To Theory One Class Linear Model Noclassical Volterra Type Integral Equation with Left Boundary Singular Point

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 integral equation, the solution found cane represented in generalized power series. Some results obtained in the general model case.


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x a x b     be a set of point on the real axis and consider an integral equation where is given constants, In what follows we in detail go into case n = 3.In this case the Equation (1) accepts the following form Integral Equation (1) at p 2 = 0, p 3 = 0 is model second kind Volterra type singular integral equation with left boundary singular point, theory construction in [1][2][3][4][5].In the case, when in (1) p 3 = 0 Equation (1) investigates in [6].
As [4,5] the solution to this equation is sought in the In this case the integrals in the Equation (1 proper one.Moreover ) are im- . right-hand de is necessarily zero at .x a  In this case in Equation (1) 0 it investigates in [1].In this cas from signs p ) is found in explicit form.In this case at 1 0 p  homogeneous integral Equat n (1) has one solution and general solution no homogeneous (1) contains one arbitrary constant and at 1 0 p  , integral Equation (1) has unique solution.In case of, when in (1) 3 0 p  , 1 0 p  , 2 0 p Equation ( io  integral Equation (1) investigates in [6].In this case in depe aracteri ic equa on obta nd from corresponding ch st ti ined solution integral Equation (1) by two arbitrary constants, one arbitrary constant.Select the case, when integral Equation (1) has unique solution.To problems investigation one dimensional and many-dimensional Volterra type integral equation with fixed boundary and interior singular points and singular domains in kernels dedicate [1][2][3][4][5][6][7].
Support that solution integral Equation (1) function       , ,    .In this case, immediately testing we see that solution homogeneous differential Equation ( 2) is given by formula , general where x  definable by formula (5) eneous integral uation (1).So, function satisfy homog Eq   x  determined by formula ( 5) is given general solutio ogeneous integral Equation (1).For obtained the solution non homogeneous inte n hom gral Eq uation (1), first time use the variation arbitrary constants methods, we use the general solution of the differential 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 where The solution of the type (6) obtained in the case, when , then function (5) satisfied Equation (1).Be valid the following confirmation.
(1) parameters Theorem 1.Let in integral Equation such that, the roots of the p algebraic Equation (4) real, different and positive, function   C  ,   0 with asymptotic behavior (7).Then integral Equation (1) x a  form vanishing in point is always solvability and its solution is given by ula (6),   , , , at If, the roots of the characteristic Equation (4) real,ferent and where -are arbitrary constants.Th of the type (8) exist, if , , min , at  is always solvability and its solution is given by formula ined and in the following cases: a) 1 0 6 , , tant.The solution of the type (10) exist, if where C 6 are arbitrary cons So, we proof.The following confirmation.uation (1) parameters Theorem 3. Let in integral Eq   such that, the roo j ts of the -4) real, different and also 1 algebraic Equ ation ( 0 tion similar to m 3, obtained and in the following cases: a) 1 0 Confirma theore If the roots of the characteristic Equation (4) real, different and   , then from integral representation (6 er that ) follows, in ord uation (1) in this c ssary C 3 = 0.In this case, if exist solution integral Equation (1), then its will be represented in form

The Case, When the Roots of the Characteristic Equation Real and Equal
Let in integral Equation (1) parameters   Non homogeneous integral Equation (1), always solvable.Its general solution contain three arbitrary constant and given by formula The solution of the type (15

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 o in the case, when S such th ots of the ch eristic Equation (4) real, equal and negative, that is 1 2 at, the all ro aract ion Real and Two   In this c ase, if solution integral Equation (1) exist, then it will be represented in form The solution of the type (18) exist, if 1 0 , with the following asymptotic behavior Assume that a funct From integral representation (18) follow of the algebraic Equation ( 4) satisfy condition of the theorem 7, besides s, if the roots , , .
In this case for convergence integrals in right part (20), necessary

 
C  class is given by formula (20), where -arbitrary constants.
  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 Now suppose, that the roots of the algebraic Equation (4) satisfy condition of the theorem 7, besides 1 0 In this case for convergence integrals in right part So, we proof.the following confirmation.be Theorem 9. Let in integral Equation (1) arameters p   In the case, when 1 0   , 0 A  , then from integral representation (18) follows, that, if exidt solution integral Equation (1) in this case, then it is possible in following form In this case for convergence integrals in right part (24), it is sufficient Characteristics 10.In the case, when fulfillment any condition theorem 10, then solution integral equati in point . the following confir 10.Let in integral Equation So, we proof mation.

Theorem
(1) parameters  , olution, which given by formula (24).have unique s on (1) x a  vanish and its behavior determined from following asymptotic formula

Property of the Solution
Let fulfillment any condition of the theorem 1. Differentiating the solution of the type (6), imm diate verification, we can easily convince to correctness of the following eq e uality: In an analogous way differentiating the expression (26), we have From Equality ( 6), ( 26), (27) we find Differentiating the solution of the type (8), immediate v lowing equality:   eri correctness of the fol-fication, we can easily convince to From equality (8) and (31) we find From integral representation (10) it follows that if parameters   1) satisfy all condition of theorem 3, then the solution of the type (10) has the property From integral representation (14) it follows that (34) 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 From integral representation (18) it follows that cos ln 2 sin ln 2 sin ln cos ln 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: 1 lim sin ln cos ln sin ln 2 cos ln .
Differentiating the solution of the type (20), immediate verification, we can easily convince to correctness of the following equality: cos ln sin ln sin ln cos ln 1 sin ln cos ln 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: 1 lim cos ln sin ln cos ln 1 lim .
From integral representation (22) it follows that if parameters  

Boundary Value Problems
When, the general solution constants, arbitrary constants higher mentioned properties of the solution the integral Equation ( 1 C Γ , when the roots the algebraic Equation ( 4) real, different and also 1 0 where A 21 , A 22 -are given constants.
Problem N 3 .Is required found the solution of the integral Equation (1) from class   C Γ , fferent when the roots the algebraic Equation (4) real, di and also 1 0

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where A 31 -are gi stant.Problem N 4 .Is required found the solution of the integral Equation (1) from class   C , when the roots the algebraic Equation (4) real, equal and positive, that is where A 41 , A 42 , A 43 -are given constants.x where A 71 -are given constant.Solution problem N 1 .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 Substituting obtained valued C 1 , C 2 and C 3 in formula (6), we find the solution of problem N 1 in form  . Let fulfillment eorem 7. Then using solution of the type (18) and its properties (42)-(44), and condit 3) we have: . Substituting this valued C 1 , C 2 and C 3 in formula (18) we findthe solution of problem N 5 in form where constant 0    and f k , 0,1, 2, k   , are given nstants.We attempt to ind a solution of (1) in the form where the coefficients, ,    , putting the found coefficients back into (64), we arrive at the particular solution of (1).
en the solution to integral Equation ( 1) can be repreform (64) necessary and sufficiently that 0 , that is, it is necessary and sufficiently that function satisfies olv ility condition the following three s ab x a

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In this case the solution of the integral Equation (1) in the cla ss of function can be represented in form (64) is given by formula Immediately testing it we see that, if converges radius of the series (63) is defined by formula the existence of the solution of Equation (1) can be represented 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

General Ca
In general case to integral Equation (I) corresponding th following algebraic equation According to the mentioned above, writing the solution integral Equation (VI) in depend to the roots of the characteristic Equation (V) or (II), after substituting for  

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. Besides, let in Equation (1) function   f     3 x C   two.Then differentiating integral Equa- mes, we obtained the following third orn differential equation tion (1) three ti der degeneratio suchthat, the roots of the characteristic Equat and different.Its denote by 1 2 3

stics 5 .
In this case, when in integral Equation (1) parameters  of theorem 5, then so l (1) in point x a  vanish and its asymptotic the one roots of characteristic and two the roots of the characteristic equation com conjugate.Correspondingly its denote by  , 2 A iB    ,

1 . 2 .
) give possibility for integral Equation (1) put and investigate the following boundary value problems: Problem N Is required found the solution of the integral Equation (1) from class   C Γ , when the roots the algebraic Equation (4) real, different and positive by boundary conditions (49) where A 11 , A 12 , A 13 -are given constants.Problem N Is required found the solution of the integral Equation (1) from class

Problem N 5 . 6
Is required found the solution of the integral Equation (1) from class   C Γ , when the one roots of the algebraic Equation (4) real positive, two out of its complexwhere A 51 , A 52 , A 53 -are given Problem N

7 .
where A 61 , A 62 -are given constants.Problem N Is required found the solution of the integral Equation (1) from class se

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case, when the roots of the Equation (II) real, different and positive have the following confirmation.Theorem 19.Let in integral Equation (I) parmeters Some results obtained in the general case to.(II) real, such that, the roots of the algebraic Equadifferent and positive, function   Then integral Equation (I) in class of function  

F
x from formula (VII) we arrive at the solution of the new type integral equation.At specific condition to functions In this basis the problem investigation in l Equation (IV), reduce to problem investigation Volterra type integral equation with weak singul n po x a .

Presenta n the Sol Equation (1) in the Generalized Power hat tio ution of the Integral Series
5.  : then converges radius of the series [5]that is for this type integral equation, homogeneous integral equation may have non-zero solution.In particular in certain cases (Example, roots of the characteristic Equation (4) or (II) real, diffe is type integral equation coincides to the theory Fredholm 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 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   2 L 