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A Discontinuous Galerkin Method for Fractional Differential Equations
Author(s) -
Franz Sebastian,
Rubisch Johanna
Publication year - 2016
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201610369
Subject(s) - discontinuous galerkin method , mathematics , convergence (economics) , galerkin method , fractional calculus , dimension (graph theory) , mathematical analysis , order (exchange) , partial differential equation , work (physics) , finite element method , pure mathematics , physics , finance , economics , thermodynamics , economic growth
We investigate the numerical treatment of fractional derivatives using a discontinuous Galerkin approach. We consider fractional differential equations in one dimension. The most common definitions, the Riemann‐Liouville and Caputo derivatives, as well as a third definition are considered. Different types of initial condition statements are investigated for the extended discontinuous Galerkin method. We obtain numerical results on the order of convergence in the L 2 and H 1 norms. We compare the classical discontinuous Galerkin method with our adapted approach and show results for the different types of fractional derivatives. Further work was carried out on parabolic differential equations with fractional time derivatives, which is to be published cf. [2]. A keystone in the further research is the development of analytical tools for theoretical convergence results. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)