Representation of Solutions of Linear Homogeneous Caputo Fractional Differential Equations with Continuous Variable Coefficients
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1 Repor Nuber: KSU MATH 3 E R 6 Represeo o Souos o Ler Hoogeeous puo Fro ere Equos w ouous Vrbe oees Su-Ae PAK Mog-H KM d Hog-o O * Fu o Mes K Sug Uvers Pogg P R Kore * orrespodg uor e: oogo@ooo Absr We osder e o ude sses o souos o er oogeeous puo ro dere equos w ouous vrbe oees Here we ged seres-represeo o e o ude sse b oees o e osdered equos d e represeo o souo o vue probes usg e o ude sse Aordg o our resus e o ude sse o souos o er oogeeous dere equo w puo ro dervves d ouous vrbe oees s dere represeos ordg o e dsrbuos o e owes order o e ro dervves e equo d e dse ro e ges order o s de order o e ro dervves e equo Kewords: o ude sses er oogeeous equos puo ro dervves MS : 34A8 6A33 roduo s we ow ro dervve s exee oo or desrpo o peoe udg eor d ered ees eds o see d eoog [] So ree uors re wde sudg e eor o ro dere equos d er ppo o oer eds o see d eoog d e ops udes o o e exsee uqueess d represeo o souos o er equos bu so d uer eods sb buro os d oro probes sses o oer ro equos See [] d oer revewg res d ppers o ee seo s pper we sud seres represeo represes e o ude sse o souos o er oogeeous puo ro dere equo w ouous vrbe oees o usg s oees Oe dvge o usg r uus sudg dere equos s provde osed orue seres represeo o souos eve e se w vrbe oees Bo e [] e obed exp represeo o e geer
2 Su-Ae PAK Mog-H KM d Hog-o O souo o e sse o er oogeeous puo ro dere equo w os oees e se udes oe er o ro dervve Hu e [] e osdered er puo ro dere equo w ers o ro dervves d os oees d ob souos o s d o ro dere equos b Ado deoposo eod Luo e [5] e deveoped e opero uus o Musŝ s pe or puo ro dere operors d pped o ob e ex souos o vue probes or er ro dere equos w os oees d ro dervves puo s sese Mor e [6] e obed souos o vue probes o ro dere equos w os oees usg Neu seres or e orrespodg Voerr egr equos d expressed b geerzed Mg-Leer uos [] opero uus o e Musss pe ws rodued or e geerzed Re-Louve ro dervve operor d ws pped o sove e orrespodg vue probe or geer -er er ro dere equo w os oees d GRLF o rbrr orders d pes Bu e provded vues depede o e pe o geerzed Re-Louve ro dervves -ers er ro dere equo d deed e spe o souos depede o e pe Te dd o provde eessr d sue odo or exsee o e souo o -ers er ro dere equo d us e de soe ses s eoed [9] [9] uors prese e exsee d represeo o souo or -ers er vue probe w geerzed Re-Louve ro dervves d os oees b usg opero uus Te sud o e ro dere equos w vrbe oees re so provded Kbs e [3] e vesged souos roud ordr po or er oogeeous puo ro dere equos w seque ro dervves o order < d vrbe oees [8] uors suded o exp represeos o Gree s uo or er Re-Louves ro dere operors w vrbe oees ouous [ d pped o ob exp represeos or souo o o-oogeeous ro dere equo w vrbe oees o geer pe [7] uors suded e exsee d uqueess o e souo o oer dere equo w puo ro dervve e spe o ouous derebe uos usg B xed po eore [3] e suded pproxo eod spe ooo eod o sove oer ro dere equos w odos or boudr odos s re we osder e o ude sses o souos o er oogeeous puo ro dere equo w ers d vrbe oees ouous o e erv [ T] Here we ged seres-represeo o e o ude sse b oees o e osdered equos d e represeo o souo o vue probes usg e o ude sse Our eod s e suessve pproxo Our resu s e dsover o 5 pers o dsrbuos o ro orders e equo w deere e represeos o e souos
3 Represeo o Souos o Fro ere Equos w Vrbe oees Aordg o our resus e o ude sse o souos o er oogeeous dere equo w puo ro dervves d ouous vrbe oees s dere represeos ordg o e dsrbuos o e owes order o e ro dervves e equo d e dse ro e ges order o s de order o e ro dervves e equo Te reder e pper s orgzed s oows seo we se our probe d provde soe preres udg bs oeps d es Furerore we gve 3 pers o dsrbuos o ro orders o e osdered equo w deere e represeos o e souos seos 3 4 d 5 we provde seresrepreseos o e o ude sses b oees o e osdered equos d e represeo o souo o vue probes usg e o ude sse ose 3 ses respeve e s seo we gve soe ousos Probes d Les Preres s pper we sud e oowg er oogeeous puo ro dere equo w vrbe oees: < < T Here T > s rbrr posve uber re oegve re ubers su > > > d ere exs oegve egers su < s evde Rer Te eger depede o e ges order s por d provdes e uber o odos o gve order o esure e exsee d uqueess o e souo o e vue probe o Now we sud vue probe o e equo w e oowg odos: b R We use e uo spes [ b] d [ b] rodued [] We < we deoe b [ b] e spe o uos [ b] d se s deed o b] prur [ b] [ ] For N we deoe b b [ b] e spe o uos w re es ouous derebe o [ b] d ve e dervve x su x [ b] Ad we deoe s oows: [ b] [ b] Te deo o er ors o ese spes re so provded [7] 3
4 Su-Ae PAK Mog-H KM d Hog-o O eo [] Le R [ T ] < Te ro egrs o order > o uo e es o Re-Louve re deed b τ τ dτ > Γ prur we deoe s eo [] We N < e puo ro dervves o order > o uo re deed b! Here s Re-Louve ro dervve: d : > d Le [4] Le N { } d R Te spe [ b] osss o ose d o ose uos w re represeed e or τ ϕ τ dτ! were ϕ [ b] d re rbrr oss eo 3 A sse o uos s ed o ude sse o souos o e oogeeous equo sses < < ; Te ouso o s pper s e souos o ve e dere represeos ordg o e dere dsrbuos o ro orders order o exp s suo er we rodue e oowg dex ses H o ep o sow e dsrbuo o gve orders H { : } 3 Rer H d H H < H φ were φ s e ep se we e H w s e ses dex o ro orders o do o exeed Te s evde s e uber o eees o H d < Tus we < represes e uber o su orders < d prur e s o su ro orders T < 4
5 Represeo o Souos o Fro ere Equos w Vrbe oees We w osder e oowg 3 possbe ses w represe dere dsrbuos o ro orders Ts s e de o s pper w ows us suessu gve represeos o souos o se : H φ s se H φ d our equo s e er er o uow uo e er o - dervve Te s ed e pe s se we ve d < < < < < < 5 < < < se : d ere exss { } su H φ H φ s se H φ d H φ T s ere s o e er o dervve o order o or ower e equo Te s ed e pe s se we ve ] d < < < < < < < < < < ** se : H φ s se φ T s ere o re e ers o dervve o order o ger e equo Te s ed e pe s se ] d us ] or Fro e deo we ow s se our equo beoes ver spe egro-dere equo w be rsored o egr equo w ooogeeous er o poo o w e degree o Now e osder e oowg pproxo sequee or : b /! Te we ve e oowg e * d 4 Le Assue > d [ T ] Te Te vue probe s uque souo [ T ] w s e [ T ] o e pproxo sequee 4 Le [ Assue d [ T ] [ T ] e e vue probe s uque souo [ T ] w s e [ T ] o e pproxo sequee 4
6 Su-Ae PAK Mog-H KM d Hog-o O Proo: Le Te b ssupo we ve [ T ] or < : [ T ] d us :[ T ] Q R R Q s ou d Lpz odo B e resu o [7] e e s proved QE Rer 3 < : [ T ] e e souo o e vue probe sses [ T ] T ] 3 Te o Fude Sse o Souos o e Equos o e Tpe Teore Le [ d Assue d [ T ] [ T ] Assue H φ s s e pe Te ere exss e uque o ude sse [ T ] o souos o d s wre b Proo Fro e - e exsee d uqueess o o ude sses s evde Now e d e o sse s e [ T ] o e pproxo sequee 6 Frs we d Le 6 e d ro 7 we ve B e deo we ve 8 Here we ue 9 Se H φ we ve H For d us 5 7 6
7 Represeo o Souos o Fro ere Equos w Vrbe oees 7 e > d us ro < we ve Se 9 e or d Subsug s o 8 e e rs pproxo o s gve b d se d we rewre s oows: Tus we ge e rs er o 5 e se o Now prove ] [ T Fro we ve Fro e ssupo ] [ T d > d us we ve ] [ T s ] [ T O e oer d ] [ T d > d us b e we ve ] [ T Tereore ] [ T Tus we proved ] [ T Nex we osder e se 7 o d e seod pproxo o Now e : d ue Fro e deo
8 Su-Ae PAK Mog-H KM d Hog-o O 8 Here se < < we ve > d us we rewre s oows: Se ] [ T we ve d ereore Tus e seod pproxo s gve s oows: Se e ] [ T d ereore Fro e ssupo d e we ve ] [ T Tus we ve ] [ T Now uder e ssupo e - pproxo o s provded b d ] [ T we d e -s pproxo o
9 Represeo o Souos o Fro ere Equos w Vrbe oees Tus e -s pproxo o s provded b e sr w s e bove we ge ] [ T B duo we proved or pproxo o s provded b d ] [ T Fro e- e sequee { } overges ] [ T d we ge e rs eee o o sse:
10 Su-Ae PAK Mog-H KM d Hog-o O ] [ T Now or we d e - eee o e o sse Fro 6 d b 7 e rs pproxo o s gve b B e deo s s oows e d d us d us we ve < e > so d us < T s we ve < So e rs pproxo o s provded b d we ve ] [ T Here : Te seod pproxo o s gve b
11 Represeo o Souos o Fro ere Equos w Vrbe oees Here oe we ve B duo e - pproxo o s gve b Tereore QE oror Uder e ssupo o eore souo ] [ T o e vue probe d uque exss d represes b b Here ] [ T s e o ude sse o gve b 5 Teore Assue > ] [ T d φ H Te ere exss e uque o ude sse ] [ T o souos o d s wre b
12 Su-Ae PAK Mog-H KM d Hog-o O Proo Fro e - e exsee d uqueess o o sses s evde Now e d e o sse s e o e pproxo sequee 6 d 7 Fxed we d e - eee o e o sse Here Noe or Se > e or we ve < 3 e < d us we ve 4 Tereore e rs pproxo o s gve b d we ve ] [ T For e seod pproxo s gve s oows: 5
13 Represeo o Souos o Fro ere Equos w Vrbe oees 3 B 3 d e we rewre 5 s oows: e e seod pproxo s gve s oows: Tus we ve e represeo o e seod pproxo o : For 4 3 we d b duo d b eg we ge e represeo d o e o ude sse s obvous ro e ] [ T QE oror Uder e ssupo o eore souo ] [ T o e vue probe d uque exss d represes b b
14 Su-Ae PAK Mog-H KM d Hog-o O Here [ T ] gve b d s e o ude sse o 4 Te o Fude Sse o Souos o e Equos o e Tpe Teore 3 Le < < d ssue d [ T ] [ T ] Assue e equo s e pe T s ssue d ere exss } su H φ d H φ Te { ere exss e uque o ude sse [ T ] o souos o d s wre b 6 Proo Fro e - e exsee d uqueess o o ude sses s evde For { } we d e - eee o e o o ude sses s e [ T ] o e pproxo sequee : 7 8 ` Te rs pproxo o or be wre b ` 9 We eed o ue Here oe d us 4
15 Represeo o Souos o Fro ere Equos w Vrbe oees Tereore order o ue b or ever xed d we us osder e reo bewee d Fro e ssupo ere exss { } su H { : } φ d H { : } φ Tus > or ever d ro < we ve Tereore or { } ere exss { } su d us Nex or we ue Le Noe < < < < d H { : } Te or we ve < O e oer d se < we ve d us or we ve < For we ve 3 Noe < < < d < Tus or { } we ve d [ ] 4 Pug 3 d 4 ogeer we we ve { 5 Nex ore geer e were { } Fro e deo o or { } we ve O e oer d ro < we ve d us or we ve < d 5
16 Su-Ae PAK Mog-H KM d Hog-o O 6 } { e < d us Tus we ve Pug ese ogeer or we ge 6 Fro d 6 e rs pproxo o s wre b 7 d ] [ T Nex we d e seod pproxo o Fro 8 d 9 we wre ` d b 7 we ve Fro d us Le } { Fro 7 we wre B 6 we ve Noe Te we ge
17 Represeo o Souos o Fro ere Equos w Vrbe oees 7 T s e seod pproxo o s gve s oows: ; B duo we ge e - pproxo o : ; Tus e o ude sse o souos o s gve b 6 d 7 d we ve ] [ T QE oror 3 Uder e ssupo o eore 3 souo ] [ T o e vue probe d uque exss d represes b b Here ] [ T s e o ude sse o gve b 6 d 7 Teore 4 Assue > d T ] [ Assue e equo s e pe T s ssue d ere exss } { su φ H d φ H Te ere exss e uque o ude sse ] [ T o souos o d s wre b 8 9 3
18 Su-Ae PAK Mog-H KM d Hog-o O Proo Fro e e exsee d uqueess o o ude sses s evde For { } we d e - eee o e o o ude sses s e o e pproxo sequee 8 d 9 Le e b 9 e rs pproxo o s gve s oows: ` Here ue Fro e deo 3 We ow Fro e ssupo H φ d H φ we ve > or O e oer d < d us Tereore or ere exss { } su Tus d us For we ge [ ] 3 s e se osdero w 6 d us we ve 33 For se we ws ve < we Tus we ws ve d ereore So we ge 34 Pu 333 d 34 ogeer we ve 8
19 Represeo o Souos o Fro ere Equos w Vrbe oees sr w s e bove we ge e seod pproxo o : B duo we ge e - pproxo o d e g we ge e o ude sse [ ] [ T ] QE [ ] d oror 4 Uder e ssupo o eore 4 souo [ T ] o e vue probe d uque exss d represes b b Here [ T ] s e o ude sse o gve b 8 9 d 3 5 Te o Fude Sse o Souos o e Equos o e Tpe Teore 5 Le < < d ssue [ T ] Assue e equo s e pe T s ssue φ Te s e uque H o ude sse { : } [ T ] o souos d s represeed s oows: 9
20 Su-Ae PAK Mog-H KM d Hog-o O 35 Proo Fro e e exsee d uqueess o o ude sse [ T ] s evde Le d ue e rs pproxo o e we ve Fro e ssupo o < or d us H { : } φ s evde [ ] d Sr we es ow d us we ge QE [ T ] oror 5 Uder e ssupo o eore 5 souo [ T ] o e vue probe d uque exss d represes b b Here [ T ] s e o ude sse o gve b 35 Rer 4 Te eore 5 d oror 5 sow e equos o pe ve e souo depede o e oees d orders rees e essee o e equos o pe 6 ousos Frs e o ude sse o souos o er oogeeous dere equo w puo ro dervves d ouous vrbe oees s dere represeos ordg o e dsrbuos o e owes order T s s dere represeos depedg o weer se or < se or se Seod e se d se e o ude sse o souos o er oogeeous dere equo w puo ro dervves d
21 Represeo o Souos o Fro ere Equos w Vrbe oees ouous vrbe oees s dere represeos ordg o e dse bewee e wo ges orders d Trd 5 pers o dsrbuos o ro orders e equo w deere e represeos o e souos re s oows: d eore d oror ; d > eore d oror ; 3 < d eore 3 d oror 3 ; 4 d > eore 4 d oror 4 ; 5 eore 5 d oror 5 Reerees [] Bo B Rvero M d Truo JJ O sses er ro dere equos w os oees Apped Mes d opuo [] Hu Y Luo Y Lu Z A Souo o e er ro dere equo b Ado deoposo eod J opu App M Vo 5 ssue 5 M 8 9 [3] Kbs AARvero MRodrgez-Ger LTruo JJ -A souos o soe er ro dere equos w vrbe oees App M d opu [4] Kbs AA Srvsv HM d Truo J J Teor d Appos o Fro ere Equos Esever Aserd-Too 6 [5] Y Luo R Goreo; A opero eod or sovg ro dere equos A e Ve [6] Mor T d So K Neu-Seres Souo o Fro ere Equo erdspr oro Sees [7] Su-Ae P d Mog-H K Exsee d Uqueess o e Souo o Noer ere Equo w puo Fro ervve e Spe o ouous erebe Fuos KSU-MATH--E-R-5 Fu o es K Sug Uvers rxv:87[-p] pp -9 [8] Mog-H K d Hog-o O O Exp Represeos o Gree s Fuo or Ler Fro ere Operor w Vrbe oees KSU-MATH- -E-R- rxv:899v[-p] pp -4 [9] Mog-H K Gu-o R Hog-o O Exsee d Represeo o Souo o e Vue Probe or e Ler -er Fro ere Equo w Geerzed Re-Louve Fro ervves d os oees b usg Opero Meod KSU-MATH-3-E-R- rxv:3873v [-p] 3 pp -4 [] J T Mdo V Krov F Mrd Ree sor o ro uus ou Noer S Nuer Su
22 Su-Ae PAK Mog-H KM d Hog-o O [] L P e YQ Vgre BM d Podub roduo o THEME SETON: Fro s d oro ero Jour o Buro d os Vo No 4 4 pges O: 4/S8746 [] R Her Y Luo Z Toovs Opero eod or e souo o ro dere equos w geerzed Re-Louve ro dervves Fr App A [3] Yog-su Kg Hu-o oe Spe ooo Meod or Noer Mu- Ter Fro ere Equo ero Sposu oeoro o e 65 Aversr o e Foudo o K Sug Uvers Mes - Sep Jue Pogg PR Kore pp74-78 rxv: [-p]
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