An algorithm for the numerical solution of differential equations of fractional order. (English) Zbl 0890.65071

Diethelm, Kai

ETNA, Electron. Trans. Numer. Anal. 5, 1-6 (1997).

The author considers the fractional differential equation \[ (D^q[x-x_0])(t)=\beta x(t)+f(t), \qquad 0\leq t \leq 1, \quad x(0)=x_0, \] where \(0<q<1\), \(f\) is a given function on the interval \([0,1]\), \(\beta \leq 0\). Here \(D^q x\) denotes the Riemann-Liouville fractional derivative of order \(q\). An implicit algorithm for the approximate solution of an important class of these equations is proposed. Error estimates and numerical examples are given.

Reviewer: S.Yanchuk (Kyïv)

MSC:

65L05	Numerical methods for initial value problems involving ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems
65L70	Error bounds for numerical methods for ordinary differential equations
26A33	Fractional derivatives and integrals

Keywords:

fractional differential equation; Riemann-Liouville fractional derivative; numerical examples; error estimates; implicit algorithm