# Local Fractional Laplace Variational Iteration Method for Nonhomogeneous Heat Equations Arising in Fractal Heat Flow

Authors: Shu Xu; Xiang Ling; Carlo Cattani; Gong-Nan Xie; Xiao-Jun Yang; Yang Zhao
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

The local fractional Laplace variational iteration method is used for solving the nonhomogeneous heat equations arising in the fractal heat flow. The approximate solutions are nondifferentiable functions and their plots are also given to show the accuracy and efficiency to implement the previous method.


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## Body

## 1. Introduction

Fractional calculus [1–4] was used to deal with the heat conduction equation in fractal media. Fractional heat conduction equation was studied by many researchers [5–17]. For example, Povstenko considered the thermoelasticity based on the fractional heat conduction equation [7]. Youssef suggested the generalized theory of fractional-order thermoelasticity [8]. Ezzat and El-Karamany presented the fractional-order conduction in thermoelastic medium [9]. Ezzat proposed the fractional-order heat transfer in thermoelectric fluid [10]. Sherief et al. reported the fractional-order generalized thermoelasticity with one relaxation time [11]. Vázquez et al. used the second law of thermodynamics to fractional heat conduction equation [12]. Hristov considered the inverse Stefan problem and nonlinear heat conduction with Jeffrey’s fading memory by using the heat balance integral method [13, 14]. Davey and Prosser gave the solutions of the heat transfer on fractal and prefractal domains [15]. Ostoja-Starzewski investigated thermoelasticity of fractal media [16]. Qi and Jiang discussed space-time fractional Cattaneo diffusion equation [17]. Bhrawy and Alghamdi applied the Legendre tau-spectral method to find time fractional heat equation with nonlocal conditions [18]. Atangana and Kılıçman suggested the Sumudu transform solving certain nonlinear fractional heat-like equations [19].Recently, the local fractional calculus [20–22] was used to deal with the discontinuous problem for heat transfer in fractal media [23–25]. The nonhomogeneous heat equations arising in fractal heat flow were considered by using the local fractional Fourier series method [26]. The local fractional heat conduction equation was investigated by the local fractional variation iteration method [27]. The nondifferentiable solution of one-dimensional heat equations arising in fractal transient conduction was found by the local fractional Adomian decomposition method [28]. Local fractional Laplace variational iteration method [29, 30] was considered to deal with linear partial differential equations. In this paper, our aim is to investigate the nonhomogeneous heat equations arising in heat flow with local fractional derivative. The paper is organized as follows. Section 2 introduces the nonhomogeneous heat equations arising in heat flow with local fractional derivative. In Section 3, local fractional Laplace variational iteration method is presented. In Section 4, the nondifferentiable solutions for nonhomogeneous heat equations arising in heat flow with local fractional derivative are investigated. Finally, conclusions are shown in Section 5.


## 2. The Nonhomogeneous Heat Equations Arising in Heat Flow with Local Fractional Derivatives

In this section we present the one-dimensional nonhomogeneous heat equations arising in heat flow with local fractional derivatives.Let the local fractional volume integral of the functionu be defined as [19]
(1)∭urPdΩγ=lim⁡N→∞∑P=1NurPΔΩPγ,
where the elements of the volume ΔΩPγ→0 as N→∞ and the fractal dimension of the volume γ. The equality ux,y,z,t is the temperature at the point x,y,z∈Ω, time t∈T, and the total amount of heat Ht is described as
(2)Ht=∭cαραux,y,z,tdΩγ,
where cα is the special heat of the fractal material and ρα is the density of the fractal material.The local fractional surface integral is defined as [19, 22]
(3)∬urP·dSβ=lim⁡N→∞∑P=1NurP·nPΔSPβ,
where N are elements of area with a unit normal local fractional vector nP, ΔSPβ→0 as N→∞ for γ=3/2β=3α.From(3) the local fractional Fourier law of the material in fractal media [19, 23] was suggested as follows:
(4)dαdtαHt=∯∂Ωβk2α∇αux,y,z,t·dSβ,l
where dSβ is the fractal surface measure over Ωγ and k2α is the thermal conductivity of the fractal material.In view of(4), the change in heat reads as follows [19, 23]:
(5)dαdtαHt=∭cαραutαx,y,z,tdΩγ,
where ∂Ωβ is the boundary of Ωγ.From(2) we suggest the following source term [23]:
(6)Gt=∭gx,y,z,tdΩγ.
Making use of (4), (5), and (6), we have
(7)∭cαραutαx,y,z,tdΩγ=∯∂Ωβk2α∇αux,y,z,t·dSβ+∭gx,y,z,tdΩγ
such that
(8)∭cαραutαx,y,z,t-∇α·k2α∇αux,y,z,t-gx,y,z,tutαdΩγ=0,
which leads to the nonhomogeneous local fractional heat equations [23]:
(9)cαραutαx,y,z,t-∇α·k2α∇αux,y,z,t=gx,y,z,t.
From (9) we obtain the nonhomogeneous heat equations in the dimensionless case:
(10)ϕtαx,y,z,t-∇2αϕx,y,z,t=φx,y,z,t.
The two-dimensional case is [23]
(11)ϕtαx,y,t-∇2αϕx,y,t=φx,y,t,
and the one-dimensional case is [26]
(12)ϕtαx,t-ϕx2αx,t=φx,t.


## 3. Local Fractional Laplace Variational Iteration Method

In this section, we give the idea of local fractional Laplace variational method [29, 30] in order to investigate the one-dimensional nonhomogeneous heat equations arising in fractal heat flow.We present the following local fractional differential operator as follows:(13)Lαu-Rαu=0,
where the linear local fractional differential operator denotes Lα=d2α/dx2α and ux is a nondifferential function.We can write the local fractional functional formula as(14)un+1x=unx+I0xαλx-tαΓ1+αLαunt-Rαun.
The local fractional Laplace transform is given as [29–32]
(15)L~αfx=fsL~,αs=1Γ1+α∫0∞Eα-sαxαfxdxα,0<α≤1,
and the inverse formula of local fractional Laplace transform is suggested as [29–32]
(16)fx=L~α-1fsL,αs=12πα∫β-i∞β+i∞EαsαxαfsL~,αsdsα,
where fx is a local fractional continuous function, sα=βα+iα∞α, Resα=βα, and the local fractional integral of fx of order α in the interval a,b is given as [23]
(17)Iabαfx=1Γ1+α∫abftdtα=1Γ1+αlim⁡Δt→0∑j=0j=N-1ftjΔtjα,
with the partitions of the interval a,b which is tj,tj+1, for Δtj=tj+1-tj, t0=a, tN=b, and Δt=max⁡Δt0,Δt1,Δtj,…, j=0,…,N-1.From(15) the local fractional convolution of two functions is defined as [29–32]
(18)f1x*f2x=1Γ1+α∫-∞∞f1tf2x-tdtα,
and we have
(19)Fαf1x*f2x=L~αlf1xlL~αlf2x.
From (19) we obtain
(20)L~αun+1x=L~αunx+L~αλxαΓ1+αL~αLαunx-Rαunx.
By the local fractional variation [23, 27, 29, 30], we obtain
(21)δαL~αun+1x=δαL~αunx+δαL~αλxαΓ1+αL~αLαunx-Rαunx,
which leads to
(22)δαL~αun+1x=δαL~αunx+L~αλxαΓ1+αδαL~αLαunx=0.
From (22) we have
(23)δαL~αLαunx=δαs2αL~αunx-sαun0-unα0=s2αδαL~αunx
such that
(24)1+L~αλxαΓ1+αs2α=0.
From (24) we get
(25)L~αλxαΓ1+α=-1s2α
such that local fractional iteration algorithm reads as
(26)L~αun+1x=L~αunx-1s2αL~αLαunx-Rαunx,
where the initial value is presented as follows:
(27)L~αu0x=u0.
Therefore, the local fractional series solution is given as
(28)L~αu=lim⁡n→∞L~αun.
From (28) we arrive at
(29)u=lim⁡n→∞L~α-1L~αun.


## 4. The Nondifferentiable Solutions

In this section, we discuss the one-dimensional nonhomogeneous heat equations arising in fractal heat flow.Example 1.
The nonhomogeneous local fractional heat equation with the nondifferentiable sink term is presented as follows:(30)∂αTx,t∂tα-∂2αTx,t∂x2α=-xαEα-tαΓ1+α,0<x<1,0<t≤1,0<α≤1,
subject to the initial-boundary value conditions
(31)∂αT0,t∂xα=Eα-tα,T0,t=0.
From (26) we obtain the local fractional iteration algorithm:
(32)L~αTn+1x,t=L~αTnx,t-1s2αL~α·∂αTnx,t∂tα-∂2αTnx,t∂x2α+xαEα-tαΓ1+α=L~αTnx,t-1s2α·Eα-tαs2α∂αTns,t∂tα-s2αL~αTnx,t+sαTn0,t+Tnα0,t+Eα-tαs2α=2L~αTnx,t-1s2α∂αTns,t∂tα-Tnα0,ts2α-Eα-tαs4α,
where the initial value is given as
(33)L~αT0x,t=Eα-tαs2α.
Using (32), we have the first approximation:
(34)L~αT1x,t=2L~αT0x,t-1s2α∂αT0s,t∂tα-T0α0,ts2α-Eα-tαs4α=2Eα-tαs2α+Eα-tαs4α-Eα-tαs2α-Eα-tαs4α=Eα-tαs2α.
In view of (32) and (34), we get the second approximation:
(35)L~αT2x,t=2L~αT1x,t-1s2α∂αT1s,t∂tα-T1α0,ts2α-Eα-tαs4α=2Eα-tαs2α-Eα-tαs4α-Eα-tαs2α-Eα-tαs4α=Eα-tαs2α.
Making use of (32) and (35), the third approximate term reads as follows:
(36)L~αT3x,t=2L~αT2x,t-1s2α∂αT2s,t∂tα-T2α0,ts2α-Eα-tαs4α=2Eα-tαs2α-Eα-tαs4α-Eα-tαs2α-Eα-tαs4α=Eα-tαs2α.
From (32) and (36), the fourth approximate term can be written as follows:
(37)L~αT4x,t=2L~αT3x,t-1s2α∂αT3s,t∂tα-T3α0,ts2α-Eα-tαs4α=2Eα-tαs2α-Eα-tαs4α-Eα-tαs2α-Eα-tαs4α=Eα-tαs2α.
Making the best of (32) and (36), we can write the fifth approximate term as
(38)L~αT5x,t=2L~αT4x,t-1s2α∂αT4s,t∂tα-T4α0,ts2α-Eα-tαs4α=2Eα-tαs2α-Eα-tαs4α-Eα-tαs2α-Eα-tαs4α=Eα-tαs2α.
Hence, we obtain the final term given as
(39)L~αTnx,t=Eα-tαs2α.
In view of (28) and (29), we suggest the exact solution of (30) as
(40)Tx,t=lim⁡n→∞L~α-1L~αTnx,t=xαΓ1+αEα-tα
and its plot is shown in Figure 1.Figure 1
The nondifferentiable solution of nonhomogeneous local fractional heat equation with nondifferentiable sink term forα=ln⁡2/ln⁡3.Example 2.
We now consider the nonhomogeneous local fractional heat equation with the nondifferentiable source term:(41)∂αTx,t∂tα-∂2αTx,t∂x2α=xαcos⁡αtαΓ1+α,0<x<1,0<t≤1,0<α≤1,
subject to the initial-boundary value conditions
(42)∂2αTx,t∂x2α=sinαtα,T0,t=0.
In view of (26), the local fractional iteration algorithm can be structured as follows:
(43)L~αTn+1x,t=L~αTnx,t-1s2αL~α·∂αTnx,t∂tα-∂2αTnx,t∂x2α-xαcos⁡αtαΓ1+α=L~αTnx,t-1s2α·cos⁡αtαs2α∂αTns,t∂tα-s2αL~αTnx,t+sαTn0,t+Tnα0,t-cos⁡αtαs2α=2L~αTnx,t-1s2α∂αTns,t∂tα-Tnα0,ts2α+cos⁡αtαs4α.
Appling (43) gives the first approximate term:
(44)L~αT1x,t=2L~αT0x,t-1s2α∂αT0s,t∂tα-T0α0,ts2α+cos⁡αtαs4α=2sinαtαs2α-cos⁡αtαs4α-sinαtαs2α+Eα-tαs4α=sinαtαs2α.
In view of (43) and (44), the second approximate term reads as
(45)L~αT2x,t=2L~αT1x,t-1s2α∂αT1s,t∂tα-T1α0,ts2α+cos⁡αtαs4α=2sinαtαs2α-cos⁡αtαs4α-sinαtαs2α+Eα-tαs4α=sinαtαs2α.
Making use of (43) and (45), we arrive at the third approximate term:
(46)L~αT3x,t=2L~αT2x,t-1s2α∂αT2s,t∂tα-T2α0,ts2α+cos⁡αtαs4α=2sinαtαs2α-cos⁡αtαs4α-sinαtαs2α+Eα-tαs4α=sinαtαs2α.
From (43) and (46) we give the fourth approximation:
(47)L~αT4x,t=2L~αT3x,t-1s2α∂αT3s,t∂tα-T3α0,ts2α+cos⁡αtαs4α=2sinαtαs2α-cos⁡αtαs4α-sinαtαs2α+Eα-tαs4α=sinαtαs2α.
In view of (43) and (47), the fifth approximate term is presented as
(48)L~αT5x,t=2L~αT4x,t-1s2α∂αT4s,t∂tα-T4α0,ts2α+cos⁡αtαs4α=2sinαtαs2α-cos⁡αtαs4α-sinαtαs2α+Eα-tαs4α=sinαtαs2α.
Hence, we finally have
(49)L~αTnx,t=sinαtαs2α
so that the exact solution of nonhomogeneous local fractional heat equation with nondifferentiable source term is
(50)Tx,t=lim⁡n→∞L~α-1L~αTnx,t=xαΓ1+αsinαtα.
For the fractal dimension α=ln⁡2/ln⁡3, the plot of the nondifferentiable solution of the nonhomogeneous local fractional heat equation with the nondifferentiable source term is shown in Figure 2.Figure 2
The nondifferentiable solution of the nonhomogeneous local fractional heat equation with the nondifferentiable sink term forα=ln⁡2/ln⁡3.


## 5. Conclusions

At the present work, the nonhomogeneous heat equations arising in the fractal heat flow were investigated. The local fractional Laplace variational iteration method was applied to obtain the nondifferentiable solutions for the nonhomogeneous local fractional heat equations with the nondifferentiable source and sink terms. Finally, the graphs of the obtained solutions are also shown.



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