Adaptive IQ and IMQ-RBFs for solving Initial Value Problems: Adam-Bashforth and Adam-Moulton methods

TL;DR

This paper introduces adaptive IQ and IMQ radial basis function techniques to improve Adam-Bashforth and Adam-Moulton methods, enhancing convergence and accuracy for solving initial value problems.

Contribution

The paper develops adaptive RBF-based modifications to classical methods, achieving better convergence and error reduction through parameter tuning.

Findings

Enhanced convergence rates demonstrated numerically.

Adaptive methods outperform classical counterparts in accuracy.

Superiority varies depending on the problem and method.

Abstract

In this paper, our objective is primarily to use adaptive inverse-quadratic (IQ) and inverse-multi-quadratic (IMQ) radial basis function (RBF) interpolation techniques to develop an enhanced Adam-Bashforth and Adam-Moulton methods. By utilizing a free parameter involved in the radial basis function, the local convergence of the numerical solution is enhanced by making the local truncation error vanish. Consistency and stability analysis is presented along with some numerical results to back up our assertions. The accuracy and rate of convergence of each proposed technique are equal to or better than the original Adam-Bashforth and Adam-Moulton methods by eliminating the local truncation error thus, the proposed adaptive methods are optimal. We conclude that both IQ and IMQ-RBF methods yield an improved order of convergence than classical methods, while the superiority of one method depends on the method and the problem considered.