On solving non-homogeneous fractional differential equations of Euler type
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On solving non-homogeneous fractional differential equations of Euler type Ayad R. Khudair
Received: 23 April 2013 / Accepted: 13 May 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract In this paper, we investigate non-homogeneous fractional differential equations of Euler type with alpha-left Riemann–Liouville fractional derivatives. In fact, these fractional differential equations are analogs of Euler ordinary differential equations. The classical power series method is used to obtain a useful formula to compute a particular solution of these equations. This formula is explicit and easy to compute by using Maple software or by setting a computer code to get explicit particular solution. The related results are proved for Euler ordinary differential equations. Keywords method
Fractional differential equations · Euler differential equations · Series solution
Mathematics Subject Classification (2010)
34A08 · 34K37 · 26A33
1 Introduction Gottfried Wilhelm Leibniz (1646–1716) first introduced the idea of a symbolic method and n used the symbol d dxy(x) = D n y(x) for the nth derivative, where n is a non-negative integer. n L’Hospital asked Leibniz about the possibility that n be a fraction. “What if n = 21 .” Leibniz (1695) replied, “It will lead to a paradox.” But he added prophetically, “From this apparent paradox, one day useful consequences will be drawn” (Debnath and Bhatta 2007). So, fractional calculus studies is a classical mathematical field as old as calculus itself . In the past, fractional calculus was considered as pure mathematics, with nearly no applications. But during these last decades fractional calculus have been applied in many context of sciences (Machado et al. 2011). Nowadays, fractional derivatives have been used in many applications in electromagnetic theory, circuit theory, biology, atmospheric physics, etc., (Kilbas et al. 2006). Communicated by Cristina Turner. A. R. Khudair (B) Department of Mathematics, Faculty of Science, Basrah University, Basrah, Iraq e-mail: [email protected]
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In fact, there are two great difficulties in the study of fractional derivatives, the first one is, fractional derivatives cannot be associated to a tangent direction as the usual first derivative. The second, but not less, difficulty is their complex integro-differential definition, which make a simple manipulation with standard integer operators a complex operation that should be made carefully. Fractional differential equations arise in fluid mechanics, mathematical biology, electrochemistry, physics, and so on. Therefore, many researchers have investigated and solved several fractional differential equation types (Mophou 2010; Rajeev and Kushwaha 2013; Eidelman and Kochubei 2004; Xue et al. 2008; Guo et al. 2012; Molliq et al. 2009). Zhukovskaya et al. (2011) studied linear homogeneous differential equations with three left Riemann– Liouville fractional derivatives and obtained a complete system of linearly independent solutions by using
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