Abstract and Applied Analysis

Volume 2014 (2014), Article ID 620529, 7 pages

http://dx.doi.org/10.1155/2014/620529

## Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets

^{1}Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4^{2}Department of Physics, Urmia Branch, Islamic Azad University, P.O. Box 969, Orumiyeh, Iran^{3}Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia^{4}Institute of Space Sciences, Magurele, 077125 Bucharest, Romania^{5}Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey^{6}Department 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.

#### Abstract

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.

#### 1. Introduction

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 [22]. Maxwell’s equations on Cantor sets with local fractional vector calculus had been considered [23]. The local fractional Fourier analysis had been adapted to find Heisenberg uncertainty principle [24]. A family of local fractional Fredholm and Volterra integral equations was investigated in [25]. Local fractional variational iteration and decomposition methods for wave equation on Cantor sets were reported in [26]. 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 1 (see [20, 21, 24–26]). *Let the function , if there are
where , for and .

*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

*Definition 3 (see [20, 21, 24–26]). *Let . The local fractional integral of of order in the interval is defined as
where the partitions of the interval are denoted as , , , and with and .

Theorem 4 (local fractional Taylor’ theorem (see [20, 21])). *Suppose that , for and . Then, one has
**
with , , 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:

*Definition 5. *The local fractional Laplace transform of of order is defined as [27–30]
If , the inverse formula of (42) is defined as [27–30]
where is local fractional continuous, , and .

*Theorem 6 (see [21]). If , then one has
*

*Proof. *See [21].

*Theorem 7 (see [21]). If , then one has
*

*Proof. *
See [21].

*Theorem 8 (see [21]). If and , then one has
where
*

*Proof. *See [21].

*3. Local Fractional Sumudu Transform*

*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
*

*Proof. *From (17) and (26), the local fractional Sumudu transform of the local fractional derivative of read as
where
This completes the proof.

*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
*

*Proof. *From (18) and (26), we have
so that
where
This completes the proof.

*Theorem 14 (local fractional convolution). If and , then one has
where
*

*Proof. *From (19) and (26), we have
where
This completes the proof.

*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*

*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.

*5. Conclusions*

*5. Conclusions**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*

*Conflict of Interests**The authors declare that there is no conflict of interests regarding the publication of this paper.*

*References*

*References*

- B. B. Mandelbrot,
*The Fractal Geometry of Nature*, W. H. Freeman and Company, San Francisco, Calif, USA, 1982. View at MathSciNet - K. Falconer,
*Fractal Geometry: Mathematical Foundations and Applications*, John Wiley & Sons, New York, NY, USA, 1990. View at MathSciNet - K. Falconer,
*Techniques in Fractal Geometry*, John Wiley & Sons, Chichester, UK, 1997. View at MathSciNet - G. A. Edgar,
*Integral, Probability, and Fractal Measures*, Springer, New York, NY, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - K. B. Oldham and J. Spanier,
*The Fractional Calculus*, Academic Press, New York, NY, USA, 1974. View at MathSciNet - K. S. Miller and B. Ross,
*An Introduction to the Fractional Integrals and Derivatives Theory and Applications*, John Wiley and Sons, New York, NY, USA, 1993. - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives Theory and Applications*, Gordon and Breach Science, Yverdon, Switzerland, 1993. View at MathSciNet - R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at MathSciNet - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at MathSciNet - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, vol. 3 of*Series on Complexity, Nonlinearity and Chaos*, World Scientific, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - V. V. Uchaikin,
*Fractional Derivatives for Physicists and Engineers, Background and Theory*, Nonlinear Physical Science, Springer, Berlin, Germany, 2013. View at MathSciNet - B. J. West, M. Bologna, and P. Grigolini,
*Physics of Fractal Operators*, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - W. Chen, H. Sun, X. Zhang, and D. Korošak, “Anomalous diffusion modeling by fractal and fractional derivatives,”
*Computers & Mathematics with Applications*, vol. 59, no. 5, pp. 1754–1758, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,”
*Physical Review Letters*, vol. 80, no. 2, pp. 214–217, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Parvate and A. D. Gangal, “Calculus on fractal subsets of real line. I. Formulation,”
*Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society*, vol. 17, no. 1, pp. 53–81, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Parvate, S. Satin, and A. D. Gangal, “Calculus on fractal curves in ${\mathbf{R}}^{n}$,”
*Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society*, vol. 19, no. 1, pp. 15–27, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. K. Golmankhaneh, A. K. Golmankhaneh, and D. Baleanu, “About Maxwell's equations on fractal subsets of R3,”
*Central European Journal of Physics*, vol. 11, no. 6, pp. 863–867, 2013. View at Google Scholar - A. K. Golmankhaneh, V. Fazlollahi, and D. Baleanu, “Newtonian mechanics on fractals subset of real-line,”
*Romanian Reports in Physics*, vol. 65, no. 1, pp. 84–93, 2013. View at Google Scholar · View at Scopus - X.-J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science, New York, NY, USA, 2012. - X.-J. Yang,
*Local Fractional Functional Analysis and Its Applications*, Asian Academic, Hong Kong, 2011. - X.-J. Yang, D. Baleanu, and J. A. Tenreiro Machado, “Systems of Navier-Stokes equations on Cantor sets,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 769724, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Zhao, D. Baleanu, C. Cattani, D.-F. Cheng, and X.-J. Yang, “Maxwell's equations on Cantor sets: a local fractional approach,”
*Advances in High Energy Physics*, vol. 2013, Article ID 686371, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X.-J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis,”
*Boundary Value Problems*, vol. 2013, 131, 2013. View at Publisher · View at Google Scholar - X.-J. Ma, H. M. Srivastava, D. Baleanu, and X.-J. Yang, “A new Neumann series method for solving a family of local fractional Fredholm and Volterra integral equations,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 325121, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - D. Baleanu, J. A. Tenreiro Machado, C. Cattani, M. C. Baleanu, and X.-J. Yang, “Local fractional variational iteration and decomposition methods for wave equation on Cantor sets within local fractional operators,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 535048, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - J.-H. He, “Asymptotic methods for solitary solutions and compactons,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 916793, 130 pages, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. F. Liu, S. S. Kong, and S. J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,”
*Thermal Science*, vol. 17, no. 3, pp. 715–721, 2013. View at Google Scholar - C.-G. Zhao, A.-M. Yang, H. Jafari, and A. Haghbin, “The Yang-Laplace transform for solving the IVPs with local fractional derivative,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 386459, 5 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - S.-Q. Wang, Y.-J. Yang, and H. K. Jassim, “Local fractional function decomposition method for solving inhomogeneous wave equations with local fractional derivative,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 176395, 7 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - G. K. Watugala, “Sumudu transform: a new integral transform to solve differential equations and control engineering problems,”
*International Journal of Mathematical Education in Science and Technology*, vol. 24, no. 1, pp. 35–43, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Weerakoon, “Application of Sumudu transform to partial differential equations,”
*International Journal of Mathematical Education in Science and Technology*, vol. 25, no. 2, pp. 277–283, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. B. Deakin, “The Sumudu transform and the Laplace transform,”
*International Journal of Mathematical Education in Science and Technology*, vol. 28, no. 1, pp. 159–160, 1997. View at Google Scholar - M. A. Asiru, “Sumudu transform and the solution of integral equations of convolution type,”
*International Journal of Mathematical Education in Science and Technology*, vol. 32, no. 6, pp. 906–910, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Aşiru, “Further properties of the Sumudu transform and its applications,”
*International Journal of Mathematical Education in Science and Technology*, vol. 33, no. 3, pp. 441–449, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Eltayeb and A. Kılıçman, “A note on the Sumudu transforms and differential equations,”
*Applied Mathematical Sciences*, vol. 4, no. 21–24, pp. 1089–1098, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. B. M. Belgacem, A. A. Karaballi, and S. L. Kalla, “Analytical investigations of the Sumudu transform and applications to integral production equations,”
*Mathematical Problems in Engineering*, no. 3-4, pp. 103–118, 2003. View at Publisher · View at Google Scholar · View at MathSciNet

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