The Use of Homotopy Regularization Method for Linere and Nonlinner Fredholm Integral Equations of the First Kind

Recently, Wazwaz has studied the regularization method to the one-dimensional linear Fredholm integral equations of the first kind [Wazwaz, 2011]. In this work, we develop this method for the linear and nonlinear two-dimensional Fred-holm integral equations of the first kind. Indeed, the regularization method is used for linear integral equations directly. But nonlinear integral equations of the first kind are transformed to linearintegral equations of the first kind by a change of variable; then, The Regularization-Homotopy Method is applied. The combination of the regularization method and the homotopy perturbation method, or shortly, the regularization-homotopy method, is used to find a solution to the equation. Some examples will be used to highlight the reliability of the generalized of Regularization-Homotopy Method.


Introduction
Integral equations of the first kind in the one-dimensional case have been studied in many papers . But although these equations in two-dimensional cases have many interesting applications in Mechanical engineering, Physical sciences and other applied sciences [Wzwaz et al. 2011], only a few papers have been written about them (Molabahrami et al. 2013).In this paper, we consider the general form of the two-dimensional Fredholm integral equations of the first kind.
( , ) = ∫ ∫ ( , , , ) ( Where f and K are continuous functions and is a constant. Also, F is a continuous function which has a continuous inverse, and finally, u is the unknown function of the equation (1) to be found. Obviously, if G is linear, then Eq. (1) will be linear. As mentioned above, we develop the regularization method of [Molabahrami, 2013] to the linear case directly and in the nonlinear case, we first set u ( x , t ) = F (h ( x , t )) to convert (1 ) to linear form.

The homotopy perturbation method
The homotopy perturbation method was introduced and developed by Ji-Huan He in (PtL, 1962) and was used recently in the literature for solving linear and nonlinear problems. The homotopy perturbation method couples a homotopy technique of topology and a perturbation technique. A homotopy with embedding parameters is constructed, and the impeding parameter p is considered a small parameter. The method was derived and illustrated in [PtL 1962], and several differential equations were examined. The coupling of the perturbation method and the homotopy method has eliminated the limitations of the traditional perturbation technique [PtL, 1962]. In what follows, we illustrate the homotopy perturbation method to handle Fredholm integral Page | 20 equations of the second kind and the first kind.
( , ) = 1 ( , ) − ∫ ∫ ( , , , ) ( , ) (2) obtained above in (2). The homotopy is now constructed where the embedding parameter p monotonically increases from 0 to 1. The homotopy perturbation method permits the use of the expansion and consequently Substituting (4 ) into both sides of (3) and equating the terms with the same powers of the embedding parameter p, the recurrence relation is obtained Having determined the components ( , ), ≥ 0, we then use The series (7) Assuming that F ( u ( x ) ) is invertible, then we can write The non-linear Fredholm integral equations of the first kind are often considered an ill-posed problem, and this may lead to several difficulties. In this work, we will limit ourselves only to cases where ( , ) = ( )ℎ( ). We will now examine the illustrative linear and non-linear Fredholm integral equations of the first kind.

LinearTwo-dimensional Fredholm integral equation of the first kind
Now, we are going to apply the regularization-homotopy method to illustrate the earlier presented analysis for the linear case. Significantly, a necessary condition to guarantee a solution is that the data function ( ) must contain components which match the corresponding We next construct the homotopy Proceeding as before, we find the recurrence relation

Nonlinear Tow-dimensional Fredholm integral equation of the first kind
The regularization-homotopy method is applied to illustrate the analysis presented before for the non-linear case, as given below. However, our focus will be limited to the separable kernel ( ) = ( )ℎ( ).

Conclusion
In this work, a combination of the regularization method and the homotopy perturbation method was proposed as a reliable treatment of the Two-dimensional linear and non-linear Fredholm integral equations of the first kind. The proposed method showed reliability in handling these ill-posed problems. Three examples, linear and non-linear, were examined to illustrate the analyses which were presented. The exact solutions were formally derived if the exact solutions existed, as these equations were ill-posed. We pointed out that the corresponding analytical solutions are obtained using Mathematica.
Funding: This research received no external funding.