Variational formulation of problems involving fractional order differential operators

Abstract

Abstract. In this work, we consider boundary value problems involving Caputo and Riemann-Liouville fractional derivatives of order α ∈ (1, 2) on the unit interval (0, 1). These fractional derivatives lead to non-symmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space H α/2 0 (0, 1) but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. Finally, numerical results are presented to illustrate the error estimates.

An efficient collocation method for a Caputo two-point boundary value problem

“…Fractional-order derivatives have recently risen to prominence in the modelling of various processes; see [7,9] for several applications. The mathematical analysis of problems involving these derivatives has also attracted much attention-a survey of recent activity is given in [8].…”

“…The problem (1.5) models superdiffusion of particle motion when convection is present; see the discussion and references in [7,Section 1]. It is a member of the general class of boundary value problems that is analysed in [10].…”

Boundary Layers in a Two-Point Boundary Value Problem with a Caputo Fractional Derivative

“…Boundary value problems whose di erential operators involve fractional derivatives are of great interest, as these non-classical derivatives can model some physical processes where integer-order derivatives are unsuitable; see [6,8] for an extensive list of recent applications and mathematical developments in this area. Thus the precise behaviour of solutions to fractional-derivative boundary value problems is of fundamental importance.…”

“…The problem (1.2) models superdi usion of particle motion when convection is present; see the discussion and references in [6,Section 1]. It is a member of the general class of boundary value problems that is analysed in [10].…”

“…It is well known that, when computing numerical solution of problems with integer-order di erential operators, such layers can cause a deterioration in accuracy [13]. This is also the case for the fractional-derivative problem (1.2): see [5,6,15], where computed solutions of (1.2) become less accurate when is near 1. This loss of accuracy appears in only some numerical examples in these papers, and no explanation is given there, but it is in fact con ned to problems whose solutions exhibit a boundary layer at = 1.…”