New Model for Solving Mixed Integral Equation of the First Kind with Generalized Potential Kernel

New technique model is used to solve the mixed integral equation (MIE) of the first kind, with a position kernel contains a generalized potential function multiplying by a continuous function and continuous kernel in time, in the space L2(Ω)× C[0,T ], 0 ≤ T < 1, Ω is the domain of integration and T is the time. The integral equation arises in the treatment of various semi-symmetric contact problems, in one, two, and three dimensions, with mixed boundary conditions in the mechanics of continuous media. The solution of the MIE when the kernel of position takes the potential function form, elliptic function form, Carleman function and logarithmic kernel are discussed and obtain in this work. Moreover, many special cases are derived.


Introduction
The field of integral and integro-differential equations is a very important subject in applied mathematics, because mathematical formulation of many physical phenomena contains integral and integro-differential equations (Saeedi, L., Tari, A. & Masuleh, S. H., 2013).Integral equations are very important branch of mathematics, which come in application in many physical problems.Now, the integral equations have received considerable interest of many applications in different mathematical areas of sciences, for example see (Saeedi, L., Tari, A. & Masuleh, S. H., 2013;Geng, F. Z. & Qian, S. P. 2015;Mohammadi, M. & Mokhtari, R., 2011;Mohammadi, M. & Mokhtari, R., 2013;Diego, T. & Lima, P., 2008).Therefore, different analytic and numeric methods have been established to obtain the solutions of the integral equations.For analytical methods, we state degenerate kernel method, Cauchy method, Laplace transformation method, Fourier transformation method, potential theory method, and Krien's method.More information for the analytic methods can be found in Muskhelishvili (2002), Popov (1982), Tricomi (1985), Hochstad (1971) and Green (Dzhuraev, 1992).
When the analytic solution fails, it is often necessary to use of numerical techniques, which are appropriate combination of numerical integration and interpolation (Linz, P., 1985).The authors of (Ordokhani, Y. & Dehestani, H., 2010) used Walsh functions operational matrix with Newton-Cotes nodes for solving Fredholm-Hammerstein integro-differential equations.In (Ordokhani, Y., 2013), a collocation method based on the Bessel polynomials was used for solution of the nonlinear Fredholm-Volterra Hammerstein integro-differential equations, under mixed conditions.In (Turkyilmazoglu, M., 2014) high-order linear Fredholm integro-differential equations were solved by means of an elegant and accurate effective technique.In (Gouyandeh, Z., Allahviranloo, T. & Armand, A., 2016), the approximate solution for the nonlinear Volterra-Fredholm-Hammerste in integral equations was obtained by using the Tau-collocation method.In (Parand, K. & Rad, J. A., 2012), a numerical technique based on the spectral method was presented to solve the nonlinear Volterra-Fredholm-Hammerstein integral equations The interested reader should consult the fine exposition by Atkinson ( 1976), Delves and Mohamed (Delves, L. M. & Mohamed, J. L., 1985), and Golberg (Golberg, M. A., 1990) for some different numerical methods.
In (Abdou, M. A., 2002); Abdou, M. A., 2000)),MIEs of the second kind with different singular kernels, in position are considered, under certain conditions.Moreover, using different methods the solution, in each case,is obtained.
Consider the following MIE: Under the dynamics conditions (2) The MIE (1), under the pressure and momentum conditions (2), is investigated from the axe-symmetric contact problem for an impressing stamp of angular form on a plane into a half space.The modules of elasticity of the plane is changing according to the power law σ i = K o ϵ i h ,(0 ≤ h < 1; h = µ + 0.5); where σ i , ϵ i are the stress and the strain rate intensities respectively, while K o and h are physical constants.Assume that the stamp, has the equation formula f 1 (x, y)and is impressed into the lower surface by a variable force P(t), whose eccentricity of application e(t), and a moment M(t) that cause rigid displacements γ(t) and β(t), where γ(t) and β(t) ∈ C[0,T].When the frictional forces in the domain of contact area Ω, between the two surfaces, are so small that it can be neglected.In the interval time t ∈[0, T] the unknown function of the pressure between the two surfaces is variant, and such problem leads us to the integral equation.
In order to guarantee the existence of a unique solution of (1), we assume the following: where H is a constant, where the positive functions γ(t) and β (t) are continuous and belong to the class C [0, T ], while f 1 (x, y) ∈ L 2 (Ω) .
(ii) The kernel of position satisfies in general, the discontinuity condition (iii) The positive function F (t, τ)with its partial derivatives, with respect to its variables, are continuous in the class In the remainder part of this paper, we construct a series form solution in the space L 2 (Ω)XC[0, T ], for the MIE (1) in the treatment of various two-dimensional semi-symmetrical and three dimensions problems with mixed boundary conditions.The separation of variables is used to obtain system of VIEs of the second kind and FIE of the first kind.The mechanics problem and its physical properties is explained.The FIE of the first and second kind with logarithmic kernel is derived from the work.Moreover, the integro differential equation with Cauchy kernel is derived with its special cases.This equation has appeared in both combined infrared gaseous radiation and molecular condition, and elastic contact studies.

Fundamental Theorems
To separate the variables of Eq. ( 1) with the generalized kernel L(ξ, η)K(x − ξ, y − η),we seek the general solution in the form, see (Abdou, M. A., 2002), Where, Φ 0 (x, y, t) is the complementary solution and Φ 1 (x, y, t) is the particular solution.
Using (3) in (1), we have ∫∫ And for initial time, we have (5) Hence, from 4 and 5, we obtain ∫∫ Theorem 1(without proof): If the kernel of position L(ξ, η)k (x − ξ, y − η)satisfies the discontinuity, condition (ii), then the integral operator Theorem (2) (without proof): For a symmetric and positive kernel,the integral operator (8)through the time interval 0 ≤ t ≤ T < 1 is compact and self-adjoint operator.Therefore, we can write it in the linear form whereα n and Φ n are the eigenvalues and the eigenfunctions of the integral operator respectively.?

General Solution of Mixed Integral Equation
In this section, our aim is obtaining the general solution of Eq. ( 4) that has mixed formula of Fredholm integral term and Volterra integral term.Therefore, and in view of theorem (2),the solution of Eq. ( 4) can be expanded in a closed form of Fourier series in L 2 (Ω) × C[0, T ] as a system of linear combination of eigenvalues and eigenfunctions: where Φ 2k (x, y),Φ 2k−1 (x, y)are the even and odd functions respectively.
Using the formulas ( 9) and ( 10) in ( 6), the following results can be obtained where 11) represent system ofVIEs of the second kind with continuous kernelF(t, τ),the difference only in the free term.Therefore, every equation has a unique solution in the class (C[0, T ]) depends on the free term A (1)  k (0),γ(t) and β(t).The constant value of A (1)  k (0) can be obtained directly from Eqs. (1), (4) in the form In view of Eqs.( 8),( 9) the general solution of Eq. ( 1) can be adapted in the form Where A (0) k (t) and A (1) k (t) must satisfy the inequality and Φ k (x, y) represents the eigenfunctions of the integral operator (9).Proof: Comparing the result of Eq. ( 12) with the first formula of (9) we get In addition,we have The formula ( 14) represents the convergence condition.
The general solution of the second formula of VIEs (11) can be obtained in the form: ) In ( 15), for even values of k we take w(t) = γ(t), and for odd values we take w(t) = β(t).

General Solution of Mixed Integral Equation:
From the above discussion,the general solution of (1),can be adapted in the eigenvalues and eigenfunctions in the form: Where for even and oddnumbers we use, respectively the following conditions: The formula ( 16) represents the solution of the MIE of Eq. ( 1) in the form of eigenvaluesα k A k (t), where A k (t)is the solution of Eq. ( 15),and eigenfunctions Φ k (x, y).The value of the eigenfunctions are obtain when the kernel of (1) takes the generalized potential form.The boundary condition ( 17) is derived from (2) after using ( 16) in (2).
Secondly, we consider the following famous two relations Γ (α) is called Pachmmer symbol and F 1 ( a , b , c ; z) is the Gauss hypergeometric function, see (Covalence, E. K., 1989).
Hence, using the above two famous relations and following the same technique and way of (Abdou, M. A., 2002;Abdou, M. A., 2000), on noting the difference in notation,we obtain the following integral formula In the contact problems, in the theory of elasticity the function L (u) must satisfy, see (Aleksandrov, V. M. & Covalence, E. V., 1986).
The first formula of ( 18) represents FIE of the first kind with generalized potential kernel in the form of Weber-Sonien integral formula.It is not difficult to prove that the Weber-Sonien integral formula represents a non-homogeneous wave equation: Now, replacing the known function f (r)in ( 18) by Jacobi polynomials i.e., letting f (r) = P (n,µ− 1 2 ) k (1−2r 2 ); h = µ+ 1 2 , 0 ≤ µ < 1/2;then using Krein's method, see (Ur Rehman, M. & Khan, R. A., 2012) and after some integrations and algebraic relations, we arrive to the following formula Where the eigenvalues α n k are given by The formula (21) represents the spectral relations (solution) of ( 18) in the eigenvalues and eigenfunctions form.
Finally, the solution of (1) when the kernel of position takes the generalized potential form and the free term in the form of Jacobi polynomials,P (k,−w − ) k (1 − 2r 2 ) multiplying by a function of time A k (t) of ( 15), takes the form where A k (t), is the solution of VIEs (11) and α k is given by ( 22).
The formula (23) represents the general solution of (1) of semi-symmetric Hertz contact problem of frictionless impression of a rigid surface (G, ν)having an elastic material occupying the domain } in three dimensions in prance of time and under the dynamic conditions of pressure and momentum.

Applications of Spectral Relations and Special Cases:
In this section, many applications of spectral relations, in mathematical physics problems and special cases can be derived from (23).
Case (i): Semi -symmetric contact problem, in three dimensional with potential kernel, in presence of time: The IE (24) constructs the solution of the action problem of an inclined ring stamp,a ≤ r ≤ b, on an elastic half-space that is free from external forces and the frictional forces in the domain of contact area between the stamp and half-space are neglected.
The potential kernel can be written in the form Here, in (25), Γ(x) is the Gamma function and E ( ) is called the complete elliptic integral form.
where (β is Beta function Reγ > Reδ > 0) We can rewrite (26), for any higher order harmonic, k ≥ 0, in the integral elliptic form The spectral relations (21) take the form Hence, the complete solution of the problem is: Case (ii): Three-dimensional semi symmetric Hertz contact problem with elliptic kernel: Let, in (29) n=0 we have the solution of MIE (1), under the conditions (2) with elliptic kernel in position.The importance of the elliptic kernel came from the work of Covalence (Covalence, E. K., 1989), who developed the Fredholm equation of the first kind for the mechanics mixed problem of continuous media and obtained its approximate solution.
The general solution, in this case, takes the form The general solution of the MIE with Carleman function for n = 1/2is given as While, the spectral relations when n = − 1 2 are  Many special cases can be derived from the generalized kernel (18) as the following: (i) When h = 0, and n = ±0.5, we have the logarithmic kernel, see Fig. 5,

Figure 1 .
Figure 1.The solution of the MIE with potential kernel for some different values k, u

Figure 2 .
Figure 2. The solution of Eq.(30) for different values of k

Figure 4 .
Figure 4. Solution for different values of µ