derivive of order q. f:[,t] C C nd g k :( k, ] C C (here k=,2,...,n) re ome pproprie funcion, nd g k denoe noninnneou impule. = 0 = 0 < 2 N N N+ =T. ()

2 2 Advnce in Mhemicl Phyic Firly, we only conider C-H D q + u() = f(, u()) in ech inervl (, k+ ] (k=0,,...,n)in (), nd hen C-HD q + u () =f(, u ()), for (, k+ ] = C-H D q + k u () =f(, u ()), for (, k+ ] (2) u () =u( )+ (ln )q d f (, u ()), for (, k+ ]. Subiuing (2) ino (), we obin u + u () = g k (, u ()), (ln )q f (, u ()) d, for (, ], for ( k, ], k=,2,...,n, (3) g k (,u( )) + (ln )q d f (, u ()), for (, k+ ], k=,2,...,n. In fc, u() ifie condiion of frcionl derivive nd noninnneou impule in yem (). However, we will illure h u() inonequivleninegrlequionof yem (). For yem (), we hve yem ()} g k(,u())=u +(/Γ(q)) (ln(/))q f(,u())(d/) for ( k, ], k,2,...,n} C-HD q + u () =f(, u ()), u () =u + u () =u C, (ln )q f (, u ()) d, (, k+ ], k=0,,...,n, ( k, ], k=,2,...,n (4) C-HD q u () =f(, u ()), (, T], + u () =u C, (uing Lemm 5) u () =u + (ln )q g (, u ()) d, for (, T]. (5)

3 Advnce in Mhemicl Phyic 3 On he oher hnd, leing g k (, u()) = u +(/Γ(q)) (ln(/ )) q f(, u())(d/) for ( k, ] nd ll k,2,...,n} in (3), we ge u () u + = u + (ln )q f (, u ()) d, for (, ], (ln )q f (, u ()) d for ( k, ], k=,2,...,n, (6) u + k Γ(q) [ (ln )q f (, u ()) d + (ln )q f (, u ()) d ], for (, k+ ], k=,2,...,n. If u() i equivlen o yem (), hen (5) i equivlen o (6). Therefore, ome unfi equion cn be obined + (ln )q d f (, u ()), (7) (ln )q f (, u ()) d = (ln )q d f (, u ()) nd here (, k+ ], k =,2,...,N.Therefore, u() i no equivlen wih yem (), nd u() will be regrded n pproxime oluion of yem (). Nex, conidering C-H D q + u() = f(, u()) for (, k+ ] (k=0,,...,n)inwholeinervl[, T],wehve C-HD q + u () =f(, u ()), for (, k+ ] [, T] u () =C k +u + (ln )q f (, u ()) d, for (, k+ ] [, T], here C k reomeconn. (8) Subiuing (8) ino (), we obin C 0 =0nd Hence, ubiuing (8) nd (9) ino (), we ge C k =g k (,u( )) u k (ln )q f (, u ()) d, (9) k=,2,...,n. u () u + = g k (, u ()), (ln )q f (, u ()) d, for (, ], for ( k, ], k=,2,...,n, (0) g k (,u( )) + Γ(q) [ (ln )q f (, u ()) d k (ln )q f (, u ()) d ] for (, k+ ], k=,2,...,n.

4 4 Advnce in Mhemicl Phyic Obviouly, (0) ifie he condiion in yem () nd Eq. (0)} g k (,u())=u +(/Γ(q)) (ln(/))q f(,u())(d/) for ( k, ], k,2,...,n} yem ()}. g k (,u())=u +(/Γ(q)) (ln(/))q f(,u())(d/) for ( k, ], k,2,...,n} Therefore, (0) i oluion of yem (). () Definiion 3 (ee [2, p. 4]). Le R(α) 0 nd n=[r(α)] + nd φ φ : [, b] C :δ (n ) φ(x) AC[, b]}, 0<< b<.then C-H D α +φ(x) exi everywhere on [, b] nd if α N 0, C-HD α x +φ (x) = Γ (n α) = H J n α + δn φ (x), (ln x )n α δ n φ () d (3) Remrk. Equion (0) i only priculr oluion of yem () becue i doe no conin he imporn pr (ln(/)) q f(, u())(d/) of he pproxime oluion u(). Nex, ome definiion nd concluion re inroduced in Secion 2, he equivlen inegrl equion will be given for differenil equion wih Cpuo-Hdmrd frcionl derivive nd noninnneou impule in Secion 3, nd n exmple will how h here exi he generl oluion for hi frcionl differenil equion wih noninnneou impule in Secion Preinrie Definiion 2 (ee [5, p. 0]). Le 0 b be finie or infinie inervl of he hlf-xi R +.Thelef-idedHdmrd frcionl inegrl of order α C (R(α) > 0) of funcion φ(x) i defined by H Jα +φ (x) = Γ (α) x (ln x )α φ () d, where Γ( ) i he Gmm funcion. ( <x<b), (2) The lef-ided Cpuo-Hdmrd frcionl derivive +φ(x) i preened in [2] by he following. C-HD α where differenil operor δ = x(d/dx) nd δ 0 y(x) = y(x). Lemm 4 (ee [2, p. 5]). Le R(α) > 0, n=[r(α)] +,nd φ C[, b].ifr(α) =0or α N,hen C-HD α ( + H Jα +φ) (x) =φ(x). (4) Lemm 5 (ee [2, p. 6]). Le φ AC n δ [, b] or Cn δ [, b] nd α C,hen H Jα ( C-HD α n + +φ) (x) =φ(x) δ k φ () k! k=0 (ln x )k. (5) Lemm 6 (ee [5, p 4]). Le 0<R(q) < nd ξ i conn. Afuncionu() : [, T] C igenerloluionofheyem C-HD q + u () =h(, u ()), (, T], = k (k=,2,...,m), Δu =k =u( + k ) u( k )=Δ k (u ( k )) C, u () =u, u C, k=,2,...,m, if nd only if u() ifie he frcion inegrl equion (6) u () u + = u + k i= (ln )q h d, for (, ], Δ i (u ( i )) + (ln )q h d (7) +ξ k i= Δ i (u ( i )) Γ(q) i [ (ln q i ) h d + (ln )q d h i (ln )q h d ] for ( k, k+ ], k=,2,...,m.

5 Advnce in Mhemicl Phyic 5 provided h he inegrl in (7) exi, nd here h = h(, u()). 3. Min Reul Theorem 7. Le ξ k (here k =,2,...,N)beomerbirry conn. Syem () i equivlen o g k (,u( )) k u () = u + g k (, u ()) (ln )q f d (ln )q f d + + ξ k Γ(q) [g k (,u( )) u k [ (ln )q d f + (ln )q d f (ln )q f d ] (ln )q f d (ln )q f d ] for (, ], for ( k, ], k=,2,...,n, for (, k+ ], k=,2,...,n, (8) provided h he inegrl in (8) exi, nd here f = f(, u()). Proof. Sep (Neceiy). We will verify h (8) ifie ll condiion of yem (). For convenience, we divide hi ecion ino hree ep. Sep.. Equion (8) ifie he frcionl derivive in yem (). By (8), for (, k+ ] (here k=0,,...,n), we ge C-HD q + u () = C-H D q + g k (,u( )) Γ(q) (ln )q d f + + ξ k Γ(q) (g k (,u( )) u k (ln )q f d ) (ln )q f d [ (ln )q d f + (ln )q d f (ln )q f d ]} = f (, u ()) (, k+ ] + ξ k Γ(q) (g k (,u( )) u Γ(q) (ln )q d f ) [ C-H D q ( (ln )q d f + k ) C-H D q ( (ln + )q f d )] (, k+ ] =f(, u ()) (k, k+ ]. (9) So, (8) ifie he frcionl derivive in yem (). Sep.2. We cn verify h (8) ifie noninnneou impulendiniilvlueinyem(). Sep.3. Verify h (8) ifie hidden condiion of yem (). Leing g k (, u()) = u +(/Γ(q)) (ln(/))q f(d/) for ( k, ] nd ll k,2,...,n}in (8), we obin g k (,u())=u +(/Γ(q)) (ln(/))q f(d/) k,2,...,n}, ( k, ] u () = g k (,u())=u +(/Γ(q)) (ln(/))q f(d/) k,2,...,n}, ( k, ] u + (ln )q f d, for (, ], g k (, u ()), for ( k, ], k=,2,...,n, g k (,u( )) k + ξ k Γ(q) (g k (,u( )) u (ln )q f d + k [ (ln )q d f + (ln )q d f (ln )q f d (ln )q f d ) (ln )q f d ] for (, k+ ], k=,2,...,n,

6 6 Advnce in Mhemicl Phyic u + (ln )q f d = u + (ln )q f d u + (ln )q f d for (, ], for ( k, ], k=,2,...,n, for (, k+ ], k=,2,...,n. (20) Moreover,iiureh(20)iheoluionofyem(4)by Lemm 5. Thu, (8) ifie ll condiion of yem (). Sep 2 (Sufficiency).Wewillverifyhheoluionofyem () ifie (8). For convenience, we divide hi ecion ino hree ep. Sep 2.. Verify h he oluion of yem () ifie (8) in inervl (, ] nd (, ]. By Lemm 5, he oluion of () ifie u () =u + (ln )q f d nd u() = g (, u()) for (, ]. for (, ], (2) Sep 2.2. Verify h he oluion of yem () ifie (8) in inervl (, 2 ] nd ( 2, 2 ]. For (, 2 ], he pproxime oluion (by he bove dicuion bou u())i given u () =g (,u( )) + (ln )q d f for (, 2 ], (22) u ()} = Γ(q) [ (ln )q f d (ln )q d f (ln )q d f ]. (24) Therefore, uppoe e () = Γ(q) γ(g (,u( )) u (ln )q f d ) [ (ln )q f d (ln )q d f (ln )q d f ], where γ i n undeermined funcion wih γ(0) =.Thu, (25) wih error e () = u() u() for (, 2 ].Byhepriculr oluion (0), he exc oluionu() of yem () ifie [g (,u( )) u (/Γ(q)) Thu, =u + (ln( /)) q f(d/)] 0 (ln )q f d [g (,u( )) u (/Γ(q)) = [g (,u( )) u (/Γ(q)) u () (ln( /)) q f(d/)] 0 for (, 2 ]. e () (ln( /)) q f(d/)] 0 u () (23) u () = u () +e () =g (,u( )) Γ(q) (ln )q d f + + Γ(q) [ γ (g (,u( )) u (ln )q f d )] (ln )q f d [ (ln )q d f + (ln )q d f (ln )q f d ] for (, 2 ]. (26)

7 Advnce in Mhemicl Phyic 7 On he oher hnd, leing,wege ( C-H D q + u) () =f(, u ()), (, k+ ], k=0,, u () =g (, u ()), (, ], u () =u C, ( C-H D q + u) () =f(, u ()), (, k+ ], k=0,, = u( )=g (,u( )), u () =u C, ( C-H D q + u) () =f(, u ()), (, k+ ], k=0,, = u( + ) u( )=g (,u( )) u (ln )q d f, u () =u C. (27) By uing Lemm 6 o (27), we ge γ(z)=ξ z, z C,nd here ξ i n rbirry conn. Thu, u () =g (,u( )) + u (ln )q d f (ln )q f d + ξ Γ(q) (g (,u( )) (ln )q f d ) [ (ln )q d f + (ln )q d f (ln )q f d ] for (, 2 ]. And u() = g 2 (, u()) for ( 2, 2 ]. (28) Sep 2.3. Verify h he oluion of yem () ifie (8) in inervl (, k+ ] nd ( k+,+ ]. The pproxime oluion (, k+ ] i given by u () =g k (,u( )) + (ln )q d f for (, k+ ], k=,2,...,n, (29) wih error e k () = u() u() for (, k+ ].Moreover,by he priculr oluion (0), he exc oluion of yem () ifie [g k (,u( )) u (/Γ(q)) =u + (ln(/)) q f(d/)] 0 (ln )q f d u () for (, k+ ]. (30) Thu, [g k (,u( )) u (/Γ(q)) = [g k (,x( )) u (/Γ(q)) u ()} = Γ(q) [ (ln(/)) q f(d/)] 0 e k () (ln(/)) q f(d/)] 0 (ln )q f d u () (ln )q d f (ln )q d f ]. Therefore, uppoe e k () =π(g k (,u( )) u k (ln )q f d ) [g k (,u( )) u (/Γ(q)) = Γ(q) π(g k (,u( )) u k (ln )q f d ) (ln(/)) q f(d/)] 0 e k () [ (ln )q d f + (ln )q d f (ln )q f d ], (3) (32)

8 8 Advnce in Mhemicl Phyic where π i n undeermined funcion wih π(0) =. Therefore, u () = u () +e k () =g k (,u( )) Γ(q) (ln )q d f + + Γ(q) π (g k (,u( )) u (ln )q f d k (ln )q f d )} [ (ln )q d f + (ln )q d f (ln )q f d ] for (, k+ ]. On he oher hnd, conider pecil ce of yem () (33) k C-HD q + u () =f(, u ()), ( i, i+ ], i=0,,...,k, u () =u + (ln )q f (, u ()) d, ( i, i ], i=,2,...,k, u () =g k (, u ()), ( k, ], u () =u C, ( C-H D q + u) () =f(, u ()), ( i, i+ ], i=0,,...,k, u () =u + = (ln )q f (, u ()) d, u( )=g k (,u( )), u () =u C, ( i, i ], i=,2,...,k, (34) ( C-H D q + u) () =f(, u ()), ( i, i+ ], i=0,,...,k, u () =u + = (ln )q f (, u ()) d, u( + k ) u( k )=g k (,u( )) u k u () =u C. (ln )q f (, u ()) d, ( i, i ], i=,2,...,k, Uing Lemm 6 for yem (34), we obin π(z) = ξ k z, z C,ndhereξ k i n rbirry conn. Thu By Sufficiency nd Neceiy, yem () i equivlen o (8). The proof i compleed. u () =g k (,u( )) k + u k (ln )q f d + (ln )q f d ) (ln )q d f ξ k Γ(q) (g k (,u( )) [ (ln )q d f + (ln )q d f (ln )q f d ] for (, k+ ]. (35) 4. Exmple In hi ecion, we will give n impulive frcionl yem o illure h here exi generl oluion for frcionl differenil equion wih noninnneou impule. Exmple. Le u conider he following impulive liner frcionl yem: ( C-H D /2 + u) () = ln, (, π ] (π, 2π], 2 u () = in, u () =. ( π 2,π], (36)

9 Advnce in Mhemicl Phyic 9 By Theorem 7, yem (36) h generl oluion + Γ (/2) in, (ln )/2 ln d, for (,π 2 ], for ( π 2,π], π u () = Γ (/2) (ln π )/2 ln d + Γ (/2) (ln )/2 ln d (37) π ξ + Γ (/2) [ Γ (/2) (ln π )/2 ln d ] π [ (ln π )/2 ln d + π (ln )/2 ln d (ln )/2 ln d ], for (π, 2π], where ξ i conn. Afer ome elemenry compuion, (37) cn be rewrien by in, u () = π (ln )3/2, for (, π 2 ], 4 3 π (ln π)3/ π (ln )3/2 > + 2ξ 3 π [ 4 3 π (ln π)3/2 ] for ( π 2,π], (38) [2(ln π) 3/2 +(ln π )/2 (2 ln +ln π) 2(ln ) 3/2 ], for (π, 2π]. > >π Nex, le u verify h (38) ifie ll condiion of yem (36). By Definiion 3, we hve C-HD /2 + ( 4 3 π (ln )3/2 )= 4 Γ ( /2) 3 π (ln ) /2 δ((ln ) 3/2 ) d = Γ (/2) 4 3 π (ln 3 ) /2 2 (ln d )/2 = ln C-HD /2 + [(ln π )/2 (2 ln +ln π) ] >π = Γ ( /2) (ln ) /2 δ[(ln π )/2 (2 ln +ln π) ] d >π = Γ (/2) (ln ) /2 π δ[(ln π )/2 (2 ln +ln π)] d = Γ (/2) (ln π ) /2 [2 (ln π )/2 + 2 (ln π ) /2 (2 ln +ln π)] d = Γ (/2) (ln ) /2 [2 (ln π )/2 π + 2 (ln π ) /2 (2 ln π 3 = Γ (/2) + (3/2) ln π Γ (/2) π +3ln π)] d (ln ) /2 (ln π )/2 d (ln π ) /2 (ln d π ) /2 = 3 2 Γ( 2 ) ln π Γ( 2 ) ln π= 3 2 Γ( 2 ) ln. >π (39)

10 0 Advnce in Mhemicl Phyic Therefore, for (, π/2] nd (π,2π]in (38), we hve C-HD /2 + u () = C-HD /2 + ( π (ln )3/2 )=ln, for (, π 2 ], C-HD /2 + u () = (π,2π] C-HD /2 + 4 (ln π)3/2 3 π π (ln )3/2 + 2ξ > 3 π [ 4 3 π (ln π)3/2 ] [2(ln π) 3/2 +(ln π )/2 (2 ln +ln π) >π 2(ln ) 3/2 > ]} (π,2π] = C-H D / π (ln )3/2 + 2ξ > 3 π [ 4 3 π (ln π)3/2 ] [(ln π )/2 (2 ln +ln π) >π 2(ln ) 3/2 ]} =ln > > + 2ξ 3 π [ (π,2π] 4 3 π (ln π)3/2 ][ 3 2 Γ( 2 ) ln >π 3 2 Γ( 2 ) ln =ln >]} > } (π,2π], (π,2π] for (π, 2π]. (40) Thu, (38) ifie frcionl derivive nd noninnneou impule in yem (36). Therefore, (38) i generl oluion of yem (36). Compeing Inere The uhor declre h hey hve no compeing inere. Acknowledgmen The work decribed in hi pper i finncilly uppored by he Nionl Nurl Science Foundion of Chin (Grn no , , , nd ) nd he Reerch Foundion of Educion Bureu of Jingxi Province, Chin (Grn no. GJJ4738). Reference [] E. Hernández nd D. O Regn, On new cl of brc impulive differenil equion, Proceeding of he Americn Mhemicl Sociey,vol.4,no.5,pp ,203. [2]M.Pierri,D.O Regn,ndV.Rolnik, Exienceofoluion for emi-liner brc differenil equion wih no innneou impule, Applied Mhemic nd Compuion,vol. 29, no. 2, pp , 203. [3] J. Wng, Y. Zhou, nd Z. Lin, On new cl of impulive frcionl differenil equion, Applied Mhemic nd Compuion,vol.242,pp ,204. [4] P.-L. Li nd C.-J. Xu, Mild oluion of frcionl order differenil equion wih no innneou impule, Open Mhemic,vol.3,pp ,205. [5] A. A. Kilb, H. H. Srivv, nd J. J. Trujillo, Theory nd Applicion of Frcionl Differenil Equion,Elevier,Amerdm, The Neherlnd, [6] A.A.Kilb, Hdmrd-ypefrcionlclculu, Journl of he Koren Mhemicl Sociey,vol.38,no.6,pp.9 204,200. [7]P.L.Buzer,A.A.Kilb,ndJ.J.Trujillo, Compoiionof Hdmrd-ype frcionl inegrion operor nd he emigroup propery, Journl of Mhemicl Anlyi nd Applicion,vol.269,no.2,pp ,2002. [8]P.L.Buzer,A.A.Kilb,ndJ.J.Trujillo, Mellinrnform nlyi nd inegrion by pr for Hdmrd-ype frcionl inegrl, Journl of Mhemicl Anlyi nd Applicion, vol.270,no.,pp. 5,2002. [9] P.Thirmnu,S.K.Nouy,ndJ.Triboon, Exiencend uniquene reul for Hdmrd-ype frcionl differenil equion wih nonlocl frcionl inegrl boundry condiion, Abrc nd Applied Anlyi, Aricle ID , Ar. ID , 9 pge, 204. [0] M. Kek, Sequenil frcionl differenil equion wih Hdmrd derivive, Communicion in Nonliner Science nd Numericl Simulion, vol. 6, no. 2, pp , 20. [] B. Ahmd nd S. K. Nouy, A fully Hdmrd ype inegrl boundry vlue problem of coupled yem of frcionl differenil equion, Frcionl Clculu nd Applied Anlyi, vol.7,no.2,pp ,204. [2] F. Jrd, T. Abdeljwd, nd D. Blenu, Cpuo-ype modificion of he Hdmrd frcionl derivive, Advnce in Difference Equion, 202:42, 8 pge, 202. [3]Y.Y.Gmbo,F.Jrd,D.Blenu,ndT.Abdeljwd, On Cpuo modificion of he Hdmrd frcionl derivive, Advnce in Difference Equion, vol.204,ricleno.0,2 pge, 204. [4] Y.Adjbi,F.Jrd,D.Blenu,ndT.Abdeljwd, OnCuchy problem wih CAPuo Hdmrd frcionl derivive, Journl of Compuionl Anlyi nd Applicion, vol.2,no.4, pp.66 68,206. [5] X. Zhng, The generl oluion of differenil equion wih Cpuo-Hdmrd frcionl derivive nd impulive effec, Advnce in Difference Equion,vol.205,ricle25,6pge, 205. [6] X. Zhng, Z. Liu, H. Peng, T. Shu, nd S. Yng, The generl oluion of impulive yem wih cpuo- hdmrd frcionl derivive of order q C (R(q) (, 2)), Mhemicl Problem in Engineering, vol. 206, Aricle ID 80802, 20 pge, 206. [7] X. Zhng, X. Zhng, nd M. Zhng, On he concep of generl oluion for impulive differenil equion of frcionl order q ε (0, ), Applied Mhemic nd Compuion, vol.247,pp , 204. [8] X. Zhng, On he concep of generl oluion for impulive differenil equion of frcionl order q (,2), Applied Mhemic nd Compuion,vol.268,pp.03 20,205.

11 Advnce in Mhemicl Phyic [9] X. Zhng, P. Agrwl, Z. Liu, nd H. Peng, The generl oluion for impulive differenil equion wih Riemnn-Liouville frcionl-order q(, 2), Open Mhemic, vol. 3, pp , 205. [20] X. Zhng, T. Shu, H. Co, Z. Liu, nd W. Ding, The generl oluion for impulive differenil equion wih Hdmrd frcionl derivive of order q (,2), Advnce in Difference Equion,vol.206,ricle4,206. [2] X. Zhng, X. Zhng, Z. Liu, W. Ding, H. Co, nd T. Shu, On he generl oluion of impulive yem wih Hdmrd frcionl derivive, Mhemicl Problem in Engineering, vol. 206, AricleID28430,2pge,206.

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