Formal Consistency Versus Actual Convergence Rates of Difference Schemes for Fractional-Derivative Boundary Value Problems
-
José Luis Gracia
and
Martin Stynes
Abstract
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.
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Articles in the same Issue
- Contents
- Fcaa Related News, Events and Books (Fcaa–Volume 18–2–2015)
- New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian
- Pollutant Reduction of a Turbocharged Diesel Engine Using a Decentralized Mimo Crone Controller
- Experimental Implications of Bochner-Levy-Riesz Diffusion
- Fractional Diffusion on Bounded Domains
- On a System of Fractional Differential Equations with Coupled Integral Boundary Conditions
- A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix
- Solving Fractional Delay Differential Equations: A New Approach
- Formal Consistency Versus Actual Convergence Rates of Difference Schemes for Fractional-Derivative Boundary Value Problems
- Asymptotic Stability Of Dynamic Equations With Two Fractional Terms: Continuous Versus Discrete Case
- Analysis of Natural and Artificial Phenomena Using Signal Processing and Fractional Calculus
- Fractional Approach for Estimating Sap Velocity in Trees
- Fractional Calculus: Quo Vadimus? (Where are we Going?)
Articles in the same Issue
- Contents
- Fcaa Related News, Events and Books (Fcaa–Volume 18–2–2015)
- New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian
- Pollutant Reduction of a Turbocharged Diesel Engine Using a Decentralized Mimo Crone Controller
- Experimental Implications of Bochner-Levy-Riesz Diffusion
- Fractional Diffusion on Bounded Domains
- On a System of Fractional Differential Equations with Coupled Integral Boundary Conditions
- A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix
- Solving Fractional Delay Differential Equations: A New Approach
- Formal Consistency Versus Actual Convergence Rates of Difference Schemes for Fractional-Derivative Boundary Value Problems
- Asymptotic Stability Of Dynamic Equations With Two Fractional Terms: Continuous Versus Discrete Case
- Analysis of Natural and Artificial Phenomena Using Signal Processing and Fractional Calculus
- Fractional Approach for Estimating Sap Velocity in Trees
- Fractional Calculus: Quo Vadimus? (Where are we Going?)
