Caputo Equations in the frame of fractional operators with Mittag-Leffler kernels
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1 nvenon Jounl o Reseh Tehnoloy n nneen & Mnemen JRTM SSN: wwwjemom Volume ssue 0 ǁ Ooe 08 ǁ PP 9-45 Cuo uons n he me o onl oeos wh M-ele enels on Qn Chenmn Hou* Ynn Unvesy Jln Ynj 00 ASTRACT: n hs e we s use onl nel omul o suy he onl nel oey o n AR Seonly we esen neon y s omuls o he n AR onl evves wh M-ele enels when he oeos s Fnlly we nvese he sel-jonness eenvlues n eenunon oees o he oesonn onl Sum-ouvlle euons y usn onl neon y s omuls KYWORS: M-ele enels; AR n onl evves; Sum- ouvlle euons NTROUCTON The onl eenl lys ey ole n he evelomen o mhems n sene els 4] 7] 8] eselly o he see onl lulus ws o nees mon sevel mhemns n hs een evelon ly n een yes s vey mennul o evelo new non-lol onl evves 5] 0] The oose enels e non-snul suh s hose wh M-ele enels Wh mes hose onl evves wh M-ele enels moe neesn s h he oesonn onl nels onn Remnn-ouvlle onl nels s o he suue The vne o evelo moe een lohms n solvn onl ynml sysems y onenn only on he eenl euons n hs e we esen neon y s omuls o he n AR onl eenes wh M-ele enels o oe whh wll e use o ove he sel-jonness eenvlues n eenunon oees o sule onl eene Sum-ouvlle euons wh oe ouny onons Fom ] we now he lssl Sum-ouvlle olem o lne een-l euon o seon oe s ouny vlue olem o he om: whee 0 ] 0 e onnuous unons on he nevl ] suh s 0 0 on ] The eenl euon n e wen n he om: whee Pmee o whh he ove ouny vlue olem hs nonvl soluon s lle n eenvlue Volume ssue 0 wwwjemom 9
2 Cuo uons n he me o onl oeos PRMNARS enon ] The M-ele unon o one mee s ene y n he M-ele unon o wo mees s ene y whee z z enon ] The le onl nel o oe 0 sn hs he ollown om s s s The h onl nel o oe 0 enn hs he ollown om s s s whee 0 ] e onnuous on he nevl ] enon 5] e H Then he le Cuo o- nl evve s ene y Then he le Remnn-ouvlle onl evve s ene y AR AR n on he oesonn onl nel s ene y A A A e H Then he h Cuo onl evve s ene y 4 Volume ssue 0 wwwjemom 40
3 Cuo uons n he me o onl oeos Then he h Remnn-ouvlle onl evve s ene y AR AR 5 he oesonn onl nel s ene y A A 6 Fom 5] we hve AR A AR A emm e e onnuous unons on he nevl ] hen A A AR A A AR Poo Usn enon e 0 we e A AR A AR A AR A AR A A A A ] ] enon 4 ] e H he le Cuo onl evve s ene y he le Remnn-ouvlle onl evve s ene y AR n on he h Cuo onl evve s ene y he h Remnn-ouvlle onl evve s ene y AR 0 0 whee Volume ssue 0 wwwjemom 4
4 Cuo uons n he me o onl oeos Volume ssue 0 wwwjemom 4 MAN RSUTS Fom] ell he le enelze onl nel oeo s w w 7 Anloously he h enelze onl nel oeo n e ene y w w 8 emm e H usn n hen he Cuo onl evve wh M-ele enels s ene y Then he Remnn-ouvlle onl evve wh M-ele enels s ene y AR AR Poo euse ] e n so ] e ue o 0! ] ] e so! 0 0 whee 0 n n se hen emm e H hen AR
5 Cuo uons n he me o onl oeos Volume ssue 0 wwwjemom 4 AR Poo Usn emm we e _ AR Smlly he oo o he seon s s ollows: AR Conse he onl Sum-louvlle olem wh Cuo oeo s ] 9 whee 0 0 e el vlue onnuous unons on he nevl ] n ouny onons whee 0 0 Theoem The onl Cuo oeo s sel-jon Poo We hve e su o we hve nen om o we hve ] ] ] 0 ] ]
6 Cuo uons n he me o onl oeos Hene Th s s sel-jon Conse he onl Sum-louvlle olem wh Remnn-ouvlle oeo s R whee AR AR ] 4 AR AR 0 0 e el vlue onnuous unons on he nevl ] ouny onons AR AR whee n R Theoem The onl Remnn-ouvlle oeo Poo The oo s sml o h o Theoem s sel-jon Theoem The eenvlues o he Sum-ouvlle euons wh Cuo oeo e el Poo Assume h s he eenvlue o 9 oesonn o eenunon hen n s omle onjue ssy 7 8 We mully 7 y n 8 y esevely n su o on nen om o n usn Theoem we e 0 n 0 we onlue h Theoem 4 The eenvlues o he Sum-ouvlle euons wh Cuo oeo e el Poo The oo s sml o h o Theoem Theoem 5 The eenunons oesonn o sn eenvlues o he Sum-ouvlle euons 9 e ohoonl wh ese o he weh unon on ] h s 0 When he unons oeson o eenvlues Poo e n e wo sn eenvlues o 9 oesonn o he eenunons esevely Then we hve 9 0 n Volume ssue 0 wwwjemom 44
7 Cuo uons n he me o onl oeos We mully 9 n 0 y n esevely n su he esuls o on Now nen om o we e 0 Sne ollows h 0 whh omlees he oo Theoem 6 The eenunons oesonn o sn eenvlues o he Sum-ouvlle euons 4 e ohoonl wh ese o he weh unon on ] h s When he unons 0 oeson o eenvlues Poo The oo s sml o h o Theoem 5 RFRNCS: ] AA Kls M So RK Sen Genelze M-ele unon n enelze onl lulus oeos ne TnsSeF5No00 49 ] Poluny Fonl eenl uons Aem Pess New Yo 999 ] Chunzh sene n unueness o soluons o onl ouny vlue olems wh - ln oeo 08 4] T Aeljw lenu Monoony esuls o onl eene oeos wh see eonenl enels Av e u 07: ] T Aeljw A yunov ye neuly o onl oeos wh nonsnul M-ele enel J neul Al 07:0 07 6] T Aeljw lenu see onl eenes wh nonsnul see M-ele enels Av e u 06: 06 7] R Hle Alons o Fonl Clulus n Physs Wol Sen Sn-oe 000 8] F Mn Fonl Clulus n Wves n ne Vsoelsy mel Collee Pess onon 00 9] H Gy NF Zhn On new enon o he onl eene Mh Com ] A Fee K ehelm Yu uho Fonl-oe vsoelsy FOV: Consuve evelomen usn he onl lulus NASA s Glenn Reseh Cene Oho00 ] R Hle Alons o Fonl Clulus n Physs Wol Sen Snoe 000 ] R Kles G Rons M Soolov s Anomlous Tnso: Founons n Alons Wley- VCH Wenhem 008 ] R Me T Aeljw A Peeson Sum ouvlle uons n he me o onl oeos wh M-ele enels n he see vesons MhCA08 Volume ssue 0 wwwjemom 45
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