Generalized Adams Product Quadrature Rules for Fractional Differential Equations

An initial value problem for a Fractional Differential Equation (FDE) can be reformulated as a Volterra integral equation of the second kind with weakly singular kernel. Its solution can be approximated by applying suitable convolution quadratures. Methods of this type currently available in the literature are, for example, Adams product quadrature rules or Fractional Linear Multistep Methods, [2, 3]. It is known that the order of an A-stable convolution quadrature cannot exceed two, [3]. Clearly, this result represents an extension of the second Dahlquist barrier for linear multistep methods (LMMs) for ordinary differential equations. This barrier has been overcomed by using LMMs as Boundary Value Methods (BVMs), namely by completing the discrete problem generated by a LMM with suitable boundary conditions, [1]. By virtue of this result, we have investigated if the BVM approach is successful in overcoming the barrier established in [3]. This led us to derive a generalized version of implicit Adams product quadrature rules called Fractional Generalized Adams Methods (FGAMs). We will present these new schemes and discuss their accuracy and stability properties. In particular, we will show that they are always A-stable in a generalized sense. Finally, the results of a numerical experiment confirming the valuable properties of this approach will be reported.