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.
Let f(x)∈Cα(a,b). The local fractional derivative of f(x) of order α in the interval [a,b] is defined as
(2)dαf(x0)dxα=Δα(f(x)-f(x0))(x-x0)α,
where
(3)Δα(f(x)-f(x0))≅Γ(1+α)[f(x)-f(x0)].
The local fractional partial differential operator of order α(0<α≤1) was given by [20, 21]
(4)∂α∂tαu(x0,t)=Δα(u(x0,t)-u(x0,t0))(t-t0)α,
where
(5)Δα(u(x0,t)-u(x0,t0))≅Γ(1+α)[u(x0,t)-u(x0,t0)].
Let f(x)∈Cα[a,b]. The local fractional integral of f(x) of order α in the interval [a,b] is defined as
(6)Ib(α)af(x)=1Γ(1+α)∫abf(t)(dt)α=1Γ(1+α)limΔt→0∑j=0j=N-1f(tj)(Δtj)α,
where the partitions of the interval [a,b] are denoted as (tj,tj+1), j=0,…,N-1, t0=a, and tN=b with Δtj=tj+1-tj and Δt=max{Δt0,Δt1,Δtj,…}.
Suppose that f((k+1)α)(x)∈Cα(a,b), for k=0,1,…,n and 0<α≤1. Then, one has
(7)f(x)=∑k=0nf(kα)(x0)Γ(1+kα)(x-x0)kα+f((n+1)α)(ξ)Γ(1+(n+1)α)(x-x0)(n+1)α
with a<x0<ξ<x<b, ∀x∈(a,b), where(8)f((k+1)α)(x)=Dx(α)…Dx(α)︷k+1timesf(x).
Proof (see [<xref ref-type="bibr" rid="B21">20</xref>, <xref ref-type="bibr" rid="B22">21</xref>]).
Local fractional Mc-Laurin’s series of the Mittag-Leffler functions on Cantor sets is given by [20, 21]
(9)Eα(xα)=∑k=0∞xαkΓ(1+kα),x∈R,0<α≤1,
and local fractional Mc-Laurin’s series of the Mittag-Leffler functions on Cantor sets with the parameter ζ reads as follows:
(10)Eα(ζαxα)=∑k=0∞ζkαxαkΓ(1+kα),x∈R,0<α≤1.
As generalizations of (9) and (10), we have
(11)f(x)=∑k=0∞akxαk,
where ak(k=0,1,2,…,n) are coefficients of the generalized polynomial function on Cantor sets.
Making use of (10), we get
(12)Eα(iαxα)=∑k=0∞ikαxαkΓ(1+kα),
where iα is the imaginary unit with Eα(iα(2π)α)=1.
Let us consider the polynomial function on Cantor sets in the form
(13)f(x)=∑k=0∞iαkxαk,
where |x|<1.
Hence, we have the closed form of (13) as follows:
(14)f(x)=11-iαxα.
Definition 5.
The local fractional Laplace transform of f(x) of order α is defined as [27–30]
(15)Lα{f(x)}=fsL,α(s)=1Γ(1+α)∫0∞Eα(-sαxα)f(x)(dx)α.
If Fα{f(x)}≡fωF,α(ω), the inverse formula of (42) is defined as [27–30]
(16)f(x)=Lα-1{fsL,α(s)}=1(2π)α∫β-i∞β+i∞Eα(sαxα)fsL,α(s)(ds)α,
where f(x) is local fractional continuous, sα=βα+iα∞α, and Re(s)=β>0.
Theorem 6 (see [<xref ref-type="bibr" rid="B22">21</xref>]).
If Lα{f(x)}=fsL,α(s), then one has
(17)Lα{f(α)(x)}=sαLα{f(x)}-f(0).
Proof.
See [21].
Theorem 7 (see [<xref ref-type="bibr" rid="B22">21</xref>]).
If Lα{f(x)}=fsL,α(s), then one has
(18)Lα{I0x(α)f(x)}=1sαLα{f(x)}.
Proof.
See [21].
Theorem 8 (see [<xref ref-type="bibr" rid="B22">21</xref>]).
If Lα{f1(x)}=fs,1L,α(s) and Lα{f2(x)}=fs,2L,α(s), then one has
(19)Lα{f1(x)*f2(x)}=fs,1L,α(s)fs,2L,α(s),
where
(20)f1(x)*f2(x)=1Γ(1+α)∫0∞f1(t)f2(x-t)(dt)α.
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 LFSα:f(x)→F(u), namely,
(21)LFSα{f(x)}=LFSα{∑k=0∞akxαk}=∑k=0∞Γ(1+kα)akzαk.
As typical examples, we have
(22)LFSα{Eα(iαxα)}=∑k=0∞iαkzαk,LFSα{xαΓ(1+α)}=zα.
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
(23)LFSα{f(x)}=Fα(z)=:1Γ(1+α)×∫0∞Eα(-z-αxα)f(x)zα(dx)α,0<α≤1.
Following (23), its inverse formula is defined as
(24)LFSα-1{Fα(z)}=f(x),0<α≤1.
Theorem 10 (linearity).
If LFSα{f(x)}=Fα(z) and LFSα{g(x)}=Gα(z), then one has
(25)LFSα{f(x)+g(x)}=Fα(z)+Gα(z).
Proof.
As a direct result of the definition of local fractional Sumudu transform, we get the following result.
If Lα{f(x)}=fsL,α(s) and LFSα{f(x)}=Fα(z), then one has
(26)LFSα{f(x)}=1zαLα{f(1x)},(27)Lα{f(x)}=LFSα[f(1/s)]sα.
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 LFSα{f(x)}=Fα(z), then one has
(28)LFSα{dαf(x)dxα}=Fα(z)-f(0)zα.
Proof.
From (17) and (26), the local fractional Sumudu transform of the local fractional derivative of f(x) read as
(29)LFSα{H(x)}=Lα{H(1/x)}zα=Lα{f(1/x)}/zα-f(0)zα=Fα(z)-f(0)zα,
where
(30)H(x)=dαf(x)dxα.
This completes the proof.
As the direct result of (28), we have the following results.
If LFSα{f(x)}=Fα(z), then we have
(31)LFSα{dnαf(x)dxnα}=1znα[Fα(z)-∑k=0n-1zkαf(kα)(0)].
When n=2, from (31), we get
(32)LFSα{d2αf(x)dx2α}=1z2α[Fα(z)-f(0)-zαf(α)(0)].
Theorem 13 (local fractional Sumudu transform of local fractional derivative).
IfLFSα{f(x)}=Fα(z), then one has
(33)LFSα{Ix(α)0f(x)}=zαFα(z).
Proof.
From (18) and (26), we have
(34)Lα{Ix(α)0f(x)}=1sαLα{f(x)}
so that
(35)LFSα{B(x)}=Lα{B(1/x)}zα=Lα{f(1x)}=zαFα(z),
where
(36)B(x)=Ix(α)0f(x).
This completes the proof.
Theorem 14 (local fractional convolution).
If LFSα{f(x)}=Fα(z) and LFSα{g(x)}=Gα(z), then one has
(37)LFSα{f(x)*g(x)}=zαFα(z)Gα(z),
where
(38)f(x)*g(x)=1Γ(1+α)∫0∞f(t)g(x-t)(dt)α.
Proof.
From (19) and (26), we have
(39)LFSα{f(x)*g(x)}=Lα{f(x)*g(x)}zα=Lα{f(1/x)}Lα{g(1/x)}zα=zαFα(z)Gα(z),
where
(40)Fα(z)=Lα{f(1/x)}zα,Gα(z)=Lα{g(1/x)}zα.
This completes the proof.
In the following, we present some of the basic formulas which are in 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
xαΓ(1+α)
zα
∑k=0∞akxαk
∑k=0∞Γ(1+kα)akzαk
Eα(axα)
11-azα
Eα(xα)=∑k=0∞xαkΓ(1+kα)
sinα(axα)
azα1+a2z2α
sinαxα=∑k=0∞(-1)kxα(2k+1)Γ[1+α(2k+1)]
cosα(axα)
11+a2z2α
cosαxα=∑k=0∞(-1)kx2αkΓ(1+2αk)
sinhα(axα)
azα1-a2z2α
sinhαxα=∑k=0∞xα(2k+1)Γ[1+α(2k+1)]
coshα(axα)
11-a2z2α
coshαxα=∑k=0∞x2αkΓ(1+2αk)
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:
(41)dαf(x)dxα=f(x),
subject to the initial value condition(42)f(0)=5.
Taking the local fractional Sumudu transform gives
(43)Fα(z)-f(0)zα=Fα(z),
where
(44)LFSα{f(x)}=Fα(z).
Making use of (43), we obtain
(45)Fα(z)=51-zα.
Hence, from (45), we get
(46)f(x)=5Eα(xα)
and we draw its graphs as shown in Figure 1.
The plot of nondifferentiable solution of (41) with the parameter α=ln2/ln3.
Example 2.
We consider the following initial value problems:
(47)dαf(x)dxα+f(x)=xαΓ(1+α)
and the initial boundary value reads as
(48)f(0)=-1.
Taking the local fractional Sumudu transform, from (47) and (48), we have
(49)Fα(z)-f(0)zα+Fα(z)=zα
so that
(50)Fα(z)=zα-1.
Therefore, the nondifferentiable solution of (47) is
(51)f(x)=xαΓ(1+α)-1
and we draw its graphs as shown in Figure 2.
The plot of nondifferentiable solution of (47) with the parameter α=ln2/ln3.
Example 3.
We give the following initial value problems:
(52)d2αf(x)dx2α=f(x),
together with the initial value conditions
(53)f(α)(0)=0,f(0)=2.
Taking the local fractional Sumudu transform, from (52), we obtain
(54)1z2α[Fα(z)-f(0)-zαf(α)(0)]=Fα(z),
which leads to
(55)Fα(z)=f(0)+zαf(α)(0)1-z2α=21-z2α.
Therefore, form (55), we give the nondifferentiable solution of (52)
(56)f(x)=2coshα(xα),
and we draw its graphs as shown in Figure 3.
The plot of nondifferentiable solution of (52) with the parameter α=ln2/ln3.
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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