Krein's method and mixed integral equation of Volterra Fredholm type. R. T. Matoog

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1 Krei's method d mied itegr eqtio of Voterr Fredhom type R T Mtoog Deprtmet of Mthemtics Fcty of Appied Scieces Umm A-Qr Uiersity Mkkh Sdi Arbi PO Bo 7 rmtoog@yhoocom Abstrct: Here the eistece of iqe sotio of Voterr Fredhom itegr eqtio V-FIE of the first kid is cosidered i the spce L [ ] C T T< The Fredhom itegr term is cosidered i positio with discotios kere whie the Voterr itegr term is cosidered i time with cotios kere Usig meric we he system of Fredhom itegr eqtios SFIEs of the first fid The sig Krei's method the sotio of SFIEs is obtied i the form of spectr retioships SRs Fiy my speci cses i fid mechics d cotct probems re discssed [Mtoog R T Krei's method d mied itegr eqtio of Voterr Fredhom type Life Sci J 4; : 7-4] ISSN: Keywords: Voterr Fredhom itegr eqtio Krei s method cotct probems spectr retioships Chebyshe poyomis CPsMSC: 45B5 45R Itrodctio: Sigr itegr eqtios of the first kid he receied cosiderbe iterest i the mthemtic itertres becse of their my fied of ppictios i differet res of scieces for empe see [- 4] The sotio of these IEs c be obtied yticy sig oe of the foowig methods: Cchy method [5] poteti theory method [6] orthogo poyomis method [7] itegr trsformtio methods [4-7] d Krei's method [8] Mkhitri d Abdo [9] discssed some differet methods for soig the FIE of the first kid with ogrithmic kere I this work we cosider the V-FIE of the first kid t y F t τ k ϕ y τ dy dτ πf t π r t f * - y [ ] t τ [ ] ; L cos z k z d L m + m + der the coditio ϕ t d - t L is cotios d positie The fctio for d stisfies the foowig symptotic eqities L m m + O m L + O m 4 The V-FIE der the coditio c be iestigted from the cotct probem G υ hig estic of rigid srfce mteri occpyig the domi [ - ] where f is describig the srfce bse of stmp This stmp is impressed ito estic yer srfce by ribe kow force t [ ] t < whose eccetricity of ppictio e t tht cse rigid dispcemet γ t Here G is ced the dispcemet mgitde d υ is Poisso's coefficiet I order to grtee the eistece of iqe sotio of we ssme for the two costts E d D the foowig coditios: i The kere of positio stisfies y - - k d dy E ii The positie cotios kere which represets the resistce force of the mteri F t τ C [ ] [ ] d stisfies F t τ < D iii The cotios fctio of time t C [ ] γ whie the positio 7

2 fctio f L [ ] d f t L [ ] C [ ]The orm of f t is m τ - L C t t f f d dτ i The kow poteti fctio ϕ t stisfies Höder coditio with respect to time d Lipschitz coditio with respect to positio I this work we se meric method to trsform the V-FIE ito ier SFIEs of the first kid The sig Krei s method the sotio of SFIEs c be obtied i the form of spectr retioships SRs of CPs My speci cses re deried d discssed from the work Moreoer some ppictios i cotct probems d fid mechics re cosidered System of FIEs If we diide the iter [T] t s t < t < L < t N whe t tk k K The V-FIE tkes the form see [] t k F t τ k y ϕ y τ dydτ F k y φ - y dy πf k k - p + I we egect the error term O h where h m h h t + t The costt defied s the chrcteristic mber see [] Aso we sed the foowig ottios ϕ t ϕ F t t F f t f The bodry coditio becomes d k φ - k k re costts Let i m d sch tht y is ery sm the sig the term the retio [7] cos z d + d d is costt 4 the coditios tke the form π d y d + y F + - dy y π K Ν where re the CPs of the first fid d order Proof: The proof of depeds o the foowig emms Lemm : For positie itegers we he Ι + + d β where re Jcobi poyomis JPs Proof: For proig et + g y y where y re the CPs of the first kid the 8 c be writte i the form d Ι π + d D + d D where D 4 Usig the sbsttio s s ds s t d the retio the form 4 tkes the form D t t dt 5 Usig the fmos retio betwee CPs LPs d JPs see [] π t - t y dt y + y 6 where re Legedre poyomis LPs the form 5 yieds π D 7 Aso the first deritie of 7 tkes the form 8

3 dd d π ; N 8 β for egtie iteger Fiy itrodcig 7 8 i we obti the reqired rest Corory: Pt i we he D + + d 9 Usig the fmos retio β Γ + +! Γ + the form 9 becomes + D d where Γ is the Gmm fctio Corory: The e of the secod d dd derities is gie by d d d dd d d D π + + Lemm : The e of the foowig itegr d d d s ds Α d d s tkes the form Α π Γ! + π y y y y + y y y K; K Ν Proof: For proig the emm we itrodce i to he d Α + + π 4 d Assme i 4 the sbstittio y z to he A z A + ydy + y + yz yz π π z z + π z y dy yz ydy 5 z the 5 If we pt y yieds A π z z [ ] z + π + z + d + z [ z ] π + z [ z ] 6 If we se the fmos forms [] r β z z γz dz + + Γ Γ r! Γ + Γ + r Γ R > R r > e d γ F + + β + ; + + r; e 7 d + β ; F β 8 where F ; β β ; z is the geerized hypergeometric series d F β; γ ; z is the hypergeometric Gss fctio the first itegr term of 6 becomes π! [ ] z d z Γ + 9 Aso sig the sme wy the secod d third itegr term of 6 yied d 9

4 z [ z ] d πγ z + + Γ + d [ z ] π! + z Γ + d z + + z Itrodcig the three forms 9 - i the emm is proed Fiy to proe the theorem we write i the CPs form for this prpose we mst cosider the foowig fmos forms see [ ] i ii d Γ + Γ πγ + C Retio betwee Jcobi d Gegeber poyomis Γ + Γ πγ + + C + iii im Γ C 4 Retio betwee Chebyshe d Gegeber poyomis Usig these fmos retios i oe hs A π 5 Itrodcig 5 d i 6 the theorem is proed By sig the sme wy we c proe this theorem Theorem : The spectr retioships for the SFIEs with the kere defied by 4 d the kow fctio is odd is gie by J π F + d s K Ν 6 s ds The proof of theorem c be obtied directy by foowig the sme wy of theorem 4 Cocsio d rests: From the boe rests d discssio the foowig my be cocded The cotct probem of rigid srfce of estic mteri whe stmp of egth is impressed ito estic yer srfce of strip by ribe t t < whose eccetricity of ppictio e t represets V-FIE of the first kid The meric method sed trsforms the V-FIE ito SFIEs The SFIEs depeds o the mber of F tτ with respect to time derities of tt [ ] < 4 The dispcemet probems of t pe deformtio of ifiite rigid strip with width pttig o estic yer of thickess h is cosidered s speci cse of this work whe t F t τ f t Η d ϕ ψ Here Η represets the dispcemet mgitde d ψ the kow fctio represets the dispcemet stress 5 The probems of ifiite rigid strip with width impressed i iscos iqid yer of thickess h whe the strip hs eocity restig from the impsie force iwt e i where is the costt eocity w is the gr eocity restig rottig the strip bot z-is re cosidered s speci cse of this work whe F tτ costt d t see [4] 6I the discssio 4 d 5 whe h this mes the depth of the iqid Fid mechics or the thickess of estic mteri cotct probem becomes ifiite 7The three kids of the dispcemet probem i the theory of esticity d mied cotct probems which discssed i Aeksdro et [4]Mskeishii [5] Gree [6] d Popo [7]re cosidered speci cses of this work 8 My importt retioships c be deried from 4

5 si If m si y si ; d if m + si t y t we he the t t foowig SFIEs F + d ψ d hk si 4 The boe system eds to the foowig SRs si cos d si F + d si cos cos π π m 4 d + d si m si si m m K Ν cos d cos π t m m + t m 4 t m + t F si cos + ii Differetitig with respect to we he y dy F πu y y y dy F y 44 U where re the CPs of the secod kid Aso 44 yieds F cot t t cos d cos cos cos ec cos ec U t t t t si + cos U + [ t] m m m 4 m m 45 sec cot t F cos cos t cosec sec sec U t t t 46 9 The mied itegr eqtio with Crem kere c be estbished from this work by sig the foowig retio y y h y < υ < 47 where h υ y y y υ is smooth fctio The importce of Crem kere cme from the work of Artiio [] who hs show tht the cotct probem of oier theory of psticity i its first pproimtio redce to FIE of the first kid with Crem kere The retio betwee the eigees d the correspodig Chebyshe poyomit re obtied i the foowig figres d 4

6 Fig 5 Fig: Fig: Fig46: 5 Refereces [] Costt C Itegr eqtio of the first kid i pe esticity J Qort App Mth L [] M A O symptotic methods for Fredhom Voterr itegr eqtio of the secod kid i cotct probems J Comp App Mth []Abdo M A d F A Sm Voterr Fredhom itegr eqtio of the first kid d spectr retioships App Mth Compt [4] Aeksdro V M d E V Koeko Probems i Mechics Medi with Mied Bodry Coditios Nk Moscow 986 [5] Mskeishii N I Sigr Itegr Eqtios Noordhoff Netherd 95 [6] Gree C D Itegr Eqtio Methods New York 969 [7] Popo G Y Cotct Probems for Liery Deformbe Bse Kie Odess 98 4

7 [8] Abdo M A d N Y Ezz-Edi Krei s method with certi sigr kere for soig the itegr eqtio of the first kid Period Mth Hg Vo 8 No [9] Mkhitri S M d M A Abdo O differet methods of sotio of the itegr eqtio for the per cotct probem of esticity Dk Acd Nk Arm SSR []Grdstei IS d I M Ryzhik Tbe of Itegrs Series d Prodct forth editio Egd Acdemic Press 98 [] Btem G d A Ergeyi Higher Trscedet Fctios T Nk Moscow 974 [] Artiio N K Pe Cotct Probem of the Theory of Creep J App Mth Mech