A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations

Abstract

In this paper we propose an efficient and easy-to-implement numerical method for an α-th order Ordinary Differential Equation (ODE) when α ∈ (0, 1), based on a one-point quadrature rule. The quadrature point in each sub-interval of a given partition with mesh size h is chosen judiciously so that the degree of accuracy of the quadrature rule is 2 in the presence of the singular integral kernel. The resulting time-stepping method can be regarded as the counterpart for fractional ODEs of the well-known mid-point method for 1storder ODEs. We show that the global error in a numerical solution generated by this method is of the order O(h 2), independently of α. Numerical results are presented to demonstrate that the computed rates of convergence match the theoretical one very well and that our method is much more accurate than a well-known one-step method when α is small.

On necessary optimality conditions and exact penalization for a constrained fractional optimal control problem

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Numerical blow-up analysis of the explicit L1-scheme for fractional ordinary differential equations

“…Fractional ordinary differential equations (FODEs) have attracted much attention over the past decades [2,4,11,16,18]. This is mostly because they efficiently describe many phenomena arising in physics, economics, biology, and other aspects [1,10,22,26].…”

Optimality conditions of singular controls for systems with Caputo fractional derivatives

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