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Fractional high order methods for the nonlinear fractional ordinary differential equation. (English) Zbl 1118.65079

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), 0 D t α y(t)=f(y,t),(t>0),n-1<αn,y (i) (0)=y 0 (i) ,i=0,1,2,,n-1 where f(y,t) satisfies the condition |f(y 1 ,t)-f(y 2 ,t)|L|y 1 -y 2 | in t[0,T].

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 (p-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 p-HOFLMSM.

65L05Initial value problems for ODE (numerical methods)
65L20Stability and convergence of numerical methods for ODE
26A33Fractional derivatives and integrals (real functions)
34K28Numerical approximation of solutions of functional-differential equations
34A34Nonlinear ODE and systems, general
65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations