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This paper introduces techniques for the estimation of solutions to fractional order differential equations (FODEs) and the approximation of a function's Caputo fractional derivative. These techniques are based on scattered data interpolation via reproducing kernel Hilbert spaces (RKHSs). Specifically, an RKHS is generated for the purpose of estimating fractional derivatives from the Mittag-Leffler function. The RKHS, called the Mittag-Leffler RKHS, as well as others are utilized to estimate Caputo fractional derivatives and to introduce a modified Adams-Bashforth- Moulton method for the estimation of the solution to FODEs. [ABSTRACT FROM AUTHOR]