Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 482890, 9 pages doi:10.1155/2012/482890 Research Article Finite Element Method for Linear Multiterm Fractional Differential Equations Abdallah A. Badr Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt Correspondence should be addressed to Abdallah A. Badr, badrzoo@yahoo.com Received 14 September 2012; Revised 5 October 2012; Accepted 5 October 2012 Academic Editor: Morteza Rafei Copyright q 2012 Abdallah A. Badr. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the linear multiterm fractional differential equation fDE. Existence and uniqueness of the solution of such equation are discussed. We apply the finite element method FEM to obtain the numerical solution of this equation using Galerkin approach. A comparison, through examples, between our techniques and other previous numerical methods is established. 1. Introduction Recently, many applications in numerous fields of science, engineering, viscoelastic materials, signal processing, controlling, quantum mechanics, meteorology, finance, life science, applied mathematics, and economics have been remodeled in terms of fractional calculus where derivatives and integrals of fractional order are introduced and so differential equation of fractional order are involved in these models, see 1–4. Fractional-order derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. They have been successfully used to model many problems. As an example which will give us a physical understanding of the fractional derivatives: in dynamical systems with fractional-order derivatives, fractional-order derivatives have been successfully used to model damping forces with memory effect or to describe state feedback controllers. In particular, the BagleyTorvik equation with 1/2-order derivative or 3/2-order derivative describes motion of real physical systems, an immersed plate in a Newtonian fluid, and a gas in a fluid, respectively 5. Recently, it is found in 6 that in fractional- order vibration systems of single degree of freedom, the term of fractional-order derivative whose order is between 0 and 2 acts always as damping force. In addition, almost all systems containing internal damping are not suitable to be described properly by the classical methods, but the fractional calculus represents one of the promising tools which describe