Completely monotone functions and some classes of fractional evolution equations

Abstract

Abstract. The abstract Cauchy problem for the distributed order fractional evolution equation in the Caputo and in the Riemann-Liouville sense is studied for operators generating a strongly continuous one-parameter semigroup on a Banach space. Continuous as well as discrete distribution of fractional timederivatives of order less than one are considered. The problem is reformulated as an abstract Volterra integral equation. It is proven that its kernel satisfy certain complete monotonicity properties. Based on these properties, the wellposedness of the problem is established and a series expansion of the solution is obtained. In case of ordered Banach space this representation implies positivity of the solution operator. In addition, a subordination formula is obtained.

Fractional diffusion equation with distributed-order Caputo derivative

“…In our case the distributed order fractional derivative is defined as a weighted fractional Caputo derivative, where the weight µ is supposed to be any nonnegative nontrivial function from L 1 (0, 1). This kind of problems were studied in many papers (see [1], [2], [4]- [6], [9], [10], [12], [13] ), however the authors usually impose stronger assumption on µ and consider time-independent elliptic operators. These assumptions make the analysis easier, because one can apply the Laplace transform and solution can be defined by means of Fourier series.…”

Subordination principles for the multi-dimensional space-time-fractional diffusion-wave equation

“…It is important to mention that this function is non-negative for τ ∈ R + and can be interpreted as a probability density function. Very recently, the subordination principle was extended to the case of the multiterm time-fractional diffusion-wave equations in [4] and to the case of the distributed order time-factional evolution equations in the Caputo and Riemann-Liouville sense in [3].…”

Estimates for a general fractional relaxation equation and application to an inverse source problem

“…Since g(s) ∈ , then g(s)∕s ∈  and e − g(s) ∈ . Therefore, (14) and (15) imply that (s, ) ∈  and̂(s, ) ∈  as a product of 2 completely monotone functions and, hence, (t, ) ≥ 0 and (t, ) ≥ 0 by the Bernstein's theorem. The integral identity in (12) for (t, ) can be obtained as a particular case of (10) taking = 0 and noticing that the unique solution in this case is u(t; 0) ≡ 1.…”

“…1 In Luchko and Yamamoto, 6 some uniqueness and existence results, as well as a maximum principle, are established for the initial-boundary-value problem. The particular cases of multiterm and distributed-order fractional diffusion equations are studied extensively in the last years in, eg, other studies [7][8][9][10][11][12][13][14][15] to mention only few of many recent publications.…”