I. INTRODUCTION

In recent years, the main theorem of spectral relationships plays an important role in many applications in various areas, including models of nanotechnology particles, genetic engineering, medicine, mathematical chemistry, heat condition and physical phenomena, see $[1-4]$.

On the other side, many problems of mathematical physics, contact problems in elastic media, many applications in electronic engineering applications, and mathematical biology models lead to an (IEs) in smooth or singular forms. The references $[5-7]$ discuss the methods for solving the (IEs) in smooth form antically. While the singular forms take a large aria in types of researches, see $[8-11]$. In $[10]$, a spectral technique for solving two-dimensional fractional integral equations with singular kernel was discussed by using Legendre and Chebyshev polynomials. Abdou and Salama in $[12]$ obtained the spectral relationships for the Volterra-Fredholm integral equation (V-FIE) of the first kind. Abdou $[13]$ discussed the spectral relationships that have many important applications in astrophysics, for the (F-VIE) of the first kind, when the kernel of position takes a generalized potential form. The relation between the contact problems and the (F-VIE) in three dimensions were obtained by Abdou and Moustafa in $[14]$. Abdou and Nasr in $[15]$ used Chebyshev polynomial to obtain the solution of (F-VIE) when the kernel takes a logarithmic form. The application of Orthogonal polynomials in spectral relationships of some kinds of singular contact problems is discussed by Alharbi in $[16]$. Abdou and Basseem in $[17]$ used Chyebyshev polynomial, the main theorem of spectral relationships of (FIE) to obtain the solution of (F-VIE) of the second kind numerically. Abdou and Alharbi in [18] derived a general main theorem of spectral relationships for a mixed integral equation of the first kind (MIE) in the space $L_2[-1,1]\times C[0,T]$.

Consider the integral equation,

Ref

Under the following conditions:

i. ${[{\int }_{-1}^1{\int }_{-1}^1{k}^2(x,u,y,v)dxdudydv]}^{1/2}\le c,c\text{ is constant}.$

ii. The function $f(x,y)$ with its partial derivatives with respect to $x$ and $y$ are continuous in $L_2[-1,1]\times [-1,1]$ and its norm can be defined as,

iii. The unknown function $\varphi (x,y)$ in the space $L_2[-1,1]\times [-1,1]$ behaves as the given function $f(x,y)$.

II. THE EXISTENCE OF AT LEAST ONE SOLUTION OF A TWO DIMENSIONAL IE

Theorem 1

The integral equation (1) has at least one solution under the previous condition(i)-(iii). Define the integral operator forms:

Eq. (1) can be written in operator form as:

The proof of theorem1 can be obtained automatically from the proofs of following lemmas.

Lemma1

The integral operator (3) under the conditions (i)-(iii) is bounded in the space $L_2[-1,1]\times [-1,1]$.

Taking the norm of Eq.(3) we get

Notes

Applying the Cauchy-Schwarz inequality, then using conditions (i)-(iii) to have

Eq.(5) means that the integral operator \bar{W} maps the ball \bar{S}{\alpha} into itself, where \alpha = M + |\lambda|C. Also, in the second term of inequality (5), we deduce that the integral operator W\phi(x,y) is bounded in the space L{2}([-1,1] \times [-1,1]). Therefore, \bar{W}\phi(x,y) is also bounded.

Lemma 2

The integral operator (3) under the conditions (i)-(iii) is continuous in the space $L_2[-1,1]\times [-1,1]$.

Proof:

Let ${\varphi }_1(x,y),{\varphi }_2(x,y)$ be any two functions in the space $L_2[-1,1]\times [-1,1]$ then,

After applying Cauchy-Shwarz inequality then using the conditions(i)-(iii), the previous inequality becomes,

So,

Inequality (6) implies the continuity of $\bar{W}$ in the space $L_2[-1,1]\times [-1,1]$.

Lemma 3

Suppose that a sequence of continuous functions $\{k_{n,m}(x,u,y,v)\}$ such that,

Then, there exist positive integers $n_0$, $m_0$, such that, for , , in general $n\ne m$, after neglecting the very small constants, we have

Proof:

Notes

Hence, for each using Eq.(7) and conditions(i), formula (8) is verified after neglected a small constant.

Lemma 4

Suppose that $\{{\bar{W}}_{n,m}(x,y)\}$ is a sequence of integral operators where the conditions(i)-(iii) are satisfied, then the sequence of operators

is the bounded and continuous sequence.

Taking the norm of Eq.(9) then using the conditions (i)-(iii) to get

Therefore ${\overline{W}}_{n,m}$ maps the largest ball $S_{\alpha }$ into itself.

Also to prove the continuity of ${\bar{W}}_{n,m}$ we choose any two functions ${\varphi }_1(x,y),{\varphi }_2(x,y)$ in $S_{\alpha }$ then applying Cauchy-Schwarz inequality and the conditions (i)-(iii) we get

Lemma 5

If conditions (i)-(iii) are verified then the $\overline{W}(S_{\alpha })$ is compact.

Proof:

Hence, using condition(iii) yields

Also, from Eq.(7) we have the following condition:

To prove the compactness of $\bar{W}$, we let $\{{\varphi }_{n,m}(x,y)\}$ be any sequence in $S_{\alpha }$. Then we can choose a subsequence $\{{\varphi }_{n1,m}(x,y)\}$ such that $\{{\bar{W}}_{n1,m}{\varphi }_{n1,m}(x,y)\}$ converges. From that subsequence, we can extract a new subsequence $\{{\varphi }_{n1,m1}(x,y)\}$ in which $\{{\bar{W}}_{n1,m1}{\varphi }_{n1,m1}(x,y)\}$ converges, and so on. Thus, we obtain a chain of subsequences,

Such that the sequence $\{{\bar{W}}_{ni,mk}{\varphi }_{nj,ml}(x,y)\}$ converges for all $i=1,2,\dots ,j$ and $k=1,2,\dots ,l$. Finally, we pick the sequence $\{{\varphi }_{n_n,m_m}(x,y)\}$ which is a subsequence of every ${\varphi }_{n_i,m_k}$ except for a finite number of elements, and clearly $\{{\bar{W}}_{ni,mk}{\varphi }_{nn,mm}(x,y)\}$ converges for every i,k. Now, since

For large j, k, and from(), we get

Hence, $\{\bar{W}{\varphi }_{n,m}\}$ is a Cauchy sequence, so $\bar{W}(S_{\alpha })$ is compact.

According to the previous lemmas, by Schauder fixed point theorem, see [19-20], $\bar{W}$ has at least one fixed point in $S_{\alpha }$, and Theorem 1 is proved.

III. CHEBYSHEV POLYNOMIALS AND THE SYSTEM OF THE INTEGRAL EQUATION

Suppose the approximate kernel $k_{n,m}(x,u,y,v)$ in the continuous case as,

where it satisfies the condition in Eq.(7).

Ref

Therefore Eq.(1) reduce to the algebraic system form as,

where,

is the approximate error.

To use the spectral relationships, we write the kernel of Eq.(12) in the form

where $T_l(z)$ is the Chybeshev polynomials of first kind and degree $l$.

Then Eq.(13) reduces to

Such that,

Since,

then by using the relation, see [21]

and,

Eq.(18) reduce to

Notes

Then, Eq.(13) reduce to obtaining the following algebraic system,

where,

IV. APPLICATION AND NUMERICAL DISCUSSION

Consider the IE

For fixed values of N = M = 100, we can graph the solution $\varphi (x,y)$ at different values of $\lambda$, and different shapes of surfaces $f(x,y)$, then we can graph the estimating errors

The solution ${\varphi }_{n,m}(x,y)$

Figure(1) The estimating error $E_{100}$

$f(x,y)=xy,\lambda =0.1$ The solution ${\varphi }_{n,m}(x,y)$

Figure(2)

The solution ${\varphi }_{n,m}(x,y)$

Figure(3)

Notes

The solution ${\varphi }_{n,m}(x,y)$

The estimating error $E_{100}$

Figure(4)