# Application of Local Fractional Homotopy Perturbation Method in Physical Problems

Authors: Nabard Habibi; Zohre Nouri
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

Nonlinear phenomena have important effects on applied mathematics, physics, and issues related to engineering. Most physical phenomena are modeled according to partial differential equations. It is difficult for nonlinear models to obtain the closed form of the solution, and in many cases, only an approximation of the real solution can be obtained. The perturbation method is a wave equation solution using HPM compared with the Fourier series method, and both methods results are good agreement. The percentage of error ofux,t with α=1and0.33, t =0.1 sec, between the present research and Yong-Ju Yang study for x≥0.6 is less than 10. Also, the % error for x≥0.5 in α=1and0.33, t =0.3 sec, is less than 5, whereas for α=1and0.33, t =0.8 and 0.7 sec, the % error for x≥0.4 is less than 8.


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## Body

## 1. Introduction

The homotopy perturbation (HP) method is an analytical method that can solve linear, nonlinear, and random operator equations [1, 2]. The HP method was defined by He [3–5] as a symmetrical structure using combining HP methods in topology and was presented to be used to solve various nonlinear problems in science and engineering, including the argued nonlinear equation in fluid mechanics [4]. He [5] provided a solution for the specific functional integral equations. Khan and Wu [6] have evolved a solution for homogenous and nonhomogeneous advection equation. Also, Xu [7] solved the boundary layer equation in unlimited domains. This method makes very difficult problems extremely easy, which do not need to be solved to convert nonlinear terms. The performance of nonlinear HPM has been demonstrated by many researchers. In recent years, most attention has been paid to the use of this method to solve various scientific models [6–9].Various analytical methods are used to solve equations such as nonlinear wave equations. The variance repetition method is able to reduce the size of the calculations [10]. This method is used in problems related to chaos and super chaos and hard equations [11]. The HP method is a quick technique to produce convergent responses [12].Recently, using HPM for local fractional solving of nonlinear problems in mathematical physics successfully achieved contains the following: diffusion equations on cantor space-time [13], linear and nonlinear local fractional equation of Korteweg-de Vries [14], fractal heat conduction problem [15, 16], wave equation in fractal strings [17], and the Laplace equation [18]. Finding solutions for solving differential equations of local fractional is an interesting topic. There are some approximate and analysis methods: the local fractional functional iteration method [13–17], variationally iteration method within the Yang-Laplace transforms [15], Yang-Laplace transform method [16], local fractional Adomian decomposition and function decomposition methods [18], local fractional iteration of the continuously nondifferentiable functions [19], and Sumudu transform method [20].The formulation is applied to the generalized thermoelasticity based on the fractional time derivatives under the effect of diffusion. Herzallah [21] criticized the research of Ref [a] and showed that their results are incorrect. New methodologies in fractional and fractal derivatives modeling in scientific and engineering were studied by Chen and Liang [22].Uchaikin and Sibatov [23] studied fractional derivatives on cosmic scales. The numerical calculation results demonstrated in this paper do speak well for the NoRD-model as compared with the traditional one based on integer-order operators.Chaos in a 5D hyperchaotic system with four wings in the light of nonlocal and nonsingular fractional derivatives was investigated by Bonyah [24]. The numerical simulation results depict a new chaotic behaviors with the ABC numerical scheme.Goufo and Toudjeu [25] evaluated the analysis of recent fractional evolution equations and applications. The numerical approximations of a second-order nonhomogeneous fractional Cauchy problem are performed and show regularity in the dynamics.In the study of Turkyilmazoglu [26], it is theoretically shown that under a special constraint, the HPM converges to the exact solution of sought solution of nonlinear ordinary or partial differential equation. Examples (the classical Blasius flat-plate flow problem of fluid mechanics, fourth-order parabolic partial differential equation arising in the study of the transverse vibrations of a uniform flexible beam) clearly demonstrate, why and on what interval, the corresponding homotopy series generated by the HPM converges to the exact solution. The physical problems involving stronger nonlinearities can also be dealt with the homotopy method under the assurance of convergence provided by the presented theorems. This method (HPM) is a powerful device for solving a wide variety of problems. Using the HPM, it is possible to find the exact solution or an approximate solution of the problem. Some examples such as Burgers’, Schrodinger’s, and fourth-order parabolic partial differential equations are presented [26].For this reason, the method of homeopathic disorder has been selected for some important physical issues that have received less attention in previous studies. A review of previous studies suggests that local fractional homotopy perturbation method is less commonly used for physical problems. The aim of this research is expansion of an HP method for the definition of local fractional derivation order that can solve a local fractional differential equation in usages of physical mathematics. Therefore, first, the HP method will be explained, and then the local damped wave equation, local fractional wave equation, and heat diffusion equation with several initial values are solved using this method. Also, the obtained results in applied examples are compared with the Fourier series method [27]. In the present research, the homotopy perturbation method (HPM) was applied for real physical problems in engineering which has been less studied in previous studies.


## 2. Mathematical Fundamentals

Definition 1.
Suppose that the functionfx∈Cαa,b, local fractional derivative of fx of order α, 0<α≤1, at the point x=x0 is defined as follows [28–41]:(1)fαx0=Dxαfx0=dαfxdxα∣x→x0=limx→x0∆αfx−fx0x−x0α,where ∆αfx−fx0≅Γ1+αΔfx−fx0 and fx are satisfied with the condition [22] fx−fx0≼ταx−x0α, so that [28, 36] fx−fx0<εα with U:x−x0<δ for ε,δϵ0,1 and ε,δ∈R.
However, the partial of local fractional is defined as follows [13–18]:(2)∂kα∂xkαfx.y=∂α∂xα⋯∂α∂xαfx.y.
If there existsfk+1αx=Dxk⋯Dxα⏞k+1timesfx for any x∈I⊆R, then we denote f∈Dk+1αI where k=0,1,2,⋯ [28].Definition 2.
The diffusion equation on the Cantor sets (called local fractional diffusion equation) was recently described [29] as(3)∂αux,t∂tα=a2α∂αux,y∂x2α,where a2α denotes the fractional diffusion constant which is, in essence, a measure of the efficiency of the spreading of the underlying substance, while the local fractional wave equation is written in the following form [30, 31]:(4)∂2αux,t∂t2α=a2α∂2αux,y∂x2α.
The local fractional Laplace operator is given [27, 42] as follows:(5)∇2α=∂2α∂x2α+∂2α∂y2α+∂2α∂z2α.
We notice that the local fractional diffusion equation yields(6)∇2αu=1a2α∂αux,t∂tα.
And the local fractional wave equation has the following form:(7)∇2αu=1a2α∂2αux,t∂t2α.where 1/a2α is a constant. This equation describes vibrations in a fractal medium.
The Helmholtz equation with local fractional derivative operators in the two-dimensional case was suggested in [43, 44] as follows:(8)∂2αH∂x2α+∂2αH∂y2α+w2αH=fx,y,0≺α≤1,with the initial value conditions as follows:(9)H0,y=φy,∂2αH0,y∂x2α=Ψy,where Hx,y is the unknown function and fx,y is a source term.Definition 3.
The formulas of the local fractional of special functions used in the present research are as follows [28, 30]:(10)Eαxα=∑k=0∞xαkΓ1+αk,sinαxα=∑k=0∞−1kx2k+1αΓ1+2k+1α,cosαxα=∑k=0∞−1kx2kαΓ1+2αk,sinhαxα=∑k=0∞x2k+1αΓ1+2k+1α,coshαxα=∑k=0∞x2kαΓ1+2αk.Definition 4.
The properties of local fractional derivatives and local fractional integrals of nondifferentiable functions are as follows [28, 30]:(11)dαdxαxnαΓ1+n−1α=xn−1αΓ1+n−1α,dαdxαEαxα=Eαxα,dαdxαsinαxα=cosαxα,dαdxαcosαxα=−sinαxα,dαdxαsinhαxα=coshαxα,dαdxαcoshαxα=−sinhαxα.


## 3. Analysis of HP Method

To show the ability of local fractional HP, the local fractional differential equation is written as follows:(12)Lαuα=0,u∈R,where Lα is a local fractional differential operator. Convex nondifferentiable homotopy Hαu,p is defined as follows:(13)Hαu,p=1−pαLαuα−Lαu0α+pαLαuα,u∈R,p∈0.1,that is equivalent to relationship (14):(14)Hαu,p=Lαuα−Lαu0α+pαLαuα,u∈R,p∈0.1,where p is an imbedding parameter and u0 is an initial approximation for relation (13). By substituting Hαu,p equal to zero, it can be written as(15)Hαu,p=Lαuα−Lαu0α=0,(16)Hαu,1=Lαu0α=0.In nondistinguishable homotopy, these relations are called nondistinguishable deformation, andLαuα−Lαu0α, Lαu0α are called nondistinguishable homotopy. By applying nondistinguishable perturbation method, relationship (15) is written as follows:(17)uα=u0α+Pαu1α+P2αu2α+P3αu3α+⋯=∑i=0nPiαuiα.Substituting relation (17) in relationship (13) could write(18)Hα∑i=0nPiui,P=1−PαLα∑i=0nPiui−Lαu0+PαLα∑i=0nPiui.In the other words, for the calculation of the approximated solution of relation (12), relationship Lαu is expanded as a local fractional Taylor series. Therefore,(19)Lαuα=Lα∑i=0nPiαuiα=Lαu0α+dαLαu0αduα∑i=0nPiui−u0αΓ1+α+d2αLαu0αdu2α∑i=0nPiui−u02αΓ1+2α+⋯+dnαLαu0αduα∑i=0nPiui−u0nαΓ1+nα+⋯.Now, substituting relation (17) in relationship (18) and then in equation (13), we can write(20)Hα∑i=0nPiui,P=1−PαdαLαu0αduα∑i=0nPiui−u0αΓ1+α+d2αLαu0αdu2α∑i=0nPiui−u02αΓ1+2α+dnαLαu0αduα∑i=0nPiui−u0nαΓ1+nα+PαdαLαu0αduα∑i=0nPiui−u0αΓ1+α+d2αLαu0αdu2α∑i=0nPiui−u02αΓ1+2α+dnαLαu0αduα∑i=0nPiui−u0nαΓ1+nα+⋯,that is reduceable to relations (21) and (22).(21)Hαu,1=Lαuα−Lαu0α=dαLαu0αduα∑i=0nPiui−u0αΓ1+α+d2αLαu0αdu2α∑i=0nPiui−u02αΓ1+2α+dnαLαu0αduα∑i=0nPiui−u0nαΓ1+nα+⋯=0,(22)Hαu,1=Lαuα=Lαu0α+dαLαu0αduα∑i=0nPiui−u0αΓ1+α+d2αLαu0αdu2α∑i=0nPiui−u02αΓ1+2α+dnαLαu0αduα∑i=0nPiui−u0nαΓ1+nα+⋯=0.Using relation (21) and simplification, we can write(23)p0α:Lαuα−Lαu0α=0,(24)p1α:dαLαu0αduαu1αΓ1+α+Lαu0α=0,(25)p2α:dαLαu0αduαu2αΓ1+α+d2αLαu0αdu2αu12αΓ1+2α.It can be written according to relation (24):(26)u1α=−Γ1+αLαu0αdαLαu0α/duα.Therefore, by consideringuα as the first approximation, it results to uα=u0α+pαu1α; consequently,(27)uα=u0α−pαΓ1+αLαu0αdαLαu0α/duα,Whenp=1, applying relation (27), reversible relation (28) is concluded.(28)un+1α=unα−Γ1+αLαunαdαLαunα/duα.This equation is a local fractional iteration Newton famous formula [3–11], and this relation is converged. Using relationship (28), we can calculate the relation of a local fractional iteration semi-Newton with second-order approximation parameters as follows:(29)un+1α=unα−Γ1+αLαunαdαLαunα/duα−Γ1+αΓ1+2αd2αLαunα/du2αdαLαunα/duαΓ1+αLαunαdαLαunα/duα2α.Whenp⟶1, then the approximation solution is as follows:(30)U=limP→1∑i=0nPiαui=∑i=0nui.It should be noted that the old perturbation method in [1–7] is significant in the case of the fractional domain of α equal to 1.


## 4. Illustrative Applied Examples

In this section, local damped wave equation, wave equation in the fractal strings, and heat diffusion equation in the case of local fractional using HPM were solved.Example 1.
The wave equation in the fractal strings.
At first, the wave equation in the fractal strings is also intended that is given the initial conditions [17].(31)∂2αux,t∂t2α−∂2αux,t∂x2α=0,t>0,xϵR,0<α<1,considering the boundary conditions:(32)ux,0=0,∂αux,0∂tα=Eαxα.
According to the local fractional HPM, the unrecognizable homotopy structure can be written as follows:(33)∂2αux,t∂t2α=pα∂2αux,t∂t2α−∂2αux,t∂x2α,0≤p≤1.
On the other hand, according to equation (13), we can write(34)ux,t=∑i=0npiαuix,t.
By substituting (17) in (16), we can say(35)∂2α∂t2α∑i=0npiαuix,t=pα∂2α∂t2α∑i=0npiαuix,t−∂2α∂x2α∑i=0npiαuix,t.
Now, for separating the sentences of relationship (18), just this relationship can be expanded and separately written to various powers p; therefore, linear fractional derivations of differential equations are adjusted as follows:(36)p0α:∂2αu0x,t∂t2α=0,u0x,0=0,∂αu0x,0∂tα=Eαxα,(37)p1α:∂2αu1x,t∂t2α=∂2αu0x,t∂x2α,u1x,0=0,∂αu1x,0∂tα=0,(38)p2α:∂2αu2x,t∂t2α=∂2αu1x,t∂x2α,u2x,0=0,∂αu2x,0∂tα=0,(39)p3α:∂2αu3x,t∂t2α=∂2αu2x,t∂x2α,u3x,0=0,∂αu3x,0∂tα=0,(40)p4α:∂2αu4x,t∂t2α=∂2αu3x,t∂x2α,u4x,0=0,∂αu4x,0∂tα=0.
Finally,:(41)ptα:∂2αuix,t∂t2α=∂2αui−1x,t∂x2α,ui−1x,0=0,∂αui−1x,0∂tα=0.
Evaluation of equations (36) to (41) for u0,u1,u2,u3,u4,ui, some components of the solution of local fractional HP are as follows:(42)u0x,t=tαΓ1+αEαxα,u1x,t=t3αΓ1+3αEαxα,u2x,t=t5αΓ1+5αEαxα,u3x,t=t7αΓ1+7αEαxα,u4x,t=t9αΓ1+9αEαxα.
Finally,(43)uix,t=t2i+1αΓ1+2i+1αEαxα.
Therefore, almost unrecognizable for relation (14) will be as follows:(44)ux,t=∑i=0nt2i+1αΓ1+2i+1αEαxα=sinhαtαEαxα.
According to equation (15), Eαxα is considered as boundary condition for computations as follows:(45)Eαxα=∑i=0NxiαΓ1+iα.
With these conditions using the computer programming in the MATLAB software,ux,t can be plotted as shown in Figure 1. For Figures 1 and 2, α=1 and α=ln2/ln3, respectively.
Since most references in their calculationsα equal to α=ln2/ln3 are considered, therefore, in this study, for better understanding of local fractional wave equation, α=ln2/ln3 is considered and for three different time versus x plotted in Figure 3, and it is remarkable.Example 2.
Heat diffusion equation in the case of local fractional.
The next example, heat diffusion equation in the case of local fractional, using the HPM, is solved as follows:(46)∂αux,t∂tα−∂2αux,t∂x2α=−cosαxα.
In this case, the initial condition is considered as follows:(47)ux,0=sinαxα.
Now, HPM can be used, and the above equation can be written as follows:(48)∂αux,t∂tα−∂αu0x,t∂tα=pα∂2αux,t∂x2α−cosαxα−∂αu0x,t∂tα.
According to relationshipux,t=∑i=0npiαuix,t and substitution in equation (48), consequently(49)∂α∂tα∑i=0npiαuix,t−∂αu0x,t∂tα=pα∂2α∂x2α∑i=0npiαuix,t−cosαxα−∂αu0x,t∂tα.
Therefore, linear fractional derivations of differential equations are as follows:(50)p0α:∂αu0x,t∂tα−∂αu0x,t∂tα=0,u0x,0=sinαxα,(51)p1α:∂αu1x,t∂tα=∂2αu0x,t∂x2α−cosαxα−∂αu0x,t∂tα,u1x,0=0,(52)p2α:∂αu2x,t∂tα=∂2αu1x,t∂x2α,u2x,0=0,(53)p3α:∂αu3x,t∂tα=∂2αu2x,t∂x2α,u3x,0=0,(54)p4α:∂αu4x,t∂tα=∂2αu3x,t∂x2α,u4x,0=0,and so on. By solving equations (50) to (54), it becomes clear that(55)u0x,0=sinαxα,u1x,t=−tαΓ1+αsinαxα+cosαxα,u2x,t=t2αΓ1+2αsinαxα+cosαxα,u3x,t=−t3αΓ1+3αsinαxα+cosαxα,u4x,t=t4αΓ1+4αsinαxα+cosαxα.
Whenp⟶1, then ux,t=∑i=0nuix,t; therefore,(56)ux,t=u0x,t+u1x,t+u2x,t+u3x,t+⋯,ux,t=sinαxα+∑j=1∞−1jtjαΓ1+jαsinαxα+cosαxα.
The equation can be rewritten as(57)ux,t=∑j=0∞−1jtjαΓ1+jαsinαxα+cosαxα−cosαxα,ux,t=Eα−tαsinαxα+cosαxα−cosαxα.
Consideringα=ln2/ln3, ux,t is plotted in Figure 4.
Now, equation (48) is considered as follows:(58)∂αux,t∂tα−∂2αux,t∂x2α=0,that the initial condition of the form ux,0=Eαxα is considered; applying the above operation is determined answering the following:(59)ux,t=∑i=1ntiαΓ1+iαEαxα=EαtαEαxα.
By placingα=ln2/ln3, ux,t is shown in Figure 5.Example 3.
Local damped wave equation.
The second example, local damped wave equation (60), using the HPM solved [45].(60)∂2αux,t∂t2α−∂αux,t∂tα−∂2αux,t∂x2α=0.
In this case, the initial condition is considered as follows:(61)ux,0=Eαxα=∑i=0NxiαΓ1+iα.
Now, HPM can be used, and the above equation can be written as follows:(62)∂2αux,t∂t2α−∂2αu0x,t∂t2α=pα∂2αux,t∂x2α+∂αux,t∂tα−∂2αu0x,t∂t2α.
According to relationshipux,t=∑i=0npiαuix,t and substitution in equation (62), it results to the following:(63)∂2α∂t2α∑i=0npiαuix,t−∂2αu0x,t∂t2α=pα∂2α∂x2α∑i=0npiαuix,t+∂α∂tα∑i=0npiαuix,t−∂2αu0x,t∂t2α.
Therefore, linear fractional derivations of differential equations are as follows:(64)p0α:∂2αu0x,t∂t2α−∂2αu0x,t∂t2α=0,u0x,0=Eαxα,(65)p1α:∂2αu1x,t∂t2α=∂2αu0x,t∂x2α+∂αu0x,t∂tα,u1x,0=0,(66)p2α:∂2αu2x,t∂t2α=∂2αu1x,t∂x2α+∂αu1x,t∂tα,u2x,0=0,(67)p3α:∂2αu3x,t∂t2α=∂2αu2x,t∂x2α+∂αu2x,t∂tα,u3x,0=0,(68)p4α:∂2αu4x,t∂t2α=∂2αu3x,t∂x2α+∂αu3x,t∂tα,u4x,0=0.
By solving the equations (64) to (68), it becomes clear that(69)u0x,0=Eαxα,u0x,t=∫u0x,0dtα=tαΓ1+αEαxα,∂2αu1x,t∂t2α=∂2α∂x2αtαΓ1+αEαxα+∂α∂tαtαΓ1+αEαxα,u1x,t=t3αΓ1+3αEαxα+tαΓ1+αEαxα,u2x,t=t5αΓ1+5αEαxα+t2αΓ1+2αEαxα,u3x,t=t7αΓ1+7αEαxα+t3αΓ1+3αEαxα,u4x,t=t9αΓ1+9αEαxα+t4αΓ1+4αEαxα.
Finally,(70)umx,t=tm+1αΓ1+m+1αEαxα+tmαΓ1+mαEαxα.
Whenp⟶1, then ux,t=∑i=0nuix,t; therefore,(71)ux,t=u0x,t+u1x,t+u2x,t+u3x,t+⋯ux,t=tαΓ1+αEαxα+∑j=1∞tj+1αΓ1+j+1α+tjαΓ1+jαEαxα.
By placingα=ln2/ln3, ux,t versus x,t is illustrated in Figure 6. Also, Figure 7 shows that for three different times, the variation of ux,t versus x is considerable.Figure 1
Variation ofux,t versus x,t and α=1.Figure 2
Variation ofux,t versus x,t and α=ln2/ln3.Figure 3
Variation ofux,t versus x, with α=ln2/ln3, t=0.2,0.5,0.8sec.Figure 4
Variation ofux,tversus x,t, with α=ln2/ln3.Figure 5
Variation ofux,t versus x,t with α=ln2/ln3, equation (59).Figure 6
Variation ofux,t versus x,t with α=ln2/ln3, equation (71).Figure 7
Variation ofux,t versus x with α=ln2/ln3, t=0.2,0.6,0.9sec, relation (71).


## 5. Comparison of Results of HP and Fourier Series Methods

In this section, for the validation and benchmarking of the present method, equation (60) means that the local damped wave equation by this method and the results of the Fourier series method for this equation are compared and discussed, according to the results for the following equation with initial condition:(72)∂2αux,t∂t2α−∂αux,t∂tα−∂2αux,t∂x2α=0ux,0=Eαxα=∑i=0NxiαΓ1+iα.Using HPM, the solution is determined as follows:(73)uhomotopyx,t=tαΓ1+αEαxα+∑j=1∞tj+1αΓ1+j+1α+tjαΓ1+jαEαxα.In reference paper [36], the local fractional wave equation is solved:(74)useriesx,t=∑n=1∞Eα−tα2Ancosαρtα+Bnsinαρtαsinαnαπxlα,An=2Γ1+αnπαlαΓ1+αsinαnαπxlα−lπxαcosαnαπxlα−1,Bn=−Γ1+αρnπαlαΓ1+αcosαnαπxlα−lπxαsinαnαπxlα,whereρ=2πx/l−1/2 and Eαxα=∑i=0Nxiα/Γ1+iα.In this section, for several times andα=1and0.33, the variations of ux,t are obtained using the Fourier series and HP methods. In these graphs, the uour show that the HP method is considered in the present research and uref illustrate the obtained results of the Fourier series method. Comparison of two graphs of ux for time equal to 0.1, 0.3, and 0.6 sec for two methods with α=1,0.33 is illustrated in Figures 8–13.Figure 8
Variation ofux,t versus x with α=1, t=0.6sec.Figure 9
Variation ofux,t versus x with α=1, t=0.3sec.Figure 10
Variation ofux,t versus x with α=1, t=0.1sec.Figure 11
Variation ofux,t versus x, with α=0.33, t=0.7sec.Figure 12
Variation ofux,t versus x, with α=0.33, t=0.3sec.Figure 13
Variation ofux,t versus x with α=0.33, t=0.1sec.According to Figures8–10 (α=1) and Figures 11–13 (α=0.33), it is observed that for x>0.5, the value of error for the results is very less, and by increasing time t=0.1to0.7sec for α=1,0.33, the ln2/ln3 values of convergence increased. In wave equation, by increasing time, due to damping, the wave domain decreased. Therefore, the best of case is equal to α=0.33 and t=0.6,0.7sec.Variations ofux,t versus x with α=1, t=0.1,0.3,and0.8sec are illustrated in Tables 1 and 2. According to Table 1, the percentage of error of ux,t between the present study and Ref [36] with α=1, t=0.1sec, and x≥0.6, is less than 10. Also, the % error for x≥0.5, in α=1, t=0.3sec, is less than 5, whereas for α=1, t=0.8sec, the % error for x≥0.3 is less than 10. This comparison shows that by increasing x from 0.5 to 1 for all of the three times, the % error between the two methods of homotopy perturbation and the Fourier series decreases sharply and converges to 1.Table 1
The variation ofux,t versus x with α=1, t=0.1,0.3,and0.8sec.
xux,t, α=1tsec=0.8tsec=0.3tsec=0.1Present workYong-Ju Yang [36]Percentage of differencePresent workYong-Ju Yang [36]Percentage of differencePresent workYong-Ju Yang [36]Percentage of difference0.10.26230.297513.41980.31810.391222.98100.35880.550353.37240.20.27950.318714.02500.32290.407926.32390.37460.563150.32030.30.35740.39029.17740.40680.472616.17500.43830.609339.01440.40.40230.43397.85480.44630.512214.76590.47960.634432.27690.50.69720.71893.11250.71570.75094.92170.73130.815014.57890.60.73520.74351.12890.75190.78264.08300.76940.83748.83810.70.77320.78731.82360.78150.81504.28630.80120.86297.70100.80.90840.91260.46230.91200.92521.44740.91740.94152.62700.90.95160.96321.21900.95250.96511.32280.95650.97281.70411.01.01.001.01.001.01.00Table 2
The variation ofux,t versus x with α=0.33, t=0.1,0.3,and0.7sec.
xux,t, α=0.33tsec=0.7tsec=0.3tsec=0.1Present workYong-Ju Yang [36]Percentage of differencePresent workYong-Ju Yang [36]Percentage of differencePresent workYong-Ju Yang [36]Percentage of difference0.10.09860.130131.94730.12280.183449.34850.14140.264386.91650.20.13530.171326.60750.15840.291338.44700.17710.299669.16700.30.25230.285411.59780.26780.324421.13520.28550.392137.33800.40.31420.33817.60660.33020.378014.47610.33880.431727.42030.50.66060.67542.24040.67250.69583.46470.67840.72697.14920.60.70640.71661.44390.71230.72932.38660.71950.75775.30920.70.74770.76021.67180.75230.76511.70150.76300.79293.91870.80.89650.90160.56890.89800.90751.05790.91540.91630.00980.90.94490.95420.98420.94430.95080.68830.95330.96030.00731.01.01.001.01.001.01.00According to Table1, the percentage of error of ux,t between two methods with α=0.33, t=0.1sec, and x≥0.5, is less than 8. Also, the % error for x≥0.5 in α=0.33, t=0.3sec, is less than 4, whereas for α=0.33, t=0.7sec, the % error for x≥0.4 is less than 8. It can also be concluded that by increasing x from 0.5 to 1 for the three times, the percentage of difference between the two methods decreases sharply and converges to 1.


## 6. Conclusion

The relationship between mathematics and engineering sciences (especially mechanical engineering, chemistry, and civil engineering) and the application of high-precision mathematical theoretical methods have received much attention in recent years, but less homotopy has been considered for real physical problems in engineering. Defining common projects between mathematicians and engineers can fill the gap.The obvious advantage of the method is that it can be applied to various nonlinear problems. The main disadvantage is that we should suitably choose an initial guess, or infinite iterations are required.In the present study, the analysis of HP method was conducted. This method was applied for three examples, heat diffusion equation in the case of local fractional, the wave equation in the fractional strings, the local damped wave equation. Also, the variation ofux,t for several x,t in α=1 and α=ln2/ln3 is plotted. Therefore, the local fractional for solving of partial differential equations using the HP method is down, and the obtained results in applied examples are compared to the Fourier series method (Ref. [36]); the results of two methods are in good agreement. In these examples, for x>0.5, α=0.33, a least error was concluded. According to Tables 1 and 2, the values of ux,t for α=1and0.33, t=0.1sec, the percentage of error between present research and Ref [36] for x≥0.6 is less than 10. Also, the % error for x≥0.5 in α=1and0.33, t=0.3sec, is less than 5, whereas for α=1and0.33, t=0.8and0.7sec, the % error for x≥0.4 is less than 8. Also, by increasing x, the field variable in the two methods is converged.



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