Variable-order space-fractional diffusion equations and a variable-order modification of constant-order fractional problems

ABSTRACT

Fractional diffusion equations (FDEs) were shown to provide very competitive descriptions of challenging phenomena of anomalously diffusive transport or long-range interactions. However, FDEs introduce mathematical issues that are not common in the context of integer-order diffusion equations. For instance, the homogeneous Dirichlet boundary-value problems of linear elliptic FDEs with smooth data in one space dimension may generate solutions with singularities that do not seem physically relevant, which are in sharp contrast to their integer-order analogues do. We prove the wellposedness of the Dirichlet boundary-value problem of one dimensional variable-order linear space-fractional diffusion equations (sFDEs). We further prove that their solutions have the similar regularities as their integer-order analogues if the order has an integer limit at the boundary or have the same singularity near the boundary as their constant-order sFDE analogues if the order has a non-integer limit at the boundary. In particular, we prove that constant-order sFDEs with a variable-order modification indeed generate solutions with significantly improved regularities.

KEYWORDS:

• Variable-order space-fractional diffusion equation
• variable-order modification of constant-order problem
• two-sided
• wellposedness
• regularity

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J40
• 35A01
• 35A02
• 35A20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors would like to express their most sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper. This work was funded by the Army Research Office (AFO) MURI Grant W911NF-15-1-0562, by the National Science Foundation under Grant DMS-1620194, and by a SPARC Graduate Research Grant from the Office of the Vice President for Research at the University of South Carolina.