Improved Iterative Solution of Linear Fredholm Integral Equations of Second Kind via Inverse-Free Iterative Schemes

Abstract

This work is devoted to Fredholm integral equations of second kind with non-separable kernels. Our strategy is to approximate the non-separable kernel by using an adequate Taylor’s development. Then, we adapt an already known technique used for separable kernels to our case. First, we study the local convergence of the proposed iterative scheme, so we obtain a ball of starting points around the solution. Then, we complete the theoretical study with the semilocal convergence analysis, that allow us to obtain the domain of existence for the solution in terms of the starting point. In this case, the existence of a solution is deduced. Finally, we illustrate this study with some numerical experiments.

Analysis of a discrete-time Markov process with a bounded continuous state space by the Fredholm integral equation of the second kind

“…A special use of the Fredholm equation is the creation of photorealistic images in computer graphics, in which the Fredholm equation is used to represent light transport from virtual light sources to the image plane. Due to the fact that a vast class of initial and boundary value problems can be transformed into Volterra or Fredholm integral equations, many scientific domains use Fredholm integral equations, including engineering, applied mathematics, and mathematical physics (Gutiérrez,[10]). Other literature regarding the Fredholm integral equations focuses on efficient numerical solution techniques for the Fredholm integral equations (ZhiMin et al [19], Doucet et al [4], Tian [17], Mohammad [14]).…”