This study presents a numerical solution for systems of fractional initial value problems. It play a crucial role in various scientific and engineering fields, offering more accurate models for real-world phenomena involving memory and hereditary effects. To address these challenges, the thesis employs the \(L_{3}\) type Caputo method, a powerful numerical technique for solving system of fractional differential equation. Additionally, the fractional finite difference method is utilized for discretizing the system. Lipschitz continuity is the main condition for Picard-Lindelöf’s theorem that guarantees the existence and uniqueness of a system of fractional initial value problems depending on the given discretization. Moreover, we investigate the consistency, stability, and convergence properties of the proposed method. Stability analysis is performed to examine the behavior of the \(L_{3}\) Caputo-type method. The stability properties are analyzed to ensure that the computed solution remains bounded. Consistency is demonstrated by proving that the numerical approximation error converges to zero as the step size approaches zero. Finally, some numerical examples are given to illustrate the applicability and usefulness of the approximate solution of the obtained results.
