GENERALIZED KERNEL AND MIXED INTEGRAL EQUATION OF FREDHOLM - VOLTERRA TYPE R. T. Matoog Assistat Professor. Deartet of Matheatics, Faculty of Alied Scieces,U Al-Qura Uiversity, Makkah, Saudi Arabia Abstract: I this work, the existece of a uiue solutio of ixed itegral euatio (MIE) of the secod kid is cosidered, i the sace L ( ) C [, T ], where i the doai of itegratio ad t [, T ] is the tie. The kerel of ositio is cosidered i a geeralized otetial for. A uerical ethod is used to obtai syste of Fredhol itegral euatios (SFIEs). The existece of a uiue solutio of this syste ca be roved. Fially, ay secial cases are cosidered ad established fro the work ad soe uerical results are cosidered. Keywords: Fredhol- Volterra itegral euatio (F-VIE), geeralized otetial kerel, degeerate ethod, liear algebraic syste. MSC: 45 B5, 45G, 65R I. INTRODUCTION May robles of atheatical hysics, egieerig ad cotact robles i the theory of elasticity, fluid echaics ad uatu echaics lead to oe of the for of the itegral euatios. I [], Abdou used the searatio of variables ethod to solve the F-VIE of the first kid i the sace L ( ) C[, T], T, where i the doai of itegratio i ositio, ad t [, T ] is the tie. The oograhs of [, 3] cotai ay sectral relatioshis which are obtaied, usig the orthogoal olyoial ethod ad otetial theory ethod. I [4], Abdou obtaied the sectral relatioshis for the F-VIE of the first kid i three diesioal. The kerel of FI ter is cosidered i a geeralized otetial for, while the kerel of VI ter is a cotiuous fuctio i tie. Cosider the V- FIE t, () ( x, y, t) f ( x, y, t) F( t, ) k( x, y ) (,, ) d dd (, ) ( ) ( ), k x y x y, () i the sace L ( ) C [, T ]. Here, the kerel of ositio k ( x, y ) takes the for of geeralized otetial fuctio ad the kerel of VI ter Ft (, ) is a ositive cotiuous fuctio belogs to the class C[, T ]. The free ter f ( x, y, t ) is a kow fuctio, ( x, y, t ) is the ukow otetial fuctio, is the doai of ositio, is called Poisso ratio, is a costat defies the kid of the IE, ad is a costat, ay be colex, ad has ay hysical eaig. I order to guaratee the existece of a uiue solutio of E. () we ust assue the followig coditios: (i) The discotiuous kerel i L ( ) satisfies Coyright to IJIRSET www.ijirset.co 8
k ( x y ) dx dy C ( ( x, y, z) : x y a, z, x x ( x, ), y y ( y, )). (ii) The kerel of tie F( t, ) C[, T ] satisfies F( t, ) M,M is a costat. t [, T ], T. (iii) The give fuctio f ( x, y, t ) with its artial derivatives with resect to x, y ad t are cotiuous i the sace L ( ) C [, T ] ad for a costat G its or is defied as t f ( x, y, t) ax f ( x, y, ) dx dy d G. L ( ) C[, T ] t T (iv) The ukow fuctio ( x, y, t ) satisfies Lichitz coditio for the first ad secod arguet ad Holder coditio for the third arguet Theore () (without roof): The IE () has a existece ad uiue solutio uder the coditio M CT. I the reider art of this work, we rereset the geeralized otetial fuctio i the for of Weber- Soie itegral forula (W-SIf). The, we rereset the W-SIf as a artial differetial euatio of the first order of Cauchy tye. Moreover, the artial secod derivatives are rereseted i the ohoogeeous wave euatio.. Weber-Soie itegral forula (W-SIf): I this sectio, we rereset the ositio kerel i the for of a geeralized W-SIf. I this ai, after usig the olar coordiates i Es. (), (), we obtai The, after usig the otatios t a (, ) dd d ( r, t) F( t, ) f( r, t). r r cos ( x, y, t) ( r cos, r si, t) ( r,, t), f ( x, y, t) f ( rcos, rsi, t) f ( r,, t). (3) cos cos ( r,, t) ( r, t) ; f ( r,, t) f( r, t) ( ). si si The kerel of ositio of E. (3) becoes, cos d L ( r, ). r r cos Moreover, usig the followig three forulas, see Batea ad Ergyli [5, 6], (4) (5) Coyright to IJIRSET www.ijirset.co 8
cos d ( ) F z z cos z (,,, ) 4z F, ; ; z ( z) F, ; ;, ( z ) a b ( ) ( ) J a x J b x x dx ( a b) ( ) where ( ) is the gaa fuctio, ( ) is called Pochaer sybol ad hyergeoetric fuctio, the forula (5) yields (, ) ( ) ( ), L r c u J u r J u du Substitutig fro E. (7) ad cosiderig the substitutio the forula (3) becoes, ( ) ( ), ( ) 4ab F, ; ;. (6) ( a b) ( ) c ( ) F ( a, b; c; z ) is the Gauss ( ) X (, t ) (, t ), r f ( r, t ) g ( r, t ), (8) (7) where t a X ( r, t) F( t, ) K ( r, ) X (, ) d d g ( r, t) ( ) ( ) ( ), K r c r u J r u J u du The ositio kerel () takes a geeral for of W-SIf. 3. O the discussio of the W-SIf: We derive ay secial ad ew cases fro the W-SIf of () () Logarithic kerel: Let, i (),, we have K ( r, ) r J ( r u ) J ( u ) du ( ) c ( ) ( ). (9) () () Coyright to IJIRSET www.ijirset.co 83
() Carlea kerel: Let i () to have Fig. () K ( r, ) c r u J ( r u ) J ( u ) du.5,..5,.55 () Fig.() Fro the revious figures of Carlea fuctio we deduced that as icreases the cracks i the aterial icrease. (3) Ellitic kerel: Let, i (),,, we have the ellitic kerel. The iortace of the ellitic kerel coes fro the work of Kovaleko [7], who develoed the FIE of the first kid for the echaics ixed roble of cotiuous edia ad obtaied a aroxiate solutio of it. Fig. (3) (4) Potetial kerel: Let, i E. (),, we have the otetial kerel K ( r, ) r J ( u r) J ( u ) du I geeral, we write the kerel L ( r, ) of E. () i the Legedre olyoial for. Coyright to IJIRSET www.ijirset.co 84. (3)
K ( r, ) c ( r ) ( w ) P ( r) P ( ) w ( ).( w ) where, P ( r ) is the Legedre olyoial ad w ( ). =.7, ν=,5.,.5 (4) Fig. (4) 5,.5 5,.5 Fig.(5) Geeral cases: Here, the W-SIf is reresetig geeralized otetial for, ad as secial cases we cosider the followig:,.7,.7 Fig. (6) Theore (): The structure of the kerel W SIf, reresets Cauchy roble for the first order ad ohoogeeous wave euatio for the secod order. Proof: To rove this we differetiate E. () with resect to r ad resectively, ad the addig the result to get ( ) (, ) ( ) (, ) [ ( ) ( ) ( ) ( )] r r Usig the two faous relatios, see Batea ad Ergelyi [6] K r K r c r u J ru J u J ru J u du Coyright to IJIRSET www.ijirset.co 85 (5)
to have J ( z ) J ( z ) J ( z ), J ( z ) J ( z ) J ( z ) (6) z z ( ) K, ( r, ) ( a( r) a( )) K( r, ) ( K, ( r, ) K, ( r, )) r where a ( x ) ( ), K, ( r, ) K ( r, ). x x ad, K ( r, ) c r u J ( ru ) J ( u ) du The forula (7) reresets Cauchy roble of the first order i the ohoogeeous case. The secod derivatives lead us to the followig K (, ) (, ) (, ) ( ) ( ) r K r K r c r u J ru J u du r 4r 4 r c r u J ( ru) J ( u) du c r u [ J ( ru) J ( u) J ( ru) J ( u)] du By usig the relatios (6), ad after soe algebraic relatios, we obtai K ( r, ) [ h ( r ) h ( )] K ( r, ) ( 9) r where h ( x) ( ) x ; 4 The above forula reresets a ohoogeeous wave euatio. So, the secod derivative of the geeralized otetial kerel reresets a ohoogeeous wave euatio whe.5. REFERENCES [] Abdou, M. A. " Associate Professor Fredhol-Volterra itegral euatio of the first kid ad cotact roble" J. Al. Math. Cout. Vol. 5, 77-93,. [] Abdou, M. A. " Sectral relatioshis for the itegral oerators i cotact roble of iressig sta" J. Al. Math. Cout. Vol. 8,. 95-,. [3] Abdou, M. A. "Sectral relatioshis for the itegral euatios with Macdoald kerel ad cotact roble" J. Al. Math. Cout. Vol. 8,. 93-3,. [4] Abdou, M. A. "Fredhol-Volterra itegral euatio ad geeralized otetial kerel" J. Al. Math. Cout. Vol. 3,. 8-94,. [5] Batea, G. ad Ergelyi, A. "Higher Trascedetal Fuctios" vol. 3 Mc-Graw Hill, Lodo, New York, 99. [6] Batea, G. ad Ergelyi, A. "Higher Trascedetal Fuctios" vol. Mc-Graw Hill, Lodo, New York, 989. [7] Kovaleko, E.V. "Soe aroxiate ethods of solvig itegral euatios of ixed robles" J. Al. Math. Mech. Vol. 53,. 85-9, 989. (7) (8) Coyright to IJIRSET www.ijirset.co 86