# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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), ${}_{0}{D}_{t}^{\alpha }y\left(t\right)=f\left(y,t\right),\left(t>0\right),n-1<\alpha \le n,{y}^{\left(i\right)}\left(0\right)={y}_{0}^{\left(i\right)},i=0,1,2,\cdots ,n-1$ where $f\left(y,t\right)$ satisfies the condition $|f\left({y}_{1},t\right)-f\left({y}_{2},t\right)|\le L|{y}_{1}-{y}_{2}|$ in $t\in \left[0,T\right]$.

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.

##### MSC:
 65L05 Initial value problems for ODE (numerical methods) 65L20 Stability and convergence of numerical methods for ODE 26A33 Fractional derivatives and integrals (real functions) 34K28 Numerical approximation of solutions of functional-differential equations 34A34 Nonlinear ODE and systems, general 65R20 Integral equations (numerical methods) 45J05 Integro-ordinary differential equations 45G10 Nonsingular nonlinear integral equations