Certain results associated with mixed integral equations and their comparison via numerical methods

In this article, we consider existence and unique of solutions of linear mixed integral equations of third, second and first kinds. Then, we use the collection method to discuss numerical solutions by employing Chebyshev and Legendre polynomials. To support our results, we use Mable 18 to compute errors in all existing cases.


Introduction
Several problems of mathematical physics, theory of elasticity, viscodynamic fluids and mixed problems of mechanics start in a form of linear mixed integral equation with different kinds. For example, in the displacement problem in the theory of elasticity, see Abdou et al. [1,2], in contact problems, see Georgiadis [3] and Popov [4], in the thermomechanics, see Chirita et al. [5],in thermoelastic models, see Chirita [6], in quantum mechanics, see Rehab and Sheikh [7,8], and in thermoelasticity for an elastic plate weakened by curvilinear holes, see Abdou et al. [9]. Such equations occur as reformulations of integral mathematical problems in the linear or nonlinear form, see He [10] and Abdou and Youssef [11]. Therefore, the authors have interested in studying different methods to solve the linear and nonlinear integral equations in analytic and numerical forms. In [12], Mirzaee and Samadyar used orthonormal Bernstein collocation method to solve a mixed Volterra-Fredholm integral equation in two-dimensional. In [13], Abdou et al. the orthogonal polynomial method with the aid of Chebyshev polynomial to discuss the solution of the quadratic integral equation. While Al-Bugami in [14], used the orthogonal polynomial method to discuss the solution of Hammerstein integral equation. More information for different numerical methods can be found in Gu [15,16], Elzaki, and Alamri [17], Abdou and El-Kojok [18], Alhazmi [19], and Basseem and Alalyani [20].
Consider the following integral equation of the third kind, The mixed integral Eq. (1) is investigated from the semi-symmetric contact problem of Hertz type of two rigid surfaces G i , i , i = 1, 2 having two elastic materials occupying the domain [0, 1] . If the upper surface is impressed into the lower surface by a variable force p(t) , whose eccentricity of application e(t) and a moment M(t)

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58 that case rigid displacements (t) and (t) respectively, t h ro u g h t h e t i m e i nte r va l 0 ≤ t ≤ T < 1 a n d (t), (t) ∈ C[0, T ] . In the absence of body forces, and when the frictional forces in the domain of contact between the two surfaces are so small in which it can be neglected. The unknown function (x;t) represents the unknown normal stresses between the two surfaces through the time t , t ∈ [0, T ] . The positive continuous function F(t, ) ∈ C([0, T ] × [0, T ]) represents the resistance force of the lower material against the total pressure and moment. Here is the coefficients bed of the compressible materials that depend on its geometry and called the poison's coefficients and E i are the coefficients of Young. The mixed linear integral Eq. (1) has a Fredholm integral term in position, in the space L 2 [0, 1] , with continuous k(x, y). While, the second term of Volterra is considered in time in the class C[0, T ], T < 1, with the continuous kernel F(t, ). The known function n is called the free term. While, the unknown function (x, t) is called the potential function which can be obtained in the space L 2 (0, 1) × C[0, T ]. being a constant of many physical meaning. While, (x) defines the kind of the integral equation. For the third kind, (x) is variable, the second kind, (x) = constant ≠ 0. Finally, the first kind (x) = 0.
In the remainder part of this paper especially in Sect. 2, the existence of a unique solution of mixed linear integral Eq. (1), under certain conditions is considered. In Sect. 3, when Picard method fails, we use Banach fixed point thermo to prove the existence of a unique solution of mixed linear integral equation of the first kind. In Sect. 4, Collocation method using Chebyshev and Legendre Polynomials is conceded to discuss numerically Eq. (1). In Sect. 5, some examples are considered to discuss the numerical results for the three different kind of the mixed integral equation.

The existence and uniqueness solution
In order to guarantee the existence of a unique solution of mixed linear integral Eq. (1), we assume the following conditions: (i) The kernel of the Fredholm integral term satisfies the continuous condition: Then, from Eq. (1) we can construct sequences of solution { n (x, t)} ∞ n=0 . And then we pick two functions n (x, t), n−1 (x, t). For simplicity of manipulation, it is convenient to introduce to construct the following Hence, we have Applying the norm properties for all terms, then after assuming 1 =Ã, we obtain When, n = 1 , we have By applying the Cauchy-Schwarz inequality and using the conditions (i) and (ii), and using 0 ( By induction, we obtain Finally, we have The result of (9), leads us to say that the formula (3) has a convergent solution. Therefore, the solution is stable. So, let n → ∞ , to obtain The infinite series of (10) is convergent, and (x, t) represents the convergent solution of (1). Also, each of i is continuous, therefore (x, t) is also continuous.
To show that = 1, t = 0.01, = 0.01 is a unique solution, we assume that = 1, t = 0.1, is a continuous solution of (1), then we write Rewriting (11) and using conditions (i), (ii), we can arrive at the relation: Since < 1, the inequality is true only if = 1, t = 0.99, = 0.01. This leads to the uniqueness of the solution of (1).
Note. The Picard method always fail to prove the existence and uniqueness of a solution of the homogeneous integral Eq. (1) (f(x, t) = 0) or the first kind of the same equation ( = 0). For this, the existence and uniqueness solution can be obtained by proving the continuity and normality of the integral operator.

The normality and continuity of the integral operator
To discuss the normality and continuity of (1), we rewrite it in the integral operator form as following: where Hence, Using the conditions (i-iii), after applying Cauchy-Schwarz inequality, we get, Therefore, we have.
Hence, the integral operator is bounded. For continuity, we assume the two potential functions are 1 ( x, t ), 2 ( x, t ) . Hence, we have.
This result leads us to the continuity of the operator W since, < 1 . Hence W is a Contraction operator.

Chebyshev polynomials
Chebyshev polynomials are applied to approximate the solution of (1). For this, let the unknown function (x, t) and the known function f (x, t) take the series form of Chebyshev polynomials.
Here, C i,j are unknown constants, while f i,j are known constants can be determined from the orthogonal relation of Chebyshev polynomials.
Using Eq. (18) This algebraic system can be solved for us to obtain C i,j and substitutes in Eq. 20 to get the solution.

Legendre polynomials
To represent the solution of (1) in the Legendre polynomial form of the first order, we assume t = t 0 , x = x 0 , then S 0,0 = R x 0 , t 0 , put x = x 0 , t = t 1 , we get S 0,1 = R x 0 , t 1 , S n,m = R x n , t m .
Hence, we assume This algebraic system can be solved to obtain C i,j and substitute in Eq. (22) to get the solution.

Applications
Case (1): For the third kind equation, consider = x, t = 0.1, = 0.01. and the comparison between the errors results in Chebyshev and Legendre polynomials methods is listed in Table 1. The relationship the relationship between exact solution, numerical and error solution using Chebyshev and Legendre polynomials methods for third kind is shown in Fig. 1. with respect to Case (1). t = t 0 , x = x 0 , then S 0,0 = R x 0 , t 0 put x = x 0 , t = t 1 , we get S 0,1 = R x 0 , t 1 , S n,m = R x n , t m .
Case (2): For the third kind equations we consider = x, t= 0.01, = 0.01 and the comparison between the errors results in Chebyshev and Legendre polynomials methods are displayed in Table 2. In addition, Fig. 2 describes the relationship between exact and numerical solution using Chebyshev and Legendre polynomials methods for third kind according to case 2.  Table 3 Table 4. In addition, Fig. 4 shows the behaviour the exact solution and numerical solution in Chebyshev and Legendre polynomials for second kind in accordance with case (4).        Table 5. Furthermore, Fig. 5 diplays The relationship between exact solution, numerical and error solution using Chebyshev and Legendre polynomials methods for first kind depending in case (5).  Table 6 besides Fig. 6 describes the relationship between exact solution and numerical solution using Chebyshev and Legendre polynomials methods for first kind depending on Case (6).

Exact
Numerical Numerical Fig. 5 The relationship between exact solution, numerical and error solution using Chebyshev and Legendre polynomials methods for first kind

Conclusion
In the present paper, we consider the three kinds of the mixed linear integral equation with continuous kernels. In general, we discussed the existence and uniqueness solution of the mixed integral equation of the third kind. And then, we established the second kind as special case from the integral equations. To prove the existence and uniqueness solution of the mixed integral equation of the first kind, we use Banach fixed point theorem, where the Picard method fails. We obtain a numerical solution of the MLIE using Chebyshev method and Legendre method, while, the functions of the integral equations are represented in the form of Chebyshev and Legendre rules. The error in each example is computed.
• In Table 1 At = x, t = 0.1, & = 0.01 , the error in Chebyshev is decreasing in period 0.1 ≤ x ≤ 0.2 and increasing in 0.3 ≤ x ≤ 1 , while, in Legendre, the take the same behave of the error. • In Table 2 At = x, t = 0.01, & = 0.01 , the error in Chebyshev is decreased in period 0.1 ≤ x ≤ 0.9 , and increasing in 0.9 ≤ x ≤ 1 while, in Legendre has the same behave of the error. • In Table 3 At = 1, t = 0.01, & = 0.01 , the error in Chebyshev stability is decreased in period0 ≤ x ≤ 1 , while, in Legendret, the error is stable decreasing in the period0 ≤ x ≤ 1. • In Table 4 At = 1, t = 0.1, = 0.5 the error in Chebyshev is a stability of decreasing in the period 0 ≤ x ≤ 1 , while, in Legendre, the error stable decreasing in period 0 ≤ x ≤ 1. • In Table 5 At = 0, t = 0.1, & = 0.01 , the error in Chebyshev is decreasing in the interval of 0.1 ≤ x ≤ 0.6 and increasing in the interval of 0.7 ≤ x ≤ 1. • In Table 6 At = 0, t = 0.01, & = 0.01 , the error in Chebyshev is stable decreasing in the period 0 ≤ x ≤ 1 , while, in Legendre, the error is also stable decreasing in the period 0 ≤ x ≤ 1.