# Local Fractional Adomian Decomposition and Function Decomposition Methods for Laplace Equation within Local Fractional Operators

Authors: Sheng-Ping Yan; Hossein Jafari; Hassan Kamil Jassim
Journal: Advances in Mathematical Physics
 (2014)

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## Abstract

We perform a comparison between the local fractional Adomian decomposition and local fractional function decomposition methods applied to the Laplace equation. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.


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## Body

## 1. Introduction

Many problems of physics and engineering are expressed by ordinary and partial differential equations, which are termed boundary value problems. We can mention, for example, the wave, the Laplace, the Klein-Gordon, the Schrodinger, the Advection, the Burgers, the Boussinesq, and the Fisher equations, and others [1].Several analytical and numerical techniques were successfully applied to deal with differential equations, fractional differential equations, and local fractional differential equations [1–10]. The techniques include the heat-balance integral [11], the fractional Fourier [12], the fractional Laplace transform [12], the harmonic wavelet [13, 14], the local fractional Fourier and Laplace transform [15], local fractional variational iteration [16–18], the local fractional decomposition [19], and the generalized local fractional Fourier transform [20] methods.In this paper, we investigate the application of local fractional Adomian decomposition method and local fractional function decomposition method for solving the local fractional Laplace equation [21, 22] with the different fractal conditions.This paper is organized as follows. In Section2, the basic mathematical tools are reviewed. Section 3 presents briefly the local fractional Adomian decomposition method and the local fractional function decomposition method. Section 4 presents solutions to the local fractional Laplace equation with differential fractal conditions.


## 2. Mathematical Fundamentals

We recall in this section the notations and some properties of the local fractional operators [15–20, 23, 24].Definition 1 (see [15–20, 23, 24]).
The functionf(x) is local fractional continuous at x=x0, if it is valid for
(1)|f(x)-f(x0)|<εα,0<α≤1
with |x-x0|<δ, for ε>0 and ε∈R. For x∈(a,b), it is so called local fractional continuous on the interval (a,b), denoted by f(x)∈Cα(a,b).We notice that there are existence conditions of local fractional continuities that operating functions are right-hand and left-hand local fractional continuities. Meanwhile, the right-hand local fractional continuity is equal to its left-hand local fractional continuity. For more details, see [20].Definition 2 (see [15–20, 23, 24]).
The local fractional derivative off(x) at x=x0 is defined as
(2)Dxαf(x0)=dαdxαf(x)|x=x0=f(α)(x)=lim⁡x→x0Δα(f(x)-f(x0))(x-x0)α,
where Δα(f(x)-f(x0))≅Γ(α+1)Δ(f(x)-f(x0)).Local fractional derivative of high order is written in the form(3)f(kα)(x)=DxαDxα⋯Dxα︷ktimesf(x).
And local fractional partial derivative of high order is written in the form
(4)∂kαf(x,y)xkα=∂α∂xα∂α∂xα⋯∂α∂xα︷ktimesf(x,y).Definition 3 (see [15–20, 23, 24]).
A partition of the interval[a,b] is denoted by (tj,tj+1), j=0,…,N-1, t0=a and tN=b with Δtj=tj+1-tj and Δt=max⁡{Δt0,Δt1,…}. Local fractional integral of f(x) in the interval [a,b] is given by
(5)Ib(α)af(x)=1Γ(1+α)∫abf(t)(dt)α=1Γ(1+α)lim⁡Δt→0∑j=0N-1f(tj)(Δtj)α.
If the functions are local fractional continuous, then the local fractional derivatives and integrals exist. Some properties of local fractional derivative and integrals are given in [20].Definition 4.
Letf(x) be 2l-periodic. For k∈Z and f(x)∈Cα(a,b), the local fraction Fourier series of f(x) is defined as (see [15, 25])
(6)f(x)=a02+∑k=1∞(akcos⁡απα(kx)αlα+bksinαπα(kx)αlα),
where ak=(1/lα)∫-11f(x)cos⁡α(πα(kx)α/lα)(dx)α(7)bk=1lα∫-11f(x)sinαπα(kx)αlα(dx)α
are local fractional Fourier coefficients.Definition 5.
Let(1/Γ(1+α))∫0∞|f(x)|(dx)α<k<∞. The Yang-Laplace transforms of f(x) are given by [15, 22]
(8)Lα{f(x)}=fsL,α(s)=1Γ(1+α)∫0∞Eα(-sαxα)f(x)(dx)α,aaaaaaaaaaaaaaaaaaaaa0<α≤1,
where the latter integral converges and sα∈Rα.Definition 6.
The inverse formula of the Yang-Laplace transforms off(x) is given by [15, 22]
(9)Lα-1{fsL,α(s)}=f(t)=1(2π)α∫β-iωβ+iωEα(sαxα)fsL,α(s)(ds)α,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa0<α≤1,
where sα=βα+iαωα; fractal imaginary unit iαand Re(s)=β>0.


## 3. Analytical Methods

In order to illustrate two analytical methods, we investigate the nonlinear local fractional equation of order2α as follows:
(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 and with boundary and initial conditions
(11)u(0,t)=u(l,t)=0u(x,0)=φ(x)∂αu(x,0)∂tα=ψ(x).


### 3.1. Local Fractional Adomian Decomposition Method

We rewrite (10) in the following form:
(12)Ltt(2α)u(x,t)+k1Lt(α)u(x,t)+k2Lxx(2α)u(x,t)+k3Lx(α)u(x,t)=f(x,t).
Applying the inverse operator Ltt(-2α) to both sides of (12) yields
(13)Ltt(-2α)Ltt(2α)u(x,t)=Ltt(-2α)(-k1Ls(α)u(x,s)-k2Lxx(2α)u(x,s)aaaaaaaaaa-k3Lx(α)u(x,s)+f(x,s))u(x,t)=r(x,t)+Ltt(-2α)(f(x,s))+Ltt(-2α)(-k1Ls(α)u(x,s)-k2Lxx(2α)u(x,s)aaaaaaaaaaaaaa-k3Lx(α)u(x,s)),
where the term r(x,t) is to be determined from the fractal initial conditions.Now, we decompose the unknown functionu(x,t) as a sum of components defined by the series:
(14)u(x,t)=∑n=0∞un(x,t).
The components un(x,t) are obtained by the recursive formula:
(15)u0(x,t)=r(x,t)+Ltt(-2α)(f(x,s))un+1(x,t)=Ltt(-2α)(-k1Ls(α)un(x,s)-k2Lxx(2α)un(x,s)-k3Lx(α)un(x,s)),n≥0.


### 3.2. Local Fractional Function Decomposition Method

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
(16)u(x,t)=∑n=1∞vn(t)sinαnα(πxl)αf(x,t)=∑n=1∞fn(t)sinαnα(πxl)αφ(x)=∑n=1∞Cnsinαnα(πxl)αψ(x)=∑n=1∞Dnsinαnα(πxl)α,
where
(17)fn(t)=2lα∫01f(x,t)sinαnα(πxl)α(dx)αCn=2lα∫01φ(x)sinαnα(πxl)α(dx)αDn=2lα∫01φ(x)sinαnα(πxl)α(dx)α.
Substituting (16) into (10) implies that
(18)∂2αvn(t)∂t2α+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
(19)s2αVn(s)-Cnsα-Dn+k1(sαVn(s)-Cn)+k2(nπl)2αVn(s)+k3(nπl)αVn(s)=Fn(s).
That is,
(20)Vn(s)=Dn+k1Cn+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
(21)vn(t)=Lα-1{Vn(s)}=1(2π)α∫β-iωβ+iωEα(sαxα)Vn(s)(ds)α=1(2π)α∫β-iωβ+iωEα(sαxα)×Fn(s)s2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α+1(2π)α∫β-iωβ+iωEα(sαxα)×Dn+k1Cn+Cnsαs2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α.
Let vn(t)=v1,n(t)+v2,n(t)(22)v1,n(t)=1(2π)α∫β-iωβ+iωEα(sαxα)×Fn(s)s2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α(23)v2,n(t)=1(2π)α∫β-iωβ+iωEα(sαxα)×Dn+k1Cn+Cnsαs2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α.
Hence, we get
(24)V1,n(s)=Fn(s)s2α+k1sα+k2(nπ/l)2α+k3(nπ/l)αV2,n(s)=Dn+k1Cn+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).If-(1/4)k12+k2(nπ/l)2α+k3(nπ/l)α>0, then
(25)s2α+k1s+k2(nπl)2α+k3(nπl)α=(sα+k12)2+Dn′,
where Dn′=-(1/4)k12+k2(nπ/l)2α+k3(nπ/l)α.Then, we get(26)v1,n(t)=1Γ(1+α)Dn′×∫0tEα(-k1τα2α)sinα(Dn′τα)fn(t-τ)(dτ)αv2,n(t)=CnEα(-k1tα2α)cos⁡α(Dn′tα)+(Dn+k1Cn-k12)Eα(-k1tα2α)sinα(Dn′tα).
In case -(1/4)k12+k2(nπ/l)2α+k3(nπ/l)α<0 and -(1/4)k12+k2(nπ/l)2α+k3(nπ/l)α=0, see [26].


## 3.1. Local Fractional Adomian Decomposition Method

We rewrite (10) in the following form:
(12)Ltt(2α)u(x,t)+k1Lt(α)u(x,t)+k2Lxx(2α)u(x,t)+k3Lx(α)u(x,t)=f(x,t).
Applying the inverse operator Ltt(-2α) to both sides of (12) yields
(13)Ltt(-2α)Ltt(2α)u(x,t)=Ltt(-2α)(-k1Ls(α)u(x,s)-k2Lxx(2α)u(x,s)aaaaaaaaaa-k3Lx(α)u(x,s)+f(x,s))u(x,t)=r(x,t)+Ltt(-2α)(f(x,s))+Ltt(-2α)(-k1Ls(α)u(x,s)-k2Lxx(2α)u(x,s)aaaaaaaaaaaaaa-k3Lx(α)u(x,s)),
where the term r(x,t) is to be determined from the fractal initial conditions.Now, we decompose the unknown functionu(x,t) as a sum of components defined by the series:
(14)u(x,t)=∑n=0∞un(x,t).
The components un(x,t) are obtained by the recursive formula:
(15)u0(x,t)=r(x,t)+Ltt(-2α)(f(x,s))un+1(x,t)=Ltt(-2α)(-k1Ls(α)un(x,s)-k2Lxx(2α)un(x,s)-k3Lx(α)un(x,s)),n≥0.


## 3.2. Local Fractional Function Decomposition Method

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
(16)u(x,t)=∑n=1∞vn(t)sinαnα(πxl)αf(x,t)=∑n=1∞fn(t)sinαnα(πxl)αφ(x)=∑n=1∞Cnsinαnα(πxl)αψ(x)=∑n=1∞Dnsinαnα(πxl)α,
where
(17)fn(t)=2lα∫01f(x,t)sinαnα(πxl)α(dx)αCn=2lα∫01φ(x)sinαnα(πxl)α(dx)αDn=2lα∫01φ(x)sinαnα(πxl)α(dx)α.
Substituting (16) into (10) implies that
(18)∂2αvn(t)∂t2α+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
(19)s2αVn(s)-Cnsα-Dn+k1(sαVn(s)-Cn)+k2(nπl)2αVn(s)+k3(nπl)αVn(s)=Fn(s).
That is,
(20)Vn(s)=Dn+k1Cn+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
(21)vn(t)=Lα-1{Vn(s)}=1(2π)α∫β-iωβ+iωEα(sαxα)Vn(s)(ds)α=1(2π)α∫β-iωβ+iωEα(sαxα)×Fn(s)s2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α+1(2π)α∫β-iωβ+iωEα(sαxα)×Dn+k1Cn+Cnsαs2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α.
Let vn(t)=v1,n(t)+v2,n(t)(22)v1,n(t)=1(2π)α∫β-iωβ+iωEα(sαxα)×Fn(s)s2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α(23)v2,n(t)=1(2π)α∫β-iωβ+iωEα(sαxα)×Dn+k1Cn+Cnsαs2α+k1sα+k2(nπ/l)2α+k3(nπ/l)α(ds)α.
Hence, we get
(24)V1,n(s)=Fn(s)s2α+k1sα+k2(nπ/l)2α+k3(nπ/l)αV2,n(s)=Dn+k1Cn+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).If-(1/4)k12+k2(nπ/l)2α+k3(nπ/l)α>0, then
(25)s2α+k1s+k2(nπl)2α+k3(nπl)α=(sα+k12)2+Dn′,
where Dn′=-(1/4)k12+k2(nπ/l)2α+k3(nπ/l)α.Then, we get(26)v1,n(t)=1Γ(1+α)Dn′×∫0tEα(-k1τα2α)sinα(Dn′τα)fn(t-τ)(dτ)αv2,n(t)=CnEα(-k1tα2α)cos⁡α(Dn′tα)+(Dn+k1Cn-k12)Eα(-k1tα2α)sinα(Dn′tα).
In case -(1/4)k12+k2(nπ/l)2α+k3(nπ/l)α<0 and -(1/4)k12+k2(nπ/l)2α+k3(nπ/l)α=0, see [26].


## 4. Solutions of Local Fractional Laplace Equation in Fractal Time-Space

In this section, two examples for Laplace equation are presented in order to demonstrate the simplicity and the efficiency of the above methods.The local fractional Laplace equation (see [21]) is one of the important differential equations with local fractional derivatives. In the following, we consider solutions to local fractional Laplace equations in fractal time-space.Example 7.
Consider the following local fractional Laplace equation:(27)∂2αu(x,t)∂t2α+∂2u(x,t)∂x2α=0
subject to the fractal value conditions
(28)u(x,0)=-Eα(xα),∂αu(x,0)∂tα=0.
According to formula (15), we have
(29)u0(x,t)=r(x,t)+Ltt(-2α)(f(x,s)),un+1(x,t)=Ltt(-2α)(-k1Ls(α)un(x,s)-k2Lxx(2α)un(x,s)-k3Lx(α)un(x,s)),
where
(30)u0(x,t)=-Eα(xα).
Hence, from (29) we obtain
(31)un+1(x,t)=Ltt(-2α){-Lxx(2α)un(x,s)},=It(α)0It(α)0{-∂2αun(x,s)∂x2α},n≥1,
where
(32)u0(x,t)=-Eα(xα).
Making use of (31), we present
(33)u1(x,t)=It(α)0It(α)0{-∂2αu0(x,s)∂x2α}=It(α)0It(α)0{Eα(xα)}=t2αΓ(1+2α)Eα(xα)u2(x,t)=It(α)0It(α)0{-∂2αu1(x,s)∂x2α}=It(α)0It(α)0{-t2αΓ(1+2α)Eα(xα)}=-t4αΓ(1+4α)Eα(xα)u3(x,t)=It(α)0It(α)0{-∂2αu2(x,s)∂x2α}=It(α)0It(α)0{t4αΓ(1+4α)Eα(xα)}=t6αΓ(1+6α)Eα(xα).
Proceeding in this manner, we get
(34)un(x,t)=Eα(xα)(-1)n+1t2nαΓ(1+2nα).
Thus, the final solution reads as follows:
(35)u(x,t)=∑n=0∞un(x,t)=u0(x,t)+u1(x,t)+u2(x,t)+⋯=Eα(xα)[-1+t2αΓ(1+2α)-t4αΓ(1+4α)+t6αΓ(1+6α)⋯]=-Eα(xα)[1-t2αΓ(1+2α)+t4αΓ(1+4α)-t6αΓ(1+6α)⋯]=-Eα(xα)cos⁡α(tα).Now, we solve Example7 by using the local fractional function decomposition method.We suppose that(36)u(x,t)=∑n=1∞vn(t)Eα(nαxα)f(x,t)=∑n=1∞fn(t)Eα(nαxα)φ(x)=∑n=1∞CnEα(nαxα)ψ(x)=∑n=1∞DnEα(nαxα)
which leads to
(37)fn(t)=0,∀n;Cn=0,n≠1;C1=-1;Dn=0,∀n.
Contrasting (28) with (36), we directly get k1=0, k2=1, and k3=0 and
(38)Dn′=0,n≠1;D1′=1vn(t)=0,n≠1(39)v1,1(t)=1Γ(1+α)D1′×∫0tEα(-k1τα2α)sinα(D1′τα)f1(t-τ)(dτ)α=0(40)v2,1(t)=C1Eα(-k1tα2α)cos⁡α(D1′tα)+(D1+k1C1-k12)Eα(-k1tα2α)sinα(D1′tα)=-cos⁡α(tα).
Conclusively, we get
(41)v1(t)=v1,1(t)+v2,1(t)=-cos⁡α(tα).
Thus, we obtain
(42)u(x,t)=-Eα(xα)cos⁡α(tα)
and its graph is shown in Figure 1.Figure 1
Exact solution for local fractional Laplace equation with fractal dimensionα=ln⁡2/ln⁡3.Example 8.
We consider the following local fractional Laplace equation:(43)∂2αu(x,t)∂t2α+∂2αu(x,t)∂x2α=0
subject to the fractal value conditions
(44)u(x,0)=0,∂αu(x,0)∂tα=-Eα(xα).
Now we can structure the same local fractional iteration procedure (15). Hence, we have
(45)u0(x,t)=-tαΓ(1+α)Eα(xα)u1(x,t)=It(α)0It(α)0{-∂2αu0(x,s)∂x2α}=It(α)0It(α)0{tαΓ(1+α)Eα(xα)}=t3αΓ(1+3α)Eα(xα)u2(x,t)=It(α)0It(α)0{-∂2αu1(x,s)∂x2α}=It(α)0It(α)0{-t3αΓ(1+3α)Eα(xα)}=-t5αΓ(1+5α)Eα(xα)u3(x,t)=It(α)0It(α)0{-∂2αu2(x,s)∂x2α}=It(α)0It(α)0{t5αΓ(1+5α)Eα(xα)}=t7αΓ(1+7α)Eα(xα).
Finally, we can obtain the local fractional series solution as follows:
(46)un(x,t)=(-1)n+1t(2n+1)αΓ(1+(2n+1)α)Eα(xα).
Thus, the final solution reads as follows:
(47)u(x,t)=∑n=0∞un(x,t)=u0(x,t)+u1(x,t)+u2(x,t)+⋯=Eα(xα)[-tαΓ(1+α)+t3αΓ(1+3α)-t5αΓ(1+5α)⋯]=-Eα(xα)[tαΓ(1+α)-t3αΓ(1+3α)+t5αΓ(1+5α)⋯]=-Eα(xα)sinα(tα).Now, we solve Example8 by using the local fractional function decomposition method.We suppose that(48)u(x,t)=∑n=1∞vn(t)Eα(nαxα)f(x,t)=0=∑n=1∞fn(t)Eα(nαxα)φ(x)=0=∑n=1∞CnEα(nαxα)ψ(x)=-Eα(xα)=∑n=1∞DnEα(nαxα)
which leads to
(49)fn(t)=0,∀n;Dn=0,n≠1;D1=-1;Cn=0,∀n.
Contrasting (28) with (36), we directly get k1=0, k2=1, and k3=0 and
(50)Dn′=0,n≠1;D1′=1vn(t)=0,n≠1v1,1(t)=1Γ(1+α)D1′×∫0tEα(-k1τα2α)sinα(D1′τα)f1(t-τ)(dτ)α=0v2,1(t)=C1Eα(-k1tα2α)cos⁡α(D1′tα)+(D1+k1C1-k12)Eα(-k1tα2α)sinα(tα)=-sinα(tα).
Conclusively, we get
(51)v1(t)=v1,1(t)+v2,1(t)=-sinα(tα).
Thus, we obtain
(52)u(x,t)=-Eα(xα)sinα(tα)
and its graph is given in Figure 2.Figure 2
The plot of solution to local fractional Laplace equation with fractal dimensionα=ln⁡2/ln⁡3.


## 5. Conclusions

In this work solving the Laplace equations using the local fractional function decomposition method with local fractional operators is discussed in detail. Two examples of applications of the local fractional Adomian decomposition method and local fractional function decomposition method to the local fractional Laplace equations are investigated in detail. The reliable obtained results are complementary with the ones presented in the literature.



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