Radial Basis Function Collocation With L-BFGS Optimization for Higher Order Initial Value Problems

This paper describes a radial basis function collocation method that uses limited-memory Broyden–Fletcher–Goldfarb–Shanno optimization to solve one-dimensional ordinary differential initial value problems. Numerical tests include first-order problems and manufactured fourth-, sixth-, and eighth-order problems. The method constructs the trial solution from a polynomial part that satisfies the initial data exactly, with the remaining component expressed as a weighted Gaussian RBF expansion. The centers are located on the computational interval, and the selection of shape parameters happens prior to the optimization of coefficients. Therefore, the only trainable degrees of freedom are those represented by RBF coefficients. This formulation in the coefficient space is analytically differentiable, enables high-order derivatives to be computed directly, and transforms the finite-dimensional solution procedure into a simple residual minimization problem with explicit control over conditioning. The convergence interpretation is deliberately restricted to the applicable quasi-Newton setting: L-BFGS is used as a practical curvature-based accelerator for the smooth coefficient-space loss, whereas no global or unconditional superlinear-convergence claim is made for nonlinear RBF parameterizations. Five benchmark problems are investigated: a first-order linear IVP, a fourth-order polynomial IVP, a sixth-order exponential manufactured IVP, an eighth-order trigonometric manufactured IVP, and a nonlinear Van der Pol oscillator. The additional sixth- and eighth-order tests support the use of the weighted RBF trial form beyond fourth order, whereas the conditioning data confirm that accuracy must be interpreted together with shape-parameter selection and regularization. The results indicate that RBF-L-BFGS collocation is a flexible meshless strategy for smooth left-end IVPs when degree-of-freedom accounting, residual minimization, and numerical conditioning are treated explicitly.