Some Analytical Techniques in Fractional Calculus: Realities and Challenges
In the last decades, much effort has been dedicated to analytical aspects of the fractional differential equations. The Adomian decomposition method and the variational iteration method have been developed from ordinary calculus and become two frequently
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Some Analytical Techniques in Fractional Calculus: Realities and Challenges Dumitru Baleanu, Guo-Cheng Wu, and Jun-Sheng Duan
Abstract In the last decades, much effort has been dedicated to analytical aspects of the fractional differential equations. The Adomian decomposition method and the variational iteration method have been developed from ordinary calculus and become two frequently used analytical methods. In this article, the recent developments of the methods in the fractional calculus are reviewed. The realities and challenges are comprehensively encompassed. Keywords Fractional differential equations • Adomian decomposition method • Variational iteration method • Riemann–Liouville derivative • Caputo derivative • Adomian polynomials • One-step numeric algorithms • Approximate solutions
D. Baleanu () Department of Mathematics and Computer Sciences, Cankaya University, 06530 Balgat, Ankara, Turkey Institute of Space Sciences, Magurele-Bucharest, Romania Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia e-mail: [email protected] G.-C. Wu College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641112, P.R. China e-mail: [email protected] J.-S. Duan School of Mathematics and Information Sciences, Zhaoqing University, Zhaoqing, Guang Dong 526061, P.R. China e-mail: [email protected] J.A.T. Machado et al. (eds.), Discontinuity and Complexity in Nonlinear Physical Systems, Nonlinear Systems and Complexity 6, DOI 10.1007/978-3-319-01411-1__3, © Springer International Publishing Switzerland 2014
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3.1 Introduction We review the basic definitions of the Riemann–Liouville (R–L) and the Caputo derivatives. For additional details readers can refer to references [19, 21, 30, 58, 60, 61, 65, 70]. Definition 1. Let f .t/ be a function of class C, i.e. piecewise continuous on .t0 ; C1/ and integrable on any finite subinterval of .t0 ; C1/. Then for t > t0 , the Riemann–Liouville integral of f .t/ of ˇ order is defined as Z t 1 ˇ I f .t/ D .t /ˇ1 f ./d ; (3.1) t0 t .ˇ/ t0 where ˇ is a positive real number and ./ is Euler’s Gamma function. The fractional integral satisfies the following equalities, ˇ t0 It t0 It f .t/ t0 It .t
D
t0 / D
ˇC f .t/; t 0 It
ˇ 0; 0;
(3.2)
. C 1/ .t t0 /C ; 0; > 1: . C C 1/
(3.3)
Definition 2. Let f .t/ be a function of class C and ˛ be a positive real number satisfying m 1 < ˛ m, m 2 NC , where NC is the set of positive integers. Then, the Riemann–Liouville derivative of f .t/ of order ˛ is defined as (when it exists) ˛ t0 Dt f .t/
D
d m m˛ f .t/ ; t > t0 : t 0 It m dt
(3.4)
Defining for complementarity t0 D0t D I; the identity operator, then t0 D˛t f .t/ D f .t/ if ˛ D m; m D 0; 1; 2; : : : . Note that the Riemann–Liouville fractional derivative t0 D˛t f .t/ is not zero for the constant function f .t/ C if ˛ > 0 and ˛ … NC . For the power functions, the following holds .˛/
˛ t0 Dt .t
t0 /
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