Adapted Newton-Kantorovich Methods for Nonlinear Integral Equations

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.

On the Existence of Solutions of Nonlinear Fredholm Integral Equations from Kantorovich’s Technique

“…As x 0 (s) = sin(πs) is a reasonable choice as a starting point for Newton's method, as we can see in [12][13][14], the last inequality holds, since 1 0 sin(πt) cos(πt)x 0 (t) 2 dt = 0, and condition (6) is omitted.…”

“…with s ∈ [0, 1], that has been used by other authors as a numerical test [13,21]. Observe that, in this case, kernel K(s, t) = cos(πs) sin(πt) is separable.…”

“…In Table 2, we show the real errors for n = 10 and n = 20 when the adapted Newton's method is used in [13] to solve (10) and some points of the interval involved are chosen. In Table 3, we show the real errors when a combination of Newton's method and quadrature methods [14] and an iterative scheme based on the homotopy analysis method [12] are applied.…”

“…Table 2. Real errors for n = 10 and n = 20 when the adapted Newton's method given in [13] is applied. Table 3.…”

“…In particular, by means of the aforementioned method, we can construct a continuous mapping of an initial guess approximation to the exact solution of the equation to be solved. In [13], the authors present an adapted modification to the Newton-Kantorovich method. Finally, in [14], the Newton-Kantorovich method and quadrature methods are combined to develop a new method for solving Equation (1).…”