ABSTRACT
In the case of the Cauchy problem, the space-time fractional diffusion equation is derived from the standard diffusion equation by replacing the second-order space derivative with a Riemann-Liouville derivative of order β, 0 < β ≤ 2, and the firstorder time derivative with Caputo derivative of orderα, 0 < α ≤ 1. These derivatives are defined as follows:
Riemann-Liouville derivative of order β, 0 < β ≤ 2, for a differentiable function f on the interval [a, b]:
Dβ [f(x)] = 1
Γ(1− β) d
dx
∫ x
f(s) (x− s)β ds, a < x < b, 0 < β ≤ 2.
