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.

翻译：目前工作的根本目的是形成一种强化的Euler方法,具有适应性的反赤道和反多赤道辐射基功能(RBF)内插技术,以解决初始价值问题,这些强化方法通过使用一个自由的辐射基函数参数,改善了数字解决办法的当地趋同。提供了一致、趋同和稳定性分析,以支持我们对每种方法的索赔。数字结果显示,每种拟议方法的准确性和趋同率与最初的Euler方法相同,或者通过使当地脱轨错误消失而得到改善;因此,适应方法是最佳的。