Wellposedness of Neumann boundary-value problems of space-fractional differential equations

TL;DR

This paper analyzes the wellposedness of Neumann boundary-value problems for space-fractional differential equations, revealing that some combinations are well posed while others are ill posed, highlighting the complexity of FDE boundary conditions.

Contribution

It provides a comprehensive mathematical analysis of different Neumann boundary conditions for FDEs, identifying which are well posed and which are not.

Findings

Five out of nine FDE and boundary condition combinations are well posed.

Some boundary conditions lead to ill-posed problems.

The study demonstrates the subtlety in formulating boundary conditions for FDEs.

Abstract

Fractional differential equation (FDE) provides an accurate description of transport processes that exhibit anomalous diffusion but introduces new mathematical difficulties that have not been encountered in the context of integer-order differential equation. For example, the wellposedness of the Dirichlet boundary-value problem of one-dimensional variable-coefficient FDE is not fully resolved yet. In addition, Neumann boundary-value problem of FDE poses significant challenges, partly due to the fact that different forms of FDE and different types of Neumann boundary condition have been proposed in the literature depending on different applications. We conduct preliminary mathematical analysis of the wellposedness of different Neumann boundary-value problems of the FDEs. We prove that five out of the nine combinations of three different forms of FDEs that are closed by three types of Neumann boundary conditions are well posed and the remaining four do not admit a solution. In particular, for each form of the FDE there is at least one type of Neumann boundary condition such that the corresponding boundary-value problem is well posed, but there is also at least one type of Neumann boundary condition such that the corresponding boundary-value problem is ill posed. This fully demonstrates the subtlety of the study of FDE.