Fractional Diffusion based on Riemann-Liouville Fractional Derivatives

摘要

A fractional diffusion equation based on Riemann-Liouville fractionalderivatives is solved exactly. The initial values are given as fractionalintegrals. The solution is obtained in terms of $H$-functions. It differs fromthe known solution of fractional diffusion equations based on fractionalintegrals. The solution of fractional diffusion based on a Riemann-Liouvillefractional time derivative does not admit a probabilistic interpretation incontrast with fractional diffusion based on fractional integrals. While thefractional initial value problem is well defined and the solution finite at alltimes its values for $t\to 0$ are divergent.