# The Approximate Solutions of Three-Dimensional Diffusion and Wave Equations within Local Fractional Derivative Operator

Authors: Hassan Kamil Jassim
Journal: Abstract and Applied Analysis
 (2016)

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## Abstract

We used the local fractional variational iteration transform method (LFVITM) coupled by the local fractional Laplace transform and variational iteration method to solve three-dimensional diffusion and wave equations with local fractional derivative operator. This method has Lagrange multiplier equal to minus one, which makes the calculations more easily. The obtained results show that the presented method is efficient and yields a solution in a closed form. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new method.


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## Body

## 1. Introduction

The diffusion equation is a partial differential equation that portrays density dynamics in a material that undertakes diffusion. It is also used to describe progression demonstrating diffusive-like performance, for example, the transmission of alleles in a population genetics [1–3]. The three-dimensional diffusion equation in fractal heat transfer involving local fractional derivatives was presented as(1)∇2αφx,y,z,t=1Kα∂αφx,y,z,t∂tα,subject to the initial condition(2)φx,y,z,0=ηx,y,z,where the local fractional Laplace operator is defined as follows (see [4–8]):(3)∇2α=∂2α∂x2α+∂2α∂y2α+∂2α∂z2α.Kα is a nondifferentiable diffusion coefficient, and φ(x,y,z,t) is satisfied with the nondifferentiable temperature distribution, while the three-dimensional wave equation involving local fractional derivatives was presented as(4)∇2αφx,y,z,t=1Kα∂αφx,y,z,t∂t2α,subject to the initial conditions(5)φx,y,z,0=η1x,y,z,∂∂φx,y,z,0∂tα=η2x,y,z.Many physical problems are governed by partial differential equations (PDEs), and the solution of these equations has been a subject of many investigators in recent years. The diffusion and wave equations have been successfully modeled for many physical and engineering phenomena such as seismic analysis, rheology, fluid flow, viscous damping, viscoelastic materials, and polymer physics [9–11].Recently, the diffusion and wave problems were studied by several authors by using local fractional decomposition method [12–15], local fractional variational iteration [15–17], local fractional series expansion [18], local fractional functional decomposition method [19, 20], local fractional Laplace decomposition method [21], local fractional homotopy perturbation method [22], local fractional similarity solution [23], and local fractional differential transform method [24, 25]. In this paper, our aims are to present the coupling method of local fractional Laplace transform and variational iteration method, which is called the local fractional variational iteration transform method, and to use it to solve three-dimensional diffusion and wave equations with local fractional derivative.


## 2. Mathematical Fundamentals

In this section, we present the basic theory of local fractional calculus and concepts of local fractional Laplace transform (see [12–15]).Definition 1.
One says that a functionf(x) is local fractional continuous at x=x0; if it holds,(6)fx-fx0<εα,0<α≤1with x-x0<δ, for ε,δ>0 and ε,δ∈R. For x∈(a,b), it is called local fractional continuous on (a,b), denoted by f(x)∈Cα(a,b).Definition 2.
Settingf(x)∈Cα(a,b), the local fractional derivative of f(x) at x=x0 is defined as(7)Dxαfx0=dαdxαfxx=x0=fαx0=limx→x0⁡Δαfx-fx0x-x0α,where Δαfx-fx0≅Γ(α+1)fx-fx0.Definition 3.
Let one denote a partition of the interval[a,b] as (tj,tj+1), j=0,…,N-1, and tN=b with Δtj=tj+1-tj and Δt=max⁡{Δt0,Δt1,…}. The local fractional integral of f(x) in the interval [a,b] is given by(8)Ibαafx=1Γ1+α∫abftdtα=1Γ1+αlimΔt→0⁡∑j=0N-1ftjΔtjα.Definition 4.
Let(1/Γ1+α)∫0∞fxdxα<k<∞. The Yang-Laplace transform of fx is given by(9)Lαfx=fsL,αs=1Γ1+α∫0∞Eα-sαxαfxdxα,0<α≤1,where the latter integral converges and sα∈Rα.Definition 5.
The inverse formula of the Yang-Laplace transforms offx is given by(10)Lα-1fsL,αs=fx=12πα∫β-iωβ+iωEαsαxαfsL,αsdsα,0<α≤1,where sα=βα+iαωα; fractal imaginary unit is iα, and Re(s)=β>0.The properties for local fractional Laplace transform used in the paper are given as(11)Lαafx+bgx=afsL,αs+bgsL,αs,LαEαcαxαfx=fsL,αs-c,Lαfkαx=skαfsL,αs-sk-1αf0-sk-2αfα0-⋯-fk-1α0,LαEαaαxα=1sα-aα,Lαsinα⁡aαxα=aαs2α+a2α,Lαxkα=Γ1+kαsk+1α.


## 3. LFVITM for Three-Dimensional Diffusion Problems

We first rewrite problem (1) in the local fractional operator form(12)Ltαφx,y,z,t=KαLxx2αφx,y,z,t+Lyy2αφx,y,z,t+Lzz2αφx,y,z,t,where the local fractional differential operators Lt(α), Lxx(2α), Lyy(2α), and Lzz(2α) are defined by(13)Ltα·=∂α∂tα·,Lxx2α·=∂2α∂x2α·,Lyy2α·=∂2α∂y2α·,Lzz2α·=∂2α∂z2α·.Adopting the local fractional Laplace transform (denoted in this paper by Łα) to both sides of (12) and using the initial condition leads to(14)Łαφx,y,z,t=1sαφx,y,z+1sαŁαKαLxx2αφx,y,z,t+Lyy2αφx,y,z,t+Lzz2αφx,y,z,t.Operating with the inverse of local fractional Laplace transform on both sides of (14) gives(15)φx,y,z,t=ηx,y,z+Łα-1KαsαŁαLxx2αφ+Lyy2αφ+Lzz2αφ.Deriving both sides of (15) with respect to t, we have(16)Ltαφx,y,z,t=LtαŁα-1KαsαŁαLxx2αφ+Lyy2αφ+Lzz2αφ.By the correction function of the irrational method(17)
φ
n
+
1
x
,
y
,
z
,
t
=
φ
n
x
,
y
,
z
,
t
-
1
Γ
1
+
α
∫
0
t
L
τ
α
φ
n
-
L
τ
α
Ł
α
-
1
K
α
s
α
Ł
α
L
x
x
2
α
φ
n
+
L
y
y
2
α
φ
n
+
L
z
z
2
α
φ
n
d
τ
α
,finally, the solution φ(x,y,z,t) is given by(18)φx,y,z,t=limn→∞⁡φnx,y,z,t.We now consider the initial conditions of (2); namely, (19)φx,y,z,0=Eαxα+yα+zα,K=1,we have(20)
φ
0
x
,
y
,
z
,
t
=
E
α
x
α
+
y
α
+
z
α
,
φ
n
+
1
x
,
y
,
z
,
t
=
φ
n
x
,
y
,
z
,
t
-
1
Γ
1
+
α
∫
0
t
L
τ
α
φ
n
τ
-
L
τ
α
Ł
α
-
1
1
α
s
α
Ł
α
L
x
x
2
α
φ
n
τ
+
L
y
y
2
α
φ
n
τ
+
L
z
z
2
α
φ
n
τ
d
τ
α
.Consequently, we obtain(21)
φ
0
x
,
y
,
z
,
t
=
E
α
x
α
+
y
α
+
z
α
,
φ
1
x
,
y
,
z
,
t
=
φ
0
x
,
y
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z
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t
-
1
Γ
1
+
α
∫
0
t
L
τ
α
φ
0
τ
-
L
τ
α
Ł
α
-
1
1
α
s
α
Ł
α
L
x
x
2
α
φ
0
τ
+
L
y
y
2
α
φ
0
τ
+
L
z
z
2
α
φ
0
τ
d
τ
α
=
E
α
x
α
+
y
α
+
z
α
1
+
3
t
α
Γ
1
+
α
,
φ
2
x
,
y
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z
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t
=
φ
1
x
,
y
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z
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t
-
1
Γ
1
+
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∫
0
t
L
τ
α
φ
1
τ
-
L
τ
α
Ł
α
-
1
1
α
s
α
Ł
α
L
x
x
2
α
φ
1
τ
+
L
y
y
2
α
φ
1
τ
+
L
z
z
2
α
φ
1
τ
d
τ
α
=
E
α
x
α
+
y
α
+
z
α
1
+
3
t
α
Γ
1
+
α
+
9
t
2
α
Γ
1
+
2
α
,and so on.The solution in a nondifferentiable series form(22)φx,y,z,t=Eαxα+yα+zα1+3tαΓ1+α+9t2αΓ1+2α+⋯is readily obtained.Therefore, the exact solution can be written as(23)φx,y,z,t=Eαxα+yα+zα+3tα.


## 4. LFVITM for Three-Dimensional Wave Problems

We first rewrite the problem (4) in the local fractional operator form(24)Lttαφx,y,z,t=KαLxx2αφx,y,z,t+Lyy2αφx,y,z,t+Lzz2αφx,y,z,t.Applying the local fractional Laplace transform to both sides of (24) and using the initial condition leads to(25)Łαφx,y,z,t=1sαη1x,y,z+1s2αη2x,y,z+1s2αŁαKαLxx2αφx,y,z,t+Lyy2αφx,y,z,t+Lzz2αφx,y,z,t.Operating with the inverse of local fractional Laplace transform on both sides of (25) gives(26)φx,y,z,t=η1x,y,z+tαΓ1+αη2x,y,z+Łα-1Kαs2αŁαLxx2αφ+Lyy2αφ+Lzz2αφ.Deriving both sides of (26) with respect to t, we obtain(27)Ltαφx,y,z,t=η2x,y,z+LtαŁα-1Kαs2αŁαLxx2αφ+Lyy2αφ+Lzz2αφ.By the correction function of the irrational method,(28)φn+1x,y,z,t=φnx,y,z,t-1Γ1+α∫0tLταφn-LταŁα-1Kαs2αŁαLxx2αφn+Lyy2αφn+Lzz2αφn-η2x,y,zdτα.Finally, the solution φx,y,z,t is given by(29)φx,y,z,t=limn→∞⁡φnx,y,z,t.We now consider the initial conditions of (5); namely,(30)φx,y,z,0=0,∂αφx,y,z,0∂tα=3sinα⁡xαsinα⁡yαsinα⁡zα,K=3.Starting with the zeroth approximation,(31)φ0x,y,z,t=3tαΓ1+αsinα⁡xαsinα⁡yαsinα⁡zα.Substituting (31) in (28) we obtain the following successive approximations:(32)φ1x,y,z,t=φ0x,y,z,t-1Γ1+α∫0tLταφ0τ-LταŁα-13αs2αŁαLxx2αφ0τ+Lyy2αφ0τ+Lzz2αφ0τ-3sinα⁡xαsinα⁡yαsinα⁡zαdτα=sinα⁡xαsinα⁡yαsinα⁡zα3tαΓ1+α-27t3αΓ1+3α,φ2x,y,z,t=φ1x,y,z,t-1Γ1+α∫0tLταφ1τ-LταŁα-13αs2αŁαLxx2αφ1τ+Lyy2αφ1τ+Lzz2αφ1τ-3sinα⁡xαsinα⁡yαsinα⁡zαdτα=sinα⁡xαsinα⁡yαsinα⁡zα3tαΓ1+α-27t3αΓ1+3α+243t5αΓ1+5α,and so on.The solution in a nondifferentiable series form(33)φx,y,z,t=sinα⁡xαsinα⁡yαsinα⁡zα3tαΓ1+α-27t3αΓ1+3α+243t5αΓ1+5α⋯is readily obtained.Therefore, the exact solution can be written as(34)φx,y,z,t=sinα⁡xαsinα⁡yαsinα⁡zαsinα⁡3tα.


## 5. Conclusion

In this work, we studied the local fractional variational iteration transform method to solve three-dimensional diffusion and wave equations involving local fractional derivative operator and their nondifferentiable solutions were obtained. This method can also be applied to a large class of system of partial differential equations with approximations that converges rapidly to accurate solutions.



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