Diethelm, K. Efficient solution of multi-term fractional differential equations using P(EC)\(^m\)E methods. (English) Zbl 1035.65066 Computing 71, No. 4, 305-319 (2003). Summary: We investigate strategies for the numerical solution of the initial value problem \(y^{(\alpha_\nu)}(x) = f(x,y(x),y^{(\alpha_1)}(x),\ldots,y^{(\alpha_{\nu-1})}(x))\) with initial conditions \[ y^{(k)}(0) = y_0^{(k)}(k=0,1,\dots,\lceil\alpha_\nu\rceil-1), \] where \(0<\alpha_1<\alpha_2<\cdots<\alpha_\nu\). Here \(y^{(\alpha_j)}\) denotes the derivative of order \(\alpha_j>0\) (not necessarily \(\alpha_j \in \mathbb N\)) in the sense of Caputo. The methods are based on numerical integration techniques applied to an equivalent nonlinear and weakly singular Volterra integral equation. The classical approach leads to an algorithm with very high arithmetic complexity. Therefore we derive an alternative that leads to lower complexity without sacrificing too much precision. Cited in 54 Documents MSC: 65L05 Numerical methods for initial value problems involving ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 26A33 Fractional derivatives and integrals 65Y20 Complexity and performance of numerical algorithms 34A34 Nonlinear ordinary differential equations and systems Keywords:Fractional differential equation; multi-term equation; predictor-corrector method; numerical example; initial value problem; nonlinear and weakly singular Volterra integral equation; algorithm; arithmetic complexity PDFBibTeX XMLCite \textit{K. Diethelm}, Computing 71, No. 4, 305--319 (2003; Zbl 1035.65066) Full Text: DOI