Fractional conformable differential time-delayed problems: Theoretical analysis and numerical implementation using reproducing Hilbert's approximate scheme

The intention of this analysis is to declare and demonstrate the complete foundational theorem of existence and uniqueness for fractional conformable initial value time-delayed models incorporating time delays. The utilized proof is based on Picard's iterative method and the fixed point theorem. Thereafter, to showcase the outcomes of our theoretical research and as an application for numerical solutions, we propose an effective numerical method based on the reproducing kernel Hilbert approximation to solve some fractional time-delayed problems. Afterward, the construction of appropriate Hilbert spaces, derivation of the kernel functions, representation of the unique delayed solution and algorithm solution steps are also discussed and utilized. Indeed, the series solution, the convergence, and the error analysis outcomes are debated and discussed. For comparative analysis, we describe our gained results in the constructed tables and figures which help us to compare the numerical solution with the exact one. Concluding notes, keynotes and future are listed in the closing portion.