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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 620529, 7 pages
Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets
1Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4
2Department of Physics, Urmia Branch, Islamic Azad University, P.O. Box 969, Orumiyeh, Iran
3Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
4Institute of Space Sciences, Magurele, 077125 Bucharest, Romania
5Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey
6Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China
Received 13 February 2014; Accepted 10 May 2014; Published 26 May 2014
Academic Editor: Jordan Hristov
Copyright © 2014 H. M. Srivastava et al. 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.
Local fractional derivatives were investigated intensively during the last few years. The coupling method of Sumudu transform and local fractional calculus (called as the local fractional Sumudu transform) was suggested in this paper. The presented method is applied to find the nondifferentiable analytical solutions for initial value problems with local fractional derivative. The obtained results are given to show the advantages.
Fractals are sets and their topological dimension exceeds the fractal dimensions. Mathematical techniques on fractal sets are presented (see, e.g., [1–4]). Nonlocal fractional derivative has many applications in fractional dynamical systems having memory properties. Fractional calculus has been applied to the phenomena with fractal structure [5–12]. Because of the limit of fractional calculus, the fractal calculus as a framework for the model of anomalous diffusion [13–16] had been constructed. The Newtonian mechanics, Maxwell’s equations, and Hamiltonian mechanics on fractal sets [17–19] were generalized. The alternative definitions of calculus on fractal sets had been suggested in [20, 21] and the systems of Navier-Stokes equations on Cantor sets had been studied in . Maxwell’s equations on Cantor sets with local fractional vector calculus had been considered . The local fractional Fourier analysis had been adapted to find Heisenberg uncertainty principle . A family of local fractional Fredholm and Volterra integral equations was investigated in . Local fractional variational iteration and decomposition methods for wave equation on Cantor sets were reported in . The local fractional Laplace transforms were developed in [27–30].
The Sumudu transforms (ST) had been considered for application to solve differential equations and to deal with control engineering [31–37]. The aims of this paper are to couple the Sumudu transforms and the local fractional calculus (LFC) and to give some illustrative examples in order to show the advantages.
The structures of the paper are as follows. In Section 2, the local fractional derivatives and integrals are presented. In Section 3, the notions and properties of local fractional Sumudu transform are proposed. In Section 4, some examples for initial value problems are shown. Finally, the conclusions are given in Section 5.
2. Local Fractional Calculus and Polynomial Functions on Cantor Sets
In this section, we give the concepts of local fractional derivatives and integrals and polynomial functions on Cantor sets.
Definition 2 (see [20, 21, 24]). Let . The local fractional derivative of of order in the interval is defined as where The local fractional partial differential operator of order was given by [20, 21] where
Proof (see [20, 21]). Local fractional Mc-Laurin’s series of the Mittag-Leffler functions on Cantor sets is given by [20, 21]
and local fractional Mc-Laurin’s series of the Mittag-Leffler functions on Cantor sets with the parameter reads as follows:
As generalizations of (9) and (10), we have
where are coefficients of the generalized polynomial function on Cantor sets.
Making use of (10), we get where is the imaginary unit with .
Let us consider the polynomial function on Cantor sets in the form where .
Hence, we have the closed form of (13) as follows:
Theorem 6 (see ). If , then one has
Proof. See .
Theorem 7 (see ). If , then one has
Proof. See .
Theorem 8 (see ). If and , then one has where
Proof. See .
3. Local Fractional Sumudu Transform
In this section, we derive the local fractional Sumudu transform (LFST) and some properties are discussed.
If there is a new transform operator , namely, As typical examples, we have As the generalized result, we give the following definition.
Definition 9. The local fractional Sumudu transform of of order is defined as Following (23), its inverse formula is defined as
Theorem 10 (linearity). If and , then one has
Proof. As a direct result of the definition of local fractional Sumudu transform, we get the following result.
Theorem 11 (local fractional Laplace-Sumudu duality). If and , then one has
Proof. Definitions of the local fractional Sumudu and Laplace transforms directly give the results.
Theorem 12 (local fractional Sumudu transform of local fractional derivative). If , then one has
As the direct result of (28), we have the following results.
If , then we have When , from (31), we get
Theorem 13 (local fractional Sumudu transform of local fractional derivative). If, then one has
Theorem 14 (local fractional convolution). If and , then one has where
In the following, we present some of the basic formulas which are in Table 1.
The above results are easily obtained by using local fractional Mc-Laurin’s series of special functions.
4. Illustrative Examples
In this section, we give applications of the LFST to initial value problems.
Example 1. Let us consider the following initial value problems: subject to the initial value condition Taking the local fractional Sumudu transform gives where Making use of (43), we obtain Hence, from (45), we get and we draw its graphs as shown in Figure 1.
Example 2. We consider the following initial value problems: and the initial boundary value reads as Taking the local fractional Sumudu transform, from (47) and (48), we have so that Therefore, the nondifferentiable solution of (47) is and we draw its graphs as shown in Figure 2.
Example 3. We give the following initial value problems: together with the initial value conditions Taking the local fractional Sumudu transform, from (52), we obtain which leads to Therefore, form (55), we give the nondifferentiable solution of (52) and we draw its graphs as shown in Figure 3.
In this work, we proposed the local fractional Sumudu transform based on the local fractional calculus and its results were discussed. Applications to initial value problems were presented and the nondifferentiable solutions are obtained. It is shown that it is an alternative method of local fractional Laplace transform to solve a class of local fractional differentiable equations.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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