ENHANCING SOLUTIONS FOR NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS VIA COMBINED LAPLACE TRANSFORM AND REPRODUCING KERNEL METHOD

Ordinary differential equations (ODEs) describe diverse phenomena in engineering and physics, such as electrical networks, oscillating systems, satellite orbits, and chemical reactions. Solving these equations, particularly the non-linear ones, is often challenging due to their complexity. This study aims to innovate by integrating the Laplace transform with the reproducing kernel Hilbert space method (RKHSM), introducing an enhanced approach that surpasses classical RKHSM. The combined Laplace-RKHSM method simplifies the original non-linear ODEs, allowing for the construction of novel numerical solutions. These solutions are systematically obtained in series form, providing both analytic and approximate results. The effectiveness and efficacy of the Laplace-RKHSM are demonstrated through three applications, each showcasing the method's superior performance in terms of accuracy and computational efficiency. This new approach not only enhances the existing RKHSM but also broadens its applicability to a wider range of non-linear problems in physics and engineering.