  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.
