Formal consistency versus actual convergence rates of difference schemes for fractional-derivative boundary value problems

Gracia Lozano, José Luis (Universidad de Zaragoza) ; Stynes, Martin

Resumen: Finite difference methods for approximating fractional derivatives are often analyzed by determining their order of consistency when applied to smooth functions, but the relationship between this measure and their actual numerical performance is unclear. Thus in this paper several wellknown difference schemes are tested numerically on simple Riemann-Liouville and Caputo boundary value problems posed on the interval [0, 1] to determine their orders of convergence (in the discrete maximum norm) in two unexceptional cases: (i) when the solution of the boundary-value problem is a polynomial (ii) when the data of the boundary value problem is smooth. In many cases these tests reveal gaps between a method’s theoretical order of consistency and its actual order of convergence. In particular, numerical results show that the popular shifted Gr¨unwald-Letnikov scheme fails to converge for a Riemann-Liouville example with a polynomial solution p(x), and a rigorous proof is given that this scheme (and some other schemes) cannot yield a convergent solution when p(0)¿ 0.

Idioma: Inglés

DOI: 10.1515/fca-2015-0027

Año: 2015

Publicado en: Fractional Calculus and Applied Analysis 18 (2015), 419-436
ISSN: 1311-0454

Factor impacto JCR: 2.246 (2015)

Categ. JCR: MATHEMATICS rank: 10 / 312 = 0.032 (2015) - Q1 - T1
Categ. JCR: MATHEMATICS, INTERDISCIPLINARY APPLICATIONS rank: 15 / 101 = 0.149 (2015) - Q1 - T1
Categ. JCR: MATHEMATICS, APPLIED rank: 9 / 254 = 0.035 (2015) - Q1 - T1

Factor impacto SCIMAGO: 1.551 - Applied Mathematics (Q1) - Analysis (Q1)

Financiación: info:eu-repo/grantAgreement/ES/MEC/MTM2010-16917

Tipo y forma: Artículo (Versión definitiva)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)