On the particular solution of constant coefficient fractional differential equations Manuel D. Ortigueira 1 Campus da FCT, Quinta da Torre, 2829-516 Caparica, Portugal article info Keywords: Fractional differential equations Constant coefficient Particular solution Eigenfunction Transfer function abstract The eigenfunction approach to compute the particular solution of constant coefficient ordinary differential equations is extended to the fractional case. It is shown that the expo- nentials are also the eigenfunctions of such equations. Solutions corresponding to products of powers and exponentials are presented. The singular case is studied and a matricial algo- rithm for its implementation is presented. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction The computation of particular solutions of integer order constant coefficient ordinary linear equations was considered in some published papers mainly in [1,3]. In a previous paper [5] we studied the problem and proposed an approach based on the concept of eigenfunction. We showed how to compute the solution when the input is an exponential or the product of a power and an exponential. We studied and solved also the singular cases. In this paper we are going to enlarge those results to the fractional linear equations case. Normally they are written in the general format [4] X N k¼0 a k D a k yðtÞ¼ X M k¼0 b k D b k xðtÞ ð1Þ with t 2 R; the symbol D represents the derivative operator. The parameters a k and b k are the derivative orders that we assume to form strictly increasing sequences of positive numbers. In the so-called commensurate case we write a k ¼ b k ¼ ka. In current applications we assume that b M 6 a N for stability reasons. We will assume that xðtÞ¼ e bt t K t 2 R ð2Þ preventing the use of the Laplace transform (even two-sided) or the Fourier transform [6] to find the particular solution of (1). The treatment of the fractional case has similarities but also differences with the integer order due to the peculiarities of the fractional derivatives. Knowing that the matter of derivative definition is not pacific we make a brief introduction to the subject. We will use the Grünwald–Letnikov fractional derivative, since the derivative of an exponential is again an exponential. http://dx.doi.org/10.1016/j.amc.2014.07.070 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. E-mail address: mdo@fct.unl.pt 1 The author is with UNINOVA and Department of Electrical Engineering of Faculty of Sciences and Technology of Universidade Nova de Lisboa and INESC-ID. Applied Mathematics and Computation 245 (2014) 255–260 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc