The paper begins by referring to applications of fractional order equations, along with a brief summary of the main results achieved for this type of equation in the last decade.
The authors consider the nonlinear fractional-order order differential equation (NFOODE), where satisfies the condition in .
Existence and uniqueness theorems for the NFOODE, by K. Diethelm and N. J. Ford [J. Math. Anal. Appl. 265, No. 2, 229–248 (2002; Zbl 1014.34003)], are stated. High order fractional linear multi-step methods (-HOFLMSM) are introduced. Definitions pertaining to their consistency and stability are stated. New results relating to the consistence, convergence and stability of these methods are presented and proved. The paper concludes with numerical examples which demonstrate the computational efficiency of the -HOFLMSM.