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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 (α ν ) (x)=f(x,y(x),y (α 1 ) (x),...,y (α ν-1 ) (x)) with initial conditions

y (k) (0)=y 0 (k) (k=0,1,,α ν -1),

where 0<α 1 <α 2 <<α ν . Here y (α j ) denotes the derivative of order α j >0 (not necessarily α j ) 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:
65L05Initial value problems for ODE (numerical methods)
65L06Multistep, Runge-Kutta, and extrapolation methods
65L20Stability and convergence of numerical methods for ODE
26A33Fractional derivatives and integrals (real functions)
65Y20Complexity and performance of numerical algorithms
34A34Nonlinear ODE and systems, general