Three Methods to Solve Two Classes of Integral Equations of the Second Kind

abstract: Three methods to solve two classes of integral equations of the second kind are introduced in this paper. Firstly, two Kantorovich methods are proposed and examined to numerically solving an integral equation appearing from mathematical modeling in biology. We use a sequence of orthogonal finite rank projections. The first method is based on general grid projections. The second one is established by using the shifted Legendre polynomials. We present a new convergence analysis results and we prove the associated theorems. Secondly, a new Nyström method is introduced for solving Fredholm integral equation of the second kind.


Introduction
Over the last decades, lots of important problems in applied mathematics, science, physics, engineering , biology, electrodynamics, mechanics, economics and other different fields of computer, science and engineering are written and modeled in the form of integral equations. However, there are many obstacles to directly solve these equations . Thus, we should solve these equations by using numerical methods. Recently, several numerical results have been developed for solving integral equations. Among the most approximation schemes, Kantorovich method is the most efficient.
More recently, Mennouni established an improved convergence analysis via Kulkarni method (cf. [7]) to approximate the solution of integro-differential equation in L 2 ([−1, 1], C) by using the Legendre polynomials. In [8], the author introduced an efficient Galerkin method for a class of Cauchy singular integral equations of the second kind with constant coefficients in L 2 ([0, 1], C), the author used a sequence of orthogonal finite rank projections. The aim of [6] is to applied the Kulkarni method and a Galerkin method for solving second kind noncompact bounded operator equations. Moreover, the author used a sequence of orthogonal finite rank projections to approximate the solution of singular integral equations of the second kind with Cauchy kernel. The main idea of [10] is to propose a collocation method for solving singular integro-differential equations with logarithmic kernel using airfoil polynomials. The goal of [9] is to numerically solve the Cauchy integro-differential equations using the projection method based on the Legendre polynomials. In this paper, we introduce three methods to solve two classes of integral equations of the second kind. The first main idea of this work is to extend and improve the results of previous works via two Kantorovich methods for solving an integral equation arising from a problem in mathematical biology. The second one is to develop new Nyström method to solve a Fredholm integral equation of the second kind. In the first Kantorovich method we use a general grid projections. In the second one we exploit the shifted Legendre polynomials in our approach. The convergence analysis and new results are presented in this work. Let us consider the following integral equation of the second kind where k(·, ·) is a Fredholm kernel, and g is a known function. Equation (2.1) reads as Define the integral operator T : , k > 0 almost everywhere. We recall that for each f ∈ H, T is compact from H into itself, ( see [4]). Hence, the integral equation (2.1) has a unique solution x ∈ H. Let I denote the identity operator on H. Eq. (2.2) can be rewritten in operator form as follows: The purpose of this work is to approximate x through the solution x n of the Kantorovich equation

Projection approximations using general grids
Let (s n,j ) n j=0 be a grid on [0, 1] such that 0 < s n,0 < s n,1 < . . . < s n,n < 1. Let us consider (π n ) n≥1 , a sequence of bounded projections each one of finite rank, such that π n x := n j=1 x, e n,j e n,j , Define the modulus of continuity of the function ψ ∈ H relative to h n as follows: All functions are extended by 0 outside [0, 1]. We recall that and that, for all ψ ∈ H (cf. [2]),

First Kantorovich method via general grids
We have π n T x := n j=1 T x, e n,j e n,j .
Applying T to both sides of equation (

Convergence analysis of the first Kantorovich method
For all x ∈ H, lim n→∞ π n T x − T x = 0, and since T is compact, lim n→∞ (π n T − T ) T = 0, lim n→∞ (π n T − T ) π n T = 0.
Theorem 2.1. There exists a positive constant M , such that Proof. In fact π n x = π n T x + π n f.
and since T is compact, the M := sup is finite. Using (2.4), we get the desired result.

Second Kantorovich method via shifted Legendre polynomials
The aim of this section is to use Kantorovich method for solving (2.1) via shifted Legendre polynomials. For this purpose, let (L n ) n≥0 denote the sequence of Legendre polynomials. The Shifted Legendre Polynomial L n (s) is defined as L n (s) := L n (2s − 1).
Let us consider e n,j := 2j + 1 L j , the corresponding normalized sequence. Let (π n ) n≥0 be the sequence of bounded finite rank orthogonal projections defined by x, e n,j e n,j .
We recall that (cf. [3]) there exists C > 0 such that, for all y ∈ H r ([0, 1], C), It follows from (2.3) that Once the above system is solved, x n is recovered as Proof. We have and hence x n − x ≤ M C 0 n −r T x r , for some positive constant C 0 , so that x n − x x r ≤ α T n −r , α := M C 0 . Theorem 2.3. Assume that f ∈ H r ([0, 1], C) for some r > 0. Then, there exists β > 0 such that Proof. Recall that π n x = π n T x + π n f.
In this section, we introduce a new Nyström method for solving the integral equation (