Journal of American Science, 2012; 8(5); Spectral Relationships of Some Mixed Integral Equations of the First Kind

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1 Joural of Aerica Sciece ; 8(5); Spectral Relatioships of Soe Mixed Itegral Equatios of the First Kid S. J. Moaquel Departet of Matheatics Faculty of Sciece Kig Abdul Aziz Uiersity Saudi Arabia soaquel@au.edu.sa Abstract: Here the existece of a uique solutio of ixed itegral equatio (MIE) of the first id i three diesios is discussed i the space L C T T ; Ω is the doai of itegratio with respect to positio. A uerical ethod is used to obtai syste of Fredhol itegral equatios (SFIEs). May spectral relatioships (SRs) whe the erel of positio taes a logarithic for Carlea fuctio elliptic erel potetial fuctio ad geeralized potetial fuctio are obtaied i this wor. I additio ay iportat ew ad special cases are cosidered ad discussed. [S. J. Moaquel. Spectral Relatioships of Soe Mixed Itegral Equatios of the First Kid. J A Sci. ;8(5): 89-99]. (ISSN: 545-3). 3 Keywords: Mixed itegral equatio syste of Fredhol itegral equatios geeralized erel Weber-Soie itegral forula spectral relatioships. MSC: 45B5 45R.. Itroductio The atheatical forulatio of physical pheoea populatio geetics echaics ad cotact probles i the theory of elasticity ofte ioles sigular itegral equatio with differet erels. The oographs [-6] cotai ay differet SRs for differet ids of itegral equatios i oe two ad three diesioal. I additio i [7 8] usig Krei s ethod Mhitaria ad Abdou obtaied ay SRs for the FIE of the first id with logarithic erel ad Carlea fuctio respectiely. Cosider the MIE xy xy yt dy F y dydf xt t t - ( x ( x x x ) y y ( y y y )) x 3 3 (.) uder the coditio ( ) () Φ xt dx P t (.) Ω The itegral equatio (.) uder the coditio (.) ca be iestigated fro the ixed cotact proble of a rigid surface ( G ) G is the displaceet agitude ad is the Poisso s coefficiet haig a elastic aterial occupyig the doai Ω where Ω is the doai of itegratio with respect to positio through the tie t; t [ T ] T <. The gie fuctio f ( xt) is the su of two fuctios the first fuctio δ () t represets the displaceet of the surface uder the actio of the pressure of (.) P () t t [ T ] T < ad the secod fuctio f ( x) describes the basic forula of the surface. Here λ λ ad λ are costats ay be coplex ad haig ay physical eaigs. The uow fuctio Φ ( xt) represets the oral stresses betwee the layers of the two surfaces. The ow x - y fuctio is the erel of positio ad has a λ sigular ter while ( t τ ) F - is the erel of Volterra itegral ter i tie ad represets the resistace of the layer of the surface agaist the pressure P ( t ). I order to guaratee the existece of a uique solutio of (.) we assue the followig coditios: (i) The erel of the positio x y xx x x x3 ad y y ( y ) y y editor@aericasciece.org

2 Joural of Aerica Sciece ; 8(5); satisfies i L ( Ω ) the coditio x y dx dy A ( A: costat ) (ii) The positie cotiuous fuctio ( t τ ) C ([ T ] [ T ]) F - ad satisfies F t - τ < B B is a costat for all alues ( τ ) [ T ] t T. (iii) The gie fuctio f ( xt) with its first partial deriaties are cotiuous ad belog to the class L ( Ω ) C [ T ] where f L C ax t T t { f ( x τ ) dx } dτ (i) The uow fuctio Φ ( xt ) satisfies Hölder coditio with respect to tie ad Lipschitz coditio with respect to positio. I this wor a uerical ethod is used to trasfor the MIE (.) ito SFIEs of the first id. I additio the potetial theory ethod Fourier trasforatio ethod orthogoal polyoial ethod ad Krei s ethod will be used to establish ay theores for obtaiig the SRs of the SFIEs (.) uder the coditio (.3) i oe two ad three diesioal i the space L [ Ω ] C [ T ] T < ; Ω is the doai of itegratio with respect to positio. The erel of positio of (.) will tae the followig fors: logarithic for Carlea fuctio elliptic ad potetial erels ad geeralized potetial erel. Moreoer ay iportat ew cases will be discussed here.. Syste of Fredhol itegral equatios If we diide the iteral [T] t T as t < t < < ti T whe t t... i the MIE (.) taes the for xy xy p uf y dy y dyoh fx ( h p > ) (.) where h ax h ad h t + -t Here we used the followig otatios F ( t - t ) F Φ( yt ) Φ ( y ) f ( xt ) f ( x ) (.) The alues u ad the costat p deped o the uber of deriaties of ( t τ ) F - with respect to t see [9 ]. Also the boudary coditio (.) becoes φ ( x) dx P N Ω (.3) The forula (.) represets SIEs of the first id where it s solutio depeds o the id of the erel x y ad the doai of itegratio Ω. I the ext p+ applicatios we will eglect the error ter O( h ) 3. Theores of spectral relatioships. I this sectio we obtai the SRs of the SFIEs i oe two ad three diesioal usig differet doais ad suitable ethods. 3. SIEs with logarithic erel Cosider SFIEs of the first id xy xy u F y dy y dy f x tah u x- y () z exp( iuz) du z - u λ uder the coditios φ ( x) dx P (3.) - (3.) The erel of SFIEs (3.) ca be writte i the for (see []) tahu πz x-y ( z) exp( iuz) du - l tah z - u 4 λ (3.3) 9 editor@aericasciece.org

3 Joural of Aerica Sciece ; 8(5); If λ ad z is ery sall so that tah z z the we ay write z l tah l x y d d l 4 4 (3.4) Hece we hae uf [ l xy - d ] y dy[ l xy - d ] y dy fx Let ( ) ( T x cos cos x ) x [- ] (3.5) - deotes the Chebyshe polyoials of the first id while ( x ) si [( ) - + cos x ] si ( cos x) U - deotes the Chebyshe polyoials of the secod id. It is well ow that { T ( x )} for a orthogoal sequece of fuctios with respect to the weight - fuctio ( x ) { } - while U ( x ) for a orthogoal sequece of fuctios with respect to the x. It appears reasoable to attept a series expasio to Φ ( x ) i Eq. (3.) i weight fuctio ters of Chebyshe polyoials of the first id. This choice is ot arbitrary sice oe ca idetity a portio of the itegral as the weight fuctio associated with T ( x ). For coeiece we use the orthogoal polyoials ethod with soe well ow algebraic ad itegral relatios associated with Chebyshe polyoials see [3]. Thus i this ai we represet Φ ( x ) f ( x ) i the followig fors Theore : The SRs of the MIE (3.) uder the coditios (3.) whe the erel taes a logarithic fuctio are gie as : J l uf d l d J x y xy T y dy T y dy y y l d uf u F T x T x. (3.7) Differet ew cases ca be established fro (3.7) as the followig: () Differetiatig (3.7) with respect to x we get u F J T y dy T y dy yx y yx y uf U xu x Hece we get: u F J T y dy T y dy (3.8) y x y y x y (3.9) Thus the result of (3.8) leads to the SRs of the SFIEs with Cauchy erel. While (3.9) leads directly to the fact dy. - ( y - x ) - y ( ) f T Φ x a T ( x) f ( x ) - x - x (3.6) Usig the aboe expressios of (3.6) i (3.5) we hae the followig: () If ( ) si ξ si η x y siα siα - α ξ η α α π i (3.7) we hae the followig SRs 9 editor@aericasciece.org

4 Joural of Aerica Sciece ; 8(5); si T cos d si u F l d si cos cos J si cos T d si l d si cos cos si si l d u F l d J si si uf T T si si. (3.) (3) If u F ta ta ta y ta x J J Eq. (3.7) yields ta cos d l dt ta si cos cos ta cos d l dt ta si cos cos ta ta ta ta uf T T (3.) (4) Usig the followig relatios cos cos cos cos the forula (4.8) leads to the followig SRs cos cos (3.) ta ta ta ta cot T cos d cot T cos d u F cos cos cos cos 9 editor@aericasciece.org

5 Joural of Aerica Sciece ; 8(5); csc csc ta ta ta ta uf U U... ta si uf csc U ta cos ta 4 ta si cos U ta cos ta 4 (3.3) Here i (3.3) we obtai the SRs of the itegral operator with Hilbert erel for differet alues of ad. 3. SIEs with Carlea fuctio The iportace of Carlea fuctio cae fro the wor of Arutiuio [4] who has show that the cotact proble of the oliear theory of plasticity i its first approxiatio reduce to FIE of the first id with Carlea fuctio. If we cosider i the forula (3.) the followig sigular erel xy x y ; we hae the followig SFIEs: uf xy y dy xy y dyfx (3.4) To obtai the solutio of the forula (3.4) we assue ad represet the uow ad ow fuctios respectiely i the followig for: f x x x a C x x. f C x (3.5) Here C ( x) are Gegebauer polyoials a are the uow coefficiets ad f are the ow coefficiets. Usig the potetial theory ethod [3] ad the followig relatios [] + +. C ( x) [ x C ( x ) C ( x )] - α - -. ( - x) ( + x) C ( x) dx α+ β+ ( + α) Γ( + β) Γ( + ) β Γ!Γ Γ α+ β+ - ( ) ( ) 3 ( - + α+ ; + α+ β+ ; ) ( x ) - C ( x) ( -x ) [ C ( x) ] π dx Γ F Γ() ( + ) π dx! + C - Γ( + ) ( ) [ ( )] - ( x -) ( Re > ) Γ Re > - where Γ ( x ) is the Gaa fuctio ad ( ) 3 F...;.;.; is the Geeralized hypergeoetric fuctio we hae the followig: 93 editor@aericasciece.org

6 Joural of Aerica Sciece ; 8(5); Theore : The SRs of the MIE (3.) uder the coditio (3.) whe the erel taes the Carlea fuctio for are uf C y dy C y dy xy y xy y uf C x C x (3.6) ad uf C x C x (3.7) where cos 3.3. SIEs with potetial erel i fiite doai Assue the doai of itegratio Ω i (.) i the for { ( ) Ω xyz Ω : x + y a z } ad the erel taes the potetial fuctio for - [ x - ξ + y - η ] ( ) ( ) ( ) x - ξ y - η. Hece we hae the SFIEs with potetial erel. Usig the polar coordiates ad the usig the separatio of ariables the SIFs (.) yields a a ( ) ( ) ( ) uf L ( r ) ( ) d + L ( r ) ( ) d f ( r ) (3.8) where L cos ψ dψ π ( r ρ ) - π r + ρ - C y dy - C y dy - uf xy y xy y r ρ cosθ Usig the followig relatios see [6] [ - z cosψ + z ] ad z cos ψ dψ π α < Reα > ( α) π ( α) z! F ( ) Γ + α Γ ( α) ( α+ α+ z ) 4z F z z F ; ; z ab 4ab Jax Jbxx dx F ; ; - ab ( a b ) where F ( ab;c; z) is the Gauss hypergeoetric fuctio Γ ( x) is the Gaa fuctio ad J ( x) is the Bessel fuctio the SFIEs (3.8) taes the for a a u F K r Z d K r Z d g r ( () ( ) ( ) ( ) Z r r Φ g r f ( r )) where K ( ) ( ) ( ) r ρ π r ρ J u ρ J u r du (3.9) (3.) Eq. (3.9) represets SFIEs of the first id with erel (3.) taes a for of Weber-Soie itegral forula. Assue the solutio of (3.9) at a i the for Z ( ) ( ) ( ) () r a P ( -r ) -r ( ) (3.) where P ( y ) is the Legedre polyoial. The usig potetial theory ethod [3] ad orthogoal polyoials ethod [5] we obtai the followig: 94 editor@aericasciece.org

7 Joural of Aerica Sciece ; 8(5); Theore 3: The SRs of the SFIEs (3.) uder the coditio (3.) whe the erel taes a potetial fuctio for (3.) are ( ) ( ) u F K r P d K r P d ( - ) ( - ) r u F P ( - r ) r P (3.) where i geeral u ( ) ad P x! is a Jacobi polyoial. 3.4 SIEs with geeralized potetial erel i fiite doai i i < Whe the odules of the elasticity of the cotact proble is chagig accordig to σ K ε where σi ad εi are the stress ad strai rate itesities respectiely while K ad are the physical costats see [5]. For this the erel of Eq. (.) taes the for K x y x y (3.3) The erel of Eq. (3.3) is called the geeralized potetial erel. Usig (3.3) i (.) where Ω ( x y z ) { Ω : x + y a z } we ca arrie to the followig SFIEs. ( ) ( ) (3.4) u F K r Z d K r Z d g r where K ( ) ( ) du ( c ) (3.5) r c r u J u J u r The erel of (3.5) taes a geeralized for of Weber-Soie itegral forula. () Represetig the uow fuctios Z ad the ow fuctios g ( ) r respectiely i the Jacobi polyoials for Z ( ) r r ( ) ( ) ( ) a. P r g ( ) r r ( ) ( ) ( ) g. P r (3.6) The usig Krei s ethod see [5] we ca obtai the followig: Theore 4: The SRs of the SFIEs (3.) uder the coditio (3.) whe the erel taes a geeralized potetial fuctio for are: 95 editor@aericasciece.org

8 Joural of Aerica Sciece ; 8(5); u K u ( ) ( ) ( ) ( ) u P u du u K u P u du ( ) ( ) ( ) ( ) u F P u u P u ( ) ( ) (3.7)! ( w w w ). May special cases ca be deried fro (3.5) as the followig: (i) Carlea erel ± (ii) Logarithic erel ± (iii) Elliptic erel Fig..5 ν. Fig.3 Fig..5 ν.55 Fig editor@aericasciece.org

9 Joural of Aerica Sciece ; 8(5); Theore 5: The SRs of SFIEs (3.) uder the coditio (3.) for the coplete elliptic erel ca be obtaied i the for: r r ue r ue r u F P d P d!!!!!! uf P r P r 4 4!! ( ) ( P x P x P z is a Legedre polyoial ) (3.9) The iportace of the itegral equatio with coplete elliptic erel cae fro the wor of Koaleo [7] who deeloped the FIE of the first id for the echaics ixed proble of cotiuous edia ad obtaied a approxiate solutio for the FIE of the first id with coplete elliptic erel. (i) Potetial erel ν.5 () Geeralized potetial erel Fig. 5. Fig. 7 5 ν. Fig Fig.8 νo editor@aericasciece.org

10 Joural of Aerica Sciece ; 8(5); Fig. 9 ν. Fig. νo.5 May iportat spectral relatios ca be deried ad established fro the forula (3.5) for differet alues of ad for higher order. 4. Coclusio ad results Fro the aboe results ad discussio the followig ay be cocluded () The cotact proble of a rigid surface of a elastic aterial whe a stap of legth a is ipressed ito a elastic layer surface of a strip by a ariable Ρ() t t Τ whose eccetricity of applicatio e () t see [ ] becoes special case of this wor. () The uerical ethod used trasfors the MIE ito SFIEs. (3) The SFIEs depeds o the uber of deriaties of F ( tτ) with respect to tie [ Τ ]Τ t t. (4) The displaceet probles of ati plae deforatio of a ifiite rigid strip with width a puttig o a elastic layer of thicess h is cosidered as a special case of this wor whe t F ( tτ) f ( xt) Η ad x x. Here Η represets the displaceet agitude ad ψ ( x) the uow fuctio represets the displaceet stress see [8]. (5) The probles of ifiite rigid strip with width a ipressed i a iscous liquid layer of thicess h whe the strip has a elocity resultig fro the ipulsie force -iwt V V e i V is the costat - where elocity w is the agular elocity resultig rotatig the strip about z-axis are cosidered as special case of this wor whe F ( tτ) costat ad t see [8]. (6) I the aboe discussio (4) ad (5) ad whe h this eas that the depth of the liquid (fluid echaics ) or the thicess of elastic aterial (cotact proble ) becoes a ifiite. (7) The three ids of the displaceet proble i the theory of elasticity ad ixed cotact probles which discussed i [8] are cosidered special cases of this wor. (8) The geeralized potetial erel represets a Weber-Soi itegral forula (3.5) ad represets a o hoogeeous wae equatios. The erel (3.5) ca be writte i the Legedre polyoial for as follows α K (uν) - -w (u) + Γ (+ + - w Γ (+ + )( w - )P (u)p () - - ( P ( u ) is Legedre polyoial ad ± ± α W ). (9) Taig i id the basic relatios of Bessel fuctio the geeralized potetial erel (3.5) satisfies the followig ohoogeeous wae equatio K r h r h K r r h r r 4 ) 98 editor@aericasciece.org

11 Joural of Aerica Sciece ; 8(5); () This paper is cosidered as a geeralizatio of the worer of the cotact probles i cotiuous edia for the Fredhol itegral equatio of the first ad secod id whe the erel taes the followig fors: Logarithic erel Carlea erel elliptic itegral erel ad potetial erel. Moreoer the cotact proble which leads us to the itegrodifferetial equatio with Cauchy erel is cotaied also as a special case see []. Also i this wor the cotact probles of higher-order ( ) haroic are icluded as special cases see [ ]. Refereces. Abdou M. A. ad G. M. Abd Al-Kader(8): Mixed type of itegral equatio with potetial erel Tur J. Math.3: Abdou M. A. (): Spectral relatioships for itegral operators i cotact proble of ipressig staps J. Appl. Math. Coput. 8: Abdou M. A. (): Spectral relatioships for the itegral equatio with Macdoald erel ad cotact proble J. Appl. Math. Coput. 5: Abdou M. A. (): Fredhol-Volterra itegral equatio of the first id ad cotact proble J. Appl. Math. Coput. 5: Abdou M. A. (): Fredhol-Volterra itegral equatio ad geeralized potetial erel J. App. Math. Coput. 3: Abdou M. A.. F.A. Salaa(4): Volterra- Fredhol itegral equatio of the first id ad spectral relatioships J. Appl. Math. Coput. 53: Mhitaria S. M. M. A. Abdou(99): O differet ethod of solutio of the itegral equatio for the plaer cotact proble of elasticity Dal. Acad. Nau Ar. SSR 89(): Mhitaria S. M. ad M. A. Abdou(99): O arious ethod for the solutio of the Carlea itegral equatio Dal. Acad. Nau Ar. SSR 89(3): Deles L. M.. J. L. Mohaed(985): Coputatioal Methods for Itegral Equatios Philadelphia New Yor.. Atiso K. E. (997): The Nuerical Solutio of Itegral Equatio of the Secod Kid Cabridge.. Ya G.. Popo(98): Cotact Proble for a Liearly Deforable Base Kie - Odessa.. Gradshtey I. S. ad I. M. Ryzhi(994):Tables of Itegrals Series ad Products Fifth editio Acadeic Press. Ic. 3. Abdou M. A. (3): O asyptotic ethods for Fredhol-Volterra itegral equatio of the secod id i cotact probles J. Cop. Appl. Math. 54: Arutiuio N. K. H. (959): A plae cotact proble of creep theory Appl. Math. Mech. 3(): Abdou M. A. (): Itegral equatio of ixed type ad itegrals of orthogoal polyoials J. Cop. Appl. Math. 38: Erdelyi A. W. Magus F. Oerheatig ad G. Tricoi (973): Higher Trascedetal Fuctios Vol. Mc-Graw Hill New Yor. 7. Koaleo E. K. (989): Soe approxiate ethods of solig itegral equatio of ixed proble J. Appl. Math. Mech. 53: Alesadro V. M. E. V. Koaleo(986): Probles i the Mechaics of Cotiuous Media with Mixed Boudary Coditios Naua Moscow. 9. Abdou M. A. M. Basse(): A ai theore of spectral relatioships for Fredhol- Volterra itegral equatio of the first id ad its applicatios Math. Meth. Appl. Sci. 33: /6/ editor@aericasciece.org