On the Solution of Nonlinear Partial Differential Equation of Fractional Order

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2 aacal a oaoal os Scc a r A al coo : a aac sac r s a - alco o A s a lar clos a bo oraor o a s s S { } s a aly o lar clos a bo oraors o a s s S S o s a absrac co o als a I s ass a A ras a aalyc sro s coo ls: A r k or a 3 k or > s or a k s a os cosa s Zaa 6 s sos a s orly Hölr coos or ry S ; a s k 4 or all > r k a ar os cosas W sos also a r ss a br sc a k 5 r > a k s a os cosa s 378 oc a S or ac a ac > Also s ass a co sass ollo coos : s orly Hölr coos ; a s W W 6 or all > a all W r a ar os cosas W a s or sass scz coo W W W W ; 7 or all W W a all r s a os cosa W W s a cos a 8 ollo Gla a Slo s 9 ca ral o orr > by I I ca ra o orr by rs a absrac co als o s sabl o rr acy robl or A 9 b s o all coos cos a o a or by a y a solo o acy robl a a absrac co sc a ollo coos ar sas : a a S or all b ss a s coos o r c sass q al coo o ark : b a aac sac o ls s a col rc sac r a > a : I > a δ δ δ a δ δ δ IS:

3 3 sc a qss solo o olar racoal olo qao Hr sc a qss solo o qao 9 a cosqly s qal acy robl ll b scss a ro a aac sac by r o lck o a Pcar o 3 o lcko: I s o ll ro sc a qss solo o q9 or or : I A a ar lar bo oraors a sass coos 7 a 8 or qao 9 as a q solo a aac sac Proo: K b a oraor by 3 A K Hc a A K Sc A a ar bo oraors r ss os cosas a sc a A a 4 s coos 8 a 4 abo qaly bcos 5 K s 5 6 a K Iqaly 6 sos a oraor K as ball r o sl r ; r δ δ Sc r > > ror δ Also qaly 6 ols boss o oraor K or o cos a orla 3 las o A K K s coos 4 a 7 abo qaly bcos K K Hc or > a { } a K K I o abo qaly aks or { } { } 7 a a K K r las qaly 7 ca b aa or K K I coos scly lar a ; ror s a coraco a y aac o or K as a q o c s o cors q solo o qao 9 s cols roo o or 3 Pcar's o: orla 9 s qal o ollo ral qao s 3 8 aacal a oaoal os Scc a r IS:

4 r s a robably sy co o W 9 a W or : acy robl as a q solo aac sac roo o s or cos as a rsl o ollo las a : r coos 5 a 7 ral qao 8 as a solo Proo:s o o sccss aroaos orlas 9 a la o s o coos 7 a5 oba a r > Iroc a a a W ca coos scly lar sc a 3 Hc abo qaly ca b aa or a a y a sccss alcao o abo qaly a a a r s zro aroao c ca b ak zro l sac s srs cors orly Sc ollos a sqc { } cors orly sac o a coos co W c sass qao8 or all osqly r W a 3: r coos 5 a 7 ral qao 8 as a q solo Proo: a b o solos o q8 ro orlas 9 a a o coo 7 a s sa ar o la ρ a r a ρ s ro 3 a a ρ s cols roo o la a 4: r coos 4 5 a 6 solo o q8 sass a or Hölrcoo s 3 aacal a oaoal os Scc a r IS:

5 Proo o or : y r o las -4 c a solo o qao8 rrss q solo o acy robl aac sac a S o ll ro sably o solos o acy robl I or ors ll so a acy robl s corrcly orla or 3: { } b a sqc o cos ac o c s a solo o q al coo r S I sqc { } cors o a l S sqc { } A cors a sqc { } cors orly o sqc o solos { } cors orly o o a l co c s solo o acy robl Proo:osr sqcs { } a { } r A a A s A A s coo 7 oba A A osqly A A I o coos 5 a abo qaly bcos A A G > ε ca a os r ε sc a ε or all a s abo qaly aks or ε s or scly lar ε a Sc s a col sac ollos a sqc { } cors orly o o a coos co so sqc { } cors orly o o a coos co I ca b ro a sass a or Hölr coo o s S orollary: ral qao 8 as a q solo aacsac 4 rcal solo o olar racoal olo qao I s sco ll s P s z a Dzra o oba rcally solo o acy robl aacsac r a or s r qao8 or 4 r aacal a oaoal os Scc a r IS:

6 5 6 a ba krl ; 7 Hr ko co l a ar ko cos a sasy ollo coos: ' s a coos co ' s aral ras ar coos co c r ss a cosa c sc a c 3' s a baly ba co o s ars sc a: a or ac coos co a rals a ar coos cos b s absolly rabl rsc o or all a s c c s a cosa 4' co s coos a sass scz coo 8 ; a or all r s a os cosa ark : I o orollary ral qao 4 as a q solo aac sac Aly P a s a s ollo oaos ral qao 4 ca b rasor o ollo olar albrac sys 9 r 3 A 3 ala rals o qao3 r a roc rsls als o s ' { } 3 { } { } 6 a { } 3 3 olar albrac sys 9 rrss qaos ror aacal a oaoal os Scc a r IS:

7 aacal a oaoal os Scc a r aroa solo o ca r cor or W 33 Or a ar or as 34 or 4: albrac sys 9 as a q solo aac sac r ollo coos : s q q s a cosa 35 s q q s a cosa 36 s s s a cosa 37 ko cos or cosas D > D D > D sasy s D s 38 s 39 D s albrac sys 9 as a q solo aac sac Proo: Y b s o all cos { } sc a s a cosa D oraor by W 4 ro orlas 9 a 4 a s s s s coos q q D 4 Iqaly 4 sos a oraor as s Y q o sl r Sc > q > ror Also qaly 4 ols boss o oraor or o cos a V orlas 9 a 4 la o s s s coos a 39 oba V V 4 ros qaly 4 ros a oraor s coos a r coo s corac Hc by aac o or as a q o c s q solo o olar albrac sys or 5: I coos a 39 ar sas a sqc o cos { } { } cors orly o * co { } sqc o aroa solos { } { } cors orly o ac solo { } o olar albrac sys 9 Proo: I o q9 s s s abo qaly a o coos a 39 aks or ; Sc as so a s cols roo o or IS:

8 W s ; bcos cosqly solo o albrac sys 9 s sa solo o ral qao 4 or sos corc o sqc o aroa solos o ac solo o q4 sac or 6: I sqc o coos cos { } cors orly o co a cos a sasy rscly coos '3'-b a 8 sqc o aroa solos { } cors orly o ac solo o q4 aac sac Proo: orla 4 s aroa solo s a a a I o coos 4 '3'-b a 8 abo qaly ca b aa or ; c c D D Sc soa as Do : cosscy rror o P ollo qao 43 Also q43 s 44 Wr s aroa solo o qao4 or 7: Ass a yoss o or 6 ar sas l 45 Proo: orla 44 las o s s s s coos a 39 3 D q Sc as s or 6 ollos a 5 Alcao : I qao4 l: s a olar Volrra ral qao o sco k Abl krl 46 r ac solo abl ac ArSol rror ArSol rror aacal a oaoal os Scc a r IS:

9 aacal a oaoal os Scc a r rsls ar oba rcally abl c lss aros als o 8 or als o ac a aroa solos a rror o q 46 6 oclso I ca b obsr ro abl a: rror s or As a ar cras ro 8 rror s also cras or 98 8 a 4 3 As als o ar cras als o rror ar also cras or 8 rcs: Jlayss a VScca O al so o -sros or sco-orr a qaos Ja AalAl KPcr ck Sably o arko sros a alcaos o arabolc syssjaaalal SKoyas a P saaos Global sc or slar olo qaos olocal coos JaAalAl Y- S-YSa -s Ira - sros a absrac acy roblaas J avol o arc 997 PP75-5 WAr sol os oraors a ra sros Procoo asoc WAr Vcor al alac rasors a acy robls Isral Ja HKllra Hcbr Ira srosjcaal G br olo qaos: Solos or Irrlar olo Probls a Gralz Ial Vals Alcaos o Proc Socks olsa Sara 5 o brar Ira sros a r alcaos o absrac acy robl Pacc Ja HIra sros a ra solos o absrac acy roblsjaaalal as O absrac acy robl Jc Aal ao SPloVzo - s Ira s-ros JaAalAl ora So robably ss a aal solos o racoal olo qao aos Solos& racals l-ora Sros a so olar racoal ral qaos JAla o AAbo l- Kook& SAaa Aalyc a rc Solo o ar Paral Dral qao o racoal Orr GlobalJIcVol SD Zaa Absrac Dral qaos sarc os aacs Pa Aac Pbls Prora Sa racsco oo lbor l-ora Sros a corrc robls or olo qaos 6 IAS WOD OGSS ASA As 8 l-ora O saa a H cal O corrc orlao o a olar ral qaos aac sac IJa I Gla GSlo Gralz cos Vol ka osco959 WScr WWays racoal so a a qao Ja Pys Pr z Aalyc a rcal os or Volrra qaos SIA Plala 985 A Dzra os o Slar Iral qaos oo York 99 IS:

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