Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets

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.

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H. M. Srivastava 1 and Alireza Khalili Golmankhaneh 2 and Dumitru Baleanu 3, 4, 5 and Xiao-Jun Yang 6

Academic Editor:Jordan Hristov

1, Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada
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; 26 May 2014

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.

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 f ( x ) ∈ _{C α} ( a , b ) , if there are [figure omitted; refer to PDF] where | x - _{x 0} | < δ , for [straight epsilon] > 0 and [straight epsilon] ∈ R .

Definition 2 (see [20, 21, 24]).

Let f ( x ) ∈ _{C α} ( a , b ) . The local fractional derivative of f ( x ) of order α in the interval [ a , b ] is defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The local fractional partial differential operator of order α ( 0 < α ...4; 1 ) was given by [20, 21] [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Definition 3 (see [20, 21, 24-26]).

Let f ( x ) ∈ _{C α} [ a , b ] . The local fractional integral of f ( x ) of order α in the interval [ a , b ] is defined as [figure omitted; refer to PDF] where the partitions of the interval [ a , b ] are denoted as ( _{t j} , _{t j + 1} ) , j = 0 , ... , N - 1 , _{t 0} = a , and _{t N} = b with Δ _{t j} = _{t j + 1} - _{t j} and Δ t = max ... { Δ _{t 0} , Δ _{t 1} , Δ _{t j} , ... } .

Theorem 4 (local fractional Taylor' theorem (see [20, 21])).

Suppose that ^{f ( ( k + 1 ) α )} ( x ) ∈ _{C α} ( a , b ) , for k = 0,1 , ... , n and 0 < α ...4; 1 . Then, one has [figure omitted; refer to PDF] with a < _{x 0} < ξ < x < b , ∀ x ∈ ( a , b ) , where [figure omitted; refer to PDF]

Proof (see [20, 21]).

Local fractional Mc-Laurin's series of the Mittag-Leffler functions on Cantor sets is given by [20, 21] [figure omitted; refer to PDF] and local fractional Mc-Laurin's series of the Mittag-Leffler functions on Cantor sets with the parameter ζ reads as follows: [figure omitted; refer to PDF] As generalizations of (9) and (10), we have [figure omitted; refer to PDF] where _{a k} ( k = 0,1 , 2 , ... , n ) are coefficients of the generalized polynomial function on Cantor sets.

Making use of (10), we get [figure omitted; refer to PDF] where ^{i α} is the imaginary unit with _{E α} ( ^{i α ( 2 π ) α} ) = 1 .

Let us consider the polynomial function on Cantor sets in the form [figure omitted; refer to PDF] where | x | < 1 .

Hence, we have the closed form of (13) as follows: [figure omitted; refer to PDF]

Definition 5.

The local fractional Laplace transform of f ( x ) of order α is defined as [27-30] [figure omitted; refer to PDF] If _{F α} { f ( x ) } ...1; ^{f ω F , α} ( ω ) , the inverse formula of (42) is defined as [27-30] [figure omitted; refer to PDF] where f ( x ) is local fractional continuous, ^{s α} = ^{β α} + ^{i α ∞ α} , and Re ( s ) = β > 0 .

Theorem 6 (see [21]).

If _{L α} { f ( x ) } = ^{f s L , α} ( s ) , then one has [figure omitted; refer to PDF]

Proof.

See [21].

Theorem 7 (see [21]).

If _{L α} { f ( x ) } = ^{f s L , α} ( s ) , then one has [figure omitted; refer to PDF]

Proof.

See [21].

Theorem 8 (see [21]).

If _{L α} { _{f 1} ( x ) } = ^{f s , 1 L , α} ( s ) and _{L α} { _{f 2} ( x ) } = ^{f s , 2 L , α} ( s ) , then one has [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

See [21].

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 LF _{S α} : f ( x ) [arrow right] F ( u ) , namely, [figure omitted; refer to PDF] As typical examples, we have [figure omitted; refer to PDF] As the generalized result, we give the following definition.

Definition 9.

The local fractional Sumudu transform of f ( x ) of order α is defined as [figure omitted; refer to PDF] Following (23), its inverse formula is defined as [figure omitted; refer to PDF]

Theorem 10 (linearity).

If L F _{S α} { f ( x ) } = _{F α} ( z ) and L F _{S α} { g ( x ) } = _{G α} ( z ) , then one has [figure omitted; refer to PDF]

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 _{L α} { f ( x ) } = ^{f s L , α} ( s ) and L F _{S α} { f ( x ) } = _{F α} ( z ) , then one has [figure omitted; refer to PDF]

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 L F _{S α} { f ( x ) } = _{F α} ( z ) , then one has [figure omitted; refer to PDF]

Proof.

From (17) and (26), the local fractional Sumudu transform of the local fractional derivative of f ( x ) read as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] This completes the proof.

As the direct result of (28), we have the following results.

If LF _{S α} { f ( x ) } = _{F α} ( z ) , then we have [figure omitted; refer to PDF] When n = 2 , from (31), we get [figure omitted; refer to PDF]

Theorem 13 (local fractional Sumudu transform of local fractional derivative).

If L F _{S α} { f ( x ) } = _{F α} ( z ) , then one has [figure omitted; refer to PDF]

Proof.

From (18) and (26), we have [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] This completes the proof.

Theorem 14 (local fractional convolution).

If L F _{S α} { f ( x ) } = _{F α} ( z ) and L F _{S α} { g ( x ) } = _{G α} ( z ) , then one has [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

From (19) and (26), we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] This completes the proof.

In the following, we present some of the basic formulas which are in Table 1.

Table 1: Local fractional Sumudu transform of special functions.

Mathematical operation in the t -domain	Corresponding operation in the z -domain	Remarks
a	a	a is a constant
x α Γ ( 1 + α )	z α
∑ k = 0 ∞ ... a k x α k	∑ k = 0 ∞ ... Γ ( 1 + k α ) a k z α k
E α ( a x α )	1 1 - a z α	E α ( x α ) = ∑ k = 0 ∞ x α k Γ ( 1 + k α )
sin ... α ... ( a x α )	a z α 1 + a 2 z 2 α	sin ... α ... x α = ∑ k = 0 ∞ ( - 1 ) k x α ( 2 k + 1 ) Γ [ 1 + α ( 2 k + 1 ) ]
cos α ... ( a x α )	1 1 + a 2 z 2 α	cos α ... x α = ∑ k = 0 ∞ ( - 1 ) k x 2 α k Γ ( 1 + 2 α k )
sinh α ... ( a x α )	a z α 1 - a 2 z 2 α	sinh α ... x α = ∑ k = 0 ∞ x α ( 2 k + 1 ) Γ [ 1 + α ( 2 k + 1 ) ]
cosh α ... ( a x α )	1 1 - a 2 z 2 α	cosh α ... x α = ∑ k = 0 ∞ x 2 α k Γ ( 1 + 2 α k )

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: [figure omitted; refer to PDF] subject to the initial value condition [figure omitted; refer to PDF] Taking the local fractional Sumudu transform gives [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Making use of (43), we obtain [figure omitted; refer to PDF] Hence, from (45), we get [figure omitted; refer to PDF] and we draw its graphs as shown in Figure 1.

Figure 1: The plot of nondifferentiable solution of (41) with the parameter α = ln ... 2 / ln ... 3 .

[figure omitted; refer to PDF]

Example 2.

We consider the following initial value problems: [figure omitted; refer to PDF] and the initial boundary value reads as [figure omitted; refer to PDF] Taking the local fractional Sumudu transform, from (47) and (48), we have [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] Therefore, the nondifferentiable solution of (47) is [figure omitted; refer to PDF] and we draw its graphs as shown in Figure 2.

Figure 2: The plot of nondifferentiable solution of (47) with the parameter α = ln ... 2 / ln ... 3 .

[figure omitted; refer to PDF]

Example 3.

We give the following initial value problems: [figure omitted; refer to PDF] together with the initial value conditions [figure omitted; refer to PDF] Taking the local fractional Sumudu transform, from (52), we obtain [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] Therefore, form (55), we give the nondifferentiable solution of (52) [figure omitted; refer to PDF] and we draw its graphs as shown in Figure 3.

Figure 3: The plot of nondifferentiable solution of (52) with the parameter α = ln ... 2 / ln ... 3 .

[figure omitted; refer to PDF]

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

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

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