@article{DiethelmFord, author = {Diethelm, Kai and Ford, Neville J.}, title = {Analysis of fractional differential equations}, series = {Journal of Mathematical Analysis and Applications}, volume = {265}, journal = {Journal of Mathematical Analysis and Applications}, number = {2}, doi = {10.1006/jmaa.2000.7194}, pages = {229 -- 248}, abstract = {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.}, language = {en} } @article{DiethelmFordFreed, author = {Diethelm, Kai and Ford, Neville J. and Freed, Alan D.}, title = {A predictor-corrector approach for the numerical solution of fractional differential equations}, series = {Nonlinear Dynamics}, volume = {29}, journal = {Nonlinear Dynamics}, doi = {10.1023/A:1016592219341}, pages = {3 -- 22}, abstract = {We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.}, language = {en} } @article{DiethelmFord, author = {Diethelm, Kai and Ford, Neville J.}, title = {Numerical solution of the Bagley-Torvik equation}, series = {BIT Numerical Mathematics}, volume = {42}, journal = {BIT Numerical Mathematics}, doi = {10.1023/A:1021973025166}, pages = {490 -- 507}, abstract = {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.}, language = {en} }