High-Order Accurate Local Schemes for Fractional Differential Equations
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High-order methods inspired by the multi-step Adams methods are proposed for systems of fractional differential equations. The schemes are based on an expansion in a weighted \(L^2\) space. To obtain the schemes this expansion is terminated after \(P+1\) terms. We study the local truncation error and its behavior with respect to the step-size h and P. Building on this analysis, we develop an error indicator based on the Milne device. Methods with fixed and variable step-size are tested numerically on a number of problems, including problems with known solutions, and a fractional version on the Van der Pol equation.
KeywordsFractional differential equations Volterra equations High-order methods
This work was partially supported by the NSF DMS-1115416 and by OSD/AFOSRFA9550-09-1-0613.
- 11.Brunner, H., van der Houwen, P.J.: The Numerical Solution of Volterra Equations. Elsevier Science Publishers B.V., Amsterdam (1986)Google Scholar
- 17.Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 10th printing, National Bureau of Standards, Washington (1972)Google Scholar
- 18.Bernardi, C., Maday, Y.: Spectral Methods, Handbook of Numerical Analysis, vol. V, Techniques of Scientific Computing (Part 2). Elsavier Science, Amsterdam (1997)Google Scholar