Existence and Uniqueness of Solution for a Fractional Order Integro-Differential Equation with Non-Local and Global Boundary Conditions

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1 Applied Mthetic doi:0.436/ Pulihed Online Octoer 0 ( Eitence nd Uniquene of Solution for Frctionl Order Integro-Differentil Eqution with Non-Locl nd Glol Boundry Condition Atrct Mehrn Ftei Nihn Aliev Sedght Shhord Deprtent of Mthetic Bku Stte Univerity Bku Azern Deprtent of Applied Mthetic Univerity of Triz Triz Irn E-il: ftei.ehrn@yhoo.co hhord@trizu.c.ir Received Ferury 4 0; revied Septeer 5 0; ccepted Septeer 3 0 In thi pper we prove n iportnt eitence nd uniquene theore for frctionl order Fredhol Volterr integro-differentil eqution with non-locl nd glol oundry condition y converting it to the correponding well known Fredhol integrl eqution of econd kind. The conidered prole in thi pper h een olved lredy nuericlly in []. Keyword: Frctionl Order Integro-Differentil Eqution Non-Locl Boundry Condition Fundentl Solution. Introduction Let conider prole under oundry condition contining non-locl nd glol ter for frctionl order integro-differentil eqution q D y f K t y t t K t y t t d d q j j y y Hi t y t t di j= i d () () q 0 f K t K Hi t i = t nd re continuou rel-vlued function i nd d i i = j = re rel contnt nd oundry condition () re linerly independent.. Eitence nd Uniquene of Solution Theore. Let the function f K j t j = nd Hi t i = re continuou i nd d i. = re rel contnt the oundry condition () re linerly independent nd condition (5) i tified. Then the oundry vlue prole ()-() h unique olution. Proof: Acting in Eqution () y frctionl order derivtive opertor D [] we q get ince q q q q q D D y D f D K t y t t D K t y t t q q D D y = D y d d then we get the eqution (3) D y F M t y t d t M t y t d t Copyright 0 SciRe.

2 M. FATEMI ET AL. 93 q q d F D f = f d d q! q d M t K t d d t q! q d M t K td. d q! Now we write Eqution (3) in the generl fro D y G y (4) (3.) nd ccept tht G y i known then the fundentl olution (ee [3]) i in the for! Y. (5) > 0 < i Heviide unit function. Now we try to get oe ic reltion. The firt of thee reltion i Lgrnge forul. We ultiply oth ide of Eqution (3) y fundentl olution (5) nd integrte the otined epreion on (ee [45]) to get D y Y d G y Y d (7) G y F M t y t d t M t y t dt (7.) integrting y prt on the left hnd ide of epreion (7) nd tking into ccount tht (5) i fundentl olution of (3.) give the firt ic reltion in the for (6) y n D y Y G yyd =0 = y. Hence the firt epreion for the necery condition re otined in the for y D y Y G y Y d =0 y D y Y G yy d =0 (8) (9) It i ey to ee tht the econd epreion in (9) turn into n identity. Indeed it i een fro (5)-(6) the integrl t the right ide of the econd condition contin the vlue of the function which i zero for =. For = the the ution in the econd epreion contin the Heviide function which i zero for =0. Finlly the firt und contin poitive degree of for =0 thee ter ecoe zero t = =. Here for = the the epreion of fundentl olution for = yield the Heviide function. For = = thi ecoe therefore the econd one of necery condition (9) turn into identity. Now we contruct the econd ic epreion to get the econd group of necery condition. For tht we ultiply oth ide of (3) y the derivtive of (5) nd integrte on [67]: D yy d G yy d. Integrting y prt on the left ide of the otined epreion nd tking into ccount (5) nd (6) we get the econd ic reltion follow: y D y Y G yy d ( ) (0) =0 = y nd o the econd group of the necery condition re otined Copyright 0 SciRe.

3 94 M. FATEMI ET AL. y D y Y G y Y d =0 y D yy GyY d. =0 () Siilr to the econd epreion of (9) we cn how tht the econd epreion of () turn into identity. If we continue thi proce in order to get the -th ic reltion we ultiply (3) the -th order derivtive of (5) nd integrte on to get: d D yy G y Y d. Here once integrting y prt on the left ide of the otined epreion give D y Y D yy d = G y Y d. Thu if we tke into ccount tht (5) i the fundentl olution the lt reltion (-th) will e follow: y D y Y G y Y d = ( ) y. () Therefore the lt group of necery condition will e in the for: y D yy G y Y d = y D yy G y Y d = (3) here ove the econd necery condition turn into identity. Now we join to the given linerly independent oundry condition (7) the necery condition in (9) () nd etc. (3) tht re not identitie nd write the yte of liner lgeric eqution otined with repect to the oundry vlue of the unknown function in the following wy. y y y y y y d H t y t d t y y y y y y d H tytd t y y y y Y y Y y y GyY d ( ) y( ) y y y y Y y y y GyY d y y y Y y = G y Y d (4) Copyright 0 SciRe.

4 M. FATEMI ET AL. 95 For olving the yte (4) y the Crer rule it i necery tht it ic deterinnt differ fro zero. Accept tht the following condition i tified 0 0 Y... Y 0 0 ( ) 0 0 Then fro yte (4) we get y d H t y t d t G y Y d = = k k y d d H t y t t G y Y d = = 0 k k k k (5) (6) ( p q) denote the cofctor of the eleent t the interection of p-th row nd q-th colun of the deterinnt. Clculte the following epreion: G y Y F M t y t t M t y t ty d ( ) d d d ( ) Then we get: nd o d d d d F Y Y M t y t t Y M t y t dt F Y d ytd ty M t d ytd ty M t d. G y Y d F M t y t d t (7) F F Y d (8) M t Y M t d Y M td Finlly coing ck to (8) we tke into ccount (6) nd (7) nd write the econd kind Fredhol type integrl eqution [8] for which the oundry vlue prole ()-() i reduced to: y A B t y t d t (9) 0 k 0 k k dk k A Y Y F k dk k Y Y F k FY d 0 k 0 k k (0) Copyright 0 SciRe.

5 96 M. FATEMI ET AL. 0 k 0 k 0 k 0 k Y M t d Y M t d. k = dk k B t Y Hk t Y M k t k ( ) dk k Y Hk t Y M k t () By the hypothei of theore on the function f K j t j = nd Hi t i = the integrl Eqution (9) h unique olution nd o in ll conducted opertion we cn coe ck nd we conclude tht the olution of (9) i the unique olution of oundry vlue prole ()-(). 3. Reference [] D. Nzri nd S. Shhord Appliction of Frctionl Differentil Trnfor Method to the Frctionl Order Integro-Differentil Eqution with Nonlocl Boundry Condition Journl of Coputtionl nd Applied Mthetic Vol. 34 No pp doi:0.06/j.c [] S. G. Sko A. A. Kil nd O. I. Mrichev Frctionl Integrl nd Derivtive Theory nd Appliction Cordon nd Brech Yverdon 993. [3] C. J. Trnter Integrl Trnfor in Mtheticl Phyic Methuen London nd New York 949. [4] V. S. Vldiirov Eqution of Mtheticl Phyic Mir Puliction Mocow 984. [5] G. E. Shilov Mtheticl Anlyi. The Second Specil Coure Nuk Mocow 965. [6] S. M. Hoeini nd N. A. Aliev Sufficient Condition for the Reduction of BVP for PDE with Non-Locl nd Glol Boundry Condition to Fredhol Integrl Eqution (on Rectngulr Doin) Applied Mthetic nd Coputtion Vol. 47 No pp [7] F. Bhri N. Aliev nd S. M. Hoeini A Method for the Reduction of Four Ienionl Mied Prole with Generl Boundry Condition to Syte of Second Kind Fredhol Integrl Eqution Itlin Journl of Pure nd Applied Mthetic No pp [8] N. Aliev nd M. Jhnhehi Solution of Poioin Eqution with Glol Locl nd Non-Locl Boundry Condition Interntionl Journl of Mtheticl Eduction in Science nd Technology Vol. 33 No. 00 pp doi:0.080/ Copyright 0 SciRe.