Malaysia Joural of Maheaical Scieces 7(): 3-7 (3) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural hoepage: hp://eispe.up.edu.y/joural Boudary Value Proble for he Higher Order Equaio wih Fracioal Derivaive * Djualych Aaov Isiue of Maheaics Naioal Uiversiy of Uzbeisa 9 Duro yuli sree Tashe 5 Uzbeisa E-ail: daaov@yadex.ru *Correspodig auhor ABSTRACT I his paper we sudy he boudary value proble for higher order equaio wih fracioal derivaive i he sese of Capuo. Exisece of he uique soluio of his proble ad is coiuous depedece o he iiial daa ad o he righ par of he equaio are proved. Keywords: Boudary value probles fracioal derivaive i he sese of Capuo Volerra iegral equaio of secod id Miag-Leffler ype fucio. Maheaics Subjec Classificaio: 35G5 35D6.. INTRODUCTION The Cauchy proble for parial pseudo-differeial equaios of fracioal order i he sese of Capuo i sei-space {( x y ) :( x y) R } > ivesigaed by Goreflo Lucho ad Uarov (). The boudary value probles for parial differeial equaios wih operaors of fracioal differeiaio or iegraio i he sese of Riea-Liouville were sudied by Djrbashya ad Nersesya (968) Sao Kilbas ad Marichev (987) Nahushev () Pshu (5) Vircheo ad Riba (7) Kilbas ad Repi (). The boudary value proble for hea coducio equaio wih fracioal derivaive i he sese of Capuo ivesigaed by Kadirulov ad Tureov(6). The boudary value proble for higher order parial
Djualych Aaov differeial equaio wih fracioal derivaive i he sese of Capuo i he case whe order of fracioal derivaive belogs o he ierval ( ) was sudied by Aaov (8). The boudary value proble for he fourh order equaio wih fracioal derivaive i he sese of Capuo i he case whe order of fracioal derivaive belogs o he ierval ( ) by Aaov (9). I he prese paper we sudy he boudary value proble for he higher order parial differeial equaio wih fracioal derivaive i he sese of Capuo i he case whe he order of fracioal derivaive belogs o he ierval ( ) i a spaial doai.. STATEMENT OF THE PROBLEM I he doai {( x y ) : x p y q T } cosider he followig equaio Ω = < < < < < < we u x u y ( ) с D u = f ( x y ) () where ( ) is a fixed ieger < < c D is he operaor of fracioal differeiaio wih respec o i he sese of Capuo. The boudary value proble for higher order parial differeial equaio wih operaor c D α < α < was sudied by Aaov (9). Proble. Fid he soluio u( x y ) of he equaio () saisfyig he followig codiios u( y ) u( p y ) = = =... y q T x x () u( x ) u( x q ) = = =... x p T. y y ψ u( x y) = φ( x y) u x y = x y x p y q (4) (3) i he doai Ω. 4 Malaysia Joural of Maheaical Scieces
Boudary Value Proble for he Higher Order Equaio wih Fracioal Derivaive 3. CONSTRUCTION OF FORMAL SOLUTION OF PROBLEM We search a soluio of Proble i he for of Fourier series u( x y ) = u υ ( x y) (5) = expaded i coplee orhooral syse Deoe ( ) si π π υ x y = x si y. pq p q π π Ω = Ω ( = ) = = = p q λ. We expad he fucio f ( x y ) io he Fourier series by fucios υ ( x y ) where f ( x y ) = f υ ( x y) (6) p q = f = f ( x y ) υ x y dydx. (7) Subsiuig (5) ad (6) io equaio () we obai с D u λu f T. I is ow ha (Djrbashya (966)) = (8) where D u = I u (9) '' с I f = ( τ ) f ( τ ) dτ Г( ) is Riea-Liouville iegral of fracioal order. Malaysia Joural of Maheaical Scieces 5
Djualych Aaov Usig (9) equaio (8) ca be rewrie as I u λ u = ( ) f T. '' Acig wih operaor I o boh pars of he las equaio we ge ( ) ( τ ) Г( ) u = λ u ( τ ) dτ ( ) ' ( τ ) f ( τ ) dτ u () u (). Г () This is Volerra iegral equaio of secod id. We solve i by successive approxiaios. Deoe K ( τ ) ( τ ) T < τ < = Г( ) τ ad defiig furher sequece of erels { Ks ( ) } τ K ( τ ) = K ( τ η) K ( η ) dη s =.... s s By iducio wih respec o s we fid K s ( s ) τ T < τ < τ = Г(s ) τ. Hece for he resolve of equaio () we have he forula ( τ λ ) ( ) λ s ( τ ) R = K = s= τ ( ) = τ by recurre relaios τ E λ τ T < τ < s 6 Malaysia Joural of Maheaical Scieces
Boudary Value Proble for he Higher Order Equaio wih Fracioal Derivaive where Eα ( z) is he Miag-Leffler ype fucio (Djrbashya (966) Kadirulov ad Tureov (6)). The soluio of equaio () is expressed hrough resolve so u = u() ( λ ) ( τ ) E λ ( τ ) dτ u ' () ( λ ) ( τ ) E λ ( τ ) τdτ Г( ) η ( λ ) η E λ η τ dη τ η f ( τ ) dτ ( ) ( τ ) f ( τ ) dτ. Г( ) () Applyig he followig Dirichle forula o ieraed iegral we ge b b b x dy f ( x y) dx = dx f ( x y) dy a y a a Г ( ) ( ) η E λ ( η ) dη τ η f ( τ ) dτ = τ = f ( τ ) dτ ( τ η ) ( η ) E ( λ ( η ) ) dη. Г ( ) η () Deoe I = ( λ ) τ E λ τ dτ I = ( λ ) τ E λ τ τ dτ ( ) I τ η E λ η dη. τ 3 = Г( ) Malaysia Joural of Maheaical Scieces 7
Djualych Aaov We calculae I ( ). We chage of variable τ = s dτ = ds he Usig he forula Г ( δ ) = λ ( λ ) I s E s ds. z α δ δ α Eα ( λ )( z ) d = z Eα δ ( λz ) (3) we have λ I = ( λ ) E. Now usig he forula we obai Г ( ) ze z = E z (4) α α α ( λ ) I == E. (5) Aalogously we have ad ( λ ) I = E (6) ( ) τ λ ( τ ) I = E. (7) 3 Owig o () ad subsiuig (5) (6) ad (7) io () we ge ( ) ' u = u () E λ u () E λ ( ) ( λ ) τ E λ ( τ ) f ( τ ) dτ ( ) ( τ ) f ( τ ) dτ. Г 8 Malaysia Joural of Maheaical Scieces
Boudary Value Proble for he Higher Order Equaio wih Fracioal Derivaive Uiig he las wo iegrals ad usig (4) i is received λ ' u = u () E λ u () E ( ) τ E λ τ f τ dτ. We expad he fucios ϕ ( x y) ad ( x y) fucios υ ( x y ) where (8) ψ io Fourier series by φ x y = φ υ ( x y) dydx ψ x y = ψ υ ( x y) dydx = = p q φ = φ υ ( x y) dydx (9) p q ( x y) ( x y) ψ = ψ υ ( x y) dydx () Owig (9) ad () fro codiios (4) we fid u () = ϕ u ' () = ψ. Taig io accou he las equaliy he soluio (8) has he for u = u () E λ ψ E λ ( ) τ E λ τ f τ dτ. () Subsiuig () io (5) we ge foral soluio of proble (). 4. EXISTENCE OF THE UNIQUE SOLUTION OF PROBLEM IN L SPACE. Lea. Le ϕ L ( Ω ) ψ L ( Ω ) ad f L ( Ω ) he c = cos >. ( φ ψ ) u c f () L ( Ω) L ( Ω ) L ( Ω ) L ( Ω) Malaysia Joural of Maheaical Scieces 9
Djualych Aaov Proof. Usig he followig esiae (see Kadirulov e al. (6) p. 36) M Eα ( z) M = cos > Re z < z (3) ad he Cauchy-Schwarz iequaliy () gives u M ϕ T ψ ( τ ) f ( τ ) dτ where c Furher T c ϕ T ψ f ( ) L ( T ) MT = ax M MT. ( ) ( ϕ ψ ) = L ( T ) L ( T ) T u u d 3 c T f. Usig Parseval equaliy we have u = u ( ) ( ) L υ x y uij υij x y Ω = = i j= ( φ ψ f L ( Ω ) L ( Ω ) L ( Ω) ) = c L ( Ω) = u 3cT φ ψ f = = = = = L ( T ) where c = 3 c T. Lea is proved. Malaysia Joural of Maheaical Scieces
Boudary Value Proble for he Higher Order Equaio wih Fracioal Derivaive Corollary. Fro he esiae () he covergece of series (5) i L ( Ω ) ad u L ( Ω ) follows. Corollary. Fro he esiae () coiuous depedece of he soluio (5) o iiial daa of he Proble ad he fucio f ( x y ) follows. Iegraig he iegrals (7) (3) ad (4) by pars wih respec o x ad y we have p q () () φ ϕ = ϕ φ = cos x cos ydydx (4) xy pq ψ = ψ () p q () ψ = x xy pq ψ cos cos ydydx (5) f f () () = f f p q () () = Lea. Le f C f f = cos x cos ydydx (6) xy pq p q f f = ( x y) dydx xy υ (7) ϕ C ( Ω) ϕxy L Ω Ω fxxy C ( Ω ) fxyy C ( Ω ) fxxyy C f = o [ T] ψ = o Ω esiaes hold ψ C ( Ω ) ψ xy L ( Ω ) Ω ϕ = o Ω Ω he for ay < ε < he followig φ ψ u c ε ε (8) λ φ ψ u c ε ε () () (9) Malaysia Joural of Maheaical Scieces
Djualych Aaov Proof. Fro codiios of Lea i follows ha fucios f () ( ) are bouded f () ( ) ad f N () f N (3) () where N = cos > N = cos >. Le [ T ] where < is sufficiely ior uber. Furher we have < λ. For sufficiely large values of ad he followig are rue λ < λ < λ < λ (3) l ε ε ε ε T T. Owig o (3) (6) (3) (3) fro () we ge ( λ ( τ ) ) d ϕ ψ N u M ( ) ( ) λ τ ε N l T l λ ϕ ψ ε M ( ) ( ) ϕ ψ c ε ( ) ε M M MN l T MN where c = ax. ( ) ( ) ε The iequaliy (8) is proved. Now we prove he iequaliy (9). Owig o (3) (4) (7) (3) ad (3) fro () we fid Malaysia Joural of Maheaical Scieces
Boudary Value Proble for he Higher Order Equaio wih Fracioal Derivaive λ ϕ () () N l T N N u M ε ε ε ε ( ) ( ) ψ ϕ ψ () () c ε ε ε where c M M = ax. ( ) ( ) MN l T MN ε Lea is proved. Theore. Le he codiios of Lea hold he here exiss a uique regular soluio of Proble ad i coiuously depeds of fucios ϕ ( x y) ψ ( x y) ad f ( x y ). Proof. We have o prove he uiforly ad absoluely covergece of series (5) ad u = ( ) ( ) u υ x y (3) x = u = ( ) ( ) u υ x y (33) y = ( ) D = f υ ( x y) ( ) λ u υ ( x y). c = = (34) The series u (35) pq = is ajora for he series (5). The series (35) uiforly coverges owig o (8). Ideed Malaysia Joural of Maheaical Scieces 3
Djualych Aaov ϕ ψ u с. ε ε = = Applyig he Cauchy-Schwarz iequaliy for he su ad Parseval equaliy ϕ o he series we have = aalogously / ϕ ϕ = = = p q = π / ϕ L ( Ω ) = = / = / ψ p q ϕ = π = = L ( Ω ). The series uiforly coverge for ay accordig = = o Cauchy iegral sig. Furher sice ε > he he series ε ε = = = uiforly coverge o he sae Cauchy iegral sig. Cosequely he series (5) uiforly ad absoluely coverges for ay < i he closed doai {( x y ) : x p y q T } Ω =. The series (5) coverges a = o ϕ ( x y). Owig o < λ ad < λ he series pq λ u (36) = is ajora for he series (3) (33) ad he secod series of (34). The series (36) uiforly coverges owig o (9). 4 Malaysia Joural of Maheaical Scieces
Boudary Value Proble for he Higher Order Equaio wih Fracioal Derivaive The series f (37) pq = is ajora for he firs series of (34). The series (37) uiforly coverges T. Ideed usig (7) we have for ay [ ] = = () f f N = = = p q = N = N 4 π = = p q 36. Cosequely he firs series of (34) uiforly ad absoluely coverges i Ω. Addig (3) (33) ad (34) we covice ha he soluio (5) saisfies he equaio (). Based o properies of fucios υ ( x y ) we coclude ha he soluio (5) saisfies codiios () ad (3). Siple calculaios show ha d li E λ ( ) = li E λ ( ) = d d li E λ ( ) = li E λ ( ) =. d Hece ϕ ψ ' li u = li u () =. The las equaliies show ha he soluio (5) saisfies he codiios (4). Sice Lea is rue for regular soluio of he Proble he fro () we coclude ha he regular soluio of he Proble coiuously depeds o fucios ϕ ( x y) ψ ( x y) ad f ( x y ). Theore is proved. Wih siilar argue he followig proble ca be solved. Malaysia Joural of Maheaical Scieces 5
Djualych Aaov Proble. Fid he soluio u( x y ) of he equaio () saisfyig he codiios (4) ad x y u( y ) = u( p y ) = =... y q T x u( x ) = u( x q ) = =... x p T. x 5. CONCLUSION I his paper a boudary value proble for higher order equaio wih fracioal derivaive i he sese of Capuo is sudied i spaial doai i he case whe he order of fracioal derivaive belogs o he ierval ( ). Soluio is cosruced i he for of Fourier series. The exisece ad uiqueess of he regular soluio ad is coiuously depedece o iiial daa ad he righ par of equaio are proved. ACKNOWLEDGEMENT The auhor is graeful o referees for heir isighful coes ad suggesios. REFERENCES Aaov D. 8. Solvabiliy of boudary value probles for higher order equaios wih fracioal derivaives. Boudary value probles for differeial equaios. The collecio of proceedigs. 7: 4-9. (i Russia) Aaov D. 9. Boudary value proble for fourh order equaio wih fracioal derivaive. Quesios of calculaig ad applied aheaics. : 55-6. (i Russia) Aaov D. 9. Boudary value probles for higher order equaios wih fracioal derivaive. Proceeedig of Coferece. Differeial equaios ad heir applicaios (SaDif-9) Saara 9 Jue July 9 p. 9-. 6 Malaysia Joural of Maheaical Scieces
Boudary Value Proble for he Higher Order Equaio wih Fracioal Derivaive Djrbashya M. M. Nersesya A.B. 968. Fracioal derivaives ad Cauchy proble for differeial equaios of fracioal order // Izvesia Acad. Nau Ar. SSR. 3(): 3-8. (i Russia) Djrbashya M. M. 966. Iegral rasforaios ad represeaio of fucios i coplex doai. Mosow. 67 p. (i Russia) Goreflo M. M. Lucho Y. F. ad Uarov S. R.. O he Cauchy ad ulipoi probles for parial pseudo-differeial equaios of fracioal order // Frac. Calc. Appl. Aal. 3(3). 49-75. Kilbas A. A. ad Repi O. A.. Aalogue of Tricoi proble for parial differeial equaio coaiig diffusio equaio of fracioal order. Ieraioal Russia-Bulgaria syposiu «Mixed ype equaios ad relae probles of aalysis ad iforaics». Proceedig Nalchi-Habez. 3-7. (i Russia) Kadirulov B. J. ad Tureov B. Kh. 6. Abou oe geeralizaio of hea coduciviy equaio. Uzbe Maheaical Joural. 3: 4-45. (i Russia) Mizohaa S. 979. Theory of parial differeial equaios. Mosow: Mir publishers. (i Russia) Nahushev A. M.. Elees of fracioal calculus ad heir applicaios. Nalchi. 98 p. (i Raussia) Pshu A. M. 5. Boudary value probles for parial differeial equaios of fracioal ad coiual order. Nalchi. 86 p. (i Russia) Sao S. G. Kilbas A. A. ad Marichev O. I. 987. Iegrals ad derivaives of fracioal order ad heir soe applicaios. Mis Naua I Techia. 668 p. (i Russia) Vircheo N. A. ad Riba N. A. 7. Foudaios of fracioal iegrodiffereiaios. Kiev. 36 p. (i Uraie) Malaysia Joural of Maheaical Scieces 7