EXISTENCE, REGULARITY AND REPRESENTATION OF SOLUTIONS OF TIME FRACTIONAL DIFFUSION EQUATIONS

Using regularized resolvent families, we investigate the solvability of the fractional order inhomogeneous Cauchy problem D(t)(alpha)u(t) = Au(t) + f(t), t > 0, 0 < alpha <= 1, where D-t(alpha) is the Caputo fractional derivative of order alpha, A a closed linear operator on some Banach space X, f : [0, infinity) -> X is a given function. We define an operator family associated with this problem and study its regularity properties. When A is the generator of a beta-times integrated semigroup (T-beta(t)) on a Banach space X, explicit representations of mild and classical solutions of the above problem in terms of the integrated semigroup are derived. The results are applied to the fractional diffusion equation with non-homogeneous, Dirichlet, Neumann and Robin boundary conditions and to the time fractional order Schrodinger equation D(t)(alpha)u(t, x) = e(i theta) Delta(p)u(t, x) + f(t,x), t > 0, x is an element of R-N where pi/2 <= theta < (1 - alpha/2)pi and Delta(p) is a realization of the Laplace operator on L-p(R-N), 1 <= p < infinity.