# Analysis of Fractal Wave Equations by Local Fractional Fourier Series Method

Authors: Yong-Ju Yang; Dumitru Baleanu; Xiao-Jun Yang
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

The fractal wave equations with local fractional derivatives are investigated in this paper. The analytical solutions are obtained by using local fractional Fourier series method. The present method is very efficient and accurate to process a class of local fractional differential equations.


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## Body

## 1. Introduction

Fractional calculus deals with derivative and integrals of arbitrary orders [1]. During the last four decades, fractional calculus has been applied to almost every field of science and engineering [2–6]. In recent years, there has been a great deal of interest in fractional differential equations [7]. As a result, various kinds of analytical methods were developed [8–18]. For example, there are the exp-function method [8], the variational iteration method [9, 10], the homotopy perturbation method [11], the homotopy analysis method [12], the heat-balance integral method [13], the fractional variational iteration method [14, 15], the fractional difference method [16], the finite element method [17], the fractional Fourier and Laplace transforms [18], and so on.Recently, local fractional calculus was applied to deal with problems for nondifferentiable functions; see [19–26] and the references therein. There are also analytical methods for solving the local fractional differential equations, which are referred to in [27–34]. The local fractional series method [32–34] was applied to process the local fractional wave equation in fractal vibrating [32] and local fractional heat-conduction equation [33].More recently, the wave equation on the Cantor sets was considered as [21, 28]
(1)∂2αu(x,t)∂t2α=∂2αu(x,t)∂x2α.
Local damped wave equation was written in the form [30]
(2)∂2αu(x,t)∂t2α-∂αu(x,t)∂tα-∂2αu(x,t)∂x2α=f(x,t),
and local fractional dissipative wave equation in fractal strings was [31]
(3)∂2αu(x,t)∂t2α-∂αu(x,t)∂tα-∂2αu(x,t)∂x2α-∂αu(x,t)∂xα=f(x,t).In this paper, we investigate the application of local fractional series method for solving the following local fractional wave equation:(4)∂2αu(x,t)∂t2α-∂αu(x,t)∂tα-∂2αu(x,t)∂x2α=0,
where initial and boundary conditions are presented as
(5)u(0,t)=u(l,t)=∂αu(l,0)∂xα=0,u(x,0)=f(x),∂αu(x,0)∂tα=g(x).
The organization of the paper is as follows. In Section 2, the basic concepts of local fractional calculus and local fractional Fourier series are introduced. In Section 3, we present a local fractional Fourier series solution of wave equation with local fractional derivative. Two examples are shown in Section 4. Finally, Section 5 is devoted to our conclusions.


## 2. Mathematical Tools

In this section, we present some concepts of local fractional continuity, local fractional derivative, and local fractional Fourier series.Definition 1 (see [21, 28, 30–32]).
Suppose that there is(6)|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, where ρα|x-x0|α≤|f(x1)-f(x2)|≤τα|x-x0|α with ρ,τ>0.
Suppose that the functionf(x) satisfies the above properties of the local fractional continuity. Then the condition (6) for x∈(a,b) is denoted as
(7)f(x)∈Cα(a,b),
where dimHf(x)=α.Definition 2 (see [19–21]).
Letf(x)∈Cα(a,b). Local fractional derivative of f(x) of order α at x=x0 is given by
(8)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)).Definition 3 (see [19, 20, 32–34]).
Letf(x)∈Cα(-∞,+∞), and let f(x) be 2l-periodic. For k∈Z, local fraction Fourier series of f(x) is defined as
(9)f(x)=a02+∑k=1∞(ancosαπα(kx)αlαmmm+bnsinαπα(kx)αlα),
where the local fraction Fourier coefficients are
(10)an=1lα∫-llf(x)cosαπα(kx)αlα(dx)α,bn=1lα∫-llf(x)sinαπα(kx)αlα(dx)α,
with local fractional integral given by [21, 29–34]
(11)Ib(α)af(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{Δt1,Δt2,Δtj,…} and [tj,tj+1], j=0,…,N-1,t0=a, tN=b, is a partition of the interval [a,b].
In view of (10), the weights of the fractional trigonometric functions are expressed as follows:
(12)an=1/(Γ(1+α))∫-llf(x)cosαnα(πx/l)α(dx)α1/(Γ(1+α))∫-llcosα2nα(πx/l)α(dx)α,bn=1/(Γ(1+α))∫-llf(x)sinαnα(πx/l)α(dx)α1/(Γ(1+α))∫-llsinα2nα(πx/l)α(dx)α.Lemma 4 (see [21]).
Ifm and h are constant coefficients, then local fractional differential equation with constant coefficients
(13)d2αydx2α+mdαydxα+hy=0(m2-4h<0)
has a family of solution
(14)y(x)=AEα(-m-iα4h-m22xα)+BEα(-m+iα4h-m22xα)
with two constants A and B.Proof.
See [21].


## 3. Solution to Wave Equation with Local Fractional Derivative

If we have the particular solution of (4) in the form
(15)u(x,t)=ϕ(x)T(t),
then we get the equations
(16)ϕ(2α)+λ2αϕ=0;(17)T(2α)+T(α)+λ2αT=0.
with the boundary conditions
(18)ϕ(0)=ϕ(α)(l)=0.
Equation (4) has the solution
(19)ϕ(x)=C1cosαλαxα+C2sinαλαxα,
where C1 and C2 are all constant numbers.According to (19), for x=0 and x=l we get
(20)ϕ(0)=C1=0,ϕ(l)=ϕ(x)|x=l=C2sinαλαlα=0.
Obviously C2≠0, since otherwise ϕ(x)≡0.Hence, we arrive at(21)λnαlα=nαπα,
where n is an integer.We notice(22)λnα=(nπl)α(n=0,1,2,…),ϕn(x)=sinαλnαxα=sinαnα(πxl)α(n=0,1,2,…).Forλα=λnα and 0<ρ, following (17) implies that
(23)∑n=1∞Tn(t)=∑n=1∞Eα(-tα2)×(Ancosαρtα+Bnsinαρtα),
where
(24)ρ=4(nπ/l)2α-12.
Therefore,
(25)un(x,t)=ϕn(x)Tn(t)=Ancosαρtαsinαnα(πxl)αEα(-12tα)m+Bnsinαρtαsinαnα(πxl)αEα(-12tα).
We now suppose a local fractional Fourier series solution of (4):
(26)u(x,t)=∑n=1∞un(x,t)=∑n=1∞Eα(-tα2)m×(Ancosαρtα+Bnsinαρtα)sinαnα(πxl)α.
Therefore,
(27)∂αu(x,t)∂tα=∑n=1∞∂αun(x,t)∂tα,
where
(28)∂un(x,t)∂tα=-12Eα(-tα2)(Ancosαρtα+Bnsinαρtα)sinαnα(πxl)αmm+ρEα(-tα2)(-Ansinαρtα+Bncosαρtα)sinαnα(πxl)α,
with ρ=(4(nπ/l)2α-1)/2.Submitting (26) to (5), we have
(29)u(x,0)=∑n=1∞un(x,0)=∑n=1∞Ansinαnα(πxl)α=f(x),(30)∂αu(x,t)∂tα=∑n=1∞(-12An+ρBn)sinαnα(πxl)α=g(x).
So,
(31)∑n=1∞ρBnsinαnα(πxl)α=g(x)+∑n=1∞12Ansinαnα(πxl)α=g(x)+12f(x).
Let
(32)G(x)=g(x)+12f(x).
In view of (30) and (31), we rewrite
(33)∑n=1∞Ansinαnα(πxl)α=f(x),∑n=1∞ρBnsinαnα(πxl)α=G(x).
We now find the local fractional Fourier coefficients of f(x) and G(x), respectively,
(34)An=1/(Γ(1+α))∫0lf(x)sinαnα(πx/l)α(dx)α1/(Γ(1+α))∫0lsinα2nα(πx/l)α(dx)αmmmmmmmmmmmmm(n=0,1,2,…),ρBn=1/(Γ(1+α))∫0lG(x)sinαnα(πx/l)α(dx)α1/(Γ(1+α))∫0lsinα2nα(πx/l)α(dx)αmmmmmmmmmmmmm(n=0,1,2,…).
Following (34), we have
(35)1Γ(1+α)∫0lsinα2nα(πxl)α(dx)α=lα2Γ(1+α),
such that
(36)An=2∫0lf(x)sinαnα(πx/l)α(dx)αlα,Bn=2∫0lG(x)sinαnα(πx/l)α(dx)αρlα.
Thus, we get the solution of (4):
(37)u(x,t)=∑n=1∞un(x,t),
where
(38)un(x,t)=Eα(-tα2)×(Ancosαρtα+Bnsinαρtα)sinαnα(πxl)α,
with
(39)An=2∫0lf(x)sinαnα(πx/l)α(dx)αlα(n=0,1,2,…),Bn=2∫0lG(x)sinαnα(πx/l)α(dx)αρlα(n=0,1,2,…),
with
(40)G(x)=g(x)+12f(x).


## 4. Illustrative Examples

In order to illustrate the above result in this section, we give two examples.Let us consider (4) subject to initial and boundary conditions
(41)u(0,t)=u(l,t)=∂αu(l,0)∂xα=0,u(x,0)=f(x)=xαΓ(1+α),∂αu(x,0)∂tα=g(x)=xαΓ(1+α).
In view of (40), we have
(42)G(x)=g(x)+12f(x)=32xαΓ(1+α),
such that
(43)An=2∫0l(xα/Γ(1+α))sinαnα(πx/l)α(dx)αlα=2Γ(1+α)lαIl(α)0xαΓ(1+α)sinαnα(πxl)α=2Γ(1+α)(nπ)α{lαΓ(1+α)cosαnα(πxl)αmmmmmmmM-(lnπ)α[cosαnα(πxl)α-1]},(44)Bn=3∫0l(xα/Γ(1+α))sinαnα(πx/l)α(dx)αρlα=3Γ(1+α)ρlαIl(α)0xαΓ(1+α)sinαnα(πxl)α=-3Γ(1+α)ρ(nπ)α[lαΓ(1+α)cosαnα(πxl)αmmmmmmmmm-(lnπ)αsinαnα(πxl)α].
Hence,
(45)u(x,t)=∑n=1∞Eα(-tα2)m×(Ancosαρtα+Bnsinαρtα)sinαnα(πxl)α,
where
(46)An=-2Γ(1+α)ρ(nπ)α[lαΓ(1+α)cosαnα(πxl)αmmmmmm-(lnπ)αsinαnα(πxl)α],Bn=-3Γ(1+α)ρ(nπ)α[lαΓ(1+α)cosαnα(πxl)αmmmmmm-(lnπ)αsinαnα(πxl)α].
In view of (4), our second example is initial and boundary conditions as follows:
(47)u(0,t)=u(l,t)=∂αu(l,0)∂xα=0,u(x,0)=f(x)=xαΓ(1+α),∂αu(x,0)∂tα=g(x)=0.
Following (40), we get
(48)G(x)=12xαΓ(1+α).
Hence, we obtain
(49)An=2∫0l(xα/Γ(1+α))sinαnα(πx/l)α(dx)αlα=2Γ(1+α)lαIl(α)0xαΓ(1+α)sinαnα(πxl)α=-2Γ(1+α)ρ(nπ)α[lαΓ(1+α)cosαnα(πxl)αmmmmmmmmm-(lnπ)αsinαnα(πxl)α],Bn=∫0l(xα/Γ(1+α))sinαnα(πx/l)α(dx)αρlα=Γ(1+α)ρlαIl(α)0xαΓ(1+α)sinαnα(πxl)α=-Γ(1+α)ρ(nπ)α[lαΓ(1+α)cosαnα(πxl)αmmmmmmmm-(lnπ)αsinαnα(πxl)α].
So,
(50)u(x,t)=∑n=1∞Eα(-tα2)×(Ancosαρtα+Bnsinαρtα)sinαnα(πxl)α,
with
(51)An=2Γ(1+α)(nπ)α{lαΓ(1+α)sinαnα(πxl)αmmmmmmm-(lnπ)α[cosαnα(πxl)α-1]},Bn=-Γ(1+α)ρ(nπ)α[lαΓ(1+α)cosαnα(πxl)αmmmmmmm-(lnπ)αsinαnα(πxl)α].We notice that fraction boundary condition is expressed as a Lebesgue-Cantor staircase function [21, 32]; namely,
(52)u(x,0)=f(x)=Hα(C∩(0,x))=Ix(α)01=xαΓ(1+α),
where C is any fractal set and the fractal dimension of xα/Γ(1+α) is α. For x∈[0,1] the graph of the Lebesgue-Cantor staircase function (52) is shown in Figure 1 when fractal dimension is α=ln2/ln3.Figure 1
Forα=ln2/ln3, graph of a Lebesgue-Cantor staircase function shown at x∈[0,1].


## 5. Conclusions

The present work expresses the local fractional Fourier series solution to wave equations with local fractional derivative. Two examples are given to illustrat approximate solutions for wave equations with local fractional derivative resulting from local fractional Fourier series method. The results obtained from the local fractional analysis seem to be general since the obtained solutions go back to the classical one when fractal dimensionα=1; namely, it is a process from fractal geometry to Euclidean geometry. Local fractional Fourier series method is one of very efficient and powerful techniques for finding the solutions of the local fractional differential equations. It is also worth noting that the advantage of the local fractional differential equations displays the nondifferential solutions, which show the fractal and local behaviors of moments. However, the classical Fourier series is used to handle the continuous functions.



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