Symmetry (Jul 2021)

Numerical Solution of Two-Dimensional Fredholm–Volterra Integral Equations of the Second Kind

  • Sanda Micula

DOI
https://doi.org/10.3390/sym13081326
Journal volume & issue
Vol. 13, no. 8
p. 1326

Abstract

Read online

The paper presents an iterative numerical method for approximating solutions of two-dimensional Fredholm–Volterra integral equations of the second kind. As these equations arise in many applications, there is a constant need for accurate, but fast and simple to use numerical approximations to their solutions. The method proposed here uses successive approximations of the Mann type and a suitable cubature formula. Mann’s procedure is known to converge faster than the classical Picard iteration given by the contraction principle, thus yielding a better numerical method. The existence and uniqueness of the solution is derived under certain conditions. The convergence of the method is proved, and error estimates for the approximations obtained are given. At the end, several numerical examples are analyzed, showing the applicability of the proposed method and good approximation results. In the last section, concluding remarks and future research ideas are discussed.

Keywords