Block implicit Adams methods for fractional differential equations

• Published in: Chaos, solitons and fractals Vol. 81; pp. 365 - 377
• Main Authors: Biala, T.A., Jator, S.N.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.2015
• Subjects: Adams methods; Caputo derivatives; Fractional differential equations; Fractional partial differential equations; Initial value problems; 65L06; Fractional differential equations; 26A33; 65L20; Initial value problems; Adams methods; Fractional partial differential equations; Caputo derivatives
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2015.10.007

In this paper, we present a family of Implicit Adams Methods (IAMs) for the numerical approximation of Fractional Initial Value Problems (FIVP) with derivatives of the Caputo type. A continuous representation of the k-step IAM is developed via the interpolation and collocation techniques and adapted to cope with the integration of FIVP. This is achieved by combining the k-step IAM with (k−1)additional methods obtained from the same continuous scheme and applying them as numerical integrators in a block-by-block fashion. We also investigate the stability properties of the block methods and the regions of absolute stability of the methods are plotted in the complex plane. The block methods are tested on numerical examples including large systems resulting from the semi-discretization of one-dimensional fractional heat-like partial differential equations.