Computational Modeling of High-Order Differential Equations Using Radial Basis Functions and Numerical Optimization

الملخص

This paper presents a meshless computational framework for approximating solutions of ordinary differential equations, with a formulation suitable for extension to higher-order problems. The framework is based on the Radial Basis Functions and optimization techniques. The approach uses radial basis functions (RBFs) as a meshless approximation scheme to obtain accurate numerical solutions to initial and boundary value problems in a grid-free manner. In addition to shortening convergence behavior and enhancing computational performance, the L-BFGS scheme is introduced into the framework to derive stable numerical approximations to the governing differential equations. A structured meshless formulation for approximating ordinary differential equations is provided by the present framework which lays basis for its extension to higher-order problems. According to numerical experiments, the method accurately approximates solutions in the computational domain and provides stable extrapolations at the boundary. The numerical results give the first validation of the framework on smooth benchmark problems with known exact solutions. Furthermore, the formulation reduces dependence on predefined discretization strategies, supports flexible placement of collocation points, and enables analytical derivative evaluation, making it suitable for smooth benchmark problems and future extensions to more complex nonlinear and multidimensional differential systems in engineering.