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Applied Mathematics, 2013, 4, 1301-1312 http://dx.doi.org/10.4236/am.2013.49176 Published Online September 2013 (http://www.scirp.org/journal/am) To Theory One Class Linear Model Noclassical Volterra Type Integral Equation with Left Boundary Singular Point Nusrat Rajabov Tajik National University, Dushanbe, Tajikistan Email: nusrat38@mail.ru Received April 22, 2013; revised May 22, 2013; accepted June 1, 2013 Copyright © 2013 Nusrat Rajabov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this work, we investigate one class of Volterra type integral equation, in model case, when kernels have first order fixed singularity and logarithmic singularity. In detail study the case, when n = 3. In depend of the signs parameters solution to this integral equation can contain three arbitrary constants, two arbitrary constants, one constant and may have unique solution. In the case when general solution of integral equation contains arbitrary constants, we stand and investigate different boundary value problems, when conditions are given in singular point. Besides for considered in- tegral equation, the solution found cane represented in generalized power series. Some results obtained in the general model case. Keywords: Neoclassical Volterra Type Integral Equation; Left Boundary Singular Point; Boundary Value Problems 1. Introduction Let : x axb be a set of point on the real axis and consider an integral equation 1 1ln d xnk k k a t xa x p ta ta tfx n , (I) where is given constants, 1 j pj f x is given function in and x to be found. In what follows we in detail go into case n = 3. In this case the Equation (1) accepts the following form 2 12 3 ln lnd . x a t xa xa x pp pt tata ta fx (1) Integral Equation (1) at p2 = 0, p3 = 0 is model second kind Volterra type singular integral equation with left boundary singular point, theory construction in [1-5]. In the case, when in (1) p3 = 0 Equation (1) investigates in [6]. As [4,5] the solution to this equation is sought in the class of function , x Cab , 0a with fol- lowing asymptotic behavior t,0a x oxax a . (2) In this case the integrals in the Equation (1 proper one. Moreover ) are im- 0afa i.e si . right-hand de is necessarily zero at . x a In this case in Equation (1) 0 it investi- gates in [1]. In this casfrom signs p 23 pp depend 1 e, in p 11 0,0 ,p solution integral1) is found in explicit form. In this case at 10p homogeneous integral Equatn (1) has one solution and general solu- tion no homogeneous (1) contains one arbitrary constant and at 10p, integral Equation (1) has unique solution. In case of, when in (1) 30p, 10p, 20p Equation ( io in- tegral Equation (1) investigates in [6]. In this case in de- pearacteriic equaon obtand from corresponding chsttiined solution integral Equation (1) by two arbitrary constants, one arbitrary constant. Select the case, when integral Equation (1) has unique solution. To problems investiga- tion one dimensional and many-dimensional Volterra type integral equation with fixed boundary and interior singular points and singular domains in kernels dedicate [1-7]. Support that solution integral Equation (1) function 3 xC . Besides, let in Equation (1) function f 3 x C two. Then differentiating integral Equa- mes, we obtained the following third or- n differential equation tion (1) three ti der degeneratio C opyright © 2013 SciRes. AM N. RAJABOV 1302 3 Dx 2 123 3, xx x x pDxpDxpx Df x (3) d d x Dxa x . where Homogeneous differential Equation (3) is correspond- e following characteristic equation . (4) tion 2.1. The Case, When the Roots of th ing to th 3220pp p 12 3 2. Representation the General Solu e Characteristic Equation Real and Different Let in differential Equation (3) parameters 13 jjp ion (4) real suchthat, the roots of the characteristic Equat and different. Its denote by 123 ,, . In this case, im- mediately testing we see that solution homoge- neous differential Equation (2) is given by formula 12 3 , general 123 x xa CxaCxaC , (5) where 13 j Cj n, arbts. itrary constan Whe 01 3j , function j x definable by formula (5) eneous integraluation (1). So, function satisfy homog Eq x determined by formula (5) is given general solutioogeneous integral Equation (1). For obtained the solution non homogeneous inte n hom gral Eq uation (1), first time use the variation arbitrary con- stants methods, we use the general solution of the differ- ential Equation (3). After transformation, we see that, if solution integral Equation (1) in this case exist, then we its my be represented in the following form 123 12 333 3 12312 3 0 1123 1d ,,, x a f t xa xaxa x xa CxaCxa C fxt ta tata ta KCCCfx (6) where 13 j Cj arbitrary constants, 0 123 2332 12211313 222 123 1, 1, 1 ,, . ,, The solution of the type (6) obtained in the case, when , 3 fx C 0fa, solution integral equation (1), function x exist and belong to Class 3 C . esting, ee that, of Immediately twe s 01 3 jj , fx C 3, 0fa with asymptotic b ehavior 1 1 ,xo xa 123 max ,,at f, x a (7) then function (5) satisfied Equation (1). Be valid the following confirmation. (1) parameters Theorem 1. Let in integral Equation 13 jj such that, the roots of the p algebraic Equ- ation (4) real, different and positive, function f x C, 0 with asymptotic behavior (7). Then integral Equation (1) in class of function fa xC x a form vanishing in point is always solvability and its solution is given by ula (6), 13 j Cj are ar- gral Equation bitrary constants. Characteristics 1. Let in inte (1) pa- rameters 13 j pj , function f x satisfy any con- dition of theorem 1.Then, from (6) it follows, the solu- tion integr1) al Equation ( xC , 0a with following asymptotic behavior 1 1 , min 123 ,,, at x oxa x a . If, the roots of the characteristic Equation (4) real, - ferent and dif 10 , 20 , 30 , then it follows, from formula (6) 10C . In this case,exist th solutio integral Equation (1), then it is posllowing form if en sible is represent in fo 3 4512 3 0 245 1 ,, x a 123 33 d 23 f t taxa xa x xaCxaCfxt x atatata KCCfx (8) where -are arbitrary constants. Th of the type (8) exist, if 4 C, e solu 5 C tion fx C , 0fa 2 21 123 ,, min ,at fxo xa xa (9) with asymptotic behavior Copyright © 2013 SciRes. AM N. RAJABOV 1303 So, in this case have the following confirmation. re 2. (1) parameters following asymptotic behavior Theo m Let in integral Equation 13 jj such that, the roots of the algeb- real, different and also praic Equ ation (4) 10 , 20 , 30 , fx C , 0fa with asym ) in c pt Then integral Equation (1 otic be- havior (9). tion lass of func- x C va nishing inpoint x a is al- ways solvability and its solution is given by formula (8), 4, 5j are arbitrary constants. j Characteristics 2. Let in integral Equation (1) pa- rameters C 13 j pj, function f x satisfy any con- dition of thm 2en, (8) it follows, the solu- tion integral Equation (1) eore. Thfrom xC , 0a with 2,m 223 i atn,, x oxa x a Remark 1. Confirmation similar to theorem 2 ob- ta ined and in the following cases: a) 10 , 20 , 30 ; b) 10 , 0 2 , 0 3 . he o of the charatiIf totsterisa) ndrcc eqution (4real a different, 10 , 20 , 30 , then from integral representation), follows, that in order that (6 x is solution integral Equation (1) in this case, it is necessary 12 0CC . In this case, if exist solution integral Equa- tio it will be represented in following form n (1), then 123 3 3 23 d 0 26 ,, a 33 6 1x f t ta xata x xa Cfxt x axatata (10) tant. The solution of the type (10) exist, if KC fx where C6 are arbitrary cons fx C , 0fa with asymptotic behavior , 3 fxox a 33 at x a. (11) So, we proof. The following confirmation. uation (1) parameters Theorem 3. Let in integral Eq 13pj such that, the roo jts of the - 4) real, different and also 1 algebraic Equ ation (0 , 20 , 30 , fx C , 0fa with asympt- integrn otic be havior (11). Then tion al Euatioq(1) in class of func- x C va nishing inpoint x a is al- rmula (10) given by foways solvability and wh its solution is ere C are arbitrary consta , rameters 3nt. Characteristics 3. Let in integral Equation (1) pa- 13 j pj , function f x lution satisfy any conditio Then the soof the integral Equation n of theorem 3. (1) in point x a vanishs asymptotic behavior determined from formula and it 3at . x oxax a Remark 2.tion similar tom 3, ob- tained and in the following cases: a) 10 Confirma theore , 20 , 30 ; b) 10 , 20 , 30 . If the roots of the characteristic Equation (4) real, dif- ferent and 013 jj , then from integral repre- sentation (6er that ) follows, in ord x nece is solution integral Eqase, it isC1 = C2 = uation (1) in this cssary C3 = 0. In this case, if exist solution integral Equation (1), then its will be represented in form 23 3 3 4 d ft ta 1 33 12 0 x a ta ta1 x fx xa x The solution of the type (12) exist, if t Kfx x (12) aata fx C , 0r fa with asymptotic behavio ,0at f xoxa xa (13) So we proof the following confirmation. Theorem 4. Let in integral Equation (1) parameters 13 j pj ati such that, the roots of the algebraic Equ- The function on (4) real, different and positive. f x C, 0fa with asymptotic behavior tegrn (1) in class of function (13). Then in al Equatio xC vapoint x = a have give b(12). ation (1) pa- nishing in y formula unique so- lution, which Characteristics 4. Let in integral Equ rameters 13 j pj , function f x satisfy any con- dition of theorem 4. Then the solution of the integral Equation (1) in point x a vanish and its asymptotic behavior determined from formula ,0at x oxa x a . 2.2. The Case, When the Roots of the Characteristic Equation Real and Equal Let in integral Equation (1) parameters 13 j pj , aracteristic Equation (4) real and equal. such that, the roots of the ch In this case we have the following confirmation: Copyright © 2013 SciRes. AM N. RAJABOV 1304 Theorem 5. Let in integral Equation (1) parameters 13 j pj such that, the roots of the characteristic Equation (4) real, equal and positive, that is 12 30 that a fun . Assume ction fx C , 0fa with the following asymptotic behavior 4 4at,,. f xoxa xa Then homogeneous integral Equation (1) class of in function xC vanishing in point x a , have three linear independent solutions the type 1 x xa , 2ln x xa xa 2 ln . , 3 x xa xa Non homogeneous integral Equation (1), always solv- able. Its general solution contain three arbitrary con- stant and given by formula 6 2 ,,, , a 2 3 22 5123 ln ln 6 lnlnd xaCxaCfx 12 x xxaC f t axaxat tatata ta (14) x KC CCfx Were 13 j Cj -arbitrary constants. stics 5. In this case, when in integral Equa- tion (1) parameters Characteri 13 j pj , function f x ion inte sat- isfy any clut gra Equation ondition of theorem 5, then sol (1) in point x a vanish and its asymptotic , 0, .at xoxa xa From integral representation (14) follows. If solution integral Equation (1) at 123 0 d it’s in form behavior determined from formula exist, then we may be represente 22 66 lnln xxa 6 d . 2a ft xa ta x fx t Kfx tataxa ta (15) The solution of the type (15) exist, if fx C , 0fa 2.3. The Case, When One Roots of the Characteristic Equat the Roots of the Characteristic Equation Complex and Conjugate Let in integral Equation (1) parameters with the following asymptotic behavior , at 0, . f xoxaxa (16) o in the case, when S123 0 , proof the following confirmation. Theorem 6. Let in integral Equation (1) parameters 13 j pj such thots of the cheris- tic Equation (4) real, equal and negative, that is 12 at, the all roaract 30 . Assume hat a function t fx C , 0fa with the asymptotic behavior (16). Then, in- gral Equation (1) in class te C have mula (15). Characteristics 6. In this case, when fulfillment any condition theorem 6, then solution integral equation in po unique solution and give by for int x a vanish and its asymptotic behavior deter- mined from formula , at 0, . x oxaxa ion Real and Two 13 j pj Equation (4) real plex 1 such that, the one roots of characteristic and two the roots of the characteristic equation com conjugate. Correspondingly its denote by , 2 A iB , 3 A iB . When 10, 0A , then by this roots cor- responding following parmogeneous integral Equation (1): ticular solution ho 1 1 2 3 cos ln sin ln , . A A xxa x xaB xa x xaB xa (17) In this c ase, if solution integral Equation (1) exist, then it will be represented in form 1 3 11 0 71 23 1sin ,,, a 1 12 A A x Cxa Ca3 2 sin ln lncoslnd CB xafxcos lnxxaB x f t x xa DBt ta (18) xa xa BD ta ta KCCCfx a Btata Copyright © 2013 SciRes. AM N. RAJABOV 1305 where 222 011 20ABBB AB , 44 3DBAA 22 BA 11 2222 21 23DABABBBA . The solution of the type (18) exist, if 10 , , 0A, fx C , 0 fa with the following asymptotic behavior 5 51 , max, at . f xoxaA xa (19) So in this case we have the following confirmation. Theorem 7. Let in integral Equation (1) parameters 13 j pj tic Equation (2 such that, one the roots of the character- (4) real positive, two out of its complex is conjugate A iB , 3 A iB ). Besides let 2 RealA0 . Assume that a funct fx C , 0fa ion with asymptotic behavior (19). Then homogeegral Equation (1), in class neous int C vanishing in point x a , lvable, s and has three linear Independent solution of Non homogenous integral Equation (1) a its general solution contain three arbitrar gi the lways so y constant type (17). ven by formula (18), where 13 j Cj -arbitrary constants. Characteristics 7. In the case, when fulfillment any condition theorem 7, then solution integral Equation (1) in point x a vanish and its behavior determined from following asymptotic formula 6 61 a, min,At . x oxax a From integral representation (18) follow of the algebraic Equation (4) satisfy condition of the theorem 7, besides s, if the roots 10 , 0A. If 10 , ) in t 0A, is th on hen if exist solution integ (1case, then its represented in following form ral Equati 2 3 ,, . 1 23 3 11 0 8 cos lnsin ln 1sin A A x a xxaCBxaC B 2 lncos lnd xa fx f t xa xa BDBt ta xa BD x ata ta K In this case for convergence integrals in right part (20), necessary ta ta (20) C Cfx fx C , 0fa with asym beha ptotic vior 7 7 , at . f xoxa Axa (21) So, we proof. the following confirmation. Theorem 8. integation (1) parameters Let inral Equ 1 j pj 3 sndition theatisfy coorem 7, besides 1 0, 0.A Let 10 , 0A. Function fx C , 0fa with geneous integr asympt al Equat otic be on (1), havior (21) i in class . Then homo- C vanishing in point x a, has two linear Independlution ent so 2cos ln A x xaB xa , 3sin ln A x xaB xa . Non homogenous integral Equation (1) always solv- able and its general solution from C class is given by formula (20), where -arbitrary con- stants. 23 j Cj Characteristics 8. In the case , when fulfillment any condition theorem 8, then solution integral Equation (1) in point x a vanish and its behavior determined from following asymptotic formula A at x oxax a . Now suppose, that the roots of the algebraic Equation (4) satisfy condition of the theorem 7, besides 10 , 0A. Let 10 , 0A . Then, if exist s then its repr lowing form olution inte- gral Equation (1) in this case,esented in fol- 1 13 11 1 1 2 cos lnd 0 91 sin ln ,. A x a f t axaxa tax x xa CfxBta xa K D Bta Cfx (22 (22), ne D Bt ta ta ) cessary In this case for convergence integrals in right part fx C, 0fa with asymptotic havior 8 81 , at . f xoxa xa (23) So, we proof. the following confirmation. be Copyright © 2013 SciRes. AM N. RAJABOV 1306 Theorem 9. Let in integral Equation (1)arameters p 13 j pj satisfy condition theorem 7, besides 1 0, 0.A Let 10 , 0A. Function fx C , 0fa with geneous integr asympt al Equat otic be on (1), havior (23) i in class . Then homo- C 1 vanishing ,lin point x = a one soution 1 x xa . Non ho- mogenous integral Equation (1) always solvable and its general solution from class C is given by formula (22), where arbitrary constant. Characcs 9. In the case, when fulfillment any co 1 C- teristi ndition theorem 9, then solution integral Equation (1) in point x a vanish and its behavior deined from following asymptotic formu term la 1, at xoxa xa In the case, when 10 , 0A, then from integral representation (18) follows, that, if exidt solution integral Equation (1) in this case, then it is possible in following form 1 3 11 0 10 1sin lnl . A x a ta ta 2 cos nd ft xa xa x fxBD BDBt x axa Kfx In this case for convergence integrals in right part (24), it is sufficient ta tata (24) Characteristics 10. In the case, when fulfillment any condition theorem 10, then solution integral equati in point fx C , 0fa with asymp- totic behavior , at 0, . f xoxaxa (25) . the following confir 10. Let in integral Equation So, we proofmation. Theorem (1) parameters 13 j pj satisfy condition theorem 7, besides 1 0, 0.A Let 10 , 0A. Function Cfx , 0a with asymptotic behavior (25). T f Eq hen integral uation (1), in class C vanishing in point x a , olution, which given by formula (24). have unique s on (1) x a vanish and its behavior determined from following asymptotic formula , 0 at x oxax a 3. Property of the Solution Let fulfillment any condition of the theorem 1. Differen- tiating the solution of the type (6), immdiate verification, we can easily convince to correctness of the following eq e uality: 12 112 23 333 1 a x x Dxxa CxaC x 3 12 3 a x a CDfx 3 44 23 23 d, x ft axa xa 4 1 1 a 00 f xt atata ta (26) t where d d a x Dxa x . In an analogous way differentiating the expression (26), we have 12 3 123 333 2 2 22 2123 112233 0 444 55 5 123 123 00 1d. aaa xxx x a Dxxa CxaCxa CDfxDfx ft xa xaxa fx t tatatata (27) From Equality (6), (26), (27) we find 32 1 23 0 lim , xx xa xTx (28) 1 32 3 0 lim a xa aD Dx 2 12 a x Cxx 22 13 21313 0 0 limlim , aa xx x xa xa CxaD DxxTx (29) 13 2 x 21 3 31 212 0 lim . a x x xa DxxTx (30) 32 21 0 lim a x xa CxaDx Copyright © 2013 SciRes. AM N. RAJABOV 1307 Differentiating the solution of the type (8), immediate v lowing equality: eri correctness of the fol-fication, we can easily convince to 23 123 24 3 444 123 0 1d. x a xaC xa ft xa xat xatata ta (31) From equality (8) and (31) we find 333 123 5 0 a x CDfx fx a x Dx ta 24 43 00 1 limlim , a xx xa xa CxaDxxT 1 x (32) 35 52 0 xa 0 11 limlim . a xx xa CxaDxxTx (33) From integral representation (10) it follows that if parameters 13 j pj and function f x in Equation (1) satisfy all condition of theorem 3, then the solution of the type (10) has the property 3 6 xa x ax C . From integral representation (14) it follows that (34) 2 12 3 32 1ln2 lnln 3128lnlnd, 2 a x D xxaC a x x a xaCxaxaC Dfx ft xaxa xa fx t tatatata (35) 222 22 12 3 222342 2ln24ln ln3 8128 2lnlnd. 2 aaa xxx x a DxxaCxaCxaxaCDfxDfx ft xaxaxa fx t tatata ta (36) as the following properties: Using the formulas (14), (35) and (36), we easily see that, when fulfillment any condition of theorem 5, then solution of the type (14) h 2 2 1 22 6 limln2 ln1ln 22 lnlim x xa x x xa Cxaxaxxaxa Dxlnxa xaxx D T (37) 2lim2ln xa Cxa 2 222 22 ln 21 ln22ln x xx xaDxxa 7 lim xa x aDxxax Tx (38) x 228 3lim2 lim xx x xa xa CxaDxDx xT . (39) From integral representation (18) it follows that 1 11. A x Dx xaCxa 2 3 1 cos lnD x B BxafxDfx 1A x 23 0 4 11 221 0 coslnsin lnsinln 1sinlncosln a C ABxaB BxaCA Bxa xa xaxaxa BADDBBADDBB ta tatata d, (40) ft t ta Copyright © 2013 SciRes. AM N. RAJABOV 1308 1 1 2222 11 2 22 3 42 25 121 12 1 00 0 22 cos ln2sinln 2 sinlncosln 1 A a x A x a xx a Dx xaCxaCAB BxaABBxa CABABBxa BBxa ADBDDx ax a fxDfxD fxBtata AB 22 12 21 2sinln2cosln d. ft xa xa DADB BABD ADBBt tata ta (41) Using the formulas (18), (40) and (41), we easily see that, when fulfillment any condition of theorem 7, then solution of the type (18) has the following properties: 1222 10 1 0 0 lim , x xa T x (42) 11 lim 2 a xx xa CxaBDxABDxBABx 2 21 0 222 1 222 211 11 1 0 1limsinlncos ln sin ln2cosln. 1 sinln2cos lnlim, Aa x xa x x xa CxaDxABxaBBxa DxABBxaAB Bxax A ABBxaBAB BxaT x (43) 2 3 0 x xa 1 222 1 222 212 11 1 0 1lim[cos lnsin ln cosln2sin ln. 1 cosln2coslnlim. Aa x x xa CxaDxABxaBBxa DxABBxaABBxax A ABB xaB ABBxaTx (44) Differentiating the solution of the type (20), immediate verification, we can easily convince to correctness of the fol- lowing equality: 1 2 3 12 3 0 4 11221 0 cos lnsinln sinlncos ln 1sin lncosln A x x A x a D xxaCABxaBBxa BD CABxaBBxafx Dfx ta xaxaxa BAD BDBADBDB xa tatata d, ft t ta (45) Using the formulas (20) and (45), we easily see that, when fulfillment any condition of theorem 8, then solution of the type (20) has the following properties: 13 2cos lnsinln 1lim . x xa x xa aBBxaxBxaDx B Tx B 1limsin ln A CxaABx (46) 3 14 1limcos lnsin lncos ln 1lim . A x xa x xa CxaABxaBBxaxBxaD x B Tx B (47) Copyright © 2013 SciRes. AM N. RAJABOV Copyright © 2013 SciRes. AM 1309 From integral representation (22) it follows that if pa- rameters 13 j pj and function f x hen th in equation (1) satisfy all con of theorem 9, te solution of the type (22) has the property nditio 1 1 xa x ax C . (48) 4. Boundary Value Problems When, the general solution constants, arbitrary constants higher mentioned properties of the solution the integral Equation (1) give possibility for integral Equation (1) put and investigate the following boundary value problems: Problem N1. Is required found the solution of the in- tegral Equation (1) from class CΓ, when the roots the algebraic Equation (4) real, different and positive by boundary conditions (49) where A11, A12, A13-are given constants. Problem N2. Is required found the solution of the in- tegral Equation (1) from class 1 11 2 12 3 13 , xxa xxa xxa Tx A Tx A Tx A CΓ, when the roots the algebraic Equation (4) real, different and also 10 , 20 , 30 , by boundary conditions 4 21 5 22 , xxa xxa Tx A Tx A (50) where A21, A22-are given constants. Problem N3. Is required found the solution of the in- tegral Equation (1) from class CΓ, fferent when the roots the algebraic Equation (4) real, di and also 10 , 20 , 30 by boundary conditions 3 31 xa x ax A , (51) ven con where A31-are gistant. Problem N4. Is required found the solution of the in- tegral Equation (1) from class CΓ, when the roots the algebraic Equation (4) real, equal and positive, that is 123 0 by boundary conditions 6 41 A Tx A 42 8 43 , xxa xxa Tx A (52) where A41, A42, A43-are given constants. 7 xxa Tx Problem N5. Is required found the solution of the in- tegral Equation (1) from class CΓ, when the one roots of the algebraic Equation (4) real positive, two out of its complex-conjugate. Besides 2 Real 0A 51 52 53 , xa xa xa A A A constants. found the solution o , by bound- ary conditions (53) where A51, A52, A53-are given Problem N6. Is required f t tegral Equation (1) from class 9 10 11 x x x Tx Tx Tx he in- CΓ, when the on of the algebraic Equation itive, two out of its complex-conjugate. Besides e roots (4) real pos 10 , 2 Real 0A , by boundary conditions (54) where A61, A62-are given constants. Problem N7. Is required found the solution of the in- tegral Equation (1) from class 12 TxA 61 13 62 , xxa xxa Tx A CΓ, (4) real pos 10 when the one roots of the algebraic Equation itive, two out of its complex-conjugate. Besides, , 2 Real 0A , by boundary conditions 1 71 xa x axA , (55) where A71-are given constant. Solution problem N1. Let fulfillment any condition of theorem 1. Then using the solution of the type (6) and its properties (28)-(30) and condition (49), we have 32 1323 1112122 00 ,,СAСAС13 0 A Substituting obtained valued C1, C2 and C3 in formula (6), we find the solution of problem N1 in form 32 1323 1111213 0 ,,,. 00 x K AA Afx (56) So, we proof. Theorem 11. Let in integral Equation (1) parameters 13 j pj , function f x m N1 ha satisfy a n Probles a uniquesolution which is given by formula (56). any condition of th on (5 ny condition of theorem 1. The Solution problem N2. Let fulfillment eorem 2. Then using the solution of the type (8) and its properties (32), (33) and conditi0), we have: 421 1 С A 0 , 522 0 1 С A . Substituting this valu, ed C4 C5 in formula (8), we find the solution of problem N2 in form N. RAJABOV 1310 . 22122 00 11 ,, x KAAf So, we proof. ion (1) parameters x (57) Theorem 12. Let in integral Equat 13 jj, function p f x satisfy condition of theorem 2. Then problem N2 haue solution which is given by formula (57). ti blem N3. Let fulfillment a s uniq Soluon prony condition of theorem 3. Then using the solution of the its properties (32) and condition (51), we 1 type (10) and have: С63 A . Substitut this valued C6 in fo lu ermula (10), we find theso- tion of problem N3 in form 331 ,. x KAfx (58) So, we proof. Theorem 13. Let in integral Equation (1) parameters 13 j pj , function f x 3 has satisfy condition of Theo- problem N unique so given by formula (58). Solution problem N4. Let fulfillment any condition of theorem 5. Then using solution of the type (14) and its , we have: rem 3. Then lution, which is properties (37)-(39) and condition (52) С1 41 A , 242 С A , 343 С A Substituting this valued 1 С, 2 С and 3 С in formula (14)find thesolution of problem N4 in form , we . 5414243 ,,, x KAAAfx (59) So, we proof. Theorem 14. Let in integral Equation (1) parameters 13 j pj , function f x 4 has satisfy problem N unique solution, which is given by formula (59). any condition of th ion (5 condition of theo- rem 5. Then Solution problem N5. Let fulfillment eorem 7. Then using solution of the type (18) and its properties (42)-(44), and condit3) we have: 151 0 1 С A , 252 0 1 С A , 35 0 3 1 С A . Substituting this valued C1, C2 and C3 in formula (18) we findthe so- lution of problem N5 in form 7515253 ,,, . x KAAAfx (60) So, we proof. Theorem 15. Let in integral Equation (1) parameters 13 j pj , function f x fy condition theorem 7. Then problem N5 have unique solution, which is given by formula (60). Solution problem N. Letfuen of theorem 8. Then usin satis 6lfillmnt any conditio g solution of the type (20) and its properties (46), (47) and condition (54) we have: ting this valued C2 and C 261 С1 A B , 362 С 3 in formula (20) we find the solution of problem N6 in form 86162 ,, . x KAAfx (61) So, we proof. Theorem 16. Let in integral Equation (1) parameters 13 jjp , function f x satisfy condition theorem 8. Then problem N6 have unique solution, which is given by formula (61). Solution problem N7. Let fulfillment any condition of theorem 9. Then using solution of the type (22) and its properties (48) and condition (55) we have:1 17 С A . we findSubstituting this value C1 in formula (22) so N7 in form the lution of problem 971 ,. x KAfx (62) So, we proof. Theorem 17. Let in integral Equation (1) parameters 13 jjp , function f x satisfy condition theorem 9. Then problem N7 have unique solution, which is given by formula (62). 5. Presentan the Sol Equation (1) in the Generalized Power hat tioution of the Integral Series Suppose t f x on on has uniformly convergent power series expansi : 0 k k k f xxa f , (63) where constant 0 and fk, 0,1,2,k, are given nstants. We attempt to ind a solution of (1) in the form k x co f xa 0k k where the coefficients, , (64) 0,1, 2, kk are unknown. Substituting power series representations of v alue f x and x into (1), equating the coefficients of the corresponding function, and for , we obtain k 3 32 12 23 kk k 0,1, 2, 3,. , f kpkpk p If k 32 12 3 20kpkp (65) kp for in all 0,1,2,k , putting the found coefficients back into (64), we arrive at the particular solution of (1). 1 A B . Substitu 3 2 012 3 . 2 k k x k x 3k af If, for some values kp kpk p (66) 1 kk , 2 kk and 3 kk , con- Copyright © 2013 SciRes. AM N. RAJABOV Copyright © 2013 SciRes. AM 1311 stants ,13 j pj p in the satisfy 3 th sentedit is k 32 12 20kpkp , en the solution to integral Equation (1) can be repre- form (64) necessary and sufficiently that 0 j k f, 1,2,3j, that is, it is necessary and suf- ficiently that function satisfies olvility condition the following three sab 0, j k afx jx 1, 2,3. (67) xa In this case the solution of the integral Equation (1) in the cla f x in point x a ss of function can be represented in form (64) is given by formula 3 k 12 kk 3 k 3 3 123 3 1 32 12 3 3 32 1 12 3 2 2 k k k k kk kkk p k 11 0 k k k xx a k 11kk 2 23 3 2k f pk p k 1 k x a k f pkpk p k x a k f p kpk p (68) xa xa where 1 k xa , 2 k , 3 k arbitrary constants. Immediately testing it we see that, if converges radius of the series (63) is defined by formula 1 Rl , 1 lim n nn f l f , then converges radius of the series (66), rem 18. Let in integral Equation (1), function (68) are also defined by this formula. So, we prove the next result. Theo f x ized represent in formuniformly-conv power series type (63) and for ntegr erges general- 12 3 20kpkpkp , . Then ial Equation (1) in class of function 32 0,1,2,k x represented in form (64) has a unique- solution, which is given by formula (66). For values ary and sufficiently ful- fillment three solvability condition type integral Equation (1) in class of functio fo ays solvability and its general solution co (68). se e j k, 1, 2, 3j, 32 12 3 20kpkpkp , the existence of the solution of Equation (1) can be rep- resented in form (64) it is necess k (67). In this case n represented in rm (63) is alw ntain tree arbitrary constants and is given by formula 6. General Ca In general case to integral Equation (I) corresponding th following algebraic equation 12 3nn nn pp p 4 3! 1 !0. n n p 12 3 4 2! n p (II) Example in the case, when the roots of the Equation (II) real, dif- ferent and positive have the following confirmation. Theorem 19. Let in integral Equation (I) parmeters Some results obtained in the general case to. 1 j pjn tion (II) real, such that, the roots of the algebraic Equa- different and positive, function f x C , 0fa withpthavior 1, xa asymotic be 1 12 , max,,, at , n fxo x a Then integral Equation (I) in class of function x C vanishing in point x a is always solvability and iion i by formula ts soluts given 1 3 1 0 1123 1d ,,,,, , k n xxaC k k k xn k k a n f t xa f xt tata KCCC Cfx (III) where 1 k Ckn -arbitrary constants, 12 22 2 12 0 111 12 1,1, ,1 ,, ,,, . ,,, n n nnn n 7. Conclusions So, in this article we consider new class Volterra type integral equation, which no submitting exists Fredholm theory (Theory Volterra type integral equation in class N. RAJABOV 1312 C, 2 L), that is for this type integral equation, homogeneous integral equation may have non-zero solu- tion. In particular in certain cases (Example, roots of the characteristic Equation (4) or (II) real, dif- fe is type integral equation coincides to the theory Fred- holm integral equation. By means methods (example [5]) in the theory one dimensional singular integral equation, problem finding when all rent negative or real, equal and negative) the theory the solution general equation th 1 1,ln d , xnm m m a t xa x Kxt t ta ta fx (IV) reduces to finding solution Volterra type integral equa- tion with weak singularity. On this basis, in depend from roots of the algebraic equation 12 3 4 123 4 ,,2!,3!, 1!, nn nnn KaaK aaKaaK aanK aa 0, n (V) ,01 m K aamn lution equation contai , select cases, when general so- ns 1 arbitrary constants, and cases when Equation (IV) has unique solu- tion. In this case, integral Equation (IV), we represented to following form ,1,2,,nn n 1 1,ln xnm m t xa d , m a x Kaa (VI) where tFx ta ta 1d mt xa 1,,ln x mm m n a F xfxKxt at (VII) K at a ta . According to the mentioned above, writing the solu- tion integral Equation (VI) in depend to the roots of the characteristic Equation (V) or (II), after substituting for F x from formula (VII) we arrive at the solution of the new type integral equation. At specific condition to func- tions ,, mm K xtKaa and f x tegral equ this integr equ- ination wit al h weak ation ul arity will be Volterra type arity in poi tegra iint singnt = a. In this basis the problem inves- tigation inl Equation (IV), reduce to problem inves- tigation Volterra type integral equation with weak singu- ln po x a. REFERENCES stem of Linear Integral Equations of and Super-singular Kernels,” n-Classical Problems of Mathematical Conference, Samarkand, Uzbekistan, Kluwer, Utrecht, Boston, 11-15 September 2000, pp. 103-124. [4] e Linear I Inter thematics, Physics and Chemistry, Vol. 147, Kluwer Academic Publishers, 2004, pp. 317-326. [5] N. Rajabov, “Volterra Type Integral Equation with dary and Interior Fixed Singularity and Super-Sing Kernels and Their Application,” LAPLAMBERT Aca- demic Publishing, 2011, 282 p. Rajabov, “Aboa Type Integral uations with Bernels,” In Ad- vances in Applied Mathematics and Approximation The- ory: Contributions from AMAT2012, Springer, 2012, pp. 341-360. [7] N. Rajabov, “To Theory One Class Modeling Linear Vol- terra Type Integral Equation with Boundary Singular Ker- nels,” Theses of Reports of the 4th International Confer- ence “Function Spaces. Differential Operators. General Topology. Problems of Mathematical Education”, PFUR Publishers, Moscow, 2013, pp. 221-222. [1] N. Rajabov, “About One Three-Dimensional Volterra Type Integral Equation with Singular Boundary Surfaces in Kernels,” Russian of Sc. Dokl, Vol. 409, No. 6, 2006, pp. 749-753. [2] N. Rajabov, “On a Volterra Integral Equation,” Doclady Mathematics, Vol. 65, No. 2, 2002, pp. 217-220. [3] N. Rajabov, “Sy Volterra Type with Singular Ill-Posed and No Physics and Analysis. Proceedings of the International N. Rajabov, “About One Class of Volterra Typ ntegral Equations with anior Fixed Singular or Su- per-singular Point,” Topics in Analysisand its Applica- tions, NATO Science Series, II, Ma Boun- ularity [6] N.ut New Class of Volterr Eqoundary Singularity in K Copyright © 2013 SciRes. AM |