EXISTENCE AND UNIQUENESS THEOREMS FOR THE CAUCHY PROBLEM OF FRACTIONAL DERIVATIVE DIFFUSION EQUATIONS

This study investigates the existence and uniqueness theorems for the Cauchy problem of fractional derivative diffusion equations. Fractional diffusion equations, involving derivatives of non-integer order, provide a generalized framework for modeling anomalous diffusion and memory effects in complex systems. The Cauchy problem is reformulated as an equivalent integral equation using Caputo or Riemann–Liouville derivatives, enabling the application of functional analytic methods such as the Banach fixed-point theorem and semigroup theory. The results guarantee the well-posedness of the problem, ensuring that solutions exist, are unique, and depend continuously on initial data. These findings are fundamental for both analytical and numerical investigations of anomalous diffusion phenomena in physics, engineering, and applied mathematics.