Construction and comparative study of Euler method with adaptive IQ and IMQ-RBFs

TL;DR

This paper develops enhanced Euler methods using adaptive inverse-quadratic and inverse-multi-quadratic RBFs to improve local convergence and accuracy in solving initial value problems, supported by theoretical analysis and numerical results.

Contribution

It introduces adaptive RBF-based modifications to the Euler method, improving convergence and stability for initial value problem solutions.

Findings

Enhanced methods match or surpass original Euler accuracy.

Adaptive RBFs improve local convergence and stability.

Numerical results confirm optimality of proposed methods.

Abstract

The fundamental purpose of the present work is to constitute an enhanced Euler method with adaptive inverse-quadratic and inverse-multi-quadratic radial basis function (RBF) interpolation technique to solve initial value problems. These enhanced methods improve the local convergence of numerical solutions by utilizing a free parameter of radial basis functions. Consistency, convergence, and stability analysis are provided to support our claims for each method. Numerical results show that the accuracy and rate of convergence of each proposed method are the same as the original Euler method or improved by making the local truncation error vanish; thus, the adaptive methods are optimal.