We consider the numerical solution of (possibly nonlinear) fractional differential equations of the form y(α)(t)=f(t,y(t),y(β1)(t),y(β2)(t),…,y(βn)(t)) with α>βn>βn−1>⋯>β1 and α−βn⩽1, βj−βj−1⩽1, 0<β1⩽1, combined with suitable initial conditions. The derivatives are understood in the Caputo sense. We begin by discussing the analytical questions of existence and uniqueness of solutions, and we investigate how the solutions depend on the given data. Moreover we propose convergent and stable numerical methods for such initial value problems.
We investigate a method for the numerical solution of the nonlinear fractional differential equation D * α y(t)=f(t,y(t)), equipped with initial conditions y (k)(0)=y 0 (k), k=0,1,...,⌈α⌉−1. Here α may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.
We discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order. The differential operators are taken in the Riemann–Liouville sense and the initial conditions are specified according to Caputo's suggestion, thus allowing for interpretation in a physically meaningful way. We investigate in particular the dependence of the solution on the order of the differential equation and on the initial condition, and we relate our results to the selection of appropriate numerical schemes for the solution of fractional differential equations.
We consider the numerical solution of the Bagley-Torvik equation Ay″(t) + BD * 3/2 y(t) + Cy(t) = f(t), as a prototype fractional differential equation with two derivatives. Approximate solutions have recently been proposed in the book and papers of Podlubny in which the solution obtained with approximate methods is compared to the exact solution. In this paper we consider the reformulation of the Bagley-Torvik equation as a system of fractional differential equations of order 1/2. This allows us to propose numerical methods for its solution which are consistent and stable and have arbitrarily high order. In this context we specifically look at fractional linear multistep methods and a predictor-corrector method of Adams type.
A note on the well-posedness of terminal value problems for fractional differential equations
(2018)
This note is intended to clarify some important points about the well-posedness of terminal value problems for fractional differential equations. It follows the recent publication of a paper by Cong and Tuan in this journal, in which a counter-example calls into question the earlier results in a paper by this note's authors. Here, we show in the light of these new insights, that a wide class of terminal value problems of fractional differential equations is well posed, and we identify those cases where the well-posedness question must be regarded as open.
Many recently developed models in areas like viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context. We believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way.