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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 ${y}^{\left({\alpha }_{\nu }\right)}\left(x\right)=f\left(x,y\left(x\right),{y}^{\left({\alpha }_{1}\right)}\left(x\right),...,{y}^{\left({\alpha }_{\nu -1}\right)}\left(x\right)\right)$ with initial conditions

${y}^{\left(k\right)}\left(0\right)={y}_{0}^{\left(k\right)}\left(k=0,1,\cdots ,⌈{\alpha }_{\nu }⌉-1\right),$

where $0<{\alpha }_{1}<{\alpha }_{2}<\cdots <{\alpha }_{\nu }$. Here ${y}^{\left({\alpha }_{j}\right)}$ denotes the derivative of order ${\alpha }_{j}>0$ (not necessarily ${\alpha }_{j}\in ℕ$) 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 Initial value problems for ODE (numerical methods) 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L20 Stability and convergence 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