Abstract
This paper examines a general recurrence relation by the use of fractional reduced differential transform and then a scheme (methodology) on how to find closed solutions of one dimensional time fractional diffusion equations with initial conditions in the form of infinite fractional power series and in terms of Mittag-Leffler function in one parameter as well as their exact solutions by the use of fractional reduced differential transform method. The new general recurrence relation and the methodology of the fractional reduced differential transform method were successfully developed. The obtained new general recurrence relation helps us to solve time-fractional diffusion equations with initial conditions and various external forces by using fractional reduced differential transform method. To see its effectiveness and applicability, five test examples were presented. The results show that the general recurrence relation works successfully in solving time-fractional diffusion equations in a direct way without using linearization, transformations, perturbation, discretization or restrictive assumptions by using fractional reduced differential transform method. Key words: Time fractional diffusion equations with initial conditions, Caputo fractional derivatives, Mittag-Leffler function, Fractional reduced differential transform method (FRDTM).
Highlights
The beginning of fractional calculus is considered to be the L'Hopital’s letter that raised the question: "What does n f (x) x n mean if n 1 ?" 2 to Leibniz in the notation for differentiation of non-integer order 1 was discussed (Diethelm, 2010; Hilfer, 2000; Lazarevic et al, 2014; Millar and Ross, 1993; Ortigueira, 2011; Kumar and Saxena, 2016)
2 above, the closed solutions of Equation (2a) given that (2b) in the form of infinite fractional power series and in terms of Mittag-Leffler function in one parameter as well as their exact solutions, which were obtained by fractional reduced differential transform method (FRDTM), are in complete agreement with the results obtained by Das (2009)
The fractional reduced differential transform method was applied to five time fractional diffusion equations with initial conditions, which exist in the literature except the last one, to obtain their closed solutions in the form of infinite fractional power series and in terms of Mittag-Leffler function in one parameter as well as exact solutions
Summary
The beginning of fractional calculus is considered to be the L'Hopital’s letter that raised the question: "What does n f (x) x n mean if n 1 ?" 2 to Leibniz in the notation for differentiation of non-integer order 1 was discussed (Diethelm, 2010; Hilfer, 2000; Lazarevic et al, 2014; Millar and Ross, 1993; Ortigueira, 2011; Kumar and Saxena, 2016). Fractional calculus theory is a mathematical analysis tool to the study of integrals and derivatives of arbitrary order, which unify and generalize the notations of integerorder differentiation and n fold integration (El-Ajou et al, 2013; Millar and Ross, 1993; Oldham and Spanier, 1974; Podlubny, 1999). Fractional calculus is almost as old as the classical calculus, it was only in recent few decades that its theory and applications have rapidly developed. It was Ross who organized the first international conference on fractional calculus and its applications at the University of new Haven in June 1974, and edited the proceedings (Ross, 1975). Thereafter, because of the fact that fractional derivatives and integrals are non-local operators and this property make them a powerful instrument for the description of memory and hereditary properties of different substances (Podlubny, 1999); theory and applications of fractional calculus have attracted much interest and become a pulsating research area
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