Spectral relationships of the integral equation with logarithmic kernel in some different domains

In this work, the Fredholm integral equation ( FIE ) with logarithmic kernel is investigated from the contact problem in the plane theory of elasticity. Then, using potential theory method ( PTM ), the spectral relationships ( SRs ) of this integral equation are obtained in some different domains of the contact. Many special cases and new SRs are established and discussed from this work.


INTRODUCTION
Many problems in the half-plane of elasticity, deformation in physics and engineering are reduced to an integral equation of the first kind; see Popov [1], Covalence [2], and Alexandrov [3].In other words, different methods are established for solving the FIE with discontinuous kernel.These methods are: Cauchy method [4], potential theory method [5,6], orthogonal polynomial methods [7,8], Fourier transformation method [9, 10] and Krein's method [11,12].The FIE with logarithmic kernel is investigated from the contact problem in two dimensional problem, in the theory of elasticity from an infinite strip; occupying the region 0 yh  made of material satisfies the stress relations, lies without fraction on a rigid support.A rigid rectangular stamp is impressed into the boundary of the strip yh  by constant force p whose eccentricity of application is e .Then, using Airy function and Fourier integral forms, Abdou et al. in [12], represented the plane contact problem, in a half-plane, as a FIE of the first kind with logarithmic kernel.
Here, the PTM is used to obtain a boundary value problem (BVP) in two dimensional domain.Moreover the properties of Chebyshev polynomials of the first, second kind and the elliptic Jacobi functions are used.Then, by considering the equivalence condition between the differential equation and the integral equation we obtain many different SRs inside and outside the domain of contact (domain of integration).The importance of using spectral relationships in contact problems in the theory of elasticity can be found in Refs.[13][14][15][16][17].

Fredholm integral equation:
Consider the following FIE: where the kernel is defined by Here,

,,
x  are the dimensionless variables; while h a   is a dimensionless parameter characterizing the strip thickness.It should be noted that as  , the integral equation takes the form: In the remainder part of this paper, we will obtain the solution of the FIE (3) in the form of SRs in different domains and discuss it.

Solution of integral equation
Here, we use the PTM to obtain the SRs for the FIE of the first kind in different domains.For this purpose, consider the integral operator where, we will consider the following cases:

The first case (1) of equation (6):
Using the potential theory method, see Abdou et al. [18], we have with the equivalence condition where   x  is the Dirac-delta function.
To obtain the solution of ( 7), we use the transformation mapping function: A useful method in engineering mathematics is using a conformal mapping to transform complicated region into a simpler one, for this reason, we use equation (9).The transformation (9) maps the region in xy  plane into the region outside the unit circle  , such that   /  does not vanish or becomes infinite outside the unit circle  .The parametric equations of (9) are Using the transformation (9), we get The mapping (9) maps the upper and lower of the interval     ,, x y a a  into the lower and the upper of the semicircle 1   , respectively.Moreover the point z will be mapped onto the point 0   .
Using the transformation (9) in the BVP (7) we have Also, the equivalence condition becomes Here, we assume M a y 1 7 , 2 0 1 4 (14) to solve the BVP of (12), we assume Differentiating (15) with respect to  and , then introducing the result to satisfy the first equation of ( 12), and noting that, when r , we have 0   .Therefore, the solution of the formula (12) can be adapted in the form Where, with the aid of the second and third formulas of ( 12), we have Using the previous results in (13), we obtain Introducing (18)  The formula (19) represents the SRs for the FIE of the first kind with logarithmic kernel, where we assume the known function, in (4),  

The second case (2) of equation (6):
In order to obtain the SRs of ( 19) for xa  , we obtain from (10) that xa  for 0 Solving the BVP (12), under the condition (20), then using (17) we have the following SRs: Where,

 
Hxis the Heaviside function, see Whittaker et al. [19].M a y 1 7 , 2 0 1 4 The conformal mapping (9), for xa  maps the half plane, 0, Hence, we deduce that, outside the interval, the case for Therefore, the solution of the BVP (7) for 0   takes the form Moreover, the term of equivalence relation of ( 8) W y   will take the form Using the separation of variable method, with the aid of ( 22), in the BVP (7) and the equivalence relation ( 8), we can obtain where The formula (24) leads us to assume the general integral operator Hence, with the aid of (26), and for the interval

The third case (3) of equation (6):
When b x a  , we will seek the solution of the BVP (7) by using the transformation mapping , is the complementary elliptic modulus (see Whittaker et al. [19]), where Also, the Jacobi elliptic integral of the first kind is defined The transformation mapping (28) maps the region of the interval   Moreover, the transformation mapping (28) maps the region outside the plane Im 0 z  to the region outside the semi- ring   , ab  will be mapped inside the ring Hence, we have Using ( 32) and ( 33) in (28), with the help of properties of elliptic function, we can have the parametric equations sn u cn iv dn iv cn u dn iv sn iv x b y ib k sn u sn iv k sn u sn iv The linear coordinate uK  will cover the interval   , ab  , while uK  covers the interval   ,.
ab For this, we have   ,1 sn K k  , and the first formula of (34), after using the properties of the elliptic functions, see Whittaker et al. [19], takes the form: Also, the formula (35), with the properties of dn , can be adapted in the form M a y 1 7 , 2 0 1 4 The formula (36) is hold only for b x a  and when x is changed from b to a , v will be changed from 0 to K  .For this, we define Where,   After the above discussion, the BVP (7) can be modified as .
Using the chain rule, we can write the equivalence condition in the final form as Now, we assume the solution of (38) in the form Therefore, for determining the unknown constants , , , nn A B C and , D we assume .
From the second and third conditions of (38) and with the aid of (41), we have •Now we have the following two points of discussions: (1) The first discuss for a symmetric case, we have   , hence we get , 0,1,2,... ; 0, ln Therefore, rewrite (42) to take the form M a y 1 7 , 2 0 1 4 With the aid of the last conditions of equation ( 38) and the formula (43), we can directly determine the value of P in the form: The values of n  can be obtained, after using the famous relation, see [19]    Finally, using (44) in (39), the potential function   where, the constant P is given by (44), and 's  by (45), while the function   n TX is the CP of the first kind.

Special and new relations of SRs:
Many spectral relationships, which have many applications in astrophysics, mathematical engineering and contact problems in the theory of elasticity, can be derived and established from this work:  can be established if we consider, in (4), the following two cases: We have, directly the following: In this case, the orthogonal relation will take respectively the forms Differentiating ( 4) with respect to x , we get       The value of the integral (58) has many important applications in the contact problem when the kernel takes a Cauchy form.
Also, in  − plane we follow for this, we have where, in (65), we assumed The integral operator (II) with the SRs (65), (67) can be adapted, in the Fourier integral sine or cosine forms, as the following: The formulas (61) and ( 65)-( 68) can be used with wide applications in the displacement problems of mechanics, see Aleksandrov et al. [6], Aleksandrov [13] and Abdou [16].
In the spectral relationships (51), we assume     Finally, for the interval b x a  , after assuming a

( 28 )
M a y 1 7 , 2 0 1 4 The transformation mapping (28), z x iy  , is called the Jacobi elliptic transformation.The three basic functions of the elliptic functions are denoted     , , , cn u k du u k and   , sn u k where k is known as the elliptic modulus.They arise from the inversion of the elliptic integral of the first kind the Jacobi amplitude.The Jacobi elliptic function are periodic in   Kkand   / Kk, where   Kkis the complete elliptic integral of the first kind,

1 
 in  −plane.Moreover, the points outside the interval   , bawill be mapped outside the ring This leads us to deduce the following important relations M a y 1 into (4), and noting the definition of CP of the first kind, Also, the orthogonal relation for the CPs of the second kind takes the form