# Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative

Authors: Shun-Qin Wang; Yong-Ju Yang; Hassan Kamil Jassim
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We propose the local fractional function decomposition method, which is derived from the coupling method of local fractional Fourier series and Yang-Laplace transform. The forms of solutions for local fractional differential equations are established. Some examples for inhomogeneous wave equations are given to show the accuracy and efficiency of the presented technique.


---

## Body

## 1. Introduction

Fractional differential equations with arbitrary orders [1] have attracted more and more attention to their extensive applications in various areas, such as physics, applied mathematics, and biology [2–8]. As a result, great deal of methods for solving the fractional differential equations are developed [9–21], such as the heat balance integral method [9, 10], the homotopy analysis method [11], the variational iteration method [12], the homotopy decomposition method [13, 14], and the Adomian decomposition method [15, 16].The fractional differential equations were considered in sense of the Caputo derivative, the Riemann-Liouville derivative, and the Grünwald-Letnikov derivative [17]. However, they do not deal with the nondifferentiable functions defined on Cantor sets. Local fractional derivative [18, 19] is the best method for describing the nondifferential problems defined on Cantor sets. For example, the heat equations arising in fractal transient conduction were investigated in [19–22]. The Helmholtz and diffusion equations on the Cantor sets within local fractional derivative were discussed [23]. The Navier-Stokes equations on Cantor sets were suggested in [24]. There are some methods for solving the local fractional differential equations, such as the local fractional variational iteration method [20], the Yang-Fourier transform [21], the Yang-Laplace transform [22], the local fractional Fourier series method [25], and the local fractional Adomian decomposition method [26].In this paper, our aims are to present the coupling method of local fractional series method and Yang-Laplace transform, which is called as the local fractional function decomposition method, and to use it to solve the differential equations with local fractional derivative. The organization of the manuscript is as follows. In Section2, the basic mathematical tools are introduced. In Section 3, the local fractional function decomposition method for solving the differential equations with local fractional derivative is investigated. In Section 4, several examples are considered. Finally, in Section 5 the conclusions are given.


## 2. Mathematical Fundamentals

In this section, we introduce the basic notions of local fractional continuity, local fractional derivative, local fractional Fourier series, and special function in fractal space [18, 19], which are used in the paper.Definition 1.
Suppose that there is [19]
(1)|f(x)-f(x0)|<εα
with |x-x0|<δ, for ε,δ>0 and ε,δ∈R; then f(x) is called local fractional continuous at x=x0 and it is denoted by limx→x0f(x)=f(x0).Definition 2.
Suppose that the functionf(x) satisfies condition (1), for x∈(a,b); it is so-called local fractional continuous on the interval (a,b), denoted by
(2)f(x)∈Cα(a,b).Definition 3.
In fractal space, letf(x)∈Cα(a,b), local fractional derivative of f(x) of order α at x=x0 is given by [19]
(3)Dx(α)f(x0)=f(α)(x0)=dαf(x)dxα|x=x0=limx→x0Δα(f(x)-f(x0))(x-x0)α,
where Δα(f(x)-f(x0))≅Γ(1+α)Δ(f(x)-f(x0)).Local fractional derivative of high order and local fractional partial derivative are defined, respectively, in the following forms [18, 19]:
(4)f(kα)(x)=Dx(α)⋯Dx(α)︷ktimesf(x),∂kα∂xkαf(x,y)=∂α∂xα⋯∂α∂xα︷ktimesf(x,y).Definition 4.
In fractal space, the Mittage Leffler function, sine function, cosine function, hyperbolic sine function, and hyperbolic cosine function are, respectively, defined by [18, 19]
(5)Eα(xα)∶=∑K=0∞xαkΓ(1+kα),x∈R,0<α≤1,sinαxα∶=∑K=0∞(-1)kxα(2k+1)Γ[1+(2k+1)α],x∈R,0<α≤1,cosαxα∶=∑K=0∞(-1)kx2αkΓ(1+2kα),x∈R,0<α≤1,sinhαxα∶=Eα(xα)+Eα(-xα)2,x∈R,0<α≤1,coshαxα∶=Eα(xα)-Eα(-xα)2,x∈R,0<α≤1.Definition 5.
Letf(x) be 2l-periodic. For k∈Z and f(x)∈Cα(-∞,+∞), the local fraction Fourier series of f(x) is defined as (see [18, 25])
(6)f(x)=a02+∑k=1∞(akcosαπα(kx)αlα+bksinαπα(kx)αlα),
where
(7)ak=1lα∫-l+lf(x)cosαπα(kx)αlα(dx)α,bk=1lα∫-l+lf(x)sinαπα(kx)αlα(dx)α
are the local fraction Fourier coefficients.Definition 6.
Let(1/Γ(1+α))∫0∞|f(x)|(dx)α<k<∞. The Yang-Laplace transforms of f(x) is given by [18, 22]
(8)Lα{f(x)}=fsL,α(s)∶=1Γ(1+α)∫0∞Eα(-sαxα)f(x)(dx)α,00000000000000000000000000<α≤1,
where the latter integral converges and sα∈Rα.Definition 7.
The inverse formula of the Yang-Laplace transforms off(x) is given by [18, 22]:
(9)Lα-1{fsL,α(s)}=f(t)∶=1(2π)α∫β-i∞β+i∞Eα(sαtα)fsL,α(s)(ds)α,0000000000000000000000000<α≤1,
where sα=βα+iαωα; fractal imaginary unit iα  and  Re(s)=β>0.


## 3. Local Fractional Function Decomposition Method

In this section we will present the local fractional function decomposition method.At first, we present the local fractional differential equation(10)∂2αu(x,t)∂t2α+k1∂αu(x,t)∂tα+k2∂2αu(x,t)∂x2α+k3∂αu(x,t)∂xα=f(x,t)
with constants k1, k2, k3, 0<α≤1/2 and with boundary and initial conditions
(11)u(0,t)=u(l,t)=0,u(x,0)=φ(x),∂αu(x,o)∂tα=ψ(x).
Now we discuss the solution of (10).According to the decomposition of the local fractional function, with respect to the system{sinαnα(πx/l)α}, the following functions coefficients can be given by
(12)u(x,t)=∑n=1∞vn(t)sinαnα(πxl)α,f(x,t)=∑n=1∞fn(t)sinαnα(πxl)α,φ(x)=∑n=1∞Cnsinαnα(πxl)α,ψ(x)=∑n=1∞Dnsinαnα(πxl)α,
where
(13)fn(t)=2lα∫0lf(x,t)sinαnα(πxl)α(dx)α,Cn=2lα∫0lφ(x)sinαnα(πxl)α(dx)α,Dn=2lα∫0lφ(x)sinαnα(πxl)α(dx)α.
Substituting (12) into (10) implies that
(14)∂αvn(t)∂tα+k1∂αvn(t)∂tα+k2(nπl)2αvn(t)+k3(nπl)αvn(t)=fn(t),vn(0)=Cn,vn′(0)=Dn.
Suppose that the Yang-Laplace transforms of functions vn(t) and fn(t) are Vn(s) and Fn(s), respectively. Then we obtain
(15)s2αVn(s)-Cnsα-Dn+k1(sαVn(s)-Dn)+k2(nπl)2αVn(s)+k3(nπl)αVn(s)=Fn(s).
That is
(16)Vn(s)=Dn+k1Dn+Cnsαs2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α+Fn(s)s2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α.
Hence, we have
(17)vn(t)=Lα-1[Vn(s)]=1(2π)α∫β-i∞β+i∞Eα(sαtα)Vn(s)(ds)α=1(2π)α∫β-i∞β+i∞Eα(sαtα)000000000000×Fn(s)s2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α+1(2π)α∫β-i∞β+i∞Eα(sαtα)0000000000000×Dn+k1Dn+Cnsαs2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α.
Let
(18)vn(t)=v1,n(t)+v2,n(t),v1,n(t)=1(2π)α×∫β-i∞β+i∞Eα(sαtα)000000000×Fn(s)s2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α,v2,n(t)=1(2π)α×∫β-i∞β+i∞Eα(sαtα)000000000×Dn+k1Dn+Cnsαs2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α.
Hence, we get
(19)V1,n(s)=Fn(s)s2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α,V2,n(s)=Dn+k1Dn+Cnsαs2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α.
Then, making use of (8) and (9) and rearranging integration sequence, we have the  following  several  formulas about v1,n(t) and  v2,n(t).(I) Suppose that(20)-14k12+k2(nπl)2α+k3(nπl)α>0,s2α+k1sα+k2(nπl)2α+k3(nπl)α=(sβ+k12)2+Dn′2,
where Dn′=-(1/4)k12+k2(nπ/l)2α+k3(nπ/l)α.Then, we get(21)v1,n(t)=1Dn′Γ(1+α)×∫0tEα(-k1τα2α)sinαDn′τfn(t-τ)(dτ)α.
When
(22)V2,n(s)=Cn(sα+(k1/2))+(Dn+k1Dn-(k1/2)Cn)(sα+(k1/2))2+Dn′2,
we get
(23)v2,n(t)=CnEα(-k1tα2α)cosαD′tα+(Dn+k1Dn-k12Cn)Eα(-k1tα2α)sinαD′tα.(II) If(24)-14k12+k2(nπl)2α+k3(nπl)α=0,s2α+k1sα+k2(nπl)2α+k3(nπl)α=(sα+k12)2,
then we have
(25)v1,n(t)=1Γ(1+β)∫0tEα(-k1τα2α)ταfn(t-τ)(dτ)α.
When
(26)V2,n(s)=Cn(sα+(k1/2))+(Dn+k1Dn-(k1/2)Cn)(sα+(k1/2))2,
we arrive at
(27)v2,n(t)=CnΓ(1+α)Eα(-k1tα2α)+Dn+k1Dn-(k1/2)CnΓ(1+α)tαEα(-k1tα2α).(III)  Let(28)-14k12+k2(nπl)2α+k3(nπl)α<0,(29)s2α+k1sα+k2(nπl)2α+k3(nπl)α=(sα+k12)2-Dn′2,
where Dn′=(1/4)k12-k2(nπ/l)2α-k3(nπ/l)α.Then, we have(30)v1,n(t)=1Γ(1+α)Dn′×∫0tEα(-k1τα2α)shαDn′ταfn(t-τ)(dτ)β.
When
(31)V2,n(s)=Cn(sα+(k1/2))+(Dn+k1Dn-(k1/2)Cn)(sα+(k1/2))2-Dn′2=(k1+1)Dn+(Dn′-(k1/2))Cn2Dn′(sα+(k1/2)-Dn′)+(Dn′+(k1/2))Cn-(k1+1)Dn2Dn′(sα+(k1/2)+Dn′),
we obtain
(32)v2,n(t)=(k1+1)Dn+(Dn′-(k1/2))Cn2Dn′×Eα((Dn′-k12α)tα)+(Dn′+(k1/2))Cn-(k1+1)Dn2Dn′×Eα((-Dn′-k12α)tα).
The above results are the desired solutions.


## 4. Illustrative Examples

In order to illustrate the above results in Section3, we give the following several examples.Example 1.
The local fractional Laplace differential equation is written in the following form [18, 19]:
(33)∂2αu(x,t)∂x2α+∂2αu(x,t)∂t2α=0
subjected to the boundary and initial conditions described by
(34)u(x,0)=sinα(xα),∂αu(x,o)∂tα=sinα(xα),u(0,t)=u(π,t)=0.
From (33), the final solution can be easily deduced as follows:
(35)u(x,t)=sinα(xα)Eα(tα).Example 2.
We consider the following inhomogeneous wave equation with local fractional derivative:(36)∂2αu(x,t)∂t2α-∂2αu(x,t)∂x2α-sinαxα=0
subjected to the boundary and initial conditions
(37)u(x,0)=sinα(xα),∂αu(x,o)∂tα=0,u(0,t)=u(π,t)=0.
In order to find its solution, we suppose that
(38)u(x,t)=∑n=1∞vn(t)sinαnαxα,f(x,t)=sinαxα=∑n=1∞fn(t)sinαnαxα,φ(x)=sinαxα=∑n=1∞Cnsinαnαxα,ψ(x)=0=∑n=1∞Dnsinαnαxα,
which leads to
(39)fn(t)=0,(n≠1),f1(t)=1,Cn=0,(n≠1),C1=1,Dn=0.
Contrasting (37) with (35), we directly get
(40)k1=k3=0,k2=-1,Dn′=14k12-k2(nπl)2α-k3(nπl)α=1,(n=1),Dn′=0,(n≠1).
According to (30) and (32), we can derive
(41)vn(t)=0,(n≠1),(42)v1,1(t)=1Γ(1+α)D1′∫0tEα(-k1τα2α)shαD1′ταf1(t-τ)(dτ)α=1Γ(1+α)∫0tshατα(dτ)α=Eα(tα)+Eα(-tα)2-1,(43)v1,2(t)=Eα(tα)+Eα(-tα)2.
Conclusively, we get
(44)v1(t)=Eα(tα)+Eα(-tα)-1,vn(t)=0,(n≠1).
Thus, we obtain
(45)u(x,t)=[Eα(tα)+Eα(-tα)-1]sinα(xα).Example 3.
The inhomogeneous wave equation with local fractional differential operator is written in the following form:(46)∂2αu(x,t)∂t2α+∂2αu(x,t)∂x2α=1.
The boundary and initial conditions are described by
(47)u(x,0)=sinαxα,∂αu(x,o)∂tα=0.
In order to find the solution of (46), we set
(48)u(x,t)=∑n=1∞vn(t)sinαnαxα,f(x,t)=1=∑n=1∞fn(t)sinαnαxα,φ(x)=sinαxα=∑n=1∞Cnsinαnαxα,ψ(x)=0=∑n=1∞Dnsinαnαxα.
Hence, we get
(49)fn(t)=2[1-(-1)n]n,Cn=0(n≠1),C1=1,Dn=0.
Let
(50)s2α+n2α=(sα)2+Dn′2,Dn′=nα.
Making use of (21) and (23),we can write
(51)v1,n(t)=1Dn′Γ(1+α)2[1-(-1)n]nα∫0tsinα(nτ)α(dτ)α=(1-cosα(nτ)α)2[1-(-1)n]nα.
When
(52)V2,n(t)=Dn+k1Dn+Cnsαs2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α=sαs2α+n2α,
we obtain
(53)v2,1(t)=cosαtα,v2,n(t)=0,(n≠1).
Conclusively, we arrive at
(54)v1(t)=v1,1(t)+v2,1(t)=4-3cosαtα,vn(t)=(1-cosα(nt)α)2[1-(-1)n]nα,(n≠1).
Hence, we obtain the solution of (46) in the following form:
(55)u(x,t)=(4-3cosαtα)sinα(πxl)α+∑n=2∞(1-cosα(nt)α)2[1-(-1)n]nαsinαnα(πxl)α.Example 4.
The inhomogeneous wave equation with local fractional differential operator is written in the form(56)∂2αu(x,t)∂t2α-∂2αu(x,t)∂x2α+∂αu(x,t)∂xα-cosαtαcosαxα=0.
The boundary and initial conditions are presented as follows:
(57)u(x,0)=cosα(xα),∂αu(x,o)∂tα=0,u(-π2,t)=u(π2,t)=0.
Let
(58)u(x,t)=∑n=1∞vn(t)cosαnαxα,f(x,t)=cosαtαcosαxα=∑n=1∞fn(t)cosαnαxα,φ(x)=cosαxα=∑n=1∞Cncosαnαxα,ψ(x)=0=∑n=1∞Dncosαnαxα.
We can write
(59)fn(t)=0,(n≠1),f1(t)=cosαtα,Cn=0,(n≠1),C1=0,Dn=0.
Obviously, we have
(60)k1=0,k2=-1,k3=1,Dn′=0.
From (25) and (27) we obtain
(61)v1,1=vn(t)=0,(n≠1),v1,1(t)=1Γ(1+α)∫0tcosα(t-τ)α(dτ)α=sinαtα.
Hence, the nondifferentiable solution of (56) reads as
(62)u(x,t)=sinαtαcosα(xα).


## 5. Conclusions

In this work we proposed the local fractional function decomposition method. The applications of the methods for solving the inhomogeneous wave equations with local fractional derivative are discussed in detail. The new technique is an efficient mathematical tool for the scientists to deal with local fractional differential equations.



---

*Source: 176395-2014-01-02.xml*