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Efficient solution of multi-term fractional differential equations using P(EC)\(^m\)E methods. (English) Zbl 1035.65066

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.

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
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