International Journal of Mathematical Archive-3(2), 2012, Page: Available online through ISSN

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1 Inernaional Jornal o Mahemaical Archive- age: Available online hrogh wwwijmaino ISSN A NON-OA OUNDARY VAUE ROEM WIH INEGRA ONDIIONS OR A OURH ORDER SEUDOHYEROI EQUAION Azizbayov EI* an Y Mehraliyev Mechanics-mahemaics acly a Sae Universiy a Azerbaijan azel_azerbaijan@mailr Receive on: 6-- Accepe on: 4-- ASRA In he paper he classic solion o one-imensional bonary vale problem or a pseohyperbolic eaion wih non-classic bonary coniions is invesigae or ha he sae problem is rece o he no-sel-ajoin bonary vale problem wih eivalen bonary coniion hen sing he meho o separaion o variables by means o he nown specral problem he given no sel-ajoin bonary vale problem is rece o an inegral eaion he eisence an nieness o he inegral eaion is prove by means o he conracion mappings principle an i is shown ha his solion is a nie solion or a no-ajoin bonary vale problem inally sing he eivalence he heorem on he eisence an nieness o a non-local bonary vale problem wih inegral coniion is prove Mahemaics Sbjec lassiicaion: G Keywors an hrases: mie problem conrace mappings ie poin eisence nieness classic solion pseohyperbolic eaion INRODUION: onemporary problems o naral sciences mae necessary o sae an invesigae aliaive new problems he sriing eample o which is he class o non-local problems or parial ierenial eaions Among non-local problems we can isingish a class o problems wih inegral coniions Sch coniions appear by mahemaical simlaion o phenomena relae o physical plasma [ isribion o he hea [ process o moisre ranser in capillary-simple environmens [ wih he problems o emography an mahemaical biology HE ROEM SAEMEN AND IS REDUION O HE EQUIVAEN ROEM: onsier he eaion [4 in he omain { : } an sae or i a problem wih iniial coniions D an non-local coniions 4 ± is a given nmber are he given ncions is a sogh ncion Earlier he bonary vale problems wih non-local inegral eaions were consiere in he papers [ [ an [8 Here or we have an Ionin ype bonary coniion [ *orresponing ahor: Azizbayov EI** azel_azerbaijan@mailr Inernaional Jornal o Mahemaical Archive- eb 59

2 Azizbayov EI* / A NON-OA OUNDARY VAUE ROEM WIH INEGRA ONDIIONS / IJMA- eb- age: Deiniion: Uner he classic solion o problem -4 we nersan he ncion coninos in a close omain D ogeher wih all is erivaives conaine in eaion an saisying all coniions -4 in he orinary sense he ollowing lemma is prove similarly [7 emma : e [ D [ an he ollowing agreemen coniions be lille: 5 hen he problem on ining he classic solion o problem -4 is eivalen o he problem on eining o he ncion rom - an 6 roo: e be he solion o problem -4 Inegraing eaion wih respec o rom o we have: 7 Assming ha an allowing or we in: Hence we arrive a lillmen o 6 Now assme ha is he solion o problem - 6 hen allowing or 6 rom 8 we in: 8 rom an i is obvios ha 9 Since problem 8 9 has only a rivial solion hen ie coniion 4 is saisie he lemma is prove AUIIARY AS: Now in orer o invesigae problem - 6 we cie some nown acs onsier he ollowing specral problem [ an [5: ± IJMA All Righs Reserve 5

3 Azizbayov EI* / A NON-OA OUNDARY VAUE ROEM WIH INEGRA ONDIIONS / IJMA- eb- age: onary vale problem is no sel-ajoin he problem will be a conjgae problem Y Y Y Y Y Y We enoe he sysem o eigen an ajoin ncions o problem in he ollowing way [5: a b a bcos sin 4 π a / b / 5 We choose he sysem o eigen an ajoin ncions o he conjgae problem as ollows [5: Y Y 4cos Y 4 b asin 6 I is irecly veriie ha he biorhogonaliy coniions are lille Here δ ij is Kronecer s symbol he ollowing heorem is vali i Yj i Yj δij heorem [7: he sysem o ncions 4 orms a Riesz basis in he space an he esimaes r g g R g g g Y g Y r a b b a b c[ 4 R 8 b a [ are vali or any ncion g 7 Uner he assmpions i g [ g i s s s s g g g g s i i we esablish he valiiy o he esimaes: i i g g 8 i i i g g b a aig 9 IJMA All Righs Reserve 5

4 Azizbayov EI* / A NON-OA OUNDARY VAUE ROEM WIH INEGRA ONDIIONS / IJMA- eb- age: rher ner he assmpions i g [ g i s s s s g g g g i s i we prove he valiiy o he esimaes: i i g g i i i g g b a ai g Now enoe by α [6 an aggregae o all he ncions o he orm onsiere in D each o he ncions rom is coninos on [ an J [ α he norm in his se is eine as ollows: I is nown ha α J α is a anach space α α < [ [ EISENE AND UNIQUENESS O HE SOIION O HE OUNDARY VAUE ROEM: Since he sysem 4 orms a Riesz basis in an sysems 4 6 orm a sysem o ncions biorhogonal in each solion o problem - 6 has he orm: Y Moreover an Y are eine by relaions 4 an 6 respecively Applying he meho o separaion o variables or eermining he sogh ncions rom we have: 4 5 a 6 7 Y Y Y IJMA All Righs Reserve 5

5 Azizbayov EI* / A NON-OA OUNDARY VAUE ROEM WIH INEGRA ONDIIONS / IJMA- eb- age: Solving problem -4 we have: 8 cos sin sin 9 cos sin sin a sin sin cos a ξ sin ξ ξ sin a sin Aer sbsiion o epressions o 8 9 respecively in we have: cos sin cos sin sin a a sin sin cos sin ξ sin ξ ξsin a sin Now proceeing rom einiion o he solion o problem - 6 similar o [6 he ollowing lemma is prove emma : I is any solion o problem - 6 he ncions saisy sysem 8- heorem : e [ ± [ [ 4 D D IJMA All Righs Reserve 5

6 Azizbayov EI* / A NON-OA OUNDARY VAUE ROEM WIH INEGRA ONDIIONS / IJMA- eb- age: IJMA All Righs Reserve 54 hen problem - 6 ner small vales o has a nie classic solion roo: Denoing eal he righ han sies o 8 9 respecively an we wrie eaion in he orm: We ll sy eaion in he space I is easy o see ha / < < < < aing ino accon hese relaions we have: [ [ [ [ [ [ [ [ [ a a a a [ [ a a Here allowing or 8- we have: [ [ D a a a

7 Azizbayov EI* / A NON-OA OUNDARY VAUE ROEM WIH INEGRA ONDIIONS / IJMA- eb- age: [ [ 4 D 4 8 b a a 8 b a a [ Now consier he operaor in he sphere 8 b a a 8a D a 8 a a 5 8 D [ K K R A rom a a a 4 a D A a 4 4 a D 8 b a a 8 b a a 8 b a a 6 I is seen rom -5 ha or any K he esimaes : R A D 7 are vali 8 9 a [ hen i ollows rom esimaes 7 8 ha ner sicienly small vales o he operaor acs in he sphere K K R rom an i is conracive hereore in he sphere K K R he operaor has a nie ie poin{ } ha is a solion o eaion he ncion as an elemen o he space is coninos an has coninos erivaives on D Now prove ha an are coninos in D rom 4-6 we have: 4 [ [ [ [ 5 [ [ [ IJMA All Righs Reserve [ a 4 [

8 Azizbayov EI* / A NON-OA OUNDARY VAUE ROEM WIH INEGRA ONDIIONS / IJMA- eb- age: IJMA All Righs Reserve 56 I ollows rom esimaes 4-4 ha an are coninos in D rher i ollows rom 7 ha since by he given heorem < < < < an he more so < < hs coniions are lille I is obvios ha coniions is lille or he ncion I is easy o see ha [ [ a 4 Now i we se sysems 5-7 ealiy 4 aes he orm: 44 Where he ncions are eermine by relaion 4 an Uner he coniions o he heorem i is obvios ha < 45 hen i ollows rom 45 ha or any ie [ : [ 46 hs relaions 44 an 46 yiel

9 Azizbayov EI* / A NON-OA OUNDARY VAUE ROEM WIH INEGRA ONDIIONS / IJMA- eb- age: onseenly he ncion saisies eaion every in D So is a solion o problem - 6 an by lemma i is nie he heorem is prove y means o lemma we prove he ollowing heorem : e all he coniions o heorem an agreemen coniions 5 be lille hen or sicienly small vales o problem - has a nie classic solion 4 ONUSION: he ollowing resls have been obaine: he eisence o he solion o a no sel-ajoin bonary vale problem or a orh orer pseohyperbolic eaion has been prove he nieness o he solion o a no sel-ajoin bonary vale problem or a orh orer pseohyperbolic eaion has been shown he eisence o he classic solion o a non-classic bonary vale problem wih inegral bonary or a orh orer pseohyperbolic eaion has been prove 4 he nieness o he classic solion o a non-classic bonary vale problem wih inegral bonary or a orh orer pseohyperbolic eaion has been shown REERENES: [ eilin S Eisence o solions or one-imensional wave eaions wih non-local coniion Elecronic J o Dier Ea 76-8 [ oziani A Solion ore n probleme mie avec coniions non locales por ne classe eaions hyperbolies llein e la lasse es Sciences Acaemie Royale e elgie [ Ionin N I Solions o bonary vale problem in hea concions heory wih non-local bonary coniions Dierens Uravn [4 Gabov SA Orazov he eaion [ an several problems associae wih i USSR ompaional Mahemaics an Mahemaical hysics [5 Kasmov Mirzoyev VS On one generalizaion o Ionin s eample he absracs o scieniic conerence evoe o h anniversary o he honore scienis aca AIHseynov [6 Khaveriyev KI Azizbeov EI Invesigaion o classical solion o an one-imensional no sel-ajoin mie problem or a class o semi-linear pseopseohyperbolic eaions o orh orer roc a Sae Universiy phys-mahser 4- [7 Mehraliyev Y Ysiov MR he solion o a bonary vale problem or a secon orer parabolic eaion wih inegral coniions roceeings o IMM NAS o Azerb [8 lina S Non local problem wih inegral coniions or a pseohyperbolic eaion Di Uravnen ************************ IJMA All Righs Reserve 57