Adapted Newton-Kantorovich Methods for Nonlinear Integral Equations

Corresponding Author: Mostefa Nadir Department of Mathematics, University of Msila, 28000, Algeria Email: mostefanadir@yahoo.fr Abstract: For this work, the main idea is to make an adapted modification to the Newton-Kantorovich method destined to solve a nonlinear integral equations, so that by this technical method we obtain a simple application to this solution. Moreover, we compare the numerical results obtained by this method against ones obtained by another authors. This comparison showed the efficiency of this method.


Introduction
Nonlinear integral equation is an important branch in contemporary mathematics and arises many applied areas which include engineering problems, such as mechanics, physics, astronomy, biology, economics, potential theory and electrostatics (Golberg, 1990;Nadir and Gagui, 2014). Those equations are classified into Fredholm and Volterra equations following the upper bound of the region for the integral part is constant or variable. Many different methods are used to obtain the numerical solution of the nonlinear integral equations (Nadir and Rahmoune, 2007;Polyanin and Manzhirov, 2008): For the resolution of the functional equation P(ϕ) = 0 where P is Fréchet differentiable on a convex set of a Banach space C(Ω), Kantorovich imitates the Newton method for the equation of the tangent line given by the first two terms of Taylor's formula, written as the method of successive approximation: The explicit form to the Equation 3 is given as: In the Newton-Kantorovich method, we remark that, the kernels k(s, t, ϕ n (t)) and k ϕ (s, t, ϕ n (t)) of the righthand side of the Equation 4 are replaced by the ones k(s, t, ϕ 0 (t)) and k ϕ (s, t, ϕ 0 (t)) where ϕ 0 represents the initial value so that, the Equation 4 becomes a linear integral equation. However, in our work we treat the Equation 4 by adapted a modification, where we replace the expression (ϕ n+1 (t)-ϕ n (t)) in the right-hand side by the one (ϕ n (t)-ϕ n-1 (t)) so that, the Equation 4 becomes: The first approximation ϕ 1 (s) is obtained by substituting the initial approximation ϕ 0 (s) = f(s) into the right hand side of the integral equation, giving:

Convergence and Applications Theorem
Let P be an operator defined on a Banach space E into a Banach space F and Fréchet differentiable for ϕ∈Ω an open convex set in E, satisfies the following conditions: With the constants L,M and N satisfying LM < 1, such that, the successive approximations: Are defined for all n, ϕ n ∈ Ω 1 , n = 1, 2, ... and converge to the exact solution ϕ ∈ Ω 1 which satisfies Indeed, it is easy to see that: Using condition (A1), on Ω we obtain: Also, for ϕ ∈ Ω 1 , we get: Then, the relation (7) shows that P′(ϕ) is invertible for all ϕ ∈ Ω 1 and it comes: Or still: Given the Newton function as: With ϕ and N(ϕ) in Ω 1 and ϕ n+1 = N(ϕ n ), we get: On the other hand From the relations (9) and (10) From the relation ||ϕ q -ϕ p ||≤||ϕ q -ϕ q-1 ||+||ϕ q-1 -ϕ q-2 ||+...+||ϕ p+1 -ϕ p || it comes the sequence ϕ n is Cauchy sequence in Banach space. Thus this sequence ϕ n represents the Newton iterations are defined and converges to the solution ϕ in Ω 1 (Wouk, 1979):

Illustrating Examples
Example 1 Consider the nonlinear integral equation of Volterra:

Example 4
Consider the nonlinear integral equation of Fredholm: where, the function f(t 0 ) is chosen so that the exact solution is given by: The approximate solution ( ) t ϕ ɶ of ϕ(t) is obtained by the adapted Newton-Kantorovich method.

Example 5
Consider the nonlinear integral equation of Fredholm: where, the function f(t 0 ) is chosen so that the exact solution is given by: The approximate solution e'(t) of '(t) is obtained by the adapted Newton-Kantorovich method.

A numerical method for solving nonlinear Volterra and Fredholm integral equations, based on an adapted
Newton-Kontorovich methods is presented. The efficiency of this method is tested by solving some examples for which the exact solution is known. This allows us to estimate the exactness with our numerical results and compare those with another results. For nonlinear volterra integral equations our method is compared with the ones, the Haar wavelets and collocation, the fixed point technique with cubic Bspline scaling function, Adomian decomposition method and block pulse functions by collocation method. treated by (Babolian and Shahsavaran, 2007) Table 1, (Maleknejad et al., 2013) Table 2, (Awawdeh et al., 2009) Table 3 and (Shahsavaran, 2011) Table 4 respectively. On the other hand for nonlinear Fredholm integral equations our method is compared with the ones, the Haar wavelet method, the Urysohn form by Newton-Kantorovich-quadrature method and A numerical method treated by ((Lepik and Tamme, 2007)) Table 5, (Saberi-Nadja and Heidari, 2010) Table 6 and (Awawdeh et al., 2009) Table 7 respect1vely. Table 1. We present the exact and the approximate solutions of the equation in the example 1 in some arbitrary points, the error for N = 10 is compared with the ones treated in (Babolian and Shahsavaran, 2007) Error (Babolian and Values of t