J. Appl. Anal. 2016; 22 (2):131ś139 Research Article Gabriel Bengochea* and Manuel Ortigueira A recursive-operational approach with applications to linear diferential systems DOI: 10.1515/jaa-2016-0014 Received March 7, 2016; revised September 12, 2016; accepted September 17, 2016 Abstract: In this paper, an operational method for solving linear and nonlinear systems described by ordinary diferential equations is presented. The construction is based on the generalized derivative in the sense of distribution theory. The approach allows the response computation without needing the use of any integral transform as in the Mikusiński operational calculus. A general description of the algorithm is done and some illustrating examples are presented. The algorithm is recursive allowing to add and remove any pole or zero contribution. The extension to nonlinear systems is done by means of the Adomian polynomials. Keywords: Operational calculus, linear systems, nonlinear systems MSC 2010: 44A40, 35A24, 93C05, 93C10 1 Introduction The Laplace transform is almost surely the most important tool in studying continuous-time linear sys- tems [17], since it allows the transformation of diferential equations into algebraic equations. Frequently the one-sided Laplace transform is adopted due to its main feature that made it so useful: the direct intro- duction of the initial conditions. This creates some difculties and forces one to change its deőnition [10]. On the other hand, its usefulness is not so visible in the study of nonlinear systems. These considerations show that it would be interesting to have an alternative with the same simplicity that could solve at least the same problems without creating such difculties. In this paper, we present an operational method to compute the output of linear time-invariant difer- ential systems without using the Laplace or Fourier transform. As we will show it can be applied to some classes of nonlinear systems too. The algorithm is based on an algebraic framework which was introduced in [6] and has already been applied to several diferential equations showing its efciency, see [2ś5, 7]. In [4], an operational calculus for the fractional order Bessel operator is constructed with which the fractional Bessel equation is solved, whereas in this paper we base the construction on the integer order generalized derivative and solve linear and nonlinear systems. The operational calculus that we construct here can be considered an algebraic extension of Mikusiński’s operational calculus [14]. A good introduction to opera- tional calculus is presented in [12]; other operational methods can be consulted in [8, 9, 16, 18]. The main operation of the construction is an algebraic version of the usual convolution product. The algorithm is recursive and has another feature: it can be used to invert rational Laplace transforms. The work realized in [6] was primarily developed for usual derivatives. In this paper, we work with the generalized derivative in the sense of distribution theory [11], which makes a signiőcant diference. *Corresponding author: Gabriel Bengochea: Universidad Autónoma de la Ciudad de México, Ciudad de México, Mexico, e-mail: gabriel.bengochea@uacm.edu.mx Manuel Ortigueira: CTS-UNINOVA and Department of Electrical Engineering, Faculty of Sciences and Technology, Universidade Nova de Lisboa, Monte da Caparica, Portugal, e-mail: mdo@fct.unl.pt Brought to you by | University of South Carolina Libraries Authenticated Download Date | 3/18/17 5:51 AM