On the Decomposition of Solutions: From Fractional Diffusion to Fractional Laplacian

Abstract

This paper investigates the structure of solutions to the BVP of a class of fractional ordinary differential equations involving both fractional derivatives (R-L or Caputo) and fractional Laplacian with variable coefficients. This family of equations generalize the usual fractional diffusion equation and fractional Laplace equation. We provide a deep insight to the structure of the solutions shared by this family of equations. The specific decomposition of the solution is obtained, which consists of the “good” part and the “bad” part that precisely control the regularity and singularity, respectively. Other associated properties of the solution will be characterized as well.

Analysis of one-sided 1-D fractional diffusion operator

“…Later on, various aspects of boundary value problems involving FDARE and its variants have been extensively studied both theoretically and numerically. To name a few, see for example, [27,29,7,8,18,19,28,26].…”

“…In pioneer work [5], via defining intermediate functional spaces in terms of R-L derivatives that are essentially equivalent to the fractional Sobolev spaces, the variational formulation of (1.1) with zero boundary conditions was constructed and the weak solution was successfully established, which suggests the analogy to the classic integer-order elliptic equations. Later on in sequel work [12,18,19,15,8], the regularity of weak solution was thoroughly investigated. In particular, the decomposition of the solution was discovered in [18], consisting of a "smoother" part that is trouble-free and a "less-smooth" part that limits the regularity of the whole solution.…”

“…Later on in sequel work [12,18,19,15,8], the regularity of weak solution was thoroughly investigated. In particular, the decomposition of the solution was discovered in [18], consisting of a "smoother" part that is trouble-free and a "less-smooth" part that limits the regularity of the whole solution. This structure information reveals the fact that the solution u dramatically lacks regularity regardless of the smooth data k(x), p(x), q(x), f (x) in (1.1), which is, on the other hand, in sharp contrast to the counterpart of integer-order elliptic equations.…”