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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-Moultonmethods. By utilizing a free parameter involved in the radial basis function,the local convergence of the numerical solution is enhanced by making the localtruncation error vanish. Consistency and stability analysis is presented alongwith some numerical results to back up our assertions. The accuracy and rate ofconvergence of each proposed technique are equal to or better than the originalAdam-Bashforth and Adam-Moulton methods by eliminating the local truncationerror thus, the proposed adaptive methods are optimal. We conclude that both IQand IMQ-RBF methods yield an improved order of convergence than classicalmethods, while the superiority of one method depends on the method and theproblem considered.

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