Part I: Preliminaries
Continuous Time Random Walk
Fractional Calculus
Mittag-Leffler and Wright Functions
Part II: Fractional Ordinary Differential Equations
Cauchy Problems for Fractional ODEs
Boundary Value Problem for Fractional ODEs
Part III: Time-Fractional Diffusion
Subdiffusion: Hilbert Space Theory
Subdiffusion: Hölder Space Theory
Mathematical Preliminaries
Index

This graduate textbook provides a self-contained introduction to modern mathematical theory on fractional differential equations. It addresses both ordinary and partial differential equations with a focus on detailed solution theory, especially regularity theory under realistic assumptions on the problem data. The text includes an extensive bibliography, application-driven modeling, extensive exercises, and graphic illustrations throughout to complement its comprehensive presentation of the field. It is recommended for graduate students and researchers in applied and computational mathematics, particularly applied analysis, numerical analysis and inverse problems