Too Much Regularity May Force Too Much Uniqueness
Update: 2017-03-02 14:38:17      Author: yangjuan@csrc.ac.cn

The numerical solution of fractional-order differential equations has attracted much attention in the last 10 years; the MathSciNet database records 432 papers in this category since 2007. But there is an anomaly in almost all of this work: although it is well known that for smooth data, typical solutions of fractional-derivative initial-value and boundary value problems exhibit weak singularities at one or more boundaries of the domain, nevertheless the vast majority of published papers assume that the solutions that they approximate are very smooth globally — which of course simplifies greatly their numerical analysis. This assumption of smoothness has been “sold” as putting only a mild restriction on the problem and thereby justifying this simplification in the analysis. But now, in a new paper, we show that the smoothness assumption greatly restricts the class of problems studied and is therefore unacceptable.

The type of problem that we consider in our paper is the most popular one studied in the literature: time-dependent initial-boundary value problems on a bounded domain in one space variable, where the time derivative is a Caputo fractional derivative of order α, with 0 < α < 1 (fractional parabolic type) or 1 < α < 2 (fractional hyperbolic type). For smooth and compatible data, typical solutions u(x, t) to such problems are continuous on the closed space-time domain but are weakly singular:  when 0 < α < 1 the derivative ∂u/∂t blows up at the initial time t = 0, and when 1 < α < 2 the derivative 2u/∂t2 blows up at t = 0.

A typical solution for a fractional parabolic problem is shown in the Figure: it is continuous on the closed domain, smooth in the space variable, but its derivative ∂u/∂t becomes infinite at the initial time t =  0.

The main result in our paper is that, if one assumes even slightly more regularity than is generally true of typical solutions, then the initial condition is determined uniquely by the other data of the problem — which is completely unnatural. For example, if one considers the fractional heat equation with zero boundary data and an arbitrary smooth initial condition that vanishes at the two corners of the space-time domain, then the assumption that ∂u/∂t is continuous on the closed domain will imply that the solution of the problem must be the trivial function u(x, t) ≡   0.

Reference.  M.Stynes, Too  much regularity may force too much uniqueness, Fract.    Calc.

Appl. Anal. (to  appear).