用于求解初值问题的自适应 IQ 与 IMQ-RBF：Adam-Bashforth 和 Adam-Moulton 方法
Adaptive IQ and IMQ-RBFs for solving Initial Value Problems: Adam-Bashforth and Adam-Moulton methods
AI总结：
提出基于自适应 IQ 与 IMQ 径向基函数插值的改进型 Adam-Bashforth 和 Adam-Moulton 方法，通过自由参数消除局部截断误差，提升精度、收敛阶与稳定性。
摘要：
本文的主要目标是利用自适应逆二次（IQ）和逆多二次（IMQ）径向基函数（RBF）插值技术，开发增强型 Adam-Bashforth 与 Adam-Moulton 方法。通过利用径向基函数中包含的一个自由参数，使局部截断误差消失，从而提升数值解的局部收敛性。文中给出了一致性和稳定性分析，以及若干数值结果以支持我们的论断。通过消除局部截断误差，所提出的每种技术在精度和收敛速率上均等同于或优于原始 Adam-Bashforth 与 Adam-Mowlton 方法，因此这些自适应方法是最优的。我们得出结论：IQ 与 IMQ-RBF 方法都比经典方法产生更高的收敛阶，而其中某一方法的优越性取决于所采用的方法和所考虑的问题。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
