This book presents a comprehensive investigation into the exponential and Mittag-Leffler Euler differences for semi-linear fractional-order differential equations, a subject falling within the purview of computational mathematics. The field of exponential and Mittag-Leffler Euler differences has witnessed a period of rapid development in recent times. This has led to the emergence of new techniques such as exponential Euler integrator, exponential Runge-Kutta methods, multistep exponential integrator, exponential Rosenbrock-type methods, and more. This book puts forth several difference approaches to fractional-order differential equations and offers insights into their practical applications in the study of neural networks. Adopting a holistic approach, the book presents a foundational framework for this topic, underscoring the significance of exponential and Mittag-Leffler Euler differences in the numerical theory of fractional-order differential equations.

The book is designed for graduate students with an interest in numerical solutions of fractional-order differential equations, as well as for researchers engaged in the qualitative theory of difference equations.