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Zbl 1035.65066
Diethelm, K.
Efficient solution of multi-term fractional differential equations using P(EC)$^m$E methods.
(English)
[J] Computing 71, No. 4, 305-319 (2003). ISSN 0010-485X; ISSN 1436-5057/e

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 $\alpha_j \in \Bbb N$) 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.
MSC 2000:
*65L05 Initial value problems for ODE (numerical methods)
65L06 Multistep, Runge-Kutta, and extrapolation methods
65L20 Stability of numerical methods for ODE
26A33 Fractional derivatives and integrals (real functions)
65Y20 Complexity and performance of numerical algorithms
34A34 Nonlinear ODE and systems, general

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

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