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), \(_0D_t^\alpha y(t)=f(y,t), (t>0), n-1<\alpha \leq n, y^{(i)}(0)=y_0^{(i)}, i=0, 1, 2,\dots, n-1\) where \(f(y,t)\) satisfies the condition \(| f(y_1,t)-f(y_2,t)| \leq L| y_1-y_2| \) in \(t\in [0,T]\).
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.
Reviewer: Pat Lumb (Chester)


65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
26A33 Fractional derivatives and integrals
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
34A34 Nonlinear ordinary differential equations and systems
65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations


Zbl 1014.34003
Full Text: DOI


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