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This paper is concerned with the development of efficient algorithms for the approximate solution of fractional differential equations of the form D ^α y(t)=f(t,y(t)), α∈R ⁺−N.(†)

We briefly review standard numerical techniques for the solution of (†) and we consider how the computational cost may be reduced by taking into account the structure of the calculations to be undertaken. We analyse the fixed memory principle and present an alternative nested mesh variant that gives a good approximation to the true solution at reasonable computational cost. We conclude with some numerical examples.