Computational Mathematics and Computer Modeling with Applications (CMCMA)

Newton-Krylov generalized minimal residual algorithm in solving the nonlinear two-dimensional integral equations of the second kind on non-rectangular domains with an error estimate

https://doi.org/10.52547/CMCMA.1.2.35
Volume 1, Issue 2 - Serial Number 2
December 2022
Pages 35-45
Computational Mathematics and Computer Modeling with Applications (CMCMA)

Document Type : Regular paper

Authors

Hafez Yari 1
Mehdi Delkhosh 2

1 Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran.

2 Department of Mathematics and Computer Science, Bardaskan Branch, Islamic Azad University, Bardaskan, Iran.

Abstract
In this paper, an applicable numerical approximation has been proposed for solving nonlinear two-dimensional integral equations (2DIEs) of the second kind on non-rectangular domains. Because directly applying the collocation methods on non-rectangular domains is difficult, in this work, at first, the integral equation is converted to an equal integral equation on a rectangular domain, then the solution is approximated by applying 2D Jacobi collocation method, the implementation of these instructions reduces the integral equation to a system of nonlinear algebraic equations, therefore, solving this system has an important role to approximate the solution. In this paper, Newton-Krylov generalized minimal residual (NK-GMRes) algorithm is used for solving the system of nonlinear algebraic equations. Furthermore, an error estimate for the presented method is investigated and several examples confirm the accuracy and efficiency of the proposed instructions.

Keywords

Non-rectangular domains
2D integral equations
Jacobi polynomials
Collocation method
Newton-Krylov GMRes algorithm
  • Receive Date 31 October 2022
  • Revise Date 08 December 2022
  • Accept Date 29 December 2022

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