Efficient Solution of Multi-Term Fractional Differential Equations Using P(EC)^mE Methods

Abstract

We investigate strategies for the numerical solution of the initial value problem with initial conditions where 0<α₁<α₂<⋯<α_ν. Here y ^(α _j ) denotes the derivative of order α _j >0 (not necessarily α _j ∈ 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.