NON-LOCAL PROBLEMS WITH INTEGRAL GLUING CONDITION FOR LOADED MIXED TYPE EQUATIONS INVOLVING THE CAPUTO FRACTIONAL DERIVATIVE

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1 Elecronic Journal of Differenial Equaions, Vol. 16 (16), No. 164, pp ISSN: URL: hp://ejde.mah.sae.edu or hp://ejde.mah.un.edu NON-LOCAL PROBLEMS WITH INTEGRAL GLUING CONDITION FOR LOADED MIXED TYPE EQUATIONS INVOLVING THE CAPUTO FRACTIONAL DERIVATIVE OBIDJON KH. ABDULLAEV, KISHIN B. SADARANGANI Absrac. In his work, we sudy he eisence and uniqueness of soluions o non-local boundary value problems wih inegral gluing condiion. Mied ype equaions (parabolic-hyperbolic) involving he Capuo fracional derivaive have loaded pars in Riemann-Liouville inegrals. Thus we use he mehod of inegral energy o prove uniqueness, and he mehod of inegral equaions o prove eisence. 1. Inroducion and formulaion of a problem The models of fracional-order derivaives are more adequae han he previously used ineger-order models, because fracional-order derivaives and inegrals enable he descripion of he memory and herediary properies of differen subsances [3]. This is he mos significan advanage of he fracional-order models in comparison wih ineger-order models, in which such effecs are negleced. Fracional differenial equaions have recenly been proved o be valuable ools in he modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applicaions in viscoelasiciy, neurons, elecrochemisry, conrol, porous media, elecromagneism, ec., (see [6, 7, 8, 15, 18, 19]). There has been a significan developmen in fracional differenial equaions in recen years; see he monographs of Kilbas, Srivasava, Trujillo [14], Miller and Ross [], Podlubny [3], Samko, Kilbas, Marichev. [8] and he references herein. Very recenly, some basic heory for he iniial boundary value problems of fracional differenial equaions involving a Riemann-Liouville differenial operaor of order < α 1 has been discussed by Lakshmikanham and Vasala [16, 17]. In a series of papers (see [4, 5]) he auhors considered some classes of iniial value problems for funcional differenial equaions involving Riemann-Liouville and Capuo fracional derivaives of order < α 1: For more deails concerning geomeric and physical inerpreaion of fracional derivaives of Riemann-Liouville and Capuo ypes see [4]. Noe ha works [3, 1, 13, ] are devoed o he sudying of boundary value problems (BVP) for parabolic-hyperbolic equaions, involving fracional derivaives. BVPs for he mied ype equaions involving he Capuo and 1 Mahemaics Subjec Classificaion. 34K37, 35M1. Key words and phrases. Capuo fracional derivaives; loaded equaion; non-local problem; inegral gluing condiion; eisence; uniqueness. c 16 Teas Sae Universiy. Submied April 8, 16. Published June 8, 16. 1

2 O. KH. ABDULLAEV, K. S. SADARANGANI EJDE-16/164 he Riemann-Liouville fracional differenial operaors were invesigaed in works [1, 11]. For he firs ime i was given he mos general definiion of a loaded equaions and various loaded equaions are classified in deail by Nakhushev [1]. Noe ha wih inensive research on problem of opimal conrol of he agro-economical sysem, regulaing he label of ground waers and soil moisure, i has become necessary o invesigae BVPs for such class of parial differenial equaions.more resuls on he heory of BVP s for he loaded equaions parabolic, parabolic-hyperbolic and ellipic-hyperbolic ypes were published in works [9, 1]. Inegral boundary condiions have various applicaions in hermoelasiciy, chemical engineering, populaion dynamics, ec. Gluing condiions of inegral form were used in [, 7]. We consider he equaion: u C D α yu + p(, y) ( ) β1 u(, )d = for y > u u yy q( + y) ( y) γ1 u(, )d = for y <, +y wih he operaor CD α yf = 1 Γ(1 α) y (1.1) (y ) α f ()d, (1.) where < α, β, γ < 1, p(, y) and q( + y) are given funcions. Le Ω be domain, bounded wih segmens: A 1 A = {(, y) : = 1, < y < h}, B 1 B = {(, y) : =, < y < h}, B A = {(, y) : y = h, < < 1} a he y >, and characerisics : A 1 C : y = 1; B 1 C : + y = of he equaion (1.1) a y <, where A 1 (1; ), A (1; h), B 1 (; ), B (; h), C(1/; 1/). Le us inroduce: θ() = +1 + i 1, i = 1, D β a f() = 1 Γ(β) a ( ) β1 f()d, < β < 1. (1.3) Ω + = Ω (y > ), Ω = Ω (y < ), I 1 = { : 1 < < 1}, I = {y : < y < h}. In he domain Ω we sudy he following problem. Problem I. Find a soluion u(, y) of equaion (1.1) from he class of funcions: W = {u(, y) : u(, y) C( Ω) C (Ω ), u C(Ω + ), C D α oyu C(Ω + )}, ha saisfies he boundary condiions: u(, y) A1A = ϕ(y), y h; (1.4) u(, y) B1B = ψ(y), y h; (1.5) d d u(θ()) = a()u y(, ) + b()u (, ) + c()u(, ) + d(), I 1, (1.6) and gluing condiion lim y + y1α u y (, y) = λ 1 ()u y (, ) + λ () r()u(, )d, < < 1 (1.7) where ϕ(y), ψ(y), a(), b(), c(), d(), and λ j (), are given funcions, such ha j=1 λ j ().

3 EJDE-16/164 LOADED MIXED TYPE EQUATION WITH FRACTIONAL DERIVATIVES 3. Uniqueness of soluion for Problem I I is known ha equaion (1.1) on he characerisics coordinae ξ = + y and η = y a y has he form u ξη = q(ξ) 4 ξ ( ξ) γ1 u(, )d. (.1) We inroduce he noaion: u(, ) = τ(), 1; u y (, ) = ν (), < < 1; lim y + y1α u y (, y) = ν + (), < < 1. A soluion of he Cauchy problem for he equaion (1.1) in he domain Ω can be represened as τ( + y) + τ( y) u(, y) = y +y q(ξ)dξ y ξ dη y +y ξ ν ()d ( ξ) γ1 τ()d. Afer using condiion (1.6) and aking (1.3) ino accoun, from (.) we obain (a() 1)ν () = 1 Γ(γ)q()D γ 1 τ() + (1 b())τ () c()τ() d(). Considering above noaion and gluing condiion (1.7) we have ν + () = λ 1 ()ν () + λ () Furher from (1.1) a y + considering (1.), (.4) and we obain k1: lim y Dα1 y τ () Γ(α)λ 1 ()ν () Γ(α)λ () Main resuls. f(y) = Γ(α) lim y 1α f(y), y (.) (.3) r()τ()d, (.4) r()τ()d + Γ(β)p(, )D β 1 τ() =. Theorem.1. If he given funcions saisfy condiions ( λ () ) λ 1 ()q(), r() a() 1, p(, ), p (, ), ( (1 )q()λ1 () ) c()λ 1 (), (a() 1) (a() 1), hen, he soluion u(, y) of he Problem I is unique. ( (1 b())λ1 () (a() 1) (.5) λ () ; (.6) r() ). (.7) Proof. Known ha if homogeneous problem has only rivial soluion, hen we can sae ha original problem has unique soluion. For his aim, we assume ha he Problem I has wo soluions, hen denoing difference of hese as u(, y), we ge

4 4 O. KH. ABDULLAEV, K. S. SADARANGANI EJDE-16/164 appropriae homogenous problem. Equaion (.5) we muliply o τ() and inegrae from o 1: Γ(α) τ ()τ()d Γ(α) λ ()τ()d We invesigae he inegral I = Γ(α) Γ(β) λ 1 ()τ()ν ()d r()τ()d + Γ(β) λ 1 ()τ()ν ()d + Γ(α) τ()p(, )D β 1 τ()d. Taking (.3) ino accoun a d() =, we obain I = Γ(α)Γ(γ) + Γ(α) + Γ(α) = Γ(α) + Γ(α) Γ(α) (1 )q() a() 1 λ 1()τ()D γ 1 τ()d (1 b())λ 1 () a() 1 λ ()τ()d (1 )q() a() 1 λ 1()τ()d τ()p(, )D β 1 τ()d =. λ ()τ()d τ()τ ()d Γ(α) r()τ()d Γ(β) 1 b() a() 1 λ 1()d(τ ()) λ 1 ()c() a() 1 τ ()d Γ(α) τ()p(, )d ( ) γ1 τ()d ( ) β1 τ()d. λ () r() d( r()τ()d λ 1 ()c() a() 1 τ ()d τ()p(, )D β 1 τ()d r()τ()d) (.8) Considering τ(1) =, τ() = (which deduced from he condiions (1.4), (1.5) in homogeneous case) and on a base of he formula [9]: γ = 1 Γ(γ) cos πγ Afer some simplificaions from (.8) we obain Γ(α)q()λ 1 () I = 4(a() 1)Γ(1 γ) sin πγ Γ(α) + 4Γ(1 γ) sin πγ Γ(α) + Γ(α) λ () r() z γ dz ( τ () λ 1 () 1 b() a() 1 ( ) Γ(α) r()τ()d + z γ1 cos[z( )]dz, < γ < 1 z γ [M (, z) + N (, z)]dz [M (, z) + N (, z)]d [ λ 1 () ) d Γ(α) ( λ () r() (1 )q() ] a() 1 λ 1 ()c() a() 1 τ ()d ) ( d r()τ()d)

5 EJDE-16/164 LOADED MIXED TYPE EQUATION WITH FRACTIONAL DERIVATIVES 5 p(, ) Γ(1 β) sin πβ 1 sin πβ Γ(1 β) z β [M (, z) + N (, z)]dz z β dz [p(, )][M (, z) + N (, z)]d (.9) where M(, z) = τ() cos zd, N(, z) = τ() sin zd. Thus, owing o (.6),(.7) from (.9) i is concluded, ha τ(). Hence, based on he soluion of he firs boundary problem for he (1.1) [11, 5] owing o accoun (1.4) and (1.5) we obain u(, y) in Ω +. Furher, from funcional relaions (.3), aking ino accoun τ() we obain ha ν (). Consequenly, based on he soluion (.) we obain u(, y) in closed domain Ω. 3. Eisence of soluions for Problem I Theorem 3.1. If condiions (.6), (.7) and p(, y) C(Ω + ) C (Ω + ), q( + y) C(Ω ) C (Ω ); (3.1) ϕ(y), ψ(y) C(I ) C 1 (I ); a(), b(), c(), d() C 1 (I 1 ) C (I 1 ) (3.) hold, hen he soluion of he invesigaing problem eiss. Proof. Taking (.3) ino accoun from (.5) we obain where τ () A()τ () = f() B()τ() (3.3) f() = Γ(α)Γ(γ)(1 )λ 1()q() D γ 1 τ() Γ(β)p(, )Dβ 1 (a() 1) τ() + Γ(α)λ () r()τ()d Γ(α)λ 1()d() a() 1 A() = Γ(α)λ 1()(1 b()), B() = Γ(α)λ 1()c() a() 1 1 a() The soluion of (3.3) wih condiions (3.4) (3.5) τ() = ψ(), τ(1) = ϕ() (3.6) has he form ( τ() = ψ() + A 1 () (B()τ() f())a 1()d + where A 1() + (B()τ() f()) A 1() (B()τ() f()) A 1() A 1 () = ( ep Furher, considering (3.4) and using (1.3) from (3.7) we obain τ() ϕ() ψ() ) (3.7) ) A(z)dz d. (3.8)

6 6 O. KH. ABDULLAEV, K. S. SADARANGANI EJDE-16/164 = f 1 () + A 1 () where Γ(α) + A 1 () A 1() + Γ(α) A 1() A 1() Γ(α) f 1 () = (1 A 1() ) B()τ()d (1 )λ 1 ()q() a() 1 A 1()d A 1() B()τ()d + A p(, )d 1() + Γ(α)A 1 () A 1() d (s ) 1γ [p(, )(s ) β1 Γ(α)λ ()r(s)] A p(, )d 1() (s ) β1 [ (1 )λ 1()q()(s ) γ1 λ ()r(s)] (a() 1) (s ) β1 + A 1() B()τ()d [ (1 )λ 1()q() (s ) γ1 + λ ()r(s)] (3.9) (a() 1) Γ(α)d()λ 1 () A d 1()(a() 1) d()a 1()λ 1 () d a() 1 Γ(α)d()λ 1 () A 1()(a() 1) Afer some simplificaions we rewrie (3.9) in he form τ() = A 1 () Γ(α)A 1 () Γ(α) + + Γ(α) A 1() A 1() d A 1() (ψ() ϕ()) + ψ(). [p(, )(s ) β1 Γ(α)λ ()r(s)]a 1()d (s ) γ1 (1 )λ()q() (a() 1) A 1()d [ (1 )λ 1()q() (s ) γ1 + λ ()r(s)] A 1() (a() 1) (3.1) (s ) β1 A 1() p(, )d + A 1() A 1()B()τ()d [ (1 )λ1 ()q()(s ) γ1 ] A1 () λ ()r(s) (a() 1) (s ) β1 A 1() p(, )d A 1() A 1() B()τ()d + (s ) β1 A p(, )d + 1() A 1() B()τ()d [ (1 )λ1 ()q() ] Γ(α) (s ) γ1 A1 () + λ ()r(s) (a() 1) + f 1()

7 EJDE-16/164 LOADED MIXED TYPE EQUATION WITH FRACTIONAL DERIVATIVES 7 i.e., we have he inegral equaion: where K 1 (, s) = τ() = K(, ) = ( A1 () ) Γ(α) 1 ( A1 () ) Γ(α) + 1 ( A1 () )[ 1 K (, s) ( = A 1 () A 1(s)B(s) Γ(α) Γ(α)A 1 () K(, )τ()d + f 1 (), (3.11) { K 1 (, s),, K (, s), 1. r(s) (s ) γ1 (1 )λ()q() (a() 1) λ () A 1() (s ) β1 A 1() p(, )d + A 1(s) A 1(s) B(s) ] λ ()A 1()d + A 1 () (s ) γ1 (1 )λ()q() ) A 1()d a() 1 λ () Γ(α)r(s) A d 1() + Γ(α) A 1 () s (s ) γ1 (1 )λ()q() (a() 1) A 1() A 1 (s) s A 1(s) B(s) Γ(α)A 1() r(s) Γ(α) (1 )λ()q() A 1()(a() 1) (s )γ1 d + (1 A 1() ) (s ) β1 A p(, )d. 1() (s ) β1 A 1()p(, )d λ () A d 1() (3.1) (3.13) (3.14) Owing o class (3.1), (3.) of he given funcions and afer some evaluaions from (3.13), (3.14) and (3.1), (3.1) we conclude ha K(, ) cons, f 1 () cons. Since kernel K(, ) is coninuous and funcion in righ-side F () is coninuously differeniable, soluion of inegral equaion (3.11) we can wrie via resolven-kernel: τ() = f 1 () where R(, ) is he resolven-kernel of K(, ). ν + () we found accordingly from (.3) and (.4): ν () = ( 1)q() (a() 1) + (1 )q() (a() 1) ( ) γ1 d ( ) γ1 f 1 ()d R(, )f 1 ()d, (3.15) Unknown funcions ν () and R(, s)f 1 (s)ds

8 8 O. KH. ABDULLAEV, K. S. SADARANGANI EJDE-16/ b() a() 1 f 1() 1 b() a() 1 + c() a() 1 R(, )f 1 ()d R(, ) f 1 ()d c() a() 1 f 1() d() a() 1 and ν + () = λ()ν (). The soluion of Problem I in he domain Ω + can be wrien as [11], u(, y) = where y + G ξ (, y,, η)ψ(η)dη G ( ξ, y)τ(ξ)dξ y G ξ (, y, 1, η)ϕ(η)dη y 1 G ( ξ, y) = Γ(1 α) (y η)α/1 G(, y, ξ, η) = ( e 1,α/ 1,α/ G(, y,, η)p(ξ)dξdη y n= η α G(, y, ξ, η)dη, [ e 1,α/ 1,α/ + ξ + n )] (y η) α/ ( ξ ( ξ) β1 τ()d ξ + n ) (y η) α/ Is he Green s funcion of he firs boundary problem (1.1) in he domain Ω + wih he Riemanne-Liouville fracional differenial operaor insead of he Capuo ones [6], e 1,δ 1,δ (z) = n= z n n!γ(δ δn) is he Wrigh ype funcion [6]. Soluion of he Problem I in he domain Ω will be found by he formula (.). Hence, he proof is complee. We remark ha if a() = 1/, hen from (.3) we find τ() as a soluion Volerra ype inegral equaion. Afer ha we can find ν + () from he firs boundary value problem problem for he (1.1), and ν () will be defined from he gluing condiion (.4). References [1] O. Kh. Abdullaev; Abou a mehod of research of he non-local problem for he loaded mied ype equaion in double-conneced domain, Bullein KRASEC. Phys. Mah. Sci, 14, vol. 9, no., pp ISSN [] Abdumauvlen S. Berdyshev, Erkinjon T. Karimov, Nazgul S. Akhaeva; On a boundary-value problem for he parabolic-hyperbolic equaion wih he fracional derivaive and he sewing condiion of he inegral form. AIP Proceedings, Vol. 1611, 14, pp [3] E. Y. Arlanova; A problem wih a shif for he mied ype equaion wih he generalized operaors of fracional inegraion and differeniaion in a boundary condiion, Vesnik Samarsk Gosudarsvennogo Universiea. (8) vol. 6, no. 65, pp [4] A. Belarbi, M. Benchohra, A. Ouahab; Uniqueness resuls for fracional funcional differenial equaions wih infinie delay in Freche spaces. Appl. Anal. 85(6), No. 1, [5] M. Benchohra, J. Henderson, S. K. Nouyas, A. Ouahab; Eisence resuls for fracional order funcional differenial equaions wih infinie delay. J. Mah. Anal.Appl. 338(8), No.,

9 EJDE-16/164 LOADED MIXED TYPE EQUATION WITH FRACTIONAL DERIVATIVES 9 [6] K. Diehelm, A. D. Freed; On he soluion of nonlinear fracional order differenial equaions used in he modeling of viscoelasiciy, in: F. Keil, W. Mackens, H. Voss, J. Werher (Eds.), Scienific Compuing in Chemical Engineering IICompuaional Fluid Dynamics, Reacion Engineering and Molecular Properies, Springer-Verlag, Heidelberg.(1999), pp [7] W. G. Glockle, T. F. Nonnenmacher; A fracional calculus approach of self-similar proein dynamics, Biophys. J. 68 (1995) [8] R. Hilfer; Applicaions of Fracional Calculus in Physics, World Scienific, Singapore, (). [9] B. Islomov, U. Balaeva; Boudanry-value problems for a hird-order loaded parabolichyperbolic ype equaion wih variable coefficiens. Elecron. J. Differenial Equ., Vol. 15 (15). No. 1. pp 1-1. [1] B. J. Kadirkulov; Boundary problems for mied parabolic-hyperbolic equaions wih wo lines of changing ype and fracional derivaive. Elecron. J. Differenial Equ., Vol. 14 (14), No. 57, pp [11] E. T. Karimov, J. Akhaov; A boundary problem wih inegral gluing condiion for a parabolic-hyperbolic equaion involving he Capuo fracional derivaive. Elecron. J. Differenial Equ., Vol. 14 (14) No. 14. pp [1] A. A. Kilbas, O. A. Repin; Analogue of he Bisadze-Samarskiy problem for an equaion of mied ype wih a fracional derivaive; Differensialnye Uravneniya. (3) vol. 39, no. 5, pp , Translaion in Journal of Difference Equaions and Applicaions. (3). vol. 39, no. 5, pp [13] A. A. Kilbas, O. A. Repin; An analog of he Tricomi problem for a mied ype equaion wih a parial fracional derivaive, Fracional Calculus and Applied Analysis. (1) vol. 13, no. 1, pp [14] A. A. Kilbas, H. M. Srivasava, J. J. Trujillo; Theory and Applicaions of Fracional Differenial Equaions, in: Norh-Holland Mahemaics Sudies, vol. 4, Elsevier Science B.V., Amserdam. (6). [15] J. W. Kirchner, X. Feng, C. Neal; Fracal sreamchemisry and is implicaions for conaminan ranspor in cachmens, Naure 43 (), [16] V. Lakshmikanham, A. S. Vasala; Basic heory of fracional differenial equaions. Nonlinear Anal. 69(8), No. 8, [17] V. Lakshmikanham, A. S. Vasala; Theory of fracional differenial inequaliies and applicaions. Commun. Appl. Anal. 11(7), No. 3-4, [18] B. N. Lundsrom, M. H. Higgs, W. J. Spain, A. L. Fairhall; Fracional differeniaion by neocorical pyramidal neurons, Na. Neurosci. 11 (8), [19] F. Mainardi; Fracional calculus: some basic problems in coninuum and saisical mechanics, in: A. Carpineri, F. Mainardi (Eds.), Fracals and Fracional Calculus in Coninuum Mechanics, Springer-Verlag, Wien.(1997), pp [] K. S. Miller, B. Ross; An Inroducion o he Fracional Calculus and Differenial Equaions, John Wiley, New York, (1993). [1] A. M. Nakhushev; The loaded equaions and heir applicaions. M. Nauka, (1). 3. [] V. A. Nakhusheva; Boundary problems for mied ype hea equaion, Doklady AMAN (1) vol. 1, no., pp In Russian. [3] I. Podlubny; Fracional Differenial Equaions, Academic Press, New York, (1999). [4] I. Podlubny; Geomeric and physical inerpreaion of fracional inegraion and fracional differeniaion. Dedicaed o he 6h anniversary of Prof. Francesco Mainardi. Frac. Calc. Appl. Anal. 5(), No. 4, [5] A. V. Pskhu; Uravneniye v chasnykh proizvodnykh drobnogo poryadka. (in Russian) [Parial differenial equaion of fracional order] Nauka, Moscow, (5). pp. [6] A. V. Pskhu; Soluion of boundary value problems fracional diffusion equaion by he Green funcion mehod. Differenial equaion, 39(1) (3), pp [7] M. S. Salakhidinov, E. T. Karimov; On a nonlocal problem wih gluing condiion of inegral form for parabolic-hyperbolic equaion wih Capuo operaor. Repors of he Academy of Sciences of he Republic of Uzbekisan. (DAN RUz), No 4, (14), 6-9. [8] S. G. Samko, A. A. Kilbas, O. I. Marichev; Fracional Inegral and Derivaives: Theory and Applicaions, Gordon and Breach, Longhorne, PA, (1993). [9] M. M. Smirnov; Mied ype equaions, M. Nauka. ().

10 1 O. KH. ABDULLAEV, K. S. SADARANGANI EJDE-16/164 Obidjon Kh. Abdullaev Deparmen of Differenial equaions and Mahemaical Physics, Naional Universiy of Uzbekisan, 1114 Uzbekisan, Tashken, Uzbekisan address: Kishin S. Sadarangani Deparmen of Mahemaics, Universiy of Las-Palmas de Gran Canaria, 3517 Las Palmas, Spain address: