Numerical solution of multi-term fractional differential equations

• Published in: Zeitschrift für angewandte Mathematik und Mechanik 2009-07, Vol.89 (7), p.593-608
• Main Author: Katsikadelis, J.T.
• Format: Article
• Language: English
• Subjects: analog equation method; Exact sciences and technology; Fractional differential equations; Mathematical analysis; Mathematics; multi-term equations; Numerical analysis; Numerical analysis. Scientific computation; numerical solution; Ordinary differential equations; Sciences and techniques of general use; systems of multi-term equations
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.200900252

A numerical solution method is presented for linear multi‐term fractional differential equations (FDEs) with variable coefficients. The presented method is based on the concept of the analog equation, which converts the multi‐term FDE into a single term FDE with a fictitious source (right hand inhomogeneous term). The fictitious source is established from the integral representation of the solution of the substitute single term equation. The constructed algorithms are stable. Their accuracy depends only on the truncation and round‐off errors, which, however, negligibly influence the accuracy. The method is demonstrated by the one‐ and two‐term FDEs. The developed algorithms apply also to systems of multi‐term FDEs. Several examples are presented, which demonstrate the efficiency and the accuracy of the proposed method. The method is straightforward extended to nonlinear FDEs. A numerical solution method is presented for linear multi‐term fractional differential equations (FDEs) with variable coefficients. The presented method is based on the concept of the analog equation, which converts the multi‐term FDE into a single term FDE with a fictitious source (right hand inhomogeneous term). The fictitious source is established from the integral representation of the solution of the substitute single term equation. The constructed algorithms are stable. Their accuracy depends only on the truncation and round‐off errors, which, however, negligibly influence the accuracy. The method is demonstrated by the one‐ and two‐term FDEs. The developed algorithms apply also to systems of multi‐term FDEs. Several examples are presented, which demonstrate the efficiency and the accuracy of the proposed method. The method is straightforward extended to nonlinear FDEs.