Existence of Solutions of Three-Dimensional Fractional Differential Systems

Transcription

1 Applied Mahemaics hp://wwwscirporg/journal/am ISSN Online: 5-79 ISSN rin: Exisence of Soluions of Three-Dimensional Fracional Differenial Sysems Vadivel Sadhasivam Jayapal Kaviha Muhusamy Deepa os Graduae Research Deparmen of Mahemaics Thiruvalluvar Governmen Ars College (Affli o eriyar Universiy Rasipuram India How o cie his paper: Sadhasivam V Kaviha J Deepa M (7 Exisence of Soluions of Three-Dimensional Fracional Differenial Sysems Applied Mahemaics hps://doiorg/46/am786 Received: January 6 7 Acceped: February 4 7 ublished: February 7 7 Copyrigh 7 by auhors Scienific Research ublishing Inc This work is licensed under he Creaive Commons Aribuion Inernaional License (CC BY 4 hp://creaivecommonsorg/licenses/by/4/ Open Access Absrac In his aricle we consider he hree-dimensional fracional differenial sysem of he form D u = f v v β ( ( ( ( ( ( = ( ( ( ( ( = ( ( ( ( D v f w w D w f u u ogeher wih he Neumann boundary condiions u = u = v = v = w = w = ( ( ( ( ( ( where D D β D are he sard Capuo fracional derivaives < β A new resul on he exisence of soluions for a class of fracional differenial sysem is obained by using Mawhin s coincidence degree heory Suiable examples are given o illusrae he main resuls Keywords Fracional Differenial Equaions Boundary Value roblem Coincidence Degree Theory Inroducion Fracional calculus is a very effecive ool in he modeling of many phenomena like conrol of dynamical sysems porous media elecro chemisry viscoelasiciy elecromagneic so on The fracional heory is applicaions are menioned by many papers monographs we refer []-[9] For nonlinear fracional boundary value problem many fixed poin heorems were applied o invesigae he exisence of soluions as in references [] [] [] [] On he oher h here is anoher effecive approach Mawhin s coincidence heory which proves o be very useful for deermining he exisence of soluions for DOI: 46/am786 February 7 7

2 V Sadhasivam e al fracional order differenial equaions In recen years boundary value problems for fracional differenial equaions a resonance have been sudied in many papers (see [4]-[] The main moivaion for invesigaing he fracional boundary value problem arises from fracional advecion-dispersion equaion Hu e al [] invesigaed he wo-poin boundary value problem for fracional differenial equaions of he following form where ( = ( ( ( [ ] ( = ( = ( D x f x x x x x D is he Capuo fracional differenial operaor < [ ] f : R R is coninuous In [] Hu e al exended he above boundary value problem o he exisence of soluions for he following coupled sysem of fracional differenial equaions of he form D u = f v v ( ( ( ( ( β ( = ( ( ( ( ( = ( = ( = ( ( = ( D v g u u u v u u v v β where D D are he Capuo fracional derivaives < < β f g: [ ] R R is coninuous I seems ha here has been no work done on he boundary value problem of sysem involving hree nonlinear fracional differenial equaions Moivaed by he above observaion we invesigae he following hree-dimensional fracional differenial sysem of he form β ( = ( ( ( ( ( = ( ( ( ( ( = ( ( ( ( D u f v v D v f w w D w f u u ogeher wih he Neumann boundary condiions u = u = v = v = w = w = ( ( ( ( ( ( where D D β D are he sard Capuo fracional derivaives < β f f f : [ ] R R R is coninuous The main goal of his paper is o esablish some new crieria for he exisence of soluions of ( The mehod is based on Mawhin s coincidence degree heory The resuls in his paper are generalized of he exising ones reliminaries In his secion we give he definiions of fracional derivaives inegrals some noaions which are useful hroughou his paper There are several kinds of definiions of fracional derivaives inegrals In his paper we use he Riemann-Liouville lef sided definiion on he half-axis R he Capuo fracional derivaive Le X Y be real Banach spaces le L: dom L X Y be a Fred- ( 94

3 V Sadhasivam e al holm operaor wih index zero if dim KerL = codim Im L< ImL is closed in Y here exis coninuous projecors : X X Q: Y Y such ha Im = Ker L Ker Q = Im L X = Ker L Ker Y = Im L Im Q I follows ha L : dom L Ker Im L dom L Ker is inverible Here K denoes he inverse of L dom L Ker If Ω is an open bounded subse of X dom L Ω φ hen he map N : X Y will be called L-compac on Ω if QN ( Ω is bounded K ( I Q N : Ω X is compac where I is he ideniy operaor Lemma [4] Le L: dom L X Y be a Fredholm operaor wih index zero N : X Y be L-compac on Ω Assume ha he following condiions are saisfied Lx λnx for every ( x λ ( dom L Ker L Ω ( ; Nx Im L for every x Ker L Ω ; deg ( QN Ker L Ω Ker where Q: Y Y is a projecion such ha L Im L = Ker Q Then he operaor equaion Lx = Nx has a leas one soluion in dom L Ω Definiion [6] The Riemann-Liouville fracional inegral of order > of a funcion y: R R on he half-axis R is given by ( I y( : = ( v y( v dv for > ( provided he righ h side is poinwise defined on R Definiion [6] Assume ha x ( is ( n -imes absoluely coninuous funcion he Capuo fracional derivaive of order > of x is given by ( ( ( n n d x n ( n D : ( x = I = v x ( v dv for > n d ( n where n is he smalles ineger greaer han or equal o provided ha he righ side inegral is poinwise defined on ( n Lemma [6] Le n = If x AC hen > [ ] ( [ ] n ( : = ( I D x x c c c c ( i ( n x where ci = R i = n here n is he smalles ineger greaer i! han or equal o In his paper le us ake X = C [ ] wih he norm x = max { x x X } Y = C[ ] wih he norm y = y Y where x = max [ ] x ( Then we denoe X = X X X wih he norm ( uvw = max { u v w X X X} X Y = Y Y Y wih he norm ( xyz = max { x y z Y Y Y} Clearly Y boh X Y are Banach spaces L : dom L X Y i = by Define he operaors ( i 95

4 V Sadhasivam e al where Define he operaor where ( Lu= D u L v= D v Lw= D w β { ( ( ( } { β ( ( ( } dom L = u X D u Y u = u = dom L = v XD v Yv = v = { ( ( ( } dom L = w XD w Yw = w = L: dom L X Y by ( ( Lu L v Lw L u v w = ( { } dom L= uvw Xu dom L v dom L w dom L Le he Nemyski operaor N : X Y be defined as where N : X Y is defined by N : X Y N : X Y ( = ( Nv N wnu N uvw ( = ( ( ( Nv f v v is defined by Nw ( = f( w ( w ( is defined by N u( = f ( u( u ( Then Neumann boundary value problem ( is equivalen o he operaor equaion Main Resuls ( ( ( Luvw = N uvw uvw dom L In his secion we begin wih he following heorem on exisence of soluions for Neumann boundary value problem ( Theorem Le f f f :[ ] R R R (H here exis nonnegaive funcions ai bi ci C[ ] ( i ( ( β ( ( B C( B C( B C > ( ( β ( such ha for all ( uv R [ ] ( ( ( ( be coninuous Assume ha = wih f uv a b u c v for i= i i i i where Ai = ai Bi = bi Ci = ci ( i = ; (H here exiss a consan M > such ha for all [ ] eiher or ( ( ( uf u v > uf u v > uf u v > u > M v R 96

5 V Sadhasivam e al (H here exiss a consan m m m R saisfying { } ( ( ( uf u v < uf u v < uf u v < ; min m m m > M eiher M > such ha for every ( ( ( mn m > mn m > mn m > or ( ( ( mn m < mn m < mn m < Then Neumann boundary value problem ( has a leas one soluion Lemma Le L be defined by ( Then ( ( uvw X( uvw ( u( v( w( Ker L = Ker L Ker L Ker L { } = = ( Im L = Im L Im L Im L {( xyz Y ( s xs ( ds β ( s y( s s ( s z( s s } = = d = d = = = has he soluion roof By Lemma Lu D u( ( = ( ( u u u ( (4 From he boundary condiions we have { ( } Ker L= u Xu= u For x Im L here exiss u dom L such ha x= Lu Y By using he Lemma we ge u( = ( s x( s ds u( u ( ( Then we have u ( = ( s x( s ds u ( ( By he boundary value condiions of ( we can ge ha x saisfies ( ( s x s ds = On he oher h suppose x Y saisfies ( s x ( s ds = Le u ( = I x ( hen u dom L D u ( = x ( Hence x Im L Then we ge Similarly we have { ( ( } Im L = x Y s x s ds = { } β { ( } ( ( Ker L = v Xv= v Im L = y Y s ysds = 97

6 V Sadhasivam e al { } { ( } ( ( Ker L= w Xw= w Im L= z Y s zsds = Lemma 4 Le L be defined by ( Then L is a Fredholm operaor of index zero : X X Q: Y ors can be defined as Y are he linear coninuous projecor opera- ( = ( = ( ( ( ( u v w u v w u v w ( = ( QxQ yqz Q x y z (( ( s x( s d s β ( ( s ( ( ( ( β y s s s z s s = d d Furher more he operaor K : Im L dom L Ker can be wrien by ( β ( = ( ( ( K x y z I x I y I z roof Clearly Im = Ker L ( uvw = uvw ( I follows ha uvw = uvw uvw uvw we have X = Ker Ker L By ( (( ( ( using simple calculaion we ge ha Ker L Ker {( } For ( xyz Y we have X = Ker Ker L = Then we have ( = ( = ( Q xyz Q QxQyQz QxQ yqz By he definiion of Q we ge Similarly we can show ha ( ( = Q xyz Q xyz Le ( ( ( Q x= Qx s x s d s= Qx Qy= Qy Q z= Qz Thus we can ge ( xyz (( xyz Q( xyz Q( xyz = where ( xyz Q( xyz Ker Q Q( xyz Im Q I follows ha Ker Q = Im L Q ( xyz = Q( xyz we ge Im Q Im L {( } is clear ha Y = Im L Im Q = I Thus dim Ker L = dim Im Q = codim Im L L Hence L is a Fredholm operaor of index zero From he definiions of K we will prove ha xyz Im L we have dom L Ker Infac for ( β β ( ( ( ( ( ( K is he inverse of LK x y z = D I x D I y D I z = x y z (5 Moreover for ( uvw dom L Ker we have u( v( w( = = = 98

7 ( ( ( ( ( ( V Sadhasivam e al β β ( = ( = ( u( u( u ( v v( v ( w w( w ( K L u v w I D u I D v I D w ( ( which ogeher wih he boundary condiion ( ha ( ( uvw From (5 (6 we ge ( u = v ( = w ( = yields K Luvw = (6 K = L dom L Ker Lemma 5 Assume Ω X is an open bounded subse such ha dom L Ω φ hen N is L-compac on Ω roof By he coninuiy of f f f we can ge QN ( Ω ( ( K I Q N Ω are bounded By he Arzela-Ascoli heorem we will prove K I Q N Ω X is equiconinuous M > i = ha ( ( From he coninuiy of f f f here exis consans i ( such ha for all ( uvw Ω ( ( ( I Q N v M I Q N w M I Q N u M Furhermore for < ( uvw Ω we have ( ( ( ( ( ( ( K( I Q ( Nv ( Nw( Nu ( K( I Q ( Nv ( Nw( Nu ( β I ( I Q N v( I ( I Q N w( I ( I Q N u( ( K I QN u v w K ( I QN u ( v ( w ( = ( β ( I ( I Q Nv( I ( I Q Nw( I ( I Q Nu( = ( ( ( ( β ( I Q Nw ( I ( I Q Nw ( ( ( ( ( = I I Q Nv I I Q Nv β I I I Q Nu I I Q Nu By ( ( ( ( I I Q Nv I I Q Nv ( s ( I Q Nv ( s ds ( s ( I Q Nv ( s ds ( M M (( ( d ( = s s s s ds ( ( ( ( ( ( ( ( ( I I Q Nv I I Q Nv ( ( ( s ( I Q Nv ( s ds ( s ( I Q Nv ( s ds ( M ( ( ( ( s s ds ( s ds M = 99

8 V Sadhasivam e al Similarly we can show ha M I ( I Q Nw ( I ( I Q Nw ( β β β β ( ( ( ( ( ( ( ( β M I I Q Nw I I Q Nw β β β β M I ( I Q Nu( I ( I Q Nu( ( ( ( ( Since ( ( ( ( β ( ( M I I Q Nu I I Q Nu β β ( ( are uniformly coninuous on [ ] we K I Q N Ω X is have K ( I Q N( Ω X is equiconinuous Thus ( : compac Lemma 6 Assume ha ( H ( H hold hen he se Ω = ( uvw doml Ker LLuvw ( = λn( uvw λ ( is bounded { } \ roof Le ( uvw Ω hen N( uvw Im L ( ( ( ( s f sv s v s ds = β ( ( ( ( s f sws w s ds = ( ( ( ( s f su s u s ds = By (4 we ge Then by inegral mean value heorem here exis consans ξηζ ( such ha f( ξ v( ξ v ( ξ = f( η w( ( η w ( η = f ( ζ u( ζ u ( ζ = Then we ge v( ξ f( ξ v( ξ v ( ξ = w( η f( η w( η w ( η = u( ζ f ( ζ u( ζ u ( ζ = From ( H we ge v( ξ M w( η M u( have ( = ( ζ ( d ζ M Hence we u u u s s M u (7 ζ We obain Similarly we can show ha u M u (8 By Luvw ( λn( uvw = we ge v M v (9 w M w (

9 V Sadhasivam e al λ u( = ( s f ( sv ( s v ( s ds u( ( λ v ( = s f sws w s s v ( β β ( ( ( ( d ( Then So λ w ( = ( s f ( su ( s u ( s ds w( ( λ u ( = ( s f ( sv ( s v ( s d s ( λ β v ( = ( s f ( sws ( w ( s ds ( β λ w ( = ( s f ( su ( s u ( s d s ( u = s f sv s v s ds ( ( ( ( ( s a s b s v s c s v s ds ( ( ( ( ( ( ( A BM ( B C v ( s ds ( A BM ( B C v ( ( Similarly we have v A BM ( B C w ( β w A BM ( B C u ( ( ( Combining ( wih ( we ge v A BM B C A BM ( ( ( ( ( ( β ( B C ( B C u (4 Combining (4 wih ( we ge u A BM B C A BM [ ( ( ( ( ( ( ( ( ( β β ( B C ( B C ( A B M ( B C ( B C ( B C u

10 V Sadhasivam e al Thus from ge ( ( β ( ( B C( B C( B C ( ( β ( > (4 we u A BM ( ( ( ( ( ( ( ( β ( β B C B C B C ( B C ( A B M ( ( B C ( B C ( A B M : = K v A BM B C A BM ( ( ( ( ( ( β ( B C ( B C K : K = w A BM ( B C K : = K ( From (8 (9 ( we have ( { } u v w max K M K M K M : = K X Hence Ω is bounded Lemma 7 Assume ha ( H holds hen he se {( uvw ( uvw LN( uvw L} Ω = Ker Im is bounded roof For ( uvw Ω we have ( ( Then from N( uvw Im L From ( H imply ha ( ( s f sm d s = β ( ( s f sm ds = ( ( s f sm d s = m m m M ( uvw M u v w = m m m m m m R X Thus we ge Therefore Ω is bounded Lemma 8 Assume ha he firs par of ( H holds hen he se Ω = ( uvw Lλ( uvw ( λ QN( uvw = ( λ [ ] { } Ker is bounded roof For ( uvw Ω we have ( ( ( ( ( ( u v w = m m m m m m R λm λ s f sm ds = (5 β ( ( ( ( λm λ β s f sm ds= (6

11 V Sadhasivam e al If m m m m or m or or or ( ( ( ( λm λ s f sm ds = (7 λ = hen by ( H we ge = = = For λ ( ] > M from ( H m m m m M If λ = hen we obain m m m M Oherwise if one has ( ( ( ( λm λ s mf sm ds> β ( ( ( ( λm λ β s m f sm ds> ( ( ( ( λm λ s mf sm d s> which conradic o (5 or (6 or (7 Hence Ω is bounded Remark Suppose he second par of ( H holds hen he se {( uvw L λ( uvw ( λ QN( uvw ( λ [ ]} Ω = Ker = is bounded roof of he Theorem : Se Ω= {( uvw X ( uvw < max { KM } X } From he Lemma 4 Lemma 5 we can ge L is a Fredholm operaor of index zero N is L-compac on Ω By Lemma 6 Lemma 7 we obain ( Luvw ( λn( uvw for every (( uvw λ ( doml\ KerL Ω ( ; ( Nx Im L for every ( uvw Ker L Ω Choose (( λ = ± λ( ( λ ( H uvw uvw QN uvw By Lemma 8 (or Remark we ge H( ( uvw λ for ( uvw Ker L Ω Therefore deg ( QN Ker L Ω = deg ( H ( Ker L Ω KerL = deg ( H( Ker L Ω = deg ( ± I Ker L Ω Thus he condiion ( of Lemma is saisfied By Lemma we obain Luvw ( = N( uvw has a leas one soluion in dom L Ω Hence Neumann boundary value problem ( has a leas one soluion This complees he proof 4 Examples In his secion we give wo examples o illusrae our main resuls Example Consider he following Neumann boundary value problem of fracional differenial equaion of he form

12 V Sadhasivam e al 5 4 ( = ( ( ( ( ( D u v 6 v 8 8 ( = ( ( cos ( ( D v w 4 w ( = ( ( sin ( ( D w u 8 u u = u = v = v = w = w = ( ( ( ( ( ( (8 Here 5 7 = β = = Moreover 4 4 f ( ( ( ( ( ( ( v v = v 6 v 8 8 f ( w ( w ( = ( w ( 4 cos w ( 6 6 f ( u( u ( = ( u( 8 sin u ( Now le us compue ( ( ( a b c from f ( v ( v( f ( ( ( ( ( ( ( v v = v 6 v 8 8 = ( v ( 6 v ( 8 8 ( v ( f ( v ( v ( v ( = = = Also 8 8 From he above inequaliy we ge a ( b ( c ( f w w w w 6 6 ( ( ( = ( ( 4 cos ( 5 f ( w ( w ( w ( = = = Finally 6 6 Here a ( b ( c ( f ( u( u ( = ( u( 8 sin u ( 8 f ( u( u ( u( We ge a ( = b ( = c ( = And we ge ( B ( = B ( = C ( C ( C ( Also B = = = = Choose 8 M = M = 8 4

13 V Sadhasivam e al ( ( β ( ( B C( B C( B C ( ( β ( = > ( BBB where = π All he condi- 4 4 ions of Theorem are saisfied Hence boundary value problem (8 has a leas one soluion Example Consider he Neumann boundary value problem of fracional differenial equaion of he following form Here 4 4 ( = ( ( ( ( ( D u v 5 log v ( = ( ( ( ( ( D v w 7 w ( = ( ( ( ( D w u arcan u 5 u = u = v = v = w = w = ( ( ( ( ( ( 4 5 = β = = Moreover 4 4 f ( v ( v ( = ( v ( 5 log ( v ( 7 6 f( ( ( ( ( ( ( w w = w 7 w 9 5 (9 7 f ( u( u ( = ( u( arcan u ( 5 Now le us compue ( ( ( a b c from f ( v ( v( 4 f ( v ( v ( = ( v ( 5 log ( v ( 7 4 ( v ( = ( v ( 5 v ( 7! 5 f v v v v 7 7 ( ( ( ( ( 5 = = = Also 7 7 From he above inequaliy we ge a ( b ( c ( 6 f( ( ( ( ( ( ( w w = w 7 w = ( w ( 7 w ( 9 5 5

14 V Sadhasivam e al 7 f ( w ( w ( w ( w ( w ( w ( = = = Similarly Here a ( b ( c ( 7 f ( u( u ( = ( u( arcan u ( 7 ( u ( = ( u( u ( 5 f ( u( u ( u( u ( 5 Here a ( = b ( = c ( = We ge B ( = ( B ( = C ( = C ( = ( Also where 5 75 C = Choose 5 ( ( β ( ( B C( B C( B C ( ( β ( = > = M B = 9 = M = 5 π Hence all he condi- ions of Theorem are saisfied Therefore boundary value problem (9 has a leas one soluion 5 Conclusion We have invesigaed some exisence resuls for hree-dimensional fracional differenial sysem wih Neumann boundary condiion By using Mawhin s coin- cidence degree heory we esablished ha he given boundary value problem admis a leas one soluion We also presened examples o illusrae he main resuls Acknowledgemens The auhors would like o hank he anonymous reviewers for heir valuable commens suggesions o improve he qualiy of he manuscrip References [] Miller KS Ross B (99 An Inroducion o he Fracional Calculus Fracional Differenial Equaions Wiley New York 6

15 V Sadhasivam e al [] Samko SG Kilbas AA Marichev OI (99 Fracional Inegrals Derivaives Gordon Breach Science ublishers Yverdon [] Hilfer Z ( Appliaions of Fracional Calculus in hysics World Scienific Singapore hps://doiorg/4/779 [4] Mezler R Klafer J ( Boundary Value roblems for Fracional Diffusion Equaions hysics A hps://doiorg/6/s78-47(995-8 [5] Kilbas AA Srivasava HM Trujillo JJ (6 Theory Applicaions of Fracional Differenial Equaions Elsevier Amserdam [6] Lakshmikanham V Leela S Vasundhara Devi J (9 Theory of Fracional Dynamic Sysems Cambridge Academic ublishers Cambridge [7] Mainardi A ( Fracional Calculus Waves in Linear Viscoelasiciy Imperial College ress London hps://doiorg/4/p64 [8] Abbas S Benchora M N Guerekaa GM ( Topics in Fracional Differenial Equaions Springer New York hps://doiorg/7/ [9] Zhou Y (4 Basic Theory of Fracional Differenial Equaions World Scienific Singapore hps://doiorg/4/969 [] Ahmad B Nieo JJ (9 Exisence Resuls for a Coupled Sysem of Nonlinear Fracional Differenial Equaions wih Three-oin Boundary Condiions Compuers Mahemaics wih Applicaions hps://doiorg/6/jcamwa979 [] Liu Y Ahmad B Agarwal R ( Exisence of Soluions for a Coupled Sysem of Nonlinear Fracional Differenial Equaions wih Fracional Boundary Condiions on he Half-Line Advances in Difference Equaions 46 hps://doiorg/86/ [] Aphihana A Nouyas SK Tariboon J (5 Exisence Uniqueness of Symmeric Soluions for Fracional Differenial Equaions wih Muli-oin Fracional Inegral Condiions Boundary Value roblems 5 68 hps://doiorg/86/s [] Wang Y (6 osiive Soluions for Fracional Differenial Equaion Involving he Riemann-Sieljes Inegral Condiions wih Two arameers Journal of Nonlinear Science Applicaions [4] Mawhin J (99 Topological Degree Boundary Value roblems for Nonlinear Differenial Equaions in Topological Mehods for Ordinary Differenial Equaions Lecure Noes in Mahemaics hps://doiorg/7/bfb8576 [5] Kosmaov N ( A Boundary Value roblem of Fracional Order a Resonance Elecronic Journal of Differenial Equaions 5 - [6] Bai Z Zhang Y ( The Exisence of Soluions for a Fracional Muli-oin Boundary Value roblem Compuers Mahemaics wih Applicaions hps://doiorg/6/jcamwa8 [7] Wang G Liu W Zhu S Zheng T ( Exisence Resuls for a Coupled Sysem of Nonlinear Fracional m-oin Boundary Value roblems a Resonance Advances in Difference Equaions 44-7 hps://doiorg/55//7876 [8] Zhang Y Bai Z Feng T ( Exisence Resuls for a Coupled Sysem of Nonlinear Fracional Three-oin Boundary Value roblems a Resonance Compuers Mahemaics wih Applicaions 6-47 hps://doiorg/6/jcamwa5 [9] Jiang W ( Solvabiliy for a Coupled Sysem of Fracional Differenial Equaions a Resonance Nonlinear Analysis 85-9 hps://doiorg/6/jnonrwa 7

16 V Sadhasivam e al [] Hu Z Liu W Rui W ( Exisence of Soluions for a Coupled Sysem of Fracional Differenial Equaions Springer Berlin -5 hps://doiorg/86/ [] Hu L (6 On he Exisence of osiive Soluions for Fracional Differenial Inclusions a Resonance Springerlus hps://doiorg/86/s [] Hu Z Liu W Chen T ( Two-oin Boundary Value roblems for Fracional Differenial Equaions a Resonance Bullein of he Malaysian Mahemaical Sociey Series [] Hu Z Liu W Chen T ( Exisence of Soluions for a Coupled Sysem of Fracional Differenial Equaions a Resonance Boundary Value roblems 98 hps://doiorg/86/ Submi or recommend nex manuscrip o SCIR we will provide bes service for you: Acceping pre-submission inquiries hrough Facebook LinkedIn Twier ec A wide selecion of journals (inclusive of 9 subjecs more han journals roviding 4-hour high-qualiy service User-friendly online submission sysem Fair swif peer-review sysem Efficien ypeseing proofreading procedure Display of he resul of downloads visis as well as he number of cied aricles Maximum disseminaion of your research work Submi your manuscrip a: hp://papersubmissionscirporg/ Or conac am@scirporg 8