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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 :xaxb  be a set of point on the real axis and consider an integral equation   11ln dxnkkkatxaxpta tatfxn, (I) where is given constants, 1jpjfx 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 212 3ln lnd.xatxa xaxpp pttata tafx     (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 ,xCab, 0a with fol-lowing asymptotic behavior   t,0axoxax a . (2) In this case the integrals in the Equation (1proper one. Moreover ) are im-0afa i.esi. right-hand de is necessarily zero at .xa In this case in Equation (1) 0 it investi-gates in [1]. In this casfrom signs p 23pp depend 1 e, inp110,0 ,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 3xC. Besides, let in Equation (1) function f3x C two. Then differentiating integral Equa-mes, we obtained the following third or-n differential equation tion (1) three tider degeneratioCopyright © 2013 SciRes. AM N. RAJABOV 1302 3Dx 21233,xx xxpDxpDxpxDf x (3) ddxDxax . where Homogeneous differential Equation (3) is correspond-e following characteristic equation . (4) tion 2.1. The Case, When the Roots of thing to th3220pp p12 32. Representation the General Solue Characteristic Equation Real and Different Let in differential Equation (3) parameters 13jjp ion (4) real suchthat, the roots of the characteristic Equatand 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, general123xxa CxaCxaC  , (5) where 13jCjn, arbts. itrary constanWhe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 inten homgral 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  12312 333 312312 3011231d,,,xaftxa xaxaxxa CxaCxaC fxtta tata taKCCCfx        (6) where 13jCj arbitrary constants, 0 123 2332 122113132221231, 1, 1,, .,,    The solution of the type (6) obtained in the case, when , 3fx C0fa, solution integral equation (1), function x exist and belong to Class 3C. esting, ee that, of Immediately twe s01 3jj,  fx C3, 0fa with asymptotic b ehavior   11,xo xa123max ,,at f,xa  (7) then function (5) satisfied Equation (1). Be valid the following confirmation. (1) parameters Theorem 1. Let in integral Equation13jj such that, the roots of thep algebraic Equ-ation (4) real, different and positive, function fx C, 0 with asymptotic behavior (7). Then integral Equation (1) in class of function faxC xa form vanishing in point is always solvability and its solution is given by ula (6), 13jCj are ar-gral Equationbitrary constants. Characteristics 1. Let in inte (1) pa-rameters 13jpj, function fx satisfy any con-dition of theorem 1.Then, from (6) it follows, the solu-tion integr1) al Equation (xC, 0a with following asymptotic behavior   11, min123,,, at xoxa x a . If, the roots of the characteristic Equation (4) real, -ferent and dif10, 20, 30, then it follows, from formula (6) 10C. In this case,exist th solutio integral Equation (1), then it is posllowing form if ensible is represent in fo  34512 302451,,xa12333 d23fttaxa xaxxaCxaCfxtxatatataKCCfx   (8) where -are arbitrary constants. Th of the type (8) exist, if    4C, e solu5Ction fx C, 0fa   221123,,min ,atfxo xaxa  (9) with asymptotic behavior Copyright © 2013 SciRes. AM N. RAJABOV 1303So, in this case have the following confirmation. re 2. (1) parameters following asymptotic behavior Theo m Let in integral Equation13jj such that, the roots of the algeb- real, different and also praic Equation (4) 10, 20, 30, fx C, 0fa with asym) in cpt Then integral Equation (1otic be-havior (9).tion lass of func-x C va nishing inpoint xa is al-ways solvability and its solution is given by formula (8), 4, 5j are arbitrary constants. jCharacteristics 2. Let in integral Equation (1) pa-rameters C13jpj, function fx satisfy any con- dition of thm 2en, (8) it follows, the solu- tion integral Equation (1) eore. ThfromxC, 0a with   2,m223i atn,,xoxa x a Remark 1. Confirmation similar to theorem 2 ob-ta ined and in the following cases: a) 10, 20, 30; b) 10, 02, 03. 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 120CC. In this case, if exist solution integral Equa-tio it will be represented in following form n (1), then  1233 323d026,,a3361xftta xataxxa Cfxtxaxatata      (10) tant. The solution of the type (10) exist, if KCfx where C6 are arbitrary cons fx C, 0fa with asymptotic behavior , 3fxox a33 at xa. (11) So, we proof. The following confirmation. uation (1) parameters Theorem 3. Let in integral Eq13pj such that, the roojts of the -4) real, different and also 1algebraic Equation (0, 20, 30, fx C, 0fa with asympt- integrn otic behavior (11). Thention al Euatioq(1) in class of func-x C va nishing inpoint xa is al-rmula (10)given by foways solvability and whits solution is ere C are arbitrary consta, rameters 3nt. Characteristics 3. Let in integral Equation (1) pa- 13jpj, function fxlution satisfy any conditio Then the soof the integral Equation n of theorem 3.(1) in point xa vanishs asymptotic behavior determined from formula and it  3at .xoxax 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 013jj , 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   2333 4dftta133120xata ta1xfx xa x    The solution of the type (12) exist, if t Kfxx    (12) aata  fx C, 0r fa with asymptotic behavio  ,0atfxoxa xa (13) So we proof the following confirmation. Theorem 4. Let in integral Equation (1) parameters 13jpj atisuch that, the roots of the algebraic Equ-The function on (4) real, different and positive. fx C, 0fa with asymptotic behavior tegrn (1) in class of function (13). Then inal EquatioxC vapoint x = a havegive b(12). ation (1) pa-nishing in y formula unique so-lution, which Characteristics 4. Let in integral Equrameters 13jpj, function fx satisfy any con-dition of theorem 4. Then the solution of the integral Equation (1) in point xa vanish and its asymptotic behavior determined from formula   ,0atxoxa x a . 2.2. The Case, When the Roots of the Characteristic Equation Real and Equal Let in integral Equation (1) parameters 13jpj, aracteristic Equation (4) real and equal. such that, the roots of the chIn this case we have the following confirmation: Copyright © 2013 SciRes. AM N. RAJABOV 1304 Theorem 5. Let in integral Equation (1) parameters 13jpj 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   44at,,.fxoxa xa  Then homogeneous integral Equation (1) class of infunction xC vanishing in point xa, have three linear independent solutions the type  1xxa , 2lnxxa xa  2ln ., 3xxa xa  Non homogeneous integral Equation (1), always solv-able. Its general solution contain three arbitrary con-stant and given by formula  62,,, ,a23225123ln ln6 lnlndxaCxaCfx12xxxaCftaxaxattatata ta    (14) x KCCCfx  Were 13jCj -arbitrary constants. stics 5. In this case, when in integral Equa-tion (1) parameters Characteri13jpj , function fxion inte sat-isfy any clut graEquation ondition of theorem 5, then sol (1) in point xa vanish and its asymptotic   ,0, .atxoxaxa 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   2266 lnlnxxa6d .2aftxa taxfx   t Kfxtataxa ta   (15) The solution of the type (15) exist, if fx C, 0fa 2.3. The Case, When One Roots of the Characteristic Equatthe Roots of the Characteristic Equation Complex and Conjugate Let in integral Equation (1) parameters with the following asymptotic behavior   , at 0, .fxoxaxa  (16) o in the case, when S123 0, proof the following confirmation. Theorem 6. Let in integral Equation (1) parameters 13jpj such thots of the cheris-tic Equation (4) real, equal and negative, that is 12at, 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 pounique solution and give by forint xa vanish and its asymptotic behavior deter-mined from formula   , at 0, .xoxaxa  ion Real and Two 13jpj Equation (4) real plex 1such that, the one roots of characteristic and two the roots of the characteristic equation comconjugate. Correspondingly its denote by , 2AiB, 3AiB. When 10, 0A, then by this roots cor-responding following parmogeneous integral Equation (1): ticular solution ho   1123cos lnsin ln, .AAxxa xxaB xaxxaB xa   (17) In this c ase, if solution integral Equation (1) exist, then it will be represented in form   1311071231sin,,,a112AAxCxa Ca32sin lnlncoslndCB xafxcos lnxxaB xftx xaDBtta     (18) xa xaBDta taKCCCfxaBtata     Copyright © 2013 SciRes. AM N. RAJABOV 1305 where 22201120ABBB AB , 44 3DBAA 22BA 11 22222123DABABBBA. The solution of the type (18) exist, if 10, , 0A, fx C, 0 fa with the following asymptotic behavior  551, max, at .fxoxaA xa  (19) So in this case we have the following confirmation. Theorem 7. Let in integral Equation (1) parameters 13jpj tic Equation(2such that, one the roots of the character- (4) real positive, two out of its complex isconjugate AiB , 3AiB). Besides let 2RealA0. Assume that a functfx C, 0faion with asymptotic behavior (19). Then homogeegral Equation (1), in class neous intC vanishing in point xa,lvable,s and has three linear Independent solution ofNon homogenous integral Equation (1) a its general solution contain three arbitrargi the lways soy constanttype (17). ven by formula (18), where 13jCj -arbitrary constants. Characteristics 7. In the case, when fulfillment any condition theorem 7, then solution integral Equation (1) in point xa vanish and its behavior determined from following asymptotic formula    661 a, min,At .xoxax a  From integral representation (18) followof the algebraic Equation (4) satisfy condition of the theorem 7, besides s, if the roots 10, 0A. If 10, ) in t0A, is th on hen if exist solution integ (1case, then its represented in following form  ral Equati  23,, .12331108cos lnsin ln1sinAAxaxxaCBxaC B2lncos lndxa fxftxa xaBDBtta xaBDxata taK     In this case for convergence integrals in right part (20), necessary ta ta      (20) CCfx fx C, 0fa with asymbehaptotic vior   77, at .fxoxa Axa  (21) So, we proof. the following confirmation. Theorem 8. integation (1) parameters  Let inral Equ1jpj 3sndition theatisfy coorem 7, besides 1 0, 0.A Let 10, 0A. Function fx C, 0fa withgeneous integr asymptal Equatotic beon (1),havior (21)i in class . Then homo-C vanishing in point xa, has two linear Independlution ent so 2cos lnAxxaB xa ,  3sin lnAxxaB 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. 23jCjCharacteristics 8. In the case , when fulfillment any condition theorem 8, then solution integral Equation (1) in point xa vanish and its behavior determined from following asymptotic formula A  atxoxax 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 reprlowing form olution inte-gral Equation (1) in this case,esented in fol- 1131111 2cos lnd091sin ln,.Axaftaxaxa taxxxa CfxBta xaKD BtaCfx (22 (22), neD Btta ta   ) cessary In this case for convergence integrals in right part fx C, 0fa with asymptotic havior   881, at .fxoxa xa  (23) So, we proof. the following confirmation. beCopyright © 2013 SciRes. AM N. RAJABOV 1306 Theorem 9. Let in integral Equation (1)arameters p13jpj satisfy condition theorem 7, besides 1 0, 0.A Let 10, 0A. Function fx C, 0fa withgeneous integr asymptal Equatotic beon (1),havior (23)i in class . Then homo-C 1 vanishing ,lin point x = a one soution 1xxa.  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 co1C-teristindition theorem 9, then solution integral Equation (1) in point xa vanish and its behavior deined from following asymptotic formutermla   1,at xoxaxa 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  13110101sin lnl.Axata ta2cos ndftxa xaxfxBD BDBtxaxaKfx   In this case for convergence integrals in right part (24), it is sufficientta tata       (24) Characteristics 10. In the case, when fulfillment any condition theorem 10, then solution integral equatiin point fx C, 0fa with asymp-totic behavior   , at 0, .fxoxaxa  (25) . the following confir 10. Let in integral EquationSo, we proofmation. Theorem (1) parameters 13jpj satisfy condition theorem 7, besides 1 0, 0.A Let 10, 0A. Function Cfx, 0a with asymptotic behavior (25). TfEqhen integral uation (1), in class C vanishing in point xa, olution, which given by formula (24). have unique son (1) xa vanish and its behavior determined from following asymptotic formula   , 0 at xoxax 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 eqeuality:     12112 23333 1axxDxxa CxaCx3123axa CDfx3442323 d,xftaxa xa411a00fxtatata ta      (26) t   where ddaxDxax . In an analogous way differentiating the expression (26), we have    12 31233332 222 2123112233044455 5123123001d.aaaxxxxaDxxa CxaCxa CDfxDfxftxa xaxafx ttatatata           (27) From Equality (6), (26), (27) we find 32 1230lim ,xxxaxTx  (28) 13230lim axa aD Dx212axCxx   2213213130 0limlim ,aaxx xxa xaCxaD DxxTx     (29) 13 2x 21 3312120lim .ax xxaDxxTx  (30) 32210lim axxaCxaDx  Copyright © 2013 SciRes. AM N. RAJABOV 1307Differentiating the solution of the type (8), immediate vlowing equality: eri correctness of the fol-fication, we can easily convince to  2312324344412301d.xaxaC xaftxa xatxatata ta       (31) From equality (8) and (31) we find 33312350axCDfx fx axDxta 2443001limlim ,axxxa xaCxaDxxT   1x (32)  35520xa011limlim .axxxaCxaDxxTx   (33) From integral representation (10) it follows that if parameters 13jpj and function fx in Equation (1) satisfy all condition of theorem 3, then the solution of the type (10) has the property 36xaxaxC. From integral representation (14) it follows that (34)    212 3321ln2 lnln3128lnlnd,2axD xxaC axxaxaCxaxaC Dfxftxaxa xafx ttatatata        (35)   222 2212 32223422ln24ln ln38128 2lnlnd.2aaaxxxxaDxxaCxaCxaxaCDfxDfxftxaxaxafx ttatata 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  22122 6limln2 ln1ln22 lnlimxxax xxaCxaxaxxaxaDxlnxa xaxxDT   (37)  2lim2lnxaCxa222222 ln21 ln22lnxxxxaDxxa7limxaxaDxxax   Tx (38) x  2283lim2 limxx xxa xaCxaDxDx xT   . (39) From integral representation (18) it follows that  111.AxDx xaCxa  231cos lnDxB BxafxDfx 1Ax 2304112210coslnsin lnsinln1sinlncoslnaCABxaB BxaCA Bxaxa xaxaxaBADDBBADDBBta tatata          d, (40)ft tta  Copyright © 2013 SciRes. AM N. RAJABOV 1308    11222211 22234225121 12100 022cos ln2sinln2 sinlncosln1AaxAxaxxaDx xaCxaCAB BxaABBxaCABABBxa BBxaADBDDx ax afxDfxD fxBtataAB       2212 212sinln2cosln d.ftxa xaDADB BABD ADBBttata 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 1010 0lim ,xxaT x (42) 11lim 2axxxaCxaBDxABDxBABx      22102221222 21111 101limsinlncos lnsin ln2cosln.1sinln2cos lnlim,AaxxaxxxaCxaDxABxaBBxaDxABBxaAB BxaxAABBxaBAB BxaT x        (43)  230xxa 12221222 21211 101lim[cos lnsin lncosln2sin ln.1cosln2coslnlim.AaxxxaCxaDxABxaBBxaDxABBxaABBxaxAABB xaB ABBxaTx        (44) Differentiating the solution of the type (20), immediate verification, we can easily convince to correctness of the fol-lowing equality:    12312304112210cos lnsinlnsinlncos ln1sin lncoslnAxxAxaD xxaCABxaBBxaBDCABxaBBxafx Dfxta xaxaxaBAD BDBADBDBxa 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: 132cos lnsinln1lim .xxaxxaaBBxaxBxaDxBTxB  1limsin lnACxaABx  (46)   3141limcos lnsin lncos ln1lim .AxxaxxaCxaABxaBBxaxBxaD xBTxB   (47) Copyright © 2013 SciRes. AM N. RAJABOV Copyright © 2013 SciRes. AM 1309 From integral representation (22) it follows that if pa-rameters 13jpj and function fxhen th in equation (1) satisfy all con of theorem 9, te solution of the type (22) has the property nditio11xaxax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 111212313 ,xxaxxaxxaTx ATx ATx ACΓ, when the roots the algebraic Equation (4) real, different and also 10, 20, 30, by boundary conditions 421522 ,xxaxxaTx ATx 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 331xaxax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 641ATx A42843 ,xxaxxaTx A (52) where A41, A42, A43-are given constants. 7xxaTx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 2Real 0A515253 ,xaxaxaAAA constants. found the solution o, by bound-ary conditions (53) where A51, A52, A53-are givenProblem N6. Is required f ttegral Equation (1) from class 91011xxxTxTxTxhe in-CΓ, when the onof the algebraic Equation itive, two out of its complex-conjugate. Besides e roots (4) real pos10, 2Real 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 12TxA611362 ,xxaxxaTx A CΓ,(4) real pos10 when the one roots of the algebraic Equation itive, two out of its complex-conjugate. Besides, , 2Real 0A, by boundary conditions 171xaxaxA, (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 1323111212200,,СAСAС130A   Substituting obtained valued C1, C2 and C3 in formula (6), we find the solution of problem N1 in form 32 132311112130,,,.00xKAA Afx   (56) So, we proof. Theorem 11. Let in integral Equation (1) parameters 13jpj, function fx m N1 hasatisfy an Probles a uniquesolution which is given by formula (56). any condition of thon (5ny condition of theorem 1. TheSolution problem N2. Let fulfillmenteorem 2. Then using the solution of the type (8) and its properties (32), (33) and conditi0), we have: 4211СA0 , 52201СA. Substituting this valu, ed C4C5 in formula (8), we find the solution of problem N2 in form N. RAJABOV 1310  .221220011,,xKAAfSo, we proof. ion (1) parameters x (57) Theorem 12. Let in integral Equat13jj, function pfx satisfy condition of theorem 2. Then problem N2 haue solution which is given by formula (57). ti blem N3. Let fulfillment as uniqSoluon prony condition of theorem 3. Then using the solution of theits properties (32) and condition (51), we 1 type (10) and have: С63A. Substitut this valued C6 in foluermula (10), we find theso-tion of problem N3 in form 331,.xKAfx (58) So, we proof. Theorem 13. Let in integral Equation (1) parameters 13jpj , function fx 3 hassatisfy condition of Theo-problem N unique sogiven 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 41A, 242СA, 343СA Substituting this valued 1С, 2С and 3С in formula (14)find thesolution of problem N4 in form , we . 5414243,,,xKAAAfx (59) So, we proof. Theorem 14. Let in integral Equation (1) parameters 13jpj , function fx 4 hassatisfyproblem N unique solution, which is given by formula (59). any condition of thion (5 condition of theo-rem 5. Then Solution problem N5. Let fulfillmenteorem 7. Then using solution of the type (18) and its properties (42)-(44), and condit3) we have: 15101СA, 25201СA, 35031СA. Substituting this valued C1, C2 and C3 in formula (18) we findthe so-lution of problem N5 in form 7515253,,, .xKAAAfx (60) So, we proof. Theorem 15. Let in integral Equation (1) parameters 13jpj , function fx fy condition theorem 7. Then problem N5 have unique solution, which is given by formula (60). Solution problem N. Letfuen of theorem 8. Then usinsatis6lfillmnt any conditiog solution of the type (20) and its properties (46), (47) and condition (54) we have: ting this valued C2 and C261С1AB, 362С3 in formula (20) we find the solution of problem N6 in form 86162,, .xKAAfx (61) So, we proof. Theorem 16. Let in integral Equation (1) parameters 13jjp, function fx 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 itsproperties (48) and condition (55) we have:1 17СA. we findSubstituting this value C1 in formula (22)so N7 in form the lution of problem971,.xKAfx (62) So, we proof. Theorem 17. Let in integral Equation (1) parameters 13jjp, function fx satisfy condition theorem 9. Then problem N7 have unique solution, which is given by formula (62). 5. Presentan the SolEquation (1) in the Generalized Power hat tioution of the Integral Series Suppose tfxon on has uniformly convergent power series expansi:  0kkkfxxa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 kxco f xa0kkwhere the coefficients, , (64) 0,1, 2,kk are unknown. Substituting power series representations of v aluefx and x into (1), equating the coefficients of the corresponding function, and for , we obtain k 3321223kkk0,1, 2, 3,.,fkpkpk pIf k 3212 320kpkp (65) kp  for in all 0,1,2,k, putting the found coefficients back into (64), we arrive at the particular solution of (1).   1AB. Substitu32012 3.2kkxkx3kafIf, for some values kpkpk p  (66) 1kk, 2kk and 3kk, con-Copyright © 2013 SciRes. AM N. RAJABOV Copyright © 2013 SciRes. AM 1311stants ,13jpjp in the satisfy 3thsentedit isk 321220kpkp  , en the solution to integral Equation (1) can be repre-form (64) necessary and sufficiently that 0jkf, 1,2,3j, that is, it is necessary and suf-ficiently that function satisfies olvility condition the following three sab0, jkafx jx1, 2,3. (67) xaIn this case the solution of the integral Equation (1) in the clafx in point xass of function can be represented in form (64) is given by formula 3k    12kk3k33123313212 3332112 322kkkkkkkkkpk110kkkxxak11kk22332kfpk pk1kxakfpkpk pkxakfpkpk p     (68)   xa xawhere 1kxa, 2k, 3k arbitrary constants. Immediately testing it we see that, if converges radius of the series (63) is defined by formula 1Rl, 1lim nnnflf, 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fx ized represent in formuniformly-convpower series type (63) and for ntegrerges general- 12 320kpkpkp , . Then ial Equation (1) in class of function 320,1,2,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 functiofo ays solvability and its general solution co(68). se e jk, 1, 2, 3j,  3212 320kpkpkp , the existence of the solution of Equation (1) can be rep-resented in form (64) it is necessk(67). In this case n represented in rm (63) is alwntain tree arbitrary constants and is given by formula 6. General CaIn general case to integral Equation (I) corresponding thfollowing algebraic equation 12 3nn nnpp p43! 1 !0.nnp12 342!np   (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) parmetersSome results obtained in the general case to. 1jpjn tion (II) real,such that, the roots of the algebraic Equa- different and positive, function fx C, 0fa withpthavior 1, xa asymotic be  112,max,,, at ,nfxoxa  Then integral Equation (I) in class of function x C vanishing in point xa is always solvability and iion i by formula  ts soluts given 131011231d,,,,, ,knxxaCkkkxnkkanftxafxttataKCCC Cfx (III) where 1kCkn-arbitrary constants, 1222 2120111121,1, ,1,,,,, .,,,nnnnnn   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, 2L), 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-feis 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 11,ln d,xnmmmatxaxKxt tta tafx (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 4123 4,,2!,3!, 1!,nn nnnKaaK aaKaaK aanK aa  0,n (V) ,01mKaamn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 11,lnxnmmtxad ,maxKaa(VI) where tFxta ta   1dmtxa1,,lnxmmmnaFxfxKxt at  (VII) K ata 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 Fx from formula (VII) we arrive at the solution of the new type integral equation. At specific condition to func-tions ,,mmKxtKaa and fx tegral equthis integr equ- ination witalh weak ation ularity will be Volterra typearity in poitegraiint 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 poxa. 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 LinearI Interthematics, Physics and Chemistry, Vol. 147, Kluwer Academic Publishers, 2004, pp. 317-326. [5] N. Rajabov, “Volterra Type Integral Equation withdary and Interior Fixed Singularity and Super-SingKernels 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, “SyVolterra Type with Singular Ill-Posed and NoPhysics 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 VolterrEqoundary Singularity in KCopyright © 2013 SciRes. AM