Initial-boundary value problem for distributed order time-fractional diffusion equations

Abstract

We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency on initial value and source term. Moreover, under suitable assumption on the source term, we establish that the solution is analytic in time.

Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations

“…For diffusion equations corresponding to the case α = 1 with time independent source terms, several authors investigated the conditional stability (e.g. [5,37,38] [15,16,17,19,23,31] where some inverse coefficient problems and some related results have been considered.…”

On Time-Fractional Diffusion Equations with Space-Dependent Variable Order

“…While constant order and distributed order (DO) fractional diffusion equations have been extensively studied (see e.g. [8,10,16]) by several authors, the reference [7] is, as far as we know, the only mathematical paper dedicated to the theoretical study of VO time-fractional diffusion equations. More specifically, the time asymptotic behavior of the solution to timefractional differential equations in a bounded spatial domain was examined in [16] for CO fractional diffusion equations, and in [11] for DO fractional diffusion equations.…”

Decay of solutions to parabolic-type problem with distributed order Caputo derivative

“…Theorem 1. If µ satisfies (7), then there exists nonnegative g ∈ L 1 loc [0, ∞) such that the operator of fractional integration I (µ) , defined by the formula…”