Apple Mahema 6 7 7-9 Publhe Onlne February 6 n SRe hp://wwwrporg/journal/am hp://xoorg/46/am675 Inerval Ollaon Crera for Fraonal Paral Dfferenal Equaon wh Dampng Term Vavel Sahavam Jayapal Kavha * Po Grauae an Reearh Deparmen of Mahema Thruvalluvar Governmen Ar College Rapuram Ina Reeve 8 January 6; aepe 6 February 6; publhe 9 February 6 Copyrgh 6 by auhor an Senf Reearh Publhng In Th work lene uner he Creave Common Arbuon Inernaonal Lene (CC BY hp://reaveommonorg/lene/by/4/ Abra In h arle we wll eablh uffen onon for he nerval ollaon of fraonal paral fferenal equaon of he form D r D u x q x D u x p x f u x g D u x = a u x F x x G= R ( ( ( ( ( ( ( ( ( ( ( I bae on he nformaon only on a equene of ubnerval of he me pae [ raher han whole half lne We oner f o be monoonou an non monoonou By ung a generalze Ra ehnque negral averagng meho Phlo ype kernal an new nerval ollaon rera are eablhe We alo preen ome example o llurae our man reul Keywor Fraonal Parabol Ollaon Fraonal Dfferenal Equaon Dampng Inrouon Fraonal fferenal equaon are now reognze a an exellen oure of knowlege n moellng ynamal proee n elf mlar an porou ruure eleral nework probably an a vo elay elero hemry of orroon elero ynam of omplex meum polymer rheology nural robo * Correponng auhor How o e h paper: Sahavam V an Kavha J (6 Inerval Ollaon Crera for Fraonal Paral Dfferenal Equaon wh Dampng Term Apple Mahema 7 7-9 hp://xoorg/46/am675
V Sahavam J Kavha eonom boehnology e For he heory an applaon of fraonal fferenal equaon we refer he monograph an journal n he leraure []-[] The uy of ollaon an oher aympo propere of oluon of fraonal orer fferenal equaon ha arae a goo b of aenon n he pa few year []-[] In he la few year he funamenal heory of fraonal paral fferenal equaon wh evang argumen ha unergone nenve evelopmen [4]-[] The qualave heory of h la of equaon ll n an nal age of evelopmen In 965 Wong an Buron [] ue he fferenal equaon of he form u a f u g u = In 97 Buron an Grmer [4] ha been nvegae he qualave propere of ( ru af ( u g ( u = In 9 Nanakumaran an Pangrah [5] erve he ollaory behavor of nonlnear homogeneou fferenal equaon of he form Formulaon of he Problem ( r( y q( y p( f ( y g( y = In h arle we wh o uy he nerval ollaory behavor of non lnear fraonal paral fferenal equaon wh ampng erm of he form ( E D r( D u( x q( x D u( x p( x f ( u( x g D u( x ( ( ( = a u x F x x G= R N where a boune oman n a onan D he Remann-Louvlle fraonal ervave of orer of u wh repe o an he Laplaan operaor n N N u( x he Eulean N-pae R (e u( x = Equaon (E upplemene wh he Neumann r = x bounary onon R wh a peewe mooh bounary ( r u x B µ ( x u( x = ( x R ν where γ enoe he un exeror normal veor o an ( x an R B u x = x R In wha follow we alway aume whou menonng ha r C R R a C R R F C G R ; A ( ( A q C( GR q( = mnq( x ; p CGR ( x for ome T ( A f C( RR onvex wh uf ( u > for u g: R L onnuou where L > ( A 4 [ By a oluon of ( E ( B an ( u( x C ( GR r( D u( x C ( GR µ a non negave onnuou funon on p ( = mnp( x wh p( on any [ T x B we mean a non rval funon u( x C ( GR B a o be ollaory n G f ha ar- E alle ollaory f all oluon are ollaory To he be of our knowlege nohng known regarng he nerval ollaon rera of (E (B an (E (B upo now Movavae by []-[5] we wll eablh new nerval ollaon rera for (E (B an (E (B Our reul are eenally new ( B an ( B A oluon u( x of ( E ( B or brary large zero; oherwe nonollaory An Equaon wh an afe G an he bounary onon E 7
V Sahavam J Kavha Defnon A funon H H( where D= ( : < < whh afe H( = H an { } H on D uh ha where h h L ( DR lo Prelmnare H f = belong o a funon la P enoe by H P H C DR H > for > an ha paral ervave H = h H = h H ( ( an ( ( In h eon we wll ee he efnon of fraonal ervave an negral In h paper we ue he Remann-Louvlle lef e efnon on he half ax R The followng noaon wll be ue for he onvenene For [ U u x x x = ( where= ( = = q = q( p = p( = = Γ Γ Γ Γ enoe q Q( = h( H( q Q( = h( H( Defnon [] The Remann-Louvlle fraonal paral ervave of orer < < wh repe o of u x gven by a funon D u x : = ( v u( xv v Γ ( ( prove he rgh han e ponwe efne on R where Γ he gamma funon Defnon [] The Remann-Louvlle fraonal negral of orer > of a funon y: R R on he half-ax R gven by I y : = ( v y( v v for > Γ ( prove he rgh han e ponwe efne on R Defnon [] The Remann-Louvlle fraonal ervave of orer > of a funon y: R R on he half-ax R gven by D y : = I y for > ( ( prove he rgh han e ponwe efne on R where E an Lemma Le y be oluon of Then he elng funon of ( K : = y v v for an > ( K = Γ D y for an > ( 74
V Sahavam J Kavha Ollaon wh Monoony of f(x of (E an (B In h eon we aume ha 5 onan Theorem If he fraonal fferenal nequaly A f monoonou an afe he onon f ( u M > ( where M a D r DU q DU p f K LF (4 ha no evenually pove oluon hen every oluon of ( E an ( B ollaory n G [ where = Proof Suppoe o he onrary ha here a non ollaory oluon u( x of he problem (E an ( B whh ha no zero n [ for ome > Whou lo of generaly we may aume ha u( x > n [ Inegrang (E wh repe o x over we have x D r D u x x q x D u x p x f u x g D u x x = a u x x F x x Ung Green formula an bounary onon ( B follow ha u( x u( x x= S= µ ( x u( x S (6 γ By Jenen nequaly an ( A we ge By ung g D u( x ( D u( x x q( D q x u x x ( q( D u( x x (7 q ( DU( ( ( p x f u x g D u x x p f u x g D u x x p f u x x g D u x ( ( p f U g D u x L> we have p( x f ( u( x g D u x x p f( K L (8 In vew of ( (6-(8 (5 yel Take ( D r D U q U Lp f K F x x F = F x x herefore Lp( f ( K ( F ( D r DU q DU Therefore U( evenually pove oluon of (4 Th onra he hypohe an omplee he (5 75
V Sahavam J Kavha proof Remark Le Then DU ( U' ( = = = ( = = r r U U p p f K f K F F = we ue h ranformaon n (4 The nequaly beome ( U q U Lp f K F (9 Theorem ( an be ae a f he fferenal nequaly r U ( q U Lp f K F ha no evenually pove oluon hen every oluon of (E an (B ollaory n = [ G where Theorem Suppoe ha he onon (A - (A 5 hol Aume ha for any T here ex δ for = uh ha T < δ < < δ < [ ] [ ] afyng If here ex δ ( H P ( δ where v an φ are efne a > Γ [ ] F( = [ ] an ρ C ([ R ( H( δ ( H( δ uh ha δ H( φ H( φ H δ H ( δ δ ( v Q v Q ( for = δ Γ v = exp M ρ { } = v Lp q M Γ( ( φ ρ ρ ρ Then every oluon of ( E ( B ollaory n G Proof Suppoe o he onrary ha u( x be a non ollaory oluon of he problem ( E ( u( x n [ T T Defne he followng Ra ranformaon funon Then for T Dw for ome = Dv ( K( DU w = vr ρ ( T f ( ( w v By ung f ( K( M ( D r DU DU v( r f K D K D r f K f K ( ( ( ρ > an nequaly (4 we ge ( ( B ay 76
V Sahavam J Kavha Dw Dv w v ( ( ( q DU DU F v v( Lp( Mr( D ( K( D ( ρ ( r( f K f K f K F on he n- F on he ner- By aumpon f erval [ ] If val [ ] So herefore nequaly ( beome Dw [ ] u x > hen we an hooe T wh u x < hen we an hooe T wh ( K( Fv f ( [ ] = ( < uh ha < uh ha w q DU DU Dv v Lp Mr D K D r v f K f K = ( ( ( ρ Le w = w v = v q( = q ( U( = U ( p( = p ( K( = K ( ρ( ρ Then Dw = ( DU( = U ( D K( = K ( o ( ranforme no ρ ( ( ( ' q U U v v Lp M K ( ρ v f ( K f ( K Mv ρ v U v q ρ Lp Mr ( U ( r ρ Γ v r f ( K q M Tha Le ( v q ρ Lp M Γ( ρ ( r ρ v r q M ρ MΓ( ( v q Lp M Γ r M Γ ρ ( ρ ( ρ ( ( ( ' q φ MΓ( v δ ( v φ q MΓ( = v be an arbrary pon n ( ( = (4 ubung wh mulplyng boh e of (4 by H( 77
V Sahavam J Kavha an negrang over for δ δ = we oban H φ δ r Γ( q MΓ( v M H H q H δ δ δ v = H( δ ( h ( H( w H( w w H( δ δ MΓ( H( = H( δ ( w δ δ v M ( q h ( H( r v Γ( δ r ( ( v v Γ( Γ( q h ( H( r MΓ H v = H( δ ( ( ( δ w Q δ M Q ( v H( δ ( δ Q Γ r v δ Γ δ Leng an vng boh e by H ( δ δ δ δ 4 Γ( ( δ φ δ ( H Q v (5 H M H On he oher han ubung by mulply boh e of (4 by ( ( for δ we oban δ δ δ δ w δ M Γ( ( v ( δ ( δ ( H an negrang over Γ( H φ H H q H = H δ q M h ( H( H( H( v δ H( δ ( Q ( r v δ Γ Leng an vng boh e by H ( δ δ δ ( ( ( H φ w δ Q r v H 4 ( ( δ MΓ H δ Now we lam ha every non rval oluon of fferenal nequaly (9 ha alea one zero n ( δ (6 Suppoe he onrary By remark whou lo of generaly we may aume ha here a oluon of (9 U > Ang (5 an (6 we ge he nequaly uh ha for ( δ ( δ ( δ δ H( φ H( φ H δ H ( δ Q ( v Q ( v Γ H Γ H δ δ δ whh onra he aumpon ( Thu he lam hol 78
We oner a equene { T } [ V Sahavam J Kavha j uh ha Tj a j By he aumpon of he heorem for eah j N here ex j δj j R uh ha T j j δ j j an ( hol wh δ replae by j δ j repevely for = j N From ha every non rval oluon U ( of (9 ha j a lea one zero n j ( j Nong ha j j j T j N we ee ha every oluon U ( ha arj brary large zero Th onra he fa ha U ( non ollaory by (9 an he aumpon ux ( n [ for ome > Hene every oluon of he problem ( E ( B ollaory n G Theorem Aume ha he onon (A - (A 5 hol Aume ha here ex H P ρ C ([ ( uh ha for any = an lmup lm up H( φ v Q ( > (7 Γ( H( φ v Q ( > (8 Γ( where v ( an φ are efne a n Theorem Then every oluon of Proof For any T = ha T T le here ex > = uh ha δ δ δ In (8 ake = Then here ex E B ollaory n G = = In (7 ake = Then T H ( φ( ( ( r v Q > (9 Γ δ > = uh ha δ H( φ ( v ( Q ( > ( Γ( Dvng Equaon (9 an ( by H ( δ an H ( δ δ ( φ ( H( δ ( φ δ H δ H H ( δ repevely an ang we ge > Q ( v Q ( v Γ H Γ H δ δ δ Then follow by heorem ha every oluon of ( Coner he peal ae H( = H( hen E B ollaory n G H H = h H = h H ( ( ( ( Thu for H = H( P we have h( = h( an we noe hem by h( onanng uh H The ubla enoe by P Applyng Theorem o P we oban he followng reul Theorem 4 Suppoe ha onon (A - (A 5 hol If for eah T here ex H P C ρ an R wh < uh ha δ ([ ( δ T δ δ H( φ( φ( r v r( v( h ( δ > δ δ Γ ( δ v( δ q( v q h( δ H M Γ ( ( ( δ ( v q δ v q δ δ Γ( H ( r ( 79
V Sahavam J Kavha where v ( an φ are efne a n Theorem Then every oluon of ( E an n G Proof Le δ = for = ha = δ hen H H H ( δ = ( δ = For any w L ( we have δ w = w( δ δ δ ( φ = ( φ δ δ δ From ( we have δ H H δ ( = δ δ v q h H v q h H δ ( ( ( = δ δ v h H v h H ( ( ( ( ( δ B ollaory δ H( φ φ( r v r( v( h ( δ > δ δ Γ δ v( δ q( v q h( δ H M Γ v q δ v q δ δ H Γ( ( r δ ( φ ( φ H H > ( Γ Γ ( ( δ ( ( δ v h v h δ δ v q h( H v q h H M Γ δ δ v q H ( Γ( v q ( H δ δ ( φ ( φ H H δ q > r v h ( v q h( H v H Γ( q r v h ( v q h( H v H Γ( δ = Γ ( δ δ q v h( H ( q r v h( H( Γ( δ ne H( H( = we have 8
V Sahavam J Kavha δ ( φ ( φ H H δ > v Q ( v Q ( Γ Γ δ ( Hene every oluon of ( Le H( ( δ ( E B ollaory n G by Theorem = where > a onan Then he uffen onon (7 an (8 an be mofe n he form lm up lm up q v > Γ( r ( φ q v > Γ( r ( φ Corollary Aume ha he onon (A - (A 5 hol Aume for eah = ha an for ome > ρ C [ ( we have q lm up ( φ r v > Γ( r an q lm up ( φ r v > Γ( r( Then every oluon of ( E an B ollaory n G Theorem 5 Suppoe ha he onon (A - (A 5 hol If for eah = an for ome > afe he followng onon lmup an lm up q ( φ q > Γ( Γ q ( φ q > Γ( Γ Then every oluon of ( E an ( B ollaory n G Proof Clearly h ( ( / = h / = Noe ha an ( ( lm up h ( lm up ( Γ = Γ ( ( = Γ lm up h lm up Γ = Γ = Γ ( ( 8
V Sahavam J Kavha Coner lm up H( φ Q ( > Γ( h ( lm up H ( φ q > Γ( H( lm up ( φ ( q ( q Γ( > lm up h ( > Γ Γ ( ( q ( lm up ( φ q ( > Γ Γ Smlarly we an prove oher nequaly Nex we oner H( = R R where a onan an R = an lm R( r Theorem 6 Aume ha he onon (A - (A 5 hol If for eah = an for ome > ρ C [ ( uh ha an lm up R lm up R ( ( ( ( φ v R R q > Γ( R R( v R R q > Γ( R R ( φ Then every oluon of ( E an Proof From (7 lm up lm up lm up lm up B ollaory n G H( φ v Q ( > Γ( r v h ( r H( φ q Γ( H( > / v R R( r R R q / > Γ( r ( R R( ( ( φ lm up R v R R q > Γ( R R( ( ( φ v ( R R( φ q ( > Γ( R R( = 8
V Sahavam J Kavha Smlarly we an prove ha lm up R v ( R R φ q ( > Γ( R R n u If we hooe H( = log > > an H( = we have he followng orollare θ ( u Corollary Suppoe ha he onon (A - (A 5 hol Aume for eah = ha an for ome n > ρ C [ ( we have an lm up lm up n n q log φ ( > ( r Γ log Then every oluon of ( E an n n q log φ > Γ( log B ollaory n G Corollary Suppoe ha he onon (A - (A 5 hol Aume for eah = ha an for ome n > ρ C [ ( we have an lm up lm up n u q n φ θ ( u ( u > Γ θ θ ( u n u q n φ θ ( u ( u > Γ θ θ ( u Then every oluon of ( E an B ollaory n G 4 Ollaon whou Monoony of f(x of (E an (B We now oner non monoonou uaon f ( u ( A6 M > where M a onan u Theorem 4 Suppoe ha he onon (A - (A 4 an (A 6 hol Aume ha for any T here ex δ for = uh ha T < δ < < δ < [ ] [ ] afyng F [ ] [ ] = n (4 8
V Sahavam J Kavha If here ex δ ( H P where v an φ are efne a ( δ an C ρ ([ R > 4 Γ ( H ( δ ( H ( δ uh ha δ H( φ H( φ H δ H ( δ δ ( v Q v Q ( for = δ 4Γ ( = ρ v exp { } = v LM p q Γ( ( φ ρ ρ ρ Then every oluon of ( E ( B ollaory n G Proof Suppoe o he onrary ha u( x be a non ollaory oluon of he problem ( E ( u( x n [ T for ome T Defne he Ra ranformaon funon DU( w = vr ρ ( T K( Then for T ( r DU( D Dw Dv v r DK D r v K K f K By ung K w DU = ( ρ ( M an nequaly (4 we ge w Dv v K( Dw ( K( q DU DU F v v MLp( r( D ( K( D ( ρ ( r( K By aumpon f erval [ ] If erval [ ] So u x > hen we an hooe T wh u x < hen we an hooe T wh Therefore nequaly (6 beome [ ] K( Fv K [ ] = < uh ha < uh ha ( (5 B ay (6 F on he n- F On he n- w q DU DU Dw Dv v MLp r( D K D r v K = ( ( ρ Le w = w v = v q( = q ( U( = U ( p( = p ( K( = K ( r( r( Then Dw = ( DU( = U ( D K( = K ( D r( ρ( = ρ forme no ( (7 = o (7 ran- 84
V Sahavam J Kavha ( ρ v v ( q ρ v w q U U v v MLp K ( ρ v K K w w U v q ρ MLp ρ Γ( ρ v r r v K ( v q ρ MLp Γ( ρ ( r ρ r v q ρ Γ ( v q LM p r r Γ Γ ρ ( ρ ( ρ ( ' q φ Γ( v where ha ( v φ v LM p q ρ r ( ρ ( r ρ( = Γ φ q Γ( = v ( The remanng par of he proof he ame a ha of heorem n eon an hene ome Corollary 4 Suppoe ha he onon (A - (A 4 an (A 6 hol Aume for eah = ha an for ome > ρ C [ ( we have an lm up lm up q ( φ v > 4Γ( r q ( φ v > 4Γ( r Then every oluon of ( E an B ollaory n G 5 Ollaon wh an whou Monoony of f(x of (E an (B In h eon we eablh uffen onon for he ollaon of all oluon of ( E ( B nee he followng: The malle egen value β of he Drhle problem ( x βω ( x n ω = ω ( x = on For h we 85
V Sahavam J Kavha pove an he orreponng egen funon ( x φ pove n Theorem 5 Le all he onon of Theorem be hol Then every oluon of (E an (B ollaory n G Proof Suppoe o he onrary ha here a non ollaory oluon u( x of he problem (E an ( B whh ha no zero n [ for ome > Whou lo of generaly we may aume ha u( x > n [ Mulplyng boh e of he Equaon (E by φ ( x > an hen negrang wh repe o x over we oban for D r( D u( x φ( x x q( x D u( x φ( x x p( x f( ( u( x g( D ( u x φ x x (8 = a( u( x φ( x x F( x φ( x x Ung Green formula an bounary onon ( B follow ha u( x φ( x x= u( x φ( x x β u( x φ( x x = (9 q( x D u( x φ( x x q( D u( x φ( x x ( By ung Jenen nequaly an ( A we ge Se Therefore φ ( φ ( p x f u x g D u x x x p f u x x g D u x x ( p( φ( x xf ( u( x φ( x x φ x x g D u x = ( φ φ ( U u x x x x x φ φ p x f u x g D u x x x p x xf U g D u x ( By ung g D u( x L> we have p( x f ( u( x g D u x x p f( K L φ ( x x ( In vew of ( (9-( ( (8 yel Take D r DU ( q DU ( Lp f( K( F( x φ ( x x φ x F = F( x φ ( x x φ x ( x herefore ( x ( D r DU q DU Lp f K F Re of he proof mlar o ha of Theorem an hene he eal are ome Remark 5 If he fferenal nequaly ( r U q U p f K LF ha no evenually pove oluon hen every oluon of ( E an ( B ollaory n = [ G 86
V Sahavam J Kavha where Theorem 5 Le he onon of Theorem hol Then every oluon of (E an (B ollaory n G Theorem 5 Le he onon of Theorem 4 hol Then every oluon of (E an (B ollaory n G Corollary 5 Le he onon of Corollary hol Then every oluon of (E an (B ollaory n G Theorem 54 Le he onon of Theorem 5 hol Then every oluon of (E an (B ollaory n G Theorem 55 Le he onon of Theorem 6 hol Then every oluon of (E an (B ollaory n G Corollary 5 Le he onon of Corollary hol Then every oluon of (E an (B ollaory n G Corollary 5 Le he onon of Corollary hol Then every oluon of (E an (B ollaory n G Theorem 56 Le all he onon of Theorem 4 be hol Then every oluon of (E (B ollaory n G Corollary 54 Le he onon of Corollary 4 hol Then every oluon of (E an (B ollaory n G 6 Example In h eon we gve ome example o llurae our reul eablhe n Seon an 4 Example 6 Coner he fraonal paral fferenal equaon D D u x D u x ( ( ( u( x ( D u x o n π ( oc ( x n S ( x n x 5 5 7 9 = u( x Γ n xo Γ 4 π π 4 n xn for ( x ( π [ wh he bounary onon ( π Here (E u = u = ( = N= r( = q( x = p( x = o n π ( oc ( x n S ( x n x where C( x an S( x are he Frenel negral namely an x x C( x = o π S( x n π = f ( K( = K( K( = u( x gd ( ( u x = D u x a = 4 7 9 F( x x x π π 4 5 5 = Γ n o Γ n n I eay o ee ha q( = q( x = Bu C( x π an π mn x = mn = x π π o p( x p ( n S x Therefore 87
V Sahavam J Kavha we ake = I lear ha he onon (A - (A 5 hol We may ob- erve ha an = an v ( = o ha ρ ( π = =Γ ( = n ( o n f K K D u x x Γ Ung he propery ab a b we ge Coner π f ( K( > ( > = M gd u( x = D u( x > = L Γ { } = v Lp q M Γ( ( φ ρ ρ ρ π = ( π π ( o n Γ q lm up ( φ r v Γ( r π = lm up ( ( π π ( o n 4 Γ Γ > lm up ( 4 Γ Γ > lm up ( 4 π Γ > lm up π = q lm up ( φ r v Γ( r π = lm up ( = ( π π ( o n 4 Γ Γ Thu all onon of Corollary are afe Hene every oluon of (E ( ollae n ( π [ 88
V Sahavam J Kavha In fa u x = n xo uh a oluon of he problem (E an ( Example 6 Coner he fraonal paral fferenal equaon D D u x D u x ( ( ( x o n ( u( x n o π ( oc ( x n S ( x ( 4 o xn o x D u( x o xn 9 = u( x Γ 4 8π o xn Γ o xo 8π 4 for ( x ( π [ wh he bounary onon Here = N= r( = q ( x = (E u = u π = (4 x x ( x o n p( x = n o π ( oc ( x n S ( x ( 4 o xn o x where C( x an S( x are a n Example an f ( K( = K( o xn = K u x gd u( x = D u( x a( = 4 9 F( x = Γ o xn Γ o xo 8π 8π 4 I eay o ee ha q( = p( = ( π [ o n ] we ake ρ = I lear ha he onon (A - (A 4 an (A 6 hol We may oberve ha ( K( f = M > = K o xn v = o ha = an 89
V Sahavam J Kavha an Coner gd u( x = D u( x > = L { } = v LM p q Γ( ( φ ρ ρ ρ π = ( π ( o n Γ q lm up ( φ r v 4Γ( r π = lm up ( π ( o n Γ 4Γ > lm up ( 4 Γ Γ > lm up ( 4 π Γ π > lm up = q lm up ( φ r v 4Γ( r π = lm up ( = ( π ( o n 4 Γ Γ Thu all he onon of Corollary 4 are afe Therefore every oluon of ( E ( π [ In fa u( x = o xn uh a oluon of he problem ( E an (4 Aknowlegemen (4 ollae n The auhor hank Prof E Thanapan for h uppor o omplee he paper Alo he auhor expre her nere hank o he referee for valuable uggeon Referene [] Abba S Benhohra M an N Guerekaa GM ( Top n Fraonal Dfferenal Equaon Sprnger New York [] Klba AA Srvaava HM an Trujllo JJ (6 Theory an Applaon of Fraonal Dfferenal Equaon 9
V Sahavam J Kavha Elever Sene BV Ameram 4 [] Mller KS an Ro B (99 An Inrouon o he Fraonal Calulu an Fraonal Dfferenal Equaon John Wley an Son New York [4] Polubny I (999 Fraonal Dfferenal Equaon Aaem Pre San Dego [5] Zhou Y (4 Ba Theory of Fraonal Dfferenal Equaon Worl Senf Publhng Co Pe L Hakenak hp://xoorg/4/969 [6] Baleanu D Dehelm K Sala E an Trujllo JJ ( Fraonal Calulu Moel an Numeral Meho Sere on Complexy Nonlneary an Chao Worl Senf Publhng Hakenak [7] Hlfer R (99 Applaon of Fraonal Calulu n Phy Worl Senf Publhng Co Hakenak [8] Jumare G (6 Mofe Remann-Louvlle Dervave an Fraonal Taylor Sere of Non Dfferenable Funon Furher Reul Compuer & Mahema wh Applaon 5 67-76 hp://xoorg/6/jamwa6 [9] Mahao JT Kryakova V an Manar F ( Reen Hory of Fraonal Calulu Communaon n Nonlnear Sene an Numeral Smulaon 6 4-5 hp://xoorg/6/jnn57 [] Manar F ( Fraonal Calulu an Wave n Lnear Voelay Imperal College Pre Lonon [] Feng Q ( Inerval Ollaon Crera for a Cla of Nonlnear Fraonal Dfferenal Equaon wh Nonlnear Dampng Term IAENG Inernaonal Journal of Apple Mahema 4 54-59 [] Feng Q an Meng F ( Ollaon of Soluon o Nonlnear Fore Fraonal Dfferenal Equaon Eleron Journal of Dfferenal Equaon 69 - [] Ogrek S (5 Inerval Ollaon Crera for Funonal Dfferenal Equaon of Fraonal Orer Avane n Dfferene Equaon -8 [4] Prakah P Harkrhnan S Neo JJ an Km JH (4 Ollaon of a Tme Fraonal Paral Dfferenal Equaon Eleron Journal of Qualave Theory of Dfferenal Equaon 5 - hp://xoorg/4/ejqe45 [5] Prakah P Harkrhnan S an Benhohra M (5 Ollaon of Ceran Nonlnear Fraonal Paral Dfferenal Equaon wh Dampng Term Apple Mahema Leer 4 7-79 hp://xoorg/6/jaml48 [6] Harkrhnan S Prakah P an Neo JJ (5 Fore Ollaon of Soluon of a Nonlnear Fraonal Paral Dfferenal Equaon Apple Mahema an Compuaon 54 4-9 hp://xoorg/6/jam474 [7] Sahavam V an Kavha J (5 Fore Ollaon of Soluon of a Neural Nonlnear Fraonal Paral Funonal Dfferenal Equaon Inernaonal Journal of Apple Engneerng Reearh 8-88 [8] Sahavam V an Kavha J (5 Fore Ollaon of Soluon of a Fraonal Neural Paral Funonal Dfferenal Equaon Apple Mahema Reearh 6-7 [9] Sahavam V an Kavha J (5 Fore Ollaon for a Cla of Fraonal Parabol Paral Dfferenal Equaon Journal of Avane n Mahema 569-58 [] L WN an Sheng WH (6 Ollaon Propere for Soluon of a Kn of Paral Fraonal Dfferenal Equaon wh Dampng Term Journal of Nonlnear Sene an Applaon 9 6-68 [] Zhang S an Zhang HQ ( Fraonal Sub-Equaon Meho an I Applaon o Nonlnear Fraonal PDE Phy Leer A 75 69-7 hp://xoorg/6/jphylea9 [] Zheng B an Feng Q (4 A New Approah for Solvng fraonal Paral Dfferenal Equaon n he Sene of he Mofe Remann-Louvlle Dervave Mahemaal Problem n Engneerng 7 p [] Wong JS an Buron TA (965 Some Propere of Soluon of u a f( ugu '' ' = Monahefe für Mahemak 69 64-674 [4] Buron TA an Grmer R (97 Sably Propere of ( ru af ( u g ( u - hp://xoorg/7/bf44 '' ' = Monahefe für Mahemak 74 [5] Nanakumaran AK an Pangrah S (9 Ollaon Crera for Dfferenal Equaon of Seon Orer Mahemaa Slovaa 59 4-454 hp://xoorg/478/75-9-8-z 9