Efficient Numerical Solutions of Tenth Order Differential Equations using Vieta-Fibonacci Wavelet and Reproducing Kernel Hilbert Space Methods

Abstract

This study presents two numerical methods for solving tenth-order differential equations: the Vieta-Fibonacci wavelet method (VFWM) and the reproducing kernel Hilbert space method (RKHSM). The VFWM approximates the unknown function using Vieta-Fibonacci wavelets, transforming differential equations into algebraic ones and solving them via the collocation method. A key contribution is the derivation of the operational matrix of derivatives for Vieta-Fibonacci wavelets, enhancing computational efficiency without sacrificing accuracy. The RKHSM generates approximate and analytical solutions in series form, effectively addressing nonlinear problems. Both methods are evaluated for convergence, accuracy, and computational efficiency. Applications to three test problems demonstrate that VFWM excels in handling higher-order derivatives and boundary conditions, while RKHSM offers flexibility for a range of nonlinear issues. These methods are reliable, precise, and efficient, with potential applications in fields such as fluid dynamics and astrophysics. The study concludes with suggestions for future extensions, including fractional-order differential equations and advanced models. (c) 2026 L&H Scientific Publishing, LLC. All rights reserved.