Algebraic Kernel Method for Solving Fredholm Integral Equations

Abstract In this paper, we study the exact solution of linear Fredholm integral equations using some classical methods including degenerate kernel method and Fredholm determinants method. We propose an analytical method for solving such integral equations. This work has some goals related to suggested technique for solving Fredholm integral equations. The primary goal gives analytical solutions of such equations with minimum steps. Another goal is to compare the suggested method used in this study with classical methods. The final goal is that the propose method is an explicit formula that can be studied in detail for non-algebraic function kernels by using Taylor series expansion and for system of Fredholm integral equations.


Introduction
A variety of analytical and numerical methods are used to handle linear Fredholm integral equations such as successive approximation method, adomain decomposition method, modified decomposition method, direct computation method, Laplace transform method, Chebyshev collection method, the Taylor series method , Galerkin method, variational iteration method, spline collection method and other methods [4,10]. Avazzadeh [2] , Bakodah [3] and Wazwaz [9] studied in particular a comparison between certain analytical and numerical methods for solving integral equations. Integral equations are one of the most useful mathematical tools in both pure and applied analysis [8]. The solutions of integral equations have a major role in the fields of science and engineering. The development of science has led to the formation of many physical laws, which, when restated in mathematical form, often appear as differential equations, an integral equations or an integro-differential equations or a system of these. Engineering problems can be mathematically described by differential equations or integral equations, and thus it play very important roles in the solution of practical problems. For example, Newton's law, stating that the rate of change of the momentum of a particle is equal to the force acting on it, can be translated into mathematical language as a differential equation. Similarly, problems arising in electric circuits, chemical kinetics, and transfer of heat in a medium can all be represented mathematically as differential equations or integral equations [6,7]. Fredholm integral equations arise in many scientific applications and can be derived from boundary value problems. Erik Ivar Fredholm (1866-1927) is the best remembered for his work on integral equations and spectral theory. Fredholm was a Swedish mathematician who established the theory of integral equations and his 1903 paper in Acta Mathematica played a major role in the establishment of operator theory [1].

Methods
There are a variety of methods to solve integral equations analytically and numerically. We solve linear Fredholm integral equations using analytical methods. Some of these methods are always known and used such as degenerate kernel method and Fredholm determinants method. These two methods were selected because they depend on the fixed laws can be applied and find the exact solution of any Fredholm integral equation.

Degenerate Kernel Method
is solved in the following manner. Rewrite the equation (2) as: and introduce the notations: Then the equation (3) becomes are unknown constants, since the function ) (x y is unknown. Thus, the solution of an integral equation with degenerate kernel reduces to finding the constants . Putting the expression (5) into the integral equation (2), the equation takes the following form after simple manipulations: For the sake of brevity, we introduce the notations Volume 7 and find that Using Cramer's rule or substituting method to solve the system (6) for finding the unknowns coefficients k C . The solution of the integral equation (2) is the function ) (x y defined by the equality : x y  and using degenerate kernel method, we rewrite the given integral equation as Solving (8) and (9)  Substituting these values of 1 c and 2 c into equation (7), we obtain the solution of the given integral equation as follows:

Fredholm Determinants Method
The solution of the Fredholm integral equation of the second kind is given by the following formula where the function ) ; , is called the Fredholm resolvent kernel of equation (10) and is defined by the equation are power series in  : Whose coefficients are given by the following equations: Knowing that the coefficients We can use the formulas (15)  analytically, [5].

Example: We apply Fredholm Determinants method (Recurrence Relations) to solve the integral equation
As we note that . and , 120

Results and Discussions
Solving linear Fredholm integral equations analytically are sometimes very difficult, and they are required more computational steps. In order to have the solutions for such problems, we propose an analytical technique for solving Fredholm integral equations. The suggested method is named by algebraic function kernel method. The method depends on the kernel of linear Fredholm integral equations. Therefore, we have two main cases based on the kernels for such equations: In this case, the variable t appear with the variable x in the kernel of the integral equation, i.e.   ( We write the matrix A of eigen values to the parameter  as follows: By comparing the coefficients of the same power of x , then we can find b a , and c as follows From equations (25) and (26)