ISSN: 2322-1666 print/2251-8436 online
AN EFFICIENT TECHNIQUE FOR SOLVING LINEAR AND NONLINEAR WAVE EQUATIONS WITHIN
LOCAL FRACTIONAL OPERATORS HASSAN KAMIL JASSIM
Abstract. In this paper, we utilize reduced differential transform
method (RDTM) to obtain approximate solutions for linear and nonlinear wave equations within local fractional differential oper-ators. The operators are taken in the local fractional sense. The efficiency of the considered method is illustrated by some examples. This method reduces significantly the numerical computations com-pare with local fractional variational iteration method. The results reveal that the suggested algorithm is very effective and simple and can be applied for other linear and nonlinear problems in sciences and engineering.
Key Words: Local fractional wave equation, Local fractional derivative operators, Reduced differential transform method.
2010 Mathematics Subject Classification:Primary: 26A33; Secondary: 34A12, 35R11.
1. Introduction
There are many physical applications in science and engineering can be represented by models using the differential equations, which are very helpful for many physical problems. These equations are represented by linear and nonlinear partial differential equations and solving such differential equations is very important. Our concern in this work is to consider the linear and nonlinear wave equation with local fractional
Received: 28 July 2016, Accepted: 29 June 2017. Communicated by Hossein Jafari;
∗Address correspondence to Hassan Kamil Jassim; E-mail: [email protected]. c
2017 University of Mohaghegh Ardabili. 136
differential operator as the following:
L(2κκϑ)ϕ(η, κ)−L(2ηηϑ)ϕ(η, κ) + Λ(ϕ) =ψ(η, κ),0< α≤1,
(1.1)
L(2κκϑ)ϕ(η, κ)−L(2ηηϑ)ϕ(η, κ) + Υ(ϕ) =ψ(η, κ),0< α≤1,
(1.2)
with the initial conditions
(1.3) ϕ(η,0) =ω(η),∂
ϑϕ(η,0)
∂κϑ =ρ(η),
where Λ(ϕ) and Υ(ϕ)are linear and nonlinear functions respectively,
ψ(η, κ) is source term of nondifferentiable function, and ω(η) andρ(η) are given functions.
There are many analytical and numerical methods used to solve local fractional partial differential equations such as, local fractional function decomposition method [1, 2, 3], local fractional Adomian decomposi-tion method [3,4], local fractional series expansion method [5,6], local fractional Laplace transform method [7, 8], local fractional Fourier se-ries method [9], local fractional homotopy perturbation method [10], local fractional varitional iteration method [11,12, 13], local fractional differential transform method [14,15], local fractional Laplace decompo-sition method [16], local fractional Laplace variational iteration method [17,18,19].The local fractional reduced differential transform technique is an iterative procedure for obtaining Taylor series solution of partial differential equations. This method reduces the size of computational work and easily applicable to many physical problems. Our aim is to extend the applications of the proposed method to obtain the analytical approximate solutions to wave equations with local fractional deriva-tive operators. The paper has been organized as follows. In Section 2, we give analysis of the method used. In Section 3, we consider several illustrative examples. Finally, in Section 4, we present our conclusions.
2. Analysis of the Method
As in [20], the basic definition of reduced differential transform with local fractional operator is proposrd as follows:
Definition 2.1. Ifϕ(η, κ) is a local fractional analytical function in the domain of interest, then the local fractional spectrum function
(2.1) Φξ(η) = 1 Γ(1 +ξϑ)
∂ξϑϕ(η, κ)
∂κξϑ
κ=κ0 ,
is reduce differential transformed of the functionϕ(η, κ) via local frac-tional operator, whereξ= 0,1, . . . , n and 0< ϑ≤1.
Definition 2.2. The inverse reduced differential transform of Φξ(η) via local fractional operator is defined as follows:
(2.2) ϕ(η, κ) = ∞ X
ξ=0
Φξ(η)(κ−κ0)ξϑ. From (2.1) and (2.2) we get
(2.3) ϕ(η, κ) = ∞ X
ξ=0
(κ−κ0)ξϑ
Γ(1 +ξϑ)
∂ξϑϕ(η, κ)
∂κξϑ
κ=κ0 .
From (2.3), it is obvious that the local fractional reduced differential transform is derived from the local fractional Taylor theorems.
Whenever κ0= 0, then (2.1) and (2.2) become (2.4) Φξ(η) =
1 Γ(1 +ξϑ)
∂ξϑϕ(η, κ)
∂κξϑ
κ=0 ,
(2.5) ϕ(η, κ) = ∞ X
ξ=0
Φξ(η)κξϑ.
The following theorems that can be deduced from (2.1) and (2.2) are presented below:
Theorem 2.3. If ϕ(η, κ) =ψ(η, κ) +θ(η, κ) then (2.6) Φξ(η) = Ψξ(η) + Θξ(η). Theorem 2.4. If ϕ(η, κ) =ψ(η, κ)θ(η, κ) then (2.7) Φξ(η) =
ξ X
l=0
Ψl(η)Θξ−l(η).
Theorem 2.5. If ϕ(η, κ) =aψ(η, κ) , where a is a constant, then (2.8) Φξ(η) =aΨξ(η).
Theorem 2.6. .If ϕ(η, κ) = ∂
nϑψ(η, κ)
∂κnϑ then (2.9) Φξ(η) = Γ(1 + (ξ+n)ϑ)
Theorem 2.7. Ifϕ(η, κ) = η nϑ
Γ(1 +nϑ)
κmϑ
Γ(1 +mϑ) then (2.10) Φξ(η) =
ηnϑ
Γ(1 +nϑ)
δϑ(ξ−m) Γ(1 +mϑ), where the local fractional Dirac-delta function is given by
δϑ(ξ−m) =
1, ξ =m,
0, ξ 6=m. Theorem 2.8. .Ifϕ(η, κ) = ∂
nϑψ(η, κ)
∂ηnϑ then (2.11) Φξ(η) =
∂nϑΨξ(η)
∂ηnϑ .
For illustration of the methodology of the presented method, we write the partial differential equation within local fractional operator as:
Lϑ[ϕ(η, κ)] +Rϑ[ϕ(η, κ)] +Nϑ[ϕ(η, κ)] =ω(η, κ), (2.12)
ϕ(η,0) =φ(η).
where Lϑ = ∂
2ϑ
∂κ2ϑ and Rϑ are linear local fractional operators, Nϑ is
a nonlinear local fractional operator and ω(η, κ) is an inhomogenuous term.
By taking the local fractional reduce differential transform on both sides of (2.12), we have
Γ(1 + (ξ+ 2)ϑ)
Γ(1 +ξϑ) Φξ+2(η) = Ωξ(η)−Rϑ[Φξ(η)] +Nϑ[Φξ(η)], (2.13)
Φ0(η) =φ(η).
where Φξ(η) and Ωξ(η) are reduce differential transformed with local fractional operators of the functionsϕ(η, κ) andω(η, κ) respectively.
3. Illustrative examples
In this section, to give a clear overview of the local fractional reduce differential transform method for linear and nonlinear wave equations within local fractional operator, we present the following examples.
Example 3.1. Let us consider the following linear wave equation within local fractional operator:
(3.1) ∂
2ϑϕ(η, κ)
∂κ2ϑ −
∂2ϑϕ(η, κ)
subject to the initial conditions given by (3.2) ϕ(η,0) = 0,∂
ϑϕ(η,0)
∂κϑ =Eϑ(η ϑ).
Implementing the RDTM via local fractional derivative to (3.1), we have the following relation
(3.3) Γ(1 + (ξ+ 2)ϑ)
Γ(1 +ξϑ) Φξ+2(η)−
∂2ϑΦξ(η)
∂η2ϑ = 0, which reduce to the following formula
(3.4) Φξ+2(η) =
Γ(1 +ξϑ) Γ(1 + (ξ+ 2)ϑ)
∂2ϑΦξ(η)
∂η2ϑ , where
(3.5) Φ0(η) = 0,Φ1(η) =
1
Γ(1 +ϑ)Eϑ(η ϑ).
Following (3.4) and (3.5), we obtain the following relations Φ2(η) =
1 Γ(1 + 2ϑ)
∂2ϑΦ0(η) ∂η2ϑ = 0, Φ3(η) =
Γ(1 +ϑ) Γ(1 + 3ϑ)
∂2ϑΦ1(η) ∂η2ϑ = Γ(1 +ϑ)
Γ(1 + 3ϑ) 1
Γ(1 +ϑ)Eϑ(η ϑ)
= 1
Γ(1 + 3ϑ)Eϑ(η ϑ),
Φ4(η) =
Γ(1 + 2ϑ) Γ(1 + 4ϑ)
∂2ϑΦ
2(η) ∂η2ϑ = 0, Φ5(η) =
Γ(1 + 3ϑ) Γ(1 + 5ϑ)
∂2ϑΦ
3(η) ∂η2ϑ = Γ(1 + 3ϑ)
Γ(1 + 5ϑ) 1
Γ(1 + 3ϑ)Eϑ(η ϑ)
= 1
Γ(1 + 5ϑ)Eϑ(η ϑ), ..
Therefore, ϕ(η, κ) is evaluated as follows
ϕ(η, κ) = ∞ X
ξ=0
Φξ(η)κξϑ
= Eϑ(ηϑ)
κϑ
Γ(1 +ϑ)+
κ3ϑ
Γ(1 + 3ϑ) +
κ5ϑ
Γ(1 + 5ϑ) +· · ·
= Eϑ(ηϑ)sinhϑ(κϑ).
(3.6)
Example 3.2. The following linear wave equation within local fractional operator:
(3.7) ∂
2ϑϕ(η, κ)
∂κ2ϑ −
η2ϑ
Γ(1 + 2ϑ)
∂2ϑϕ(η, κ)
∂η2ϑ = 0,0< ϑ61 is presented and its initial valuses are defined as follows:
(3.8) ϕ(η,0) = η
2ϑ
Γ(1 + 2ϑ),
∂ϑϕ(η,0)
∂κϑ = 0.
By utilizing the local fractional reduce differential transform on both sides of (3.7), we get
(3.9) Γ(1 + (ξ+ 2)ϑ)
Γ(1 +ξϑ) Φξ+2(η)−
η2ϑ
Γ(1 + 2ϑ)
∂2ϑΦξ(η)
∂η2ϑ = 0, or equivalently
(3.10) Φξ+2(η) =
Γ(1 +ξϑ) Γ(1 + (ξ+ 2)ϑ)
η2ϑ
Γ(1 + 2ϑ)
∂2ϑΦ ξ(η)
∂η2ϑ , where
(3.11) Φ0(η) =
η2ϑ
Following (3.10) and (3.11), we obtain the following relations
Φ2(η) =
1 Γ(1 + 2ϑ)
η2ϑ
Γ(1 + 2ϑ)
∂2ϑΦ0(η) ∂η2ϑ ,
= 1
Γ(1 + 2ϑ)
η2ϑ
Γ(1 + 2ϑ), Φ3(η) =
Γ(1 +ϑ) Γ(1 + 3ϑ)
η2ϑ
Γ(1 + 2ϑ)
∂2ϑΦ1(η) ∂η2ϑ = 0,
Φ4(η) =
Γ(1 + 2ϑ) Γ(1 + 4ϑ)
η2ϑ
Γ(1 + 2ϑ)
∂2ϑΦ2(η) ∂η2ϑ , = Γ(1 + 2ϑ)
Γ(1 + 4ϑ)
η2ϑ
Γ(1 + 2ϑ) 1 Γ(1 + 2ϑ),
= 1
Γ(1 + 4ϑ)
η2ϑ
Γ(1 + 2ϑ), Φ5(η) =
Γ(1 + 3ϑ) Γ(1 + 5ϑ)
η2ϑ
Γ(1 + 2ϑ)
∂2ϑΦ3(η) ∂η2ϑ = 0,
.. .
Therefore,ϕ(η, κ) is evaluated as follows
ϕ(η, κ) = ∞ X
ξ=0
Φξ(η)κξϑ
= η
2ϑ
Γ(1 + 2ϑ)
1 + κ
2ϑ
Γ(1 + 2ϑ)+
κ4ϑ
Γ(1 + 4ϑ) +
κ6ϑ
Γ(1 + 6ϑ) +· · ·
= η
2ϑ
Γ(1 + 2ϑ)coshϑ(κ ϑ). (3.12)
Example 3.3. Consider the following nonlinear wave equation within local fractional derivative operator:
(3.13)
∂2ϑϕ(η, κ)
∂κ2ϑ −ϕ(η, κ)
∂2ϑϕ(η, κ)
∂η2ϑ +ϕ
2(η, κ) = 1− η2ϑ
Γ(1 + 2ϑ)−
κ2ϑ
with the initial conditions
(3.14) ϕ(η,0) = η
2ϑ
Γ(1 + 2ϑ),
∂ϑϕ(η,0)
∂κϑ = 0.
By applying the local fractional reduce differential transform on both sides of (3.13), we have
(3.15)
Γ(1+(ξ+2)ϑ)
Γ(1+ξϑ) Φξ+2(η)−
Pξ
l=0Φl(η)
∂2ϑ
∂η2ϑΦξ−l(η)
=δϑ(ξ)− η
2ϑ
Γ(1+2ϑ)
δϑ(ξ)
Γ(1+ϑ)−
δϑ(ξ−2)
Γ(1+2ϑ),
which reduces to
(3.16)
Φξ+2(η) = Γ(1+(Γ(1+ξ+2)ξϑ)ϑ)
hPξ
l=0Φl(η)
∂2ϑ
∂η2ϑΦξ−l(η) +δϑ(ξ)
−Γ(1+2η2ϑϑ)δϑ(ξ)− δϑ( ξ−2) Γ(1+2ϑ),
where
(3.17) Φ0(η) =
η2ϑ
By iterative calculations on (3.16) and (3.17) we obtain Φ2(η) =
1 Γ(1 + 2ϑ)
Φ0(η)
∂2ϑ
∂η2ϑΦ0(η) + 1−
η2ϑ
Γ(1 + 2ϑ)
= 1
Γ(1 + 2ϑ)
η2ϑ
Γ(1 + 2ϑ) + 1−
η2ϑ
Γ(1 + 2ϑ)
= 1
Γ(1 + 2ϑ), Φ3(η) =
Γ(1 +ϑ) Γ(1 + 3ϑ)
Φ0(η)
∂2ϑ
∂η2ϑΦ1(η) + Φ1(η)
∂2ϑ ∂η2ϑΦ0(η)
= 0,
Φ4(η) =
Γ(1 + 2ϑ) Γ(1 + 4ϑ)
Φ0(η)
∂2ϑ
∂η2ϑΦ2(η) + Φ1(η)
∂2ϑ
∂η2ϑΦ1(η) + Φ2(η)
∂2ϑ ∂η2ϑΦ0(η)
= Γ(1 + 2ϑ) Γ(1 + 4ϑ)
1 Γ(1 + 2ϑ) −
1 Γ(1 + 2ϑ)
= 0,
.. .
Hence, the approximate solutionϕ(η, κ) is evaluted as follows
ϕ(η, κ) = ∞ X
ξ=0
Φξ(η)κξϑ
= η
2ϑ
Γ(1 + 2ϑ) +
κ2ϑ
Γ(1 + 2ϑ). (3.18)
4. Conclusions
The local fractional wave equations have been analyzed using the re-duced differential transform method within local fractional differential operators. All the examples show that the local fractional reduced differ-ential transform method is a powerful mathematical tool to solve linear and nonlinear wave equations. It is also a promising method to solve other nonlinear equations. The local fractional RDTM introduces a sig-nificant improvement in the fields over existing techniques because it takes less calculations and the number of iteration is less compared by the other methods.
The author is very grateful to the referees for their valuable suggestions and opinions.
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Hassan Kamil Jassim
Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq
