Necessary conditions for convergence of difference schemes for fractional-derivative two-point boundary value problems

作者：Stynes Martin; O'Riordan Eugene^*; Gracia Jose Luis

来源：BIT Numerical Mathematics, 2016, 56(4): 1455-1477.

DOI：10.1007/s10543-016-0602-4

• Fractional derivative
• Finite difference scheme
• Necessary condition

A fractional-derivative two-point boundary value problem of the form on (0, 1) with Dirichlet boundary conditions is studied. Here is a Caputo or Riemann-Liouville fractional derivative operator of order . The discretisation of this problem by an arbitrary difference scheme is examined in detail when u or f is a polynomial. For any convergent difference scheme, it is proved rigorously that the entries of the associated matrix must satisfy certain identities. It is shown that some of these identities are not satisfied by certain well-known schemes from the research literature; this clarifies the type of problem to which these schemes can be applied successfully. The effects of the special boundary condition and the special right-hand-side condition are also investigated. This leads, under certain circumstances, to a sharpening of a recently-published finite difference scheme convergence result of two of the authors.

• 出版日期2016-12
• 单位北京计算科学研究中心