Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets

and Applied Analysis 3 Theorem 6 (see [21]). If L α {f(x)} = fL,α s (s), then one has L α {f(α) (x)} = s αL α {f (x)} − f (0) . (17) Proof. See [21]. Theorem 7 (see [21]). If L α {f(x)} = fL,α s (s), then one has


Introduction
Fractals are sets and their topological dimension exceeds the fractal dimensions. Mathematical techniques on fractal sets are presented (see, e.g., [1][2][3][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][6][7][8][9][10][11][12]. Because of the limit of fractional calculus, the fractal calculus as a framework for the model of anomalous diffusion [13][14][15][16] had been constructed. The Newtonian mechanics, Maxwell's equations, and Hamiltonian mechanics on fractal sets [17][18][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][28][29][30].
The Sumudu transforms (ST) had been considered for application to solve differential equations and to deal with control engineering [31][32][33][34][35][36][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.

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.
Let us consider the polynomial function on Cantor sets in the form where | | < 1.

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 LFS : ( ) → ( ), namely, As typical examples, we have As the generalized result, we give the following definition.
Proof. Definitions of the local fractional Sumudu and Laplace transforms directly give the results.
This completes the proof.
As the direct result of (28), we have the following results.
Proof. From (18) and (26), we have so that where This completes the proof. where Proof. From (19) and (26) 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.

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) 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 ( ) (0) = 0, Taking the local fractional Sumudu transform, from (52), we obtain Therefore, form (55), we give the nondifferentiable solution of (52) and we draw its graphs as shown in Figure 3.

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.