# The Yang-Laplace Transform for Solving the IVPs with Local Fractional Derivative

Authors: Chun-Guang Zhao; Ai-Min Yang; Hossein Jafari; Ahmad Haghbin
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The IVPs with local fractional derivative are considered in this paper. Analytical solutions for the homogeneous and nonhomogeneous local fractional differential equations are discussed by using the Yang-Laplace transform.


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## Body

## 1. Introduction

In recent years, the ordinary and partial differential equations have found applications in many problems in mathematical physics [1, 2]. Initial value problems (IVPs) for ordinary and partial differential equations have been developed by some authors in [3–6]. There are analytical methods and numerical methods for solving the differential equations, such as the finite element method [6], the harmonic wavelet method [7–9], the Adomian decomposition method [10–12], the homotopy analysis method [13, 14], the homotopy decomposition method [15, 16], the heat balance integral method [17, 18], the homotopy perturbation method [19], the variational iteration method [20], and other methods [21].Recently, owing to limit of classical and fractional differential equations, the local fractional differential equations have been applied to describe nondifferentiable problems for the heat and wave in fractal media [22, 23], the structure relation in fractal elasticity [24], and Fokker-Planck equation in fractal media [25]. Some methods were utilized to solve the local fractional differential equations. For example, the local fractional variation iteration method was used to solve the heat conduction in fractal media [26, 27]. The local fractional decomposition method for solving the local fractional diffusion and heat-conduction equations was considered in [28, 29]. The local fractional series expansion method for solving the Schrödinger equation with the local fractional derivative was presented [30]. The Yang-Laplace transform structured in 2011 [22] was suggested to deal with local fractional differential equations [31, 32]. The coupling method for variational iteration method within Yang-Laplace transform for solving the heat conduction in fractal media was proposed in [33].In this paper, our aim is to use the Yang-Laplace transform to solve IVPs with local fractional derivative. The structure of the paper is as follows. In Section2, some definitions and properties for the Yang-Laplace transform are given. Section 3 is devoted to the solutions for the homogeneous and nonhomogeneous IVPs with local fractional derivative. Finally, conclusions are presented in Section 4.


## 2. Yang-Laplace Transform

In this section we show some definitions and properties for the Yang-Laplace transform.The local fractional integral operator is defined as [22, 23, 26–33](1)Iab(α)f(x)=1Γ(1+α)∫abf(t)(dt)α=1Γ(1+α)limΔt→0∑j=0j=N-1f(tj)(Δtj)α,
where Δtj=tj+1-tj, Δt=max{Δt0,Δt1,…,Δtj,…}, [tj,tj+1], j=0,…,N-1, t0=a,tN=b, is a partition of the interval [a,b].As the inverse operator of (1), the local fractional derivative operator is given by [22, 23, 26–33]
(2)f(α)(x0)=dαf(x)dxα|x=x0=limx→x0Δα(f(x)-f(x0))(x-x0)α,
with Δα(f(x)-f(x0))≅Γ(1+α)Δ(f(x)-f(x0)).The Yang-Laplace transform is expressed by [22, 31–33]
(3)L~α{f(x)}=fsL~,α(s)=1Γ(1+α)∫0∞Eα(-sαxα)f(x)(dx)α,0000000000000000000000000<α≤1,
where f(x) is a local fractional continuous function.The inverse Yang-Laplace transform reads as [22, 31–33]
(4)f(x)=L~α-1{fsL,α(s)}=1(2π)α×∫β-i∞β+i∞Eα(sαxα)fsL~,α(s)(ds)α,
where sα=βα+iα∞α and Re(sα)=βα.Some properties for Yang-Laplace transform are presented as follows [21, 22, 22–33]:
(5)L~α{af(x)+bg(x)}=aL~α{f(x)}+bL~α{g(x)},(6)L~α{f(nα)(x)}=snαL~α{f(x)}-∑k=1ns(k-1)αf(n-k)α(0),(7)limx→0f(x)=lims→∞sαF(s),(8)limx→∞f(x)=lims→0sαF(s),(9)L~α{f(ax)}=1aαfsL,α(sa),a>0,(10)L~α{xkαf(x)}=(-1)kdkαfsL,α(s)dskα,(11)L~α{f(x-c)}=fsL,α(s)Eα(-cαsα),(12)L~α{f(x)Eα(cαxα)}=fsL,α(s-c),(13)L~α{xkαEα(cαxα)}=Γ(1+kα)(s-c)(k+1)α,(14)L~α{sinα(cαxα)}=cαs2α+c2α,(15)L~α{cosα(cαxα)}=sαs2α+c2α,(16)L~α{xkα}=Γ(1+kα)s(k+1)α.


## 3. IVPs with Local Fractional Derivatives

In this section we handle the homogeneous and non-homogeneous IVPs with local fractional derivative.


### 3.1. Homogeneous IVPs with Local Fractional Derivative

Example 1.
The homogeneous IVPs with local fractional derivative are expressed by(17)d2αyd2αx-dαydαx+2y=0.
The initial boundary conditions are presented as(18)y(0)=1,y(α)(0)=0.
From (6) we have
(19)L~α{y(α)(x)}=sαL~α{y(x)}-y(0),(20)L~α{y(2α)(x)}=s2αL~α{y(x)}-sαy(0)-f(α)(0).
Hence, making use of (19) and (20), (19) can be written as
(21)s2αL~α{y(x)}-sαy(0)-f(α)(0)-{sαL~α{y(x)}-y(0)}+2L~α{y(x)}=0.
Hence, we obtain(22)L~α{y(x)}=1sα+2y(0)=1sα+2.
So, making use of (13), we get the solution of (17):
(23)y(x)=Eα(-2xα).
The solution of (17) for α=ln2/ln3 is shown in Figure 1.Figure 1
Graph ofy(x) for α=ln2/ln3.Example 2.
Let us consider the homogeneous IVPs with local fractional derivative in the form(24)d4αyd4αx-y=0
subject to initial boundary conditions(25)y(0)=0,y(α)(0)=0,y(2α)(0)=0,y(3α)(0)=1.
From (6) we have
(26)L~α{y(4α)(x)}=s4αL~α{y(x)}-s3αy(0)-s2αy(α)(0)-sαy(2α)(0)-f(3α)(0),
so that(27)s4αL~α{y(x)}-s3αy(0)-s2αy(α)(0)-sαy(2α)(0)-f(3α)(0)-L~α{y(x)}=0.
Hence, (27) can be written as
(28)s4αL~α{y(x)}-1-L~α{y(x)}=0,
which leads to(29)L~α{y(x)}=1s4α-1.
Therefore, we get(30)y(x)=L~α-1{1s4α-1}=L~α-1{12(121sα-1-121sα+1-1s2α+1)}=14Eα(-xα)-14Eα(xα)-12sinα(xα).
The exact solution of (24) for α=ln2/ln3 is shown in Figure 2.Figure 2
Graph ofy(x) for α=ln2/ln3.


### 3.2. Nonhomogeneous IVPs with Local Fractional Derivative

Example 3.
We now consider the non-homogeneous IVPs with local fractional derivative(31)d2αyd2αx-y=sinα(xα)
subject to initial boundary conditions
(32)y(0)=0,y(α)(0)=1.
By using (6), we have
(33)L~α{y(2α)(x)}=s2αL~α{y(x)}-sαy(0)-f(α)(0),L~α{sinα(xα)}=1s2α+1so that
(34)L~α{y(x)}=34(1sα-1-1sα+1)-121s2α+1.
So,(35)y(x)=34Eα(-xα)-34Eα(xα)-12sinα(xα).
The exact solution of (31) for α=ln2/ln3 is shown in Figure 3.Figure 3
Graph ofy(x) for α=ln2/ln3.Example 4.
The non-homogeneous IVPs with local fractional derivative are(36)d2αyd2αx+y=Eα(xα).
The initial boundary conditions are(37)y(0)=1,y(α)(0)=0.
In view of (6), we give
(38)L~α{y(x)}=1(sα+1)(s2α+1)+sαs2α+1.
So, we obtain(39)y(x)=cosα(xα)+1Γ(1+α)∫0xEα(x-t)αsinα(tα)(dt)α=cosα(xα)+1Γ(1+α)×∫0xEα(tα)((tα)sinα(xα)cosα(tα)-cosα(xα)sinα(tα))(dt)α=cosα(xα)+sinα(xα){1Γ(1+α)∫0xEα(tα)cosα(tα)(dt)α}-cosα(xα){1Γ(1+α)∫0xEα(tα)sinα(tα)(dt)α}=cosα(xα)+sinα(xα){Eα(xα)[cosα(xα)+sin(xα)]-1}2-cosα(xα){Eα(xα)[sinα(xα)-cosα(xα)]+1}2=12[cosα(xα)-sinα(xα)+Eα(xα)].
The exact solution of (36) for α=ln2/ln3 is shown in Figure 4.Figure 4
Graph ofy(x) for α=ln2/ln3.


## 3.1. Homogeneous IVPs with Local Fractional Derivative

Example 1.
The homogeneous IVPs with local fractional derivative are expressed by(17)d2αyd2αx-dαydαx+2y=0.
The initial boundary conditions are presented as(18)y(0)=1,y(α)(0)=0.
From (6) we have
(19)L~α{y(α)(x)}=sαL~α{y(x)}-y(0),(20)L~α{y(2α)(x)}=s2αL~α{y(x)}-sαy(0)-f(α)(0).
Hence, making use of (19) and (20), (19) can be written as
(21)s2αL~α{y(x)}-sαy(0)-f(α)(0)-{sαL~α{y(x)}-y(0)}+2L~α{y(x)}=0.
Hence, we obtain(22)L~α{y(x)}=1sα+2y(0)=1sα+2.
So, making use of (13), we get the solution of (17):
(23)y(x)=Eα(-2xα).
The solution of (17) for α=ln2/ln3 is shown in Figure 1.Figure 1
Graph ofy(x) for α=ln2/ln3.Example 2.
Let us consider the homogeneous IVPs with local fractional derivative in the form(24)d4αyd4αx-y=0
subject to initial boundary conditions(25)y(0)=0,y(α)(0)=0,y(2α)(0)=0,y(3α)(0)=1.
From (6) we have
(26)L~α{y(4α)(x)}=s4αL~α{y(x)}-s3αy(0)-s2αy(α)(0)-sαy(2α)(0)-f(3α)(0),
so that(27)s4αL~α{y(x)}-s3αy(0)-s2αy(α)(0)-sαy(2α)(0)-f(3α)(0)-L~α{y(x)}=0.
Hence, (27) can be written as
(28)s4αL~α{y(x)}-1-L~α{y(x)}=0,
which leads to(29)L~α{y(x)}=1s4α-1.
Therefore, we get(30)y(x)=L~α-1{1s4α-1}=L~α-1{12(121sα-1-121sα+1-1s2α+1)}=14Eα(-xα)-14Eα(xα)-12sinα(xα).
The exact solution of (24) for α=ln2/ln3 is shown in Figure 2.Figure 2
Graph ofy(x) for α=ln2/ln3.


## 3.2. Nonhomogeneous IVPs with Local Fractional Derivative

Example 3.
We now consider the non-homogeneous IVPs with local fractional derivative(31)d2αyd2αx-y=sinα(xα)
subject to initial boundary conditions
(32)y(0)=0,y(α)(0)=1.
By using (6), we have
(33)L~α{y(2α)(x)}=s2αL~α{y(x)}-sαy(0)-f(α)(0),L~α{sinα(xα)}=1s2α+1so that
(34)L~α{y(x)}=34(1sα-1-1sα+1)-121s2α+1.
So,(35)y(x)=34Eα(-xα)-34Eα(xα)-12sinα(xα).
The exact solution of (31) for α=ln2/ln3 is shown in Figure 3.Figure 3
Graph ofy(x) for α=ln2/ln3.Example 4.
The non-homogeneous IVPs with local fractional derivative are(36)d2αyd2αx+y=Eα(xα).
The initial boundary conditions are(37)y(0)=1,y(α)(0)=0.
In view of (6), we give
(38)L~α{y(x)}=1(sα+1)(s2α+1)+sαs2α+1.
So, we obtain(39)y(x)=cosα(xα)+1Γ(1+α)∫0xEα(x-t)αsinα(tα)(dt)α=cosα(xα)+1Γ(1+α)×∫0xEα(tα)((tα)sinα(xα)cosα(tα)-cosα(xα)sinα(tα))(dt)α=cosα(xα)+sinα(xα){1Γ(1+α)∫0xEα(tα)cosα(tα)(dt)α}-cosα(xα){1Γ(1+α)∫0xEα(tα)sinα(tα)(dt)α}=cosα(xα)+sinα(xα){Eα(xα)[cosα(xα)+sin(xα)]-1}2-cosα(xα){Eα(xα)[sinα(xα)-cosα(xα)]+1}2=12[cosα(xα)-sinα(xα)+Eα(xα)].
The exact solution of (36) for α=ln2/ln3 is shown in Figure 4.Figure 4
Graph ofy(x) for α=ln2/ln3.


## 4. Conclusions

In this work we have used the Yang-Laplace transform to handle the homogeneous and non-homogeneous IVPs with looselocal fractional derivative. Some illustrative examples of approximate solutions for local fractional IVPs are discussed. The nondifferentiable solutions for fractal dimensionα=ln2/ln3 are shown graphically. The obtained results illustrate that the Yang-Laplace transform is an efficient mathematical tool to solve the homogeneous and non-homogeneous IVPs with local fractional derivative.



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