On the Singular Perturbations for Fractional Differential Equation

Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 752371, 9 pages http://dx.doi.org/10.1155/2014/752371 Research Article On the Singular Perturbations for Fractional Differential Equation Abdon Atangana Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9301, South Africa Correspondence should be addressed to Abdon Atangana; Received 29 August 2013; Accepted 19 December 2013; Published 9 February 2014 Academic Editors: R. Bera and Q. Xie Copyright © 2014 Abdon Atangana. 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. The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method. 1. Introduction Singular perturbation problems have constantly been playing an outstanding character in cooperation in the theory of differential equations and in their applications to the physical world. With a loose knot speaking, singularly perturbed differential equations are pigeonholed by the presence of quite a lot of essentially different scales. The existence of these scales gives one a small parameter and thus permits one to use perturbation methods. It is, however, typical of singular perturbation problems that a straight forward perturbation fails to be uniformly valid. Traditionally, this type of behaviour, which is abundant in applications, has been tackled by a large variety of methods, such as matched asymptotic expansions and averaging. Nevertheless, many fundamental issues still remain unresolved. Singular perturbation theory is a rich and ongoing area of exploration for mathematicians, physicists, and other researchers. The methods used to tackle problems in this field are many. The more basic of these include the method of matched asymptotic expansions [1–4] and WKB approximation for spatial problems [5–7], and in time, the Poincaré-Lindstedt method [8, 9], the method of multiple scales [10, 11], and periodic averaging [12, 13]. While searching in the literature, it is observed that attention has not been paid to the extension of this class of ordinary and partial differential equations to the concept of fractional calculus. In order to accommodate readers that are not acquainted with the concept of fractional calculus, it is perhaps important to recall that fractional calculus, the art of noninteger order integrals and derivatives, has gained an interesting momentum in recent years [14–20]. The applications are ranging from pure and applied mathematics. It is easy to find experts working in this field because of its beauty, while others look for applications. The main aim of this work is to provide impetus, motivation, and to bring together researchers and scientists working in the fields of fractional calculus and perturbation method by providing some useful analytical techniques to derive exact and approximate solutions of this class of equations. The general singular perturbation fractional differential equation considered in this paper is given as follows: 𝐷𝑡𝛼 𝑢 (𝑡) + 𝜀𝐴𝑢 (𝑡) + 𝐵𝑢 (𝑡) = 𝑓𝜀 (𝑡) , 𝑢 (𝑎) = 𝑔 (𝜀) , 𝑢 (𝑏) = ℎ (𝜀) , (1) where 𝑡 ∈ [𝑎, 𝑏], 𝐴, and 𝐵 are linear and nonlinear operators, respectively, 𝐷𝑡𝛼 is the fractional order derivative, 𝑓 is known function, and 𝜀 is a small parameter. We will present the review of the fractional calculus in the next section. 2 The Scientific World Journal 2. Review of Fractional Calculus A significant tip is that the fractional derivative at a point 𝑥 is a local property only when 𝑎 is an integer; in noninteger cases we cannot say that the fractional derivative at 𝑥 of a function 𝑓 depends only on values of 𝑓 very near 𝑥, in the way that integer-power derivatives unquestionably do. For that reason, it is anticipated that the philosophy involves some class of boundary conditions, involving information on the function further out. To use a metaphor, the fractional derivative requires some tangentialvision. As far as the existence of such a philosophy is concerned, the brass tacks of the subject were laid by Liouville in a paper from 1832. 2.1. Definitions Definition 1 (Riemann-Liouville integral). The classical form of fractional calculus is given by the Riemann-Liouville integral. The theory for periodic functions therefore including the “boundary condition” of repeating after a period is the Weyl integral. The Riemann-Liouville integral of order 𝛼 of a function 𝑓(𝑥) is given by −𝛼 𝛼 𝑎 𝐷𝑡 𝑓 (𝑥) = 𝑎 𝐼𝑡 𝑓 (𝑥) = 𝑥 1 ∫ (𝑥 − 𝑡)𝛼−1 𝑓 (𝑡) 𝑑𝑡. Γ (𝛼) 𝑎 (2) −𝛼 𝑥 1 𝑡 𝛼−1 𝑓 (𝑡) 𝑑𝑡. ∫ (log ( )) Γ (𝛼) 𝑎 2 𝑡 𝛼 = 𝑑𝑛 [ 𝑎 𝐼𝑛𝑡 𝑓 (𝑥)] 𝑑𝑥𝑛 1 𝑑𝑛 𝑥 ∫ (𝑥 − 𝑡)𝛼−1 𝑓 (𝑡) 𝑑𝑡. Γ (𝑛 − 𝛼) 𝑑𝑥𝑛 𝑎 (4) 𝐶 𝛼 0 𝐷𝑡 𝑓 (𝑥) 𝑥 𝑑𝑛 1 ∫ (𝑥 − 𝑡)𝛼−1 𝑛 𝑓 (𝑡) 𝑑𝑡. Γ (𝑛 − 𝛼) 𝑎 𝑑𝑡 𝑥 1 ∫ (𝑥 − 𝑡)−𝛼−1 𝑓 (𝑡) 𝑑𝑡. Γ (𝛼) 𝑎 (5) (6) The above expression is much simpler to handle on one hand; on the other hand, it is representing only with a single integral. The function in this case does not need to be differentiable in order to compute its fractional derivative. Note that the properties of this derivative will not be presented in this present work. Definition 6. The Laplace transform is a widely used integral transform with many applications in physics and engineering. The Laplace transform of the function 𝑓 is defined as follows: ∞ L (𝑓 (𝑥)) (𝑠) = ∫ 𝑒−𝑠𝑥 𝑓 (𝑥) 𝑑𝑥. (7) 2.2. Properties of Fractional Calculus. Two properties of the Laplace transform can be used to define the fractional integral operator as follows: 𝑥 L (𝐼 (𝑓)) = L (∫ 𝑓 (𝜏) 𝑑𝜏) (𝑠) 0 1 = L (𝑓 (𝑥)) (𝑠) , 𝑠 2 𝑥 𝑥1 0 0 (8) L (𝐼 (𝑓)) = L (∫ ∫ 𝑓 (𝜏) 𝑑𝜏 𝑑𝑥1 ) (𝑠) = 1 L (𝑓 (𝑥)) (𝑠) . 𝑠2 Now using the induction method, we arrive at the following: L (𝐼𝑛 (𝑓)) = Definition 4 (Caputo fractional derivative). There is another alternative for computing fractional derivatives that is the Caputo fractional derivative. It was introduced by Caputo and Michel in his 1967 paper [22]. In contrast to the Riemann-Liouville fractional derivative, when solving differential equations using Caputo’s definition, it is not necessary to define the fractional order initial conditions. Caputo’s definition is illustrated as follows: = 𝛼 𝑎 𝐷𝑡 𝑓 (𝑥) = (3) Definition 3 (Riemann-Liouville fractional deriva (...truncated)