LFNσ(1)=1sσ | LFNσ(Cosσ(τσ))=sσs2σ+u2σ |
LFNσ(τσΓ(1+σ))=uσs2σ | LFNσ(Sinσ(τσ))=uσs2σ+u2σ |
LFNσ(τqσΓ(1+qσ))=uqσs(q+1)σ | LFNσ(Coshσ(τσ))=sσs2σ−u2σ |
After decades of rapid development, the scale and complexity of modern networks have far exceed our expectations. In many conditions, traditional traffic identification methods cannot meet the demand of modern networks. Recently, fine-grained network traffic identification has been proved to be an effective solution for managing network resources. There is a massive increase in the use of fine-grained network traffic identification in the communications industry. In this article, we propose a comprehensive overview of fine-grained network traffic identification. Then, we conduct a detailed literature review on fine-grained network traffic identification from three perspectives: wired network, mobile network, and malware traffic identification. Finally, we also draw the conclusion on the challenges of fine-grained network traffic identification and future research prospects.
Citation: Xiaolin Gui, Yuanlong Cao, Ilsun You, Lejun Ji, Yong Luo, Zhenzhen Luo. A Survey of techniques for fine-grained web traffic identification and classification[J]. Mathematical Biosciences and Engineering, 2022, 19(3): 2996-3021. doi: 10.3934/mbe.2022138
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After decades of rapid development, the scale and complexity of modern networks have far exceed our expectations. In many conditions, traditional traffic identification methods cannot meet the demand of modern networks. Recently, fine-grained network traffic identification has been proved to be an effective solution for managing network resources. There is a massive increase in the use of fine-grained network traffic identification in the communications industry. In this article, we propose a comprehensive overview of fine-grained network traffic identification. Then, we conduct a detailed literature review on fine-grained network traffic identification from three perspectives: wired network, mobile network, and malware traffic identification. Finally, we also draw the conclusion on the challenges of fine-grained network traffic identification and future research prospects.
Helmholtz and Burgers' equations play an important role in various streams of applied physics. The Helmholtz equation frequently occurs in the study of physical phenomena involving elliptic partial differential equations (PDEs) such as wave and diffusion, magnetic fields, seismology, electromagnetic radiation, transmission, vibrating lines, acoustics, and geosciences. This equation is actually derived from the wave equation. The Helmholtz equation is a transformed form of the acoustic wave equation. It is utilized in a stream of seismic wave propagation and imaging. This equation plays a significant role in estimations of acoustic propagation in shallow water at low frequencies and characteristics of geodesic sea floor [1]. Mathematically, the eigenvalue problem for the Laplace operator is called the Helmholtz equation expressed by elliptic type linear PDE ∇2ϑ=−k2ϑ, where ∇2 denotes the Laplacian differential operator, k2 signifies the eigenvalue, and ϑ is the eigen function. When this equation is used in respect of waves, k is termed as the wave number which measures the spatial frequency of waves. For the first time, Samuel and Thomas [2] suggested the Helmholtz equation with fractional order. Recently, Prakash et al. [3] presented the solution of the space-fractional Helmholtz equation with the q-homotopy analysis transform method (q-HATM). More recently, Shah et al. [4] examined the fractional Helmholtz equation also.
On the other hand, the Burgers' equations [5,6,7] characterize the nonlinear diffusion phenomenon through the simplest PDEs. Burgers' equations occur mainly in the mathematical model of turbulence, fluid mechanics, and approximation of flow in viscous fluids [5,8,9]. The coupled Burgers' equations in one-dimensional form are described as a sedimentation and/or evolution model of scaled volume concentrations in fluid suspensions. More literature about coupled Burgers' equations can be found in previous works [10,11]. In view of the development of the fractional calculus approach, the Burgers' equation with a fractional derivative was first presented in [12]. After that, many authors investigated the solution for fractional Burgers' equations in past decades using approximate analytical methods (see, for example, [13,14,15,16,17,18,19,20,21,22,23]).
From the past decade, the concept of local fractional calculus and local fractional derivatives developed in the work of Yang [24,25] has been a centre of attraction among researchers. Further, many authors investigated the equations and models appearing in fractal media through various local fractional methods, for instance, local fractional homotopy perturbation method (LFHPM) for handling local fractional PDEs (LFPDEs) [26,27], local fractional Tricomi equation arising in fractal transonic flow [28], local fractional Klein-Gordon equations [29], local fractional heat conduction equation [30], local fractional wave equation in fractal strings [31], local fractional Laplace equation [32], system of LFPDEs [33], and fractal vehicular traffic flow [34], etc. In this sequence, the 2D local fractional Helmholtz equation (LFHE) was introduced in [35]. Recently, the LFHE was solved by local fractional variational iteration method [36], local fractional series expansion method [37]. The local fractional Helmholtz and coupled Helmholtz equations were handled successfully by Baleanu and Jassim [38,39,40] through various local fractional methods. In recent years, the local fractional coupled Burgers' equations (LFCBEs) were also investigated for solutions through various techniques that can be found in [41,42,43,44,45].
The 2D local fractional coupled Helmholtz equations (LFCHEs) suggested in [38] are given as follows:
∂2σϑ1(γ,τ)∂τ2σ+∂2σϑ2(γ,τ)∂γ2σ−ω12σϑ1(γ,τ)=ℵ1(γ,τ),0<σ⩽1, | (1.1) |
∂2σϑ2(γ,τ)∂τ2σ+∂2σϑ1(γ,τ)∂γ2σ−ω22σϑ2(γ,τ)=ℵ2(γ,τ),0<σ⩽1, | (1.2) |
subject to the initial conditions:
ϑ1(γ,0)=ϕ1(γ),∂σϑ1(γ,0)∂τσ=ψ1(γ),ϑ2(γ,0)=ϕ2(γ),∂σϑ2(γ,0)∂τσ=ψ2(γ), | (1.3) |
where ϑ1(γ,τ) and ϑ2(γ,τ) are unknown local fractional continuous functions, and ℵ1(γ,τ) and ℵ2(γ,τ) are the nondifferentiable source terms.
The system of nonlinear coupled Burger's equations with local fractional derivatives can be described as:
∂σϑ1∂τσ+ξ1∂2σϑ1∂γ2σ+ξ2∂σϑ1∂γσϑ1+ρ∂σ[ϑ1ϑ2]∂γσ=0,ϑ1=ϑ1(γ,τ),0<σ⩽1, | (1.4) |
∂σϑ2∂τσ+μ1∂2σϑ2∂γ2σ+μ2∂σϑ2∂γσϑ2+η∂σ[ϑ1ϑ2]∂γσ=0,ϑ2=ϑ2(γ,τ),0<σ⩽1, | (1.5) |
subject to the initial conditions:
ϑ1(γ,0)=f1(γ),ϑ2(γ,0)=f2(γ), | (1.6) |
where ξ1, ξ2, μ1, and μ2 denote real constants, ρ and η specify arbitrary constants that depend on parameters of the system, ϑ1(γ,τ) and ϑ2(γ,τ) are local fractional continuous functions, and γ lies in the computational domain Ω.
The key purpose of this work is to establish a new coupling of local fractional homotopy analysis method (LFHAM) [43,46] and local fractional natural transform (LFNT) [47], named as local fractional natural homotopy analysis method (LFNHAM) throughout in this paper. The second goal of the paper is to explore the solutions for the LFCHEs and the LFCBEs by utilizing the newly suggested combination LFNHAM. Moreover, the numerical simulations have also been presented for the obtained solutions of LFCHEs and LFCBEs for the fractal order σ=ln2/ln3 of a local fractional derivative by using MATLAB. The originality and novelty of the paper lie in the fact that the LFCHEs and LFCBEs have never been solved by using this newly suggested combination LFNHAM. In addition, convergence and uniqueness of the LFNHAM solution are also examined for the LFNHAM solution of general LFPDE in view of Banach's fixed point theory.
The notable aspect of the LFNHAM as compared to others is that it offers an extended degree of freedom for analysis and the main ingredient is an auxiliary parameter ℏ≠0 to ensure the convergence of the acquired series solution. Furthermore, a more appropriate choice of an initial guess & effortless creation of deformation equations are the interesting attributes of this method. The LFNHAM is surely beneficial as it combines two powerful algorithms to attain the solutions for nonlinear LFDEs. The LFNHAM generates a convergent series solution that revolves around a convergence parameter without involving linearization, perturbation, or descretization phenomena. In addition, the LFNHAM also minimizes the numerical work unlike other conventional methods while still giving extremely precise results. The LFNHAM provides a more general solution as compared to LFHPM, local fractional Adomian decomposition method (LFADM) and local fractional natural homotopy perturbation method (LFNHPM) and assimilates their consequences as a special case. In addition, it does not involve the computation of complicated Adomian or He's polynomials. But there is also a point of demerit with this technique. The implementation of LFNHAM can be difficult in the situation of non-evaluation of the LFNT of a function. This work checks the LFNHAM solution regarding uniqueness and convergence for the first time and the error analysis of the LFNHAM solution is also discussed. These points surely illustrate the reliability and validity of the proposed method. The other aspect of the LFNHAM is that the coupling of LFNT with LFHAM performs fast-tracked calculations in comparison to LFHAM and consequently consumes less time and less computer memory.
Moreover, the LFNT possesses two important attributes, scale property and unit-preserving property, and hence can be utilized to handle LFPDEs without exerting new frequency range. In the light of these facts, the LFNT which possesses the linearity feature, also possesses the feature of linearity of functions, and hence does not involve the changing of units. This transform performs operation similarly as the local fractional Laplace transform (LFLT) and local fractional Sumudu transform (LFST). By virtue of these facts, the LFNT may be used to analyze some complex problems of science and engineering that may be handled hardly with other integral transforms.
The rest portion of the paper is organized as follows: Section 2 presents definitions and formulae for the local fractional derivative and LFNT. Section 3 illustrates the computational procedure for the suggested scheme LFNHAM. The convergence and uniqueness of the LFNHAM solution is discussed in Section 4. Sections 5 and 6 are devoted to the implementation of the LFNHAM to the LFCHEs and LFCBEs, respectively. In Section 7, numerical simulations have been performed in respect of a fractal value. At the end, Section 8 presents the epilogue.
The section presents a quick view of some definitions and formulae which have been utilized in this work.
Definition 2.1.1. [24,25] Let (l1,l2) be the interval and Δt=max{Δt0,Δt1,Δt2,Δt3,...} be a partition of (l1,l2) with (tj,tj+1), j=0,...,N−1, t0=l1, tN=l2 with Δtj=tj+1−tj. Now, the LFI of ϑ(γ) is formulated as
l1Iσl2ϑ(γ)=1Γ(1+σ)∫l2l1ϑ(t)(dt)σ=1Γ(1+σ)limΔt→0N−1∑j=0ϑ(tj)(Δtj)σ. | (2.1) |
Definition 2.1.2. [24,25] The Mittag-Leffler function is given as
Eσ(γσ)=∞∑q=0γqσΓ(1+qσ),0<σ⩽1. | (2.2) |
Definition 2.1.3. [24,25] The fractal sine and cosine functions are given by
Sinσ(γσ)=∞∑q=0(−1)qγ(2q+1)σΓ(1+(2q+1)σ),0<σ⩽1, | (2.3) |
Cosσ(γσ)=∞∑q=0(−1)qγ2qσΓ(1+2qσ),0<σ⩽1, | (2.4) |
Sinhσ(γσ)=∞∑q=0γ(2q+1)σΓ(1+(2q+1)σ),0<σ⩽1, | (2.5) |
Coshσ(γσ)=∞∑q=0γ2qσΓ(1+2qσ),0<σ⩽1. | (2.6) |
Definition 2.1.4. [24,25] The LFD of ϑ(γ)∈Cσ(l1,l2) of order σ at γ=γ0 is presented as
Dσγϑ(γ0)=dσϑ(γ0)dγσ=ϑσ(γ0)=Δσ(ϑ(γ)−ϑ(γ0))(γ−γ0)σ,γ∈(l1,l2), | (2.7) |
where Δσ(ϑ(γ)−ϑ(γ0))≅Γ(σ+1)(ϑ(γ)−ϑ(γ0)).
The local fractional partial derivative of ϑ(γ,τ)∈Cσ(l1,l2) of order σ was provided by Yang [24,25] as follows:
∂σ∂τσϑ(γ,τ)=Δσ(ϑ(γ,τ)−ϑ(γ,τ0))(τ−τ0)σ,γ∈(l1,l2), | (2.8) |
where Δσ(ϑ(γ,τ)−ϑ(γ,τ0))≅Γ(σ+1)(ϑ(γ,τ)−ϑ(γ,τ0)).
The LFIs and LFDs of special functions used in this study and described in [24,25] are given as follows:
Dσγaϑ(γ)=aDσγϑ(γ),Dσγ(γqσΓ(1+qσ))=γ(q−1)σΓ(1+(q−1)σ),q∈N, |
DσγEσ(γσ)=Eσ(γσ),Dσγ(Sinσ(γσ))=Cosσ(γσ),Dσγ(Cosσ(γσ))=−Sinσ(γσ), |
Iσγ(γqσΓ(1+qσ))=γ(q+1)σΓ(1+(q+1)σ),q∈N, |
where γσ signifies a Cantor function.
For the first time, Khan & Khan [48] suggested a new integral transform called N-transform. Some years later, Belgacem and Silambarasan [49,50,51] changed its name to Natural transform and also presented a comprehensive study regarding its applications. This transform performs operation similarly as the Laplace and Sumudu transforms.
Definition 2.2.1. [47] The LFNT of the function ϑ(γ,τ) of order σ is defined as
LFNσ[ϑ(γ,τ)]=ˉϑσ(γ,s,u)=1Γ(1+σ)∫∞0Eσ(−sστσuσ)ϑ(γ,τ)uσ(dτ)σ,0<σ⩽1, | (2.9) |
and the corresponding inverse LFNT LFN−1σ is formulated as
LFN−1σ[ˉϑσ(γ,s,u)]=ϑ(γ,τ)=1(2πi)σ∫ρ+i∞ρ−i∞Eσ(sστσuσ)ˉϑσ(γ,s,u)(ds)σ,0<σ⩽1, | (2.10) |
where sσ and uσ signify the LFNT variables and ρ denotes a real constant. The integral in the definition of inverse LFNT is taken along sσ=ρ in the complex plane sσ=xσ+iyσ. It is notable that the LFNT converges to LFLT for u=1 and to LFST for s=1.
Some properties of the LFNT are being mentioned here:
Proposition 2.2.2. [47] The LFNT of a LFD is defined by
LFNσ[ϑ(qσ)(γ,τ)]=sqσuqσˉϑσ(γ,s,u)−q−1∑k=0s(q−k−1)σu(q−k)σϑ(kσ)(γ,0). | (2.11) |
For q=1,2 and 3, the following expressions are generated
LFNσ[ϑ(σ)(γ,τ)]=sσuσˉϑσ(γ,s,u)−1uσϑ(γ,0), |
LFNσ[ϑ(2σ)(γ,τ)]=s2σu2σˉϑσ(γ,s,u)−sσu2σϑ(γ,0)−1uσϑ(σ)(γ,0), |
LFNσ[ϑ(3σ)(γ,τ)]=s3σu3σˉϑσ(γ,s,u)−s2σu3σϑ(γ,0)−sσu2σϑ(σ)(γ,0)−1uσϑ(2σ)(γ,0). | (2.12) |
Proposition 2.2.3. [47] The linearity property of the LFNT is defined by
LFNσ[α1ϑ1(γ,τ)+α2ϑ2(γ,τ)]=α1LFNσ[ϑ1(γ,τ)]+α2LFNσ[ϑ2(γ,τ)]=α1ˉϑ1,σ(γ,s,u)+α2ˉϑ2,σ(γ,s,u), | (2.13) |
where ˉϑ1,σ(γ,s,u) and ˉϑ2,σ(γ,s,u) denote the LFNT of ϑ1(γ,τ) and ϑ2(γ,τ), respectively.
Theorem 2.2.4. (Local fractional convolution). If LFNσ{ϑ1(γ,τ)}=ˉϑ1,σ(γ,s,u) and LFNσ{ϑ2(γ,τ)}=ˉϑ2,σ(γ,s,u), we have
LFNσ{ϑ1(γ,τ)∗ϑ2(γ,τ)}=uσˉϑ1,σ(γ,s,u)ˉϑ2,σ(γ,s,u), | (2.14) |
where
ϑ1(γ,τ)∗ϑ2(γ,τ)=1Γ(1+σ)∫∞0ϑ1(γ,η)ϑ2(γ,τ−η)(dη)σ. | (2.15) |
Some useful formulae for LFNT are listed in Table 1 [47].
LFNσ(1)=1sσ | LFNσ(Cosσ(τσ))=sσs2σ+u2σ |
LFNσ(τσΓ(1+σ))=uσs2σ | LFNσ(Sinσ(τσ))=uσs2σ+u2σ |
LFNσ(τqσΓ(1+qσ))=uqσs(q+1)σ | LFNσ(Coshσ(τσ))=sσs2σ−u2σ |
To explain the basic idea of LFNHAM, the following LFPDE is taken here
Lσϑ(γ,τ)+Pσϑ(γ,τ)+Qσϑ(γ,τ)=ω(γ,τ),0<γ<1,0<τ<1, | (3.1) |
where Lσ≡∂qσ∂τqσ denotes the linear local fractional differential operator (LFDO) of order qσ i.e., ∃ a number δ>0 such that ‖Lσϑ‖⩽δ‖ϑ‖, Pσ specifies the linear fractional differential operator of general nature in γ and τ. Here, it is also assumed that Pσ is bounded i.e., |Pσ(ϑ−ϑ∗)|⩽λ|ϑ−ϑ∗|. Qσ denotes the nonlinear differential operator which is Lipschitz continuous with ξ>0 fulfilling the criteria |Qσ(ϑ−ϑ∗)|⩽ξ|ϑ−ϑ∗|, γ and τ are independent variables, ϑ(γ,τ) and ω(γ,τ) denote local fractional unknown function and nondifferentiable source term, respectively.
Now, the suggested computational approach recommends the implementation of the LFNT operator PNϵ on Eq (3.1)
LFNσ[Lσϑ(γ,τ)]+LFNσ[Pσϑ(γ,τ)]+LFNσ[Qσϑ(γ,τ)]=LFNσ[ω(γ,τ)]. | (3.2) |
Using the property of LFNT for LFDs, it follows
ˉϑσ(γ,s,u)=uqσsqσq−1∑k=0s(q−k−1)σu(q−k)σϑ(kσ)(γ,0)+uqσsqσ(LFNσ[ω(γ,τ)])−uqσsqσ(LFNσ[Pσϑ(γ,τ)]+LFNσ[Qσϑ(γ,τ)]), | (3.3) |
whereˉϑσ(γ,s,u)=LFNσ[ϑ(γ,τ)]. | (3.4) |
After simplification, we get
ˉϑσ(γ,s,u)−uqσsqσq−1∑k=0s(q−k−1)σu(q−k)σϑ(kσ)(γ,0)−uqσsqσ(LFNσ[ω(γ,τ)])+uqσsqσ(LFNσ[Pσϑ(γ,τ)]+LFNσ[Qσϑ(γ,τ)])=0. | (3.5) |
Now on account of Eq (3.5), the nonlinear operator is constituted as
Φ[φ(γ,τ;κ)]=LFNσ[φ(γ,τ;κ)]−uqσsqσq−1∑k=0s(q−k−1)σu(q−k)σφ(kσ)(γ,0;κ)−uqσsqσ(LFNσ[ω(γ,τ)])+uqσsqσ(LFNσ[Pσφ(γ,τ;κ)]+LFNσ[Qσφ(γ,τ;κ)])=0, | (3.6) |
where κ∈[0,1] is an embedding parameter, φ(γ,τ;κ) symbolizes the local fractional unknown function of γ,τ and κ, and the symbol LFNσ represents the LFNT operator.
Now utilizing the traditional approach of LFHAM [43,46] and basic methodology of HAM [52,53,54], the zeroth-order deformation equation is developed in this way:
(1−κ)LFNσ[φ(γ,τ;κ)−ϑ0(γ,τ)]=κℏΦ[φ(γ,τ;κ)], | (3.7) |
where ℏ≠0 is a convergence regulation parameter and ϑ0(γ,τ) symbolizes an initial guess for ϑ(γ,τ).
It is observed that LFNHAM makes easy the choice of auxiliary parameters, linear operator, and initial guess. The following equations stand firmly for κ=0 and κ=1 in this manner
φ(γ,τ;0)=ϑ0(γ,τ),φ(γ,τ;1)=ϑ(γ,τ). | (3.8) |
Hence, when κ takes values from 0 to 1, φ(γ,τ;κ) deviates from ϑ0(γ,τ) to ϑ(γ,τ). Next, the Taylor's series expansion of φ(γ,τ;κ) about κ generates
φ(γ,τ;κ)=ϑ0(γ,τ)+∑∞μ=1κμϑμ(γ,τ), | (3.9) |
where
ϑμ(γ,τ)=[1Γ(μ+1)∂μ[ϕ(γ,τ;κ)]∂κμ]κ=0. | (3.10) |
The convergence controller ℏ≠0 promptly provides the convergence of the series solution (3.9). Thus the series given by Eq (3.9) converges at κ=1 with appropriate pick of ϑ0(γ,τ). Thus, we have
ϑ(γ,τ)=ϑ0(γ,τ)+∑∞μ=1ϑμ(γ,τ). | (3.11) |
Equation (3.11) provides a relationship between ϑ0(γ,τ) and the exact solution ϑ(γ,τ) through the terms ϑμ(γ,τ),(μ=1,2,3,...), that will be calculated in upcoming steps. Eq (3.11) provides the solution of Eq (3.1) in the form of a series.
The vectors are constituted as
→ϑμ={ϑ0(γ,τ),ϑ1(γ,τ),ϑ2(γ,τ),ϑ3(γ,τ),......ϑμ(γ,τ)}. | (3.12) |
Now, the μth-order deformation equation is framed as
LFNσ[ϑμ(γ,τ)−χμϑμ−1(γ,τ)]=ℏℜμ[ϑμ−1(γ,τ)]. | (3.13) |
Operating the inverse of LFNT on Eq (3.13), we get
ϑμ(γ,τ)=χμϑμ−1(γ,τ)+ℏLFN−1σ(ℜμ(ϑμ−1(γ,τ))). | (3.14) |
In Eq (3.14), the value of ℜμ(ϑμ−1(γ,τ)) can be written in a new look as
ℜμ(ϑμ−1(γ,τ))=LFNσ[ϑμ−1(γ,τ)]−(1−χμ)(uqσsqσq−1∑k=0s(q−k−1)σu(q−k)σϑ(kσ)(γ,0)+uqσsqσ(LFNσ[ω(γ,τ)]))+uqσsqσLFNσ(Pσϑμ−1(γ,τ)+P′μ−1), | (3.15) |
where the value of χμ is presented as
χμ={0,μ⩽11,μ>1. | (3.16) |
In Eq (3.15), P′μ denotes homotopy polynomial suggested in [55] in functioning of LFHAM [43,46], and is formulated as
P′μ=1Γ(μ)[∂μ∂κμQσ(φ(γ,τ;κ))]κ=0, | (3.17) |
where
φ=φ0+κφ1+κ2φ2+κ3φ3⋯. | (3.18) |
Putting the value of ℜμ(ϑμ−1) from Eq (3.15) in Eq (3.14) transforms the Eq (3.14) as follows:
ϑμ(γ,τ)=(χμ+ℏ)ϑμ−1(γ,τ)−ℏ(1−χμ)LFN−1σ(uqσsqσq−1∑k=0s(q−k−1)σu(q−k)σϑ(kσ)(γ,0)+uqσsqσ(LFNσ[ω(γ,τ)]))+ℏLFN−1σ(uqσsqσLFNσ(Pσϑμ−1(γ,τ)+P′μ−1)). | (3.19) |
From Eq (3.19), the components ϑμ(γ,τ) can be evaluated for μ⩾1 and the LFNHAM solution is presented in the following way:
ϑ(γ,τ)=limN→∞N∑μ=0ϑμ(γ,τ). | (3.20) |
The significant aspect of the LFNHAM is the auxiliary parameter ℏ≠0 which guarantees the convergence of the series solution of Eq (3.1).
Theorem 3.1. If a constant 0<ℵ<1 can be estimated such that ‖ϑμ+1(γ,τ)‖⩽ℵ‖ϑμ(γ,τ)‖ for each value of μ. Moreover, if the truncated series ∑Nμ=0ϑμ(γ,τ) is assumed as an approximate solution ϑ then the maximum absolute truncated error is computed as
‖ϑ(γ,τ)−∑Nμ=0ϑμ(γ,τ)‖⩽ℵN+1(1−ℵ)‖ϑ0(γ,τ)‖. | (3.21) |
Proof. The maximum absolute truncated error is computed in this way:
‖ϑ(γ,τ)−∑Nμ=0ϑμ(γ,τ)‖=‖∑∞μ=N+1ϑμ(γ,τ)‖⩽∑∞μ=N+1‖ϑμ(γ,τ)‖⩽∑∞μ=N+1ℵμ‖ϑ0(γ,τ)‖⩽(ℵ)N+1[1+ℵ+ℵ2+⋯]‖ϑ0(γ,τ)‖⩽ℵN+1(1−ℵ)‖ϑ0(γ,τ)‖. |
This finishes the proof.
In the upcoming Section 4, we establish the convergence and uniqueness of the LFNHAM solution.
Theorem 4.1. (Uniqueness theorem). The attainment of solution by implementation of LFNHAM for the LFPDE (3.1) is unique, wherever 0<ρ<1, where
ρ=(1+ℏ)+ℏ(λ+ξ)ς. | (4.1) |
Proof. The solution of nonlinear LFPDE (3.1) is obtained as
ϑ(γ,τ)=limN→∞N∑μ=0ϑμ(γ,τ), | (4.2) |
where
ϑμ(γ,τ)=(χμ+ℏ)ϑμ−1(γ,τ)−ℏ(1−χμ)LFN−1σ(uqσsqσq−1∑k=0s(q−k−1)σu(q−k)σϑ(kσ)(γ,0)+uqσsqσ(LFNσ[ω(γ,τ)]))+ℏLFN−1σ(uqσsqσLFNσ(Pσϑμ−1(γ,τ)+P′μ−1)). | (4.3) |
Let ϑ(γ,τ) and ϑ∗(γ,τ) be two distinct possible solutions for Eq (3.1), then we acquire
|ϑ(γ,τ)−ϑ∗(γ,τ)|=|(1+ℏ)(ϑ−ϑ∗)+ℏLFN−1σ(uqσsqσLFNσ[Pσ(ϑ−ϑ∗)+Qσ(ϑ−ϑ∗)])|⩽(1+ℏ)|ϑ−ϑ∗|+ℏLFN−1σ(uqσsqσLFNσ|Pσ(ϑ−ϑ∗)+Qσ(ϑ−ϑ∗)|). | (4.4) |
Utilization of the local fractional convolution theorem for LFNT in Eq (4.4) gives
|ϑ−ϑ∗|⩽(1+ℏ)|ϑ−ϑ∗|+ℏΓ(1+σ)∫τ0|Pσ(ϑ−ϑ∗)+Qσ(ϑ−ϑ∗)|(τ−η)(q−1)σΓ(1+(q−1)σ)(dη)σ⩽(1+ℏ)|ϑ−ϑ∗|+ℏΓ(1+σ)∫τ0(λ+ξ)|ϑ−ϑ∗|(τ−η)(q−1)σΓ(1+(q−1)σ)(dη)σ. | (4.5) |
Now, with the help of mean value theorem (MVT) of LFI calculus [56,57], inequality (4.5) transforms in the following form
|ϑ(γ,τ)−ϑ∗(γ,τ)|⩽(1+ℏ)|ϑ−ϑ∗|+ℏ(λ+ξ)|ϑ−ϑ∗|ς=[(1+ℏ)+ℏ(λ+ξ)ς]|ϑ−ϑ∗|=ρ|ϑ−ϑ∗|.∴ | (4.6) |
where \rho = \left({1 + \hbar } \right) + \hbar \, \left({\lambda + \xi } \right)\, \varsigma . (4.7)
Since 0 < \rho < 1 , therefore \left| {\vartheta - {\vartheta ^ * }\, } \right| = 0 , which provides \vartheta = {\vartheta ^ * } . This ensures the aspect of uniqueness of the solution of Eq (3.1).
Theorem 4.2. (Convergence theorem). Suppose \Xi is a Banach space and \Theta :\Xi \to \Xi is a nonlinear mapping. Assume that
\left\| {\Theta \left( \omega \right) - \Theta \left( \vartheta \right)\, } \right\| \leqslant {{\rlap{-} \lambda }} \, \left\| {\omega - \vartheta } \right\| , \forall \, \, \vartheta , \, \, \omega \in \Xi . | (4.8) |
Then Banach's fixed point theory [58,59] suggests the existence of a fixed point for \Theta . Moreover, the sequence constructed by LFNHAM converges to the fixed points of \Theta with arbitrary choices of {\omega _0}, \, \, {\vartheta _0} \in \Xi and
\left\| {{\vartheta _{{l_1}}} - {\vartheta _{{l_2}}}} \right\| \leqslant \frac{{{{{\rlap{-} \lambda }} ^{{l_2}}}}}{{1 - {{\rlap{-} \lambda }} }}\left\| {{\vartheta _1} - {\vartheta _0}} \right\|, \forall \, \, \vartheta , \, \, \omega \in \Xi . | (4.9) |
Proof. It is presumed that \left({\Pi \left[\Omega \right]\, , \, \left\| {\, .\, } \right\|} \right) , where \Pi \left[\Omega \right] signifies the Banach space of continuous functions on real line interval \Omega holding the sup norm. Now, it is sufficient to prove that \left\{ {\, {\vartheta _{{l_2}}}} \right\} is a Cauchy sequence in the Banach space \Xi .
Now, consider
\left\| {\, {\vartheta _{{l_1}}} - {\vartheta _{{l_2}}}} \right\| = \mathop {\max }\limits_{\tau \, \in \Omega } \left| {{\vartheta _{{l_1}}} - {\vartheta _{{l_2}}}} \right| \\ = \mathop {\max }\limits_{\tau \, \in \Omega } \left| {\left( {1 + \hbar } \right)\, \left( {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right) + \hbar \, {}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^{q\sigma }}}}{{{s^{q\sigma }}}}{}^{LF}{N_\sigma }\left[ {{P_\sigma }\, \left( {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right) + {Q_\sigma }\, \left( {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right)\, } \right]\, } \right)\, \, } \right|\\ \leqslant \mathop {\max }\limits_{\tau \, \in \Omega } \left\{ {\left( {1 + \hbar } \right)\left| {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right|\, + \hbar \, {}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^{q\sigma }}}}{{{s^{q\sigma }}}}{}^{LF}{N_\sigma }\left[ {\left| {{P_\sigma }\, \left( {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right)} \right| + \left| {{Q_\sigma }\, \left( {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right)\, } \right|\, } \right]\, } \right)\, \, } \right\}\, . | (4.10) |
Employing the local fractional convolution theorem for LFNT in Eq (4.10), we have
\left\| {{\vartheta _{{l_1}}} - {\vartheta _{{l_2}}}} \right\| \leqslant \mathop {\max }\limits_{\tau \, \in \Omega } \left\{ {\, \left( {1 + \hbar } \right)\, \left| {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right|} \right. \\ \left. { + \frac{\hbar }{{\Gamma \left( {1 + \sigma } \right)}}\int_0^\tau {\, \left[ {\, \left| {{P_\sigma }\, \left( {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right)} \right| + \left| {{Q_\sigma }\, \left( {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right)\, } \right|\, } \right]\frac{{{{\left( {\tau - \eta } \right)}^{\left( {q - 1} \right)\sigma }}}}{{\Gamma \left( {1 + \left( {q - 1} \right)\sigma } \right)}}{{\left( {d\eta } \right)}^\sigma }} } \right\} \\ \;\;\;\;\;\;\;\;\; \leqslant \mathop {\mathop {\max }\limits_{\tau \, \in \Omega } \left\{ {\, \left( {1 + \hbar } \right)\, \left| {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right| + \frac{\hbar }{{\Gamma \left( {1 + \sigma } \right)}}\int_0^\tau {\, \left( {\lambda + \xi } \right)\, \left| {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right|\frac{{{{\left( {\tau - \eta } \right)}^{\left( {q - 1} \right)\sigma }}}}{{\Gamma \left( {1 + \left( {q - 1} \right)\sigma } \right)}}{{\left( {d\eta } \right)}^\sigma }} } \right\}}\limits_{} . | (4.11) |
Now, application of MVT of LFI calculus [56,57] reduces the inequality (4.11) in the following form
\left\| {\, {\vartheta _{{l_1}}} - {\vartheta _{{l_2}}}} \right\| \leqslant \mathop {\max }\limits_{\tau \, \in \Omega } \left\{ {\, \left( {1 + \hbar } \right)\, \left| {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right| + \hbar \, \left( {\lambda + \xi } \right)\, \left| {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right|\, \varsigma } \right\} \\ \;\;\;\;\;\;\;\;\;\; = \mathop {\max }\limits_{t\, \in \Gamma } \left\{ {\, \left( {1 + \hbar } \right) + \hbar \, \left( {\lambda + \xi } \right)\, \varsigma } \right\}\, \, \left| {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right| \\ \;\;\;\;\;\;\;\;\;\; = {{\rlap{-} \lambda }} \left\| {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right\|\, . \\ \;\;\;\;\;\;\;\;\;\;\therefore \left\| {{\vartheta _{{l_1}}} - {\vartheta _{{l_2}}}} \right\| \leqslant {{\rlap{-} \lambda }} \left\| {{\vartheta _{{l_1} - 1}} - {\vartheta _{{l_2} - 1}}} \right\| , | (4.12) |
{\rm{where}} \; {{\rlap{-} \lambda }} = \left( {1 + \hbar } \right) + \hbar \, \left( {\lambda + \xi } \right)\, \varsigma | (4.13) |
Assume {l_1} = {l_2} + 1 , then it produces
\left\| {\, {\vartheta _{{l_2} + 1}} - {\vartheta _{{l_2}}}} \right\| \leqslant {{\rlap{-} \lambda }} \left\| {{\vartheta _{{l_2}}} - {\vartheta _{{l_2} - 1}}} \right\| \leqslant {{{\rlap{-} \lambda }} ^2}\left\| {{\vartheta _{{l_2} - 1}} - {\vartheta _{{l_2} - 2}}} \right\| \cdots \leqslant {{{\rlap{-} \lambda }} ^{{l_2}}}\left\| {{\vartheta _1} - {\vartheta _0}} \right\| . | (4.14) |
Utilizing the triangular inequality, we have
\left\| {\, {\vartheta _{{l_1}}} - {\vartheta _{{l_2}}}} \right\| \leqslant \left\| {{\vartheta _{{l_2} + 1}} - {\vartheta _{{l_2}}}} \right\| + \left\| {{\vartheta _{{l_2} + 2}} - {\vartheta _{{l_2} + 1}}} \right\| + \cdots + \left\| {\, {\vartheta _{{l_1}}} - {\vartheta _{{l_1} - 1}}} \right\|\\ \;\;\;\;\;\;\;\;\leqslant \left( {{{{\rlap{-} \lambda }} ^{{l_2}}} + {{{\rlap{-} \lambda }} ^{{l_2} + 1}} + {{{\rlap{-} \lambda }} ^{{l_2} + 2}} + \cdots + {{{\rlap{-} \lambda }} ^{{l_1} - 1}}} \right)\, \, \left\| {\, {\vartheta _1} - {\vartheta _0}} \right\|\\ \;\;\;\;\;\;\;\; = {{{\rlap{-} \lambda }} ^{{l_2}}}\left( {1 + {{\rlap{-} \lambda }} + {{{\rlap{-} \lambda }} ^2} + \cdots + {{{\rlap{-} \lambda }} ^{{l_1} - {l_2} - 1}}} \right)\, \, \left\| {\, {\vartheta _1} - {\vartheta _0}} \right\| \\ \;\;\;\;\;\;\;\; = {{{\rlap{-} \lambda }} ^{{l_2}}}\left[ {\frac{{1 - {{{\rlap{-} \lambda }} ^{{l_1} - {l_2} - 1}}}}{{1 - {{\rlap{-} \lambda }} }}} \right]\, \left\| {\, {\vartheta _1} - {\vartheta _0}} \right\| . | (4.15) |
Since 0 < {{\rlap{-} \lambda }} < 1 , thus 1 - {{{\rlap{-} \lambda }} ^{{l_1} - {l_2} - 1}} < 1 , then
\left\| {\, {\vartheta _{{l_1}}} - {\vartheta _{{l_2}}}} \right\| \leqslant \frac{{{{{\rlap{-} \lambda }} ^{{l_2}}}}}{{1 - {{\rlap{-} \lambda }} }}\, \left\| {\, {\vartheta _1} - {\vartheta _0}} \right\| . | (4.16) |
But \left\| {\, {\vartheta _1} - {\vartheta _0}} \right\| < \infty , thus \left\| {\, {\vartheta _{{l_1}}} - {\vartheta _{{l_2}}}} \right\| \to 0 as n \to \infty , hence \left\{ {\, {\vartheta _{{l_2}}}} \right\} is a Cauchy sequence in \Pi \left[\Omega \right] and so \left\{ {\, {\vartheta _{{l_2}}}} \right\} is convergent. This ensures the convergence of the solution \vartheta \left({\gamma, \, \tau } \right) of LFPDE (3.1). Hence the theorem. □
In this section, LFNHAM is implemented for deriving the solutions for LFCHEs.
The following LFCHEs on Cantor set are investigated
\frac{{{\partial ^{2\sigma }}{\vartheta _1}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\tau ^{2\sigma }}}} + \frac{{{\partial ^{2\sigma }}{\vartheta _2}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}} - {\vartheta _1}\, \left( {\gamma , \, \tau } \right) = 0 , 0 < \sigma \leqslant 1, | (5.1) |
\frac{{{\partial ^{2\sigma }}{\vartheta _2}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\tau ^{2\sigma }}}} + \frac{{{\partial ^{2\sigma }}{\vartheta _1}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}} - {\vartheta _2}\, \left( {\gamma , \, \tau } \right) = 0 , 0 < \sigma \leqslant 1, | (5.2) |
subject to the fractal initial conditions:
{\vartheta _1}\, \left( {\gamma , \, 0} \right) = 0 , \frac{{{\partial ^\sigma }{\vartheta _1}\, \left( {\gamma , \, 0} \right)}}{{\partial {\tau ^\sigma }}} = {E_\sigma }\left( {{\gamma ^\sigma }} \right) , |
{\vartheta _2}\, \left( {\gamma , \, 0} \right) = 0 , \frac{{{\partial ^\sigma }{\vartheta _2}\, \left( {\gamma , \, 0} \right)}}{{\partial {\tau ^\sigma }}} = - {E_\sigma }\left( {{\gamma ^\sigma }} \right) , | (5.3) |
where {\vartheta _1}\, \left({\gamma, \, \tau } \right) and {\vartheta _2}\, \left({\gamma, \, \tau } \right) represent the local fractional continuous functions.
On account of the initial conditions (5.3) and algorithm of LFNHAM, the initial guess are written as
{\vartheta _{1, 0}}\, \left( {\gamma , \, \tau } \right) = {\vartheta _1}\, \left( {\gamma , \, 0} \right) + \frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}}{\vartheta _{1, 0}}^{\left( \sigma \right)}\, \left( {\gamma , \, 0} \right) = {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} , |
{\vartheta _{2, 0}}\, \left( {\gamma , \, \tau } \right) = {\vartheta _2}\, \left( {\gamma , \, 0} \right) + \frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}}{\vartheta _{2, 0}}^{\left( \sigma \right)}\, \left( {\gamma , \, 0} \right) = - {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} . | (5.4) |
Employing the LFNT operator {}^{LF}{N_\sigma } on Eqs (5.1) and (5.2), we get
{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _1}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\tau ^{2\sigma }}}}} \right] + {}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _2}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - {}^{LF}{N_\sigma }\left( {{\vartheta _1}\, \left( {\gamma , \, \tau } \right)} \right) = 0 , | (5.5) |
{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _2}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\tau ^{2\sigma }}}}} \right] + {}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _1}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - {}^{LF}{N_\sigma }\left( {{\vartheta _2}\, \left( {\gamma , \, \tau } \right)} \right) = 0 . | (5.6) |
Now, the implementation of formula of LFNT for local fractional derivatives yields
\frac{{{s^{2\sigma }}}}{{{u^{2\sigma }}}}{\bar \vartheta _1}\left( {\gamma , s, u} \right) - \frac{{{s^\sigma }}}{{{u^{2\sigma }}}}{\vartheta _1}\left( {\gamma , \, 0} \right) - \frac{1}{{{u^\sigma }}}{\vartheta _1}^{\left( \sigma \right)}\left( {\gamma , \, 0} \right) + {}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _2}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - {}^{LF}{N_\sigma }\left( {{\vartheta _1}\, \left( {\gamma , \, \tau } \right)} \right) = 0 , | (5.7) |
\frac{{{s^{2\sigma }}}}{{{u^{2\sigma }}}}{\bar \vartheta _2}\left( {\gamma , s, u} \right) - \frac{{{s^\sigma }}}{{{u^{2\sigma }}}}{\vartheta _2}\left( {\gamma , \, 0} \right) - \frac{1}{{{u^\sigma }}}{\vartheta _2}^{\left( \sigma \right)}\left( {\gamma , \, 0} \right) + {}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _1}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - {}^{LF}{N_\sigma }\left( {{\vartheta _2}\, \left( {\gamma , \, \tau } \right)} \right) = 0 . | (5.8) |
After rearranging the terms, we get
{\bar \vartheta _1}\left( {\gamma , s, u} \right) = \frac{1}{{{s^\sigma }}}{\vartheta _1}\left( {\gamma , \, 0} \right) + \frac{{{u^\sigma }}}{{{s^{2\sigma }}}}{\vartheta _1}^{\left( \sigma \right)}\left( {\gamma , \, 0} \right) - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _2}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] + \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _1}\, \left( {\gamma , \, \tau } \right)} \right) , | (5.9) |
{\bar \vartheta _2}\left( {\gamma , s, u} \right) = \frac{1}{{{s^\sigma }}}{\vartheta _2}\left( {\gamma , \, 0} \right) + \frac{{{u^\sigma }}}{{{s^{2\sigma }}}}{\vartheta _2}^{\left( \sigma \right)}\left( {\gamma , \, 0} \right) - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _1}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] + \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _2}\, \left( {\gamma , \, \tau } \right)} \right) . | (5.10) |
Now, further simplification in view of initial condition (5.3) reduces Eqs (5.9) and (5.10) in the following way
{\bar \vartheta _1}\left( {\gamma , s, u} \right) - \frac{{{u^\sigma }}}{{{s^{2\sigma }}}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) + \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _2}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _1}} \right) = 0 , | (5.11) |
{\bar \vartheta _2}\left( {\gamma , s, u} \right) + \frac{{{u^\sigma }}}{{{s^{2\sigma }}}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) + \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _1}\, \left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _2}} \right) = 0 . | (5.12) |
Now in view of Eqs (5.11) and (5.12), the nonlinear operators are formed as:
{\Phi _1}\left[ {{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)} \right] = {}^{LF}{N_\sigma }\left( {{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)} \right) - \frac{{{u^\sigma }}}{{{s^{2\sigma }}}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) \\ \;\;\;\;\;\;\;\;\;\;+ \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\varphi _2}\left( {\gamma , \tau ;\kappa } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)} \right) , | (5.13) |
{\Phi _2}\left[ {{\varphi _2}\left( {\gamma , \tau ;\kappa } \right)} \right] = {}^{LF}{N_\sigma }\left( {{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)} \right) + \frac{{{u^\sigma }}}{{{s^{2\sigma }}}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) \\ \;\;\;\;\;\;\;\;\;\; + \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\varphi _2}\left( {\gamma , \tau ;\kappa } \right)} \right) , | (5.14) |
where \kappa is an embedding parameter and {\varphi _1}\left({\gamma, \tau; \kappa } \right) & {\varphi _2}\left({\gamma, \tau; \kappa } \right) are real valued functions of \gamma, \tau and \kappa .
Now using the steps of the LFHAM [43,46] and basic methodology of HAM [52,53,54], \mu \, th -order deformation equations are constructed as follows:
{\vartheta _{1, \mu }}\left( {\gamma , \, \tau } \right) = {\chi _\mu }{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right) + \hbar \, {}^{LF}N_\sigma ^{ - 1}\left( {{\Re _\mu }\left( {{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right)} \right)\, , | (5.15) |
{\vartheta _{2, \mu }}\left( {\gamma , \, \tau } \right) = {\chi _\mu }{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right) + \hbar \, {}^{LF}N_\sigma ^{ - 1}\left( {{\Re _\mu }\left( {{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right)} \right)\, . | (5.16) |
In Eqs (5.15) and (5.16), the terms {\Re _\mu }\left({{\vartheta _{1, \, (\mu - 1)}}\left({\gamma, \, \tau } \right)} \right) and {\Re _\mu }\left({{\vartheta _{2, \, (\mu - 1)}}\left({\gamma, \, \tau } \right)} \right) are expressed as
{\Re _\mu }\left( {{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right) = {}^{LF}{N_\sigma }\left[ {{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right] - \left( {1 - {\chi _\mu }} \right)\, \frac{{{u^\sigma }}}{{{s^{2\sigma }}}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) \\ \;\;\;\;\;\;\;\;\;\; + \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \tau } \right)} \right) , | (5.17) |
{\Re _\mu }\left( {{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right) = {}^{LF}{N_\sigma }\left[ {{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right] + \left( {1 - {\chi _\mu }} \right)\, \frac{{{u^\sigma }}}{{{s^{2\sigma }}}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) \\ \;\;\;\;\;\;\;\;\;\;+ \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \tau } \right)} \right) . | (5.18) |
Now implementing the LFNHAM and using Eqs (5.15) to (5.18), we have
{\vartheta _{1, \mu }}\left( {\gamma , \, \tau } \right) = \left( {{\chi _\mu } + \hbar } \right){\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right) - \hbar \, \left( {1 - {\chi _\mu }} \right)\, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} \\ \;\;\;\;\;\;\;\;\;\;+ \hbar {}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \tau } \right)} \right)} \right) , \mu \geqslant 1 , | (5.19) |
{\vartheta _{2, \mu }}\left( {\gamma , \, \tau } \right) = \left( {{\chi _\mu } + \hbar } \right){\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right) + \hbar \, \left( {1 - {\chi _\mu }} \right)\, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} \\ \;\;\;\;\;\;\;\;\;\;+ \hbar {}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \tau } \right)} \right)} \right) , \mu \geqslant 1 . | (5.20) |
On account of Eqs (5.19) and (5.20) for \mu = 1 , we have
{\vartheta _{1, 1}}\left( {\gamma , \, \tau } \right) = \hbar {\vartheta _{1, \, 0}}\left( {\gamma , \, \tau } \right) - \hbar \, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} \\ \;\;\;\;\;\;\;\;\;\;+ \hbar {}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{2, \, 0}}\left( {\gamma , \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _{1, \, 0}}\left( {\gamma , \tau } \right)} \right)} \right) , | (5.21) |
{\vartheta _{2, 1}}\left( {\gamma , \, \tau } \right) = \hbar {\vartheta _{2, \, 0}}\left( {\gamma , \, \tau } \right) + \hbar \, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} \\ \;\;\;\;\;\;\;\;\;\;+ \hbar {}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{1, \, 0}}\left( {\gamma , \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _{2, \, 0}}\left( {\gamma , \tau } \right)} \right)} \right) . | (5.22) |
Using initial guess values (5.4) and further simplification reduces Eqs (5.21) and (5.22) in this way
{\vartheta _{1, 1}}\left( {\gamma , \, \tau } \right) = - 2\hbar {E_\sigma }\left( {{\gamma ^\sigma }} \right){}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}\frac{{{u^\sigma }}}{{{s^{2\sigma }}}}} \right) = - 2\hbar {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}} , | (5.23) |
{\vartheta _{2, 1}}\left( {\gamma , \, \tau } \right) = 2\hbar {E_\sigma }\left( {{\gamma ^\sigma }} \right){}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}\frac{{{u^\sigma }}}{{{s^{2\sigma }}}}} \right) = 2\hbar {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}} . | (5.24) |
By means of Eqs (5.19) and (5.20) for \mu = 2 , we have
{\vartheta _{1, 2}}\left( {\gamma , \, \tau } \right) = \left( {1 + \hbar } \right){\vartheta _{1, \, 1}}\left( {\gamma , \, \tau } \right) + \hbar {}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{2, \, 1}}\left( {\gamma , \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _{1, \, 1}}\left( {\gamma , \tau } \right)} \right)} \right) , | (5.25) |
{\vartheta _{2, 2}}\left( {\gamma , \, \tau } \right) = \left( {1 + \hbar } \right){\vartheta _{2, \, 1}}\left( {\gamma , \, \tau } \right) + \hbar {}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{1, \, 1}}\left( {\gamma , \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - \frac{{{u^{2\sigma }}}}{{{s^{2\sigma }}}}{}^{LF}{N_\sigma }\left( {{\vartheta _{2, \, 1}}\left( {\gamma , \tau } \right)} \right)} \right) . | (5.26) |
Utilizing the values provided by Eqs (5.23) and (5.24) in Eqs (5.25) and (5.26), we obtain
{\vartheta _{1, 2}}\left( {\gamma , \, \tau } \right) = - 2\hbar \, \left( {1 + \hbar } \right)\, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}} + 4{\hbar ^2}{E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{5\sigma }}}}{{\Gamma \left( {1 + 5\sigma } \right)}} , | (5.27) |
{\vartheta _{2, 2}}\left( {\gamma , \, \tau } \right) = 2\hbar \, \left( {1 + \hbar } \right)\, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}} - 4{\hbar ^2}{E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{5\sigma }}}}{{\Gamma \left( {1 + 5\sigma } \right)}} . | (5.28) |
Following the similar procedure, we obtain the rest of the values for {\vartheta _{1, \mu }}\left({\gamma, \, \tau } \right) and {\vartheta _{2, \mu }}\left({\gamma, \, \tau } \right) for \mu \geqslant 3 .
Setting the convergence-control parameter \hbar = - 1 , we attain the following values
{\vartheta _{1, 1}}\left( {\gamma , \, \tau } \right) = 2{E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}} , |
{\vartheta _{2, 1}}\left( {\gamma , \, \tau } \right) = - 2{E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}} , |
{\vartheta _{1, 2}}\left( {\gamma , \, \tau } \right) = 4{E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{5\sigma }}}}{{\Gamma \left( {1 + 5\sigma } \right)}} , |
{\vartheta _{2, 2}}\left( {\gamma , \, \tau } \right) = - 4{E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{5\sigma }}}}{{\Gamma \left( {1 + 5\sigma } \right)}} , | (5.29) |
\vdots |
and so on.
Hence, the solutions of Eqs (5.1) and (5.2) are obtained as
{\vartheta _1}\, \left( {\gamma , \tau } \right) = \sum\limits_{n = 0}^\infty {{\vartheta _{1, n}}\, \left( {\gamma , \tau } \right)} \\ \;\;\;\;\;\;\;\;\;\; = {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( {\frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} + \frac{{2{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}} + \frac{{4{\tau ^{5\sigma }}}}{{\Gamma \left( {1 + 5\sigma } \right)}} + \frac{{8\, {\tau ^{7\sigma }}}}{{\Gamma \left( {1 + 7\sigma } \right)}} + \cdots } \right) \\ \;\;\;\;\;\;\;\;\;\; = {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( {\sum\limits_{n = 0}^\infty {\frac{{{2^n}{\tau ^{\left( {2n + 1} \right)\sigma }}}}{{\Gamma \left( {1 + \left( {2n + 1} \right)\sigma } \right)}}} } \right) \\ \;\;\;\;\;\;\;\;\;\; = \frac{1}{{\sqrt 2 }}{E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( {\sum\limits_{n = 0}^\infty {\frac{{{2^{n + \frac{1}{2}}}{\tau ^{\left( {2n + 1} \right)\sigma }}}}{{\Gamma \left( {1 + \left( {2n + 1} \right)\sigma } \right)}}} } \right)\\ \;\;\;\;\;\;\;\;\;\; = \frac{1}{{\sqrt 2 }}{E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \, {\sinh _\sigma }\left( {\sqrt 2 \, {\tau ^\sigma }} \right) . | (5.30) |
Similarly, we have
{\vartheta _2}\, \left( {\gamma , \tau } \right) = \sum\limits_{n = 0}^\infty {{\vartheta _{2, n}}\, \left( {\gamma , \tau } \right)} \\ \;\;\;\;\;\;\;\;\;\; = {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( { - \frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} - \frac{{2{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}} - \frac{{4{\tau ^{5\sigma }}}}{{\Gamma \left( {1 + 5\sigma } \right)}} - \frac{{8\, {\tau ^{7\sigma }}}}{{\Gamma \left( {1 + 7\sigma } \right)}} - \cdots } \right)\\ \;\;\;\;\;\;\;\;\;\; = - {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( {\sum\limits_{n = 0}^\infty {\frac{{{2^n}{\tau ^{\left( {2n + 1} \right)\sigma }}}}{{\Gamma \left( {1 + \left( {2n + 1} \right)\sigma } \right)}}} } \right) \\ \;\;\;\;\;\;\;\;\;\; = - \frac{1}{{\sqrt 2 }}{E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \, {\sinh _\sigma }\left( {\sqrt 2 \, {\tau ^\sigma }} \right) . | (5.31) |
Finally, the solutions of the coupled Helmholtz Eqs (5.1) & (5.2) are expressed as
{\vartheta _1}\, \left( {\gamma , \tau } \right) = {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \, \frac{{{{\sinh }_\sigma }\left( {\sqrt 2 \, {\tau ^\sigma }} \right)}}{{\sqrt 2 }} , |
{\vartheta _2}\, \left( {\gamma , \tau } \right) = - {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \, \frac{{{{\sinh }_\sigma }\left( {\sqrt 2 \, {\tau ^\sigma }} \right)}}{{\sqrt 2 }} . | (5.32) |
Thus, the LFNHAM solutions are in complete agreement with the solutions obtained by Yang and Hua [60].
In this portion, the LFNHAM is executed for deriving the solutions for LFCBEs.
\frac{{{\partial ^\sigma }{\vartheta _1}}}{{\partial {\tau ^\sigma }}} + \frac{{{\partial ^{2\sigma }}{\vartheta _1}}}{{\partial {\gamma ^{2\sigma }}}} - 2\frac{{{\partial ^\sigma }{\vartheta _1}}}{{\partial {\gamma ^\sigma }}}{\vartheta _1} + \frac{{{\partial ^\sigma }\left[ {{\vartheta _1}{\vartheta _2}} \right]}}{{\partial {\gamma ^\sigma }}} = 0 , {\vartheta _1} = {\vartheta _1}\, \left( {\gamma , \, \tau } \right) , 0 < \sigma \leqslant 1, | (6.1) |
\frac{{{\partial ^\sigma }{\vartheta _2}}}{{\partial {\tau ^\sigma }}} + \frac{{{\partial ^{2\sigma }}{\vartheta _2}}}{{\partial {\gamma ^{2\sigma }}}} - 2\frac{{{\partial ^\sigma }{\vartheta _2}}}{{\partial {\gamma ^\sigma }}}{\vartheta _2} + \frac{{{\partial ^\sigma }\left[ {{\vartheta _1}{\vartheta _2}} \right]}}{{\partial {\gamma ^\sigma }}} = 0 , {\vartheta _2} = {\vartheta _2}\, \left( {\gamma , \, \tau } \right) , 0 < \sigma \leqslant 1, | (6.2) |
subject to the fractal initial conditions:
{\vartheta _1}\, \left( {\gamma , \, 0} \right) = {\vartheta _2}\, \left( {\gamma , \, 0} \right) = {E_\sigma }\left( {{\gamma ^\sigma }} \right) , | (6.3) |
where {\vartheta _1}\, \left({\gamma, \, \tau } \right) and {\vartheta _2}\, \left({\gamma, \, \tau } \right) are local fractional continuous functions.
In view of initial conditions (6.3) and LFNHAM, the initial guess are expressed as
{\vartheta _{1, 0}}\, \left( {\gamma , \, \tau } \right) = {\vartheta _{2, 0}}\, \left( {\gamma , \, \tau } \right) = {E_\sigma }\left( {{\gamma ^\sigma }} \right) . | (6.4) |
Employing the LFNT operator {}^{LF}{N_\sigma } on Eqs (6.1) and (6.2), we get
{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\vartheta _1}}}{{\partial {\tau ^\sigma }}}} \right) + {}^{LF}{N_\sigma }\left( {\frac{{{\partial ^{2\sigma }}{\vartheta _1}}}{{\partial {\gamma ^{2\sigma }}}}} \right) - 2{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\vartheta _1}}}{{\partial {\gamma ^\sigma }}}{\vartheta _1}} \right) + {}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }\left[ {{\vartheta _1}{\vartheta _2}} \right]}}{{\partial {\gamma ^\sigma }}}} \right) = 0 , | (6.5) |
{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\vartheta _2}}}{{\partial {\tau ^\sigma }}}} \right) + {}^{LF}{N_\sigma }\left( {\frac{{{\partial ^{2\sigma }}{\vartheta _2}}}{{\partial {\gamma ^{2\sigma }}}}} \right) - 2{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\vartheta _2}}}{{\partial {\gamma ^\sigma }}}{\vartheta _2}} \right) + {}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }\left[ {{\vartheta _1}{\vartheta _2}} \right]}}{{\partial {\gamma ^\sigma }}}} \right) = 0 . | (6.6) |
Now, employing the formula of LFNT for LFDs yields
\frac{{{s^\sigma }}}{{{u^\sigma }}}{\bar \vartheta _1}\left( {\gamma , s, u} \right) - \frac{1}{{{u^\sigma }}}{\vartheta _1}\left( {\gamma , \, 0} \right) + {}^{LF}{N_\sigma }\left( {\frac{{{\partial ^{2\sigma }}{\vartheta _1}}}{{\partial {\gamma ^{2\sigma }}}}} \right) - 2{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\vartheta _1}}}{{\partial {\gamma ^\sigma }}}{\vartheta _1}} \right) + {}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }\left[ {{\vartheta _1}{\vartheta _2}} \right]}}{{\partial {\gamma ^\sigma }}}} \right) = 0 , | (6.7) |
\frac{{{s^\sigma }}}{{{u^\sigma }}}{\bar \vartheta _2}\left( {\gamma , s, u} \right) - \frac{1}{{{u^\sigma }}}{\vartheta _2}\left( {\gamma , \, 0} \right) + {}^{LF}{N_\sigma }\left( {\frac{{{\partial ^{2\sigma }}{\vartheta _2}}}{{\partial {\gamma ^{2\sigma }}}}} \right) - 2{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\vartheta _2}}}{{\partial {\gamma ^\sigma }}}{\vartheta _2}} \right) + {}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }\left[ {{\vartheta _1}{\vartheta _2}} \right]}}{{\partial {\gamma ^\sigma }}}} \right) = 0 . | (6.8) |
After rearranging the terms, we get
{\bar \vartheta _1}\left( {\gamma , s, u} \right) = \frac{1}{{{s^\sigma }}}{\vartheta _1}\left( {\gamma , 0} \right) - \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^{2\sigma }}{\vartheta _1}}}{{\partial {\gamma ^{2\sigma }}}}} \right) + 2\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\vartheta _1}}}{{\partial {\gamma ^\sigma }}}{\vartheta _1}} \right) - \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }\left[ {{\vartheta _1}{\vartheta _2}} \right]}}{{\partial {\gamma ^\sigma }}}} \right) , | (6.9) |
{\bar \vartheta _2}\left( {\gamma , s, u} \right) = \frac{1}{{{s^\sigma }}}{\vartheta _2}\left( {\gamma , 0} \right) - \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^{2\sigma }}{\vartheta _2}}}{{\partial {\gamma ^{2\sigma }}}}} \right) + 2\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\vartheta _2}}}{{\partial {\gamma ^\sigma }}}{\vartheta _2}} \right) \\ \;\;\;\;\;\;\;\;\;\;- \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }\left[ {{\vartheta _1}{\vartheta _2}} \right]}}{{\partial {\gamma ^\sigma }}}} \right) . | (6.10) |
Now, further simplification in view of initial conditions (6.3) reduces Eqs (6.9) and (6.10) as follows:
{\bar \vartheta _1}\left( {\gamma , s, u} \right) - \frac{1}{{{s^\sigma }}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^{2\sigma }}{\vartheta _1}}}{{\partial {\gamma ^{2\sigma }}}}} \right) \\ \;\;\;\;\;\;\;\;\;\;- 2\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\vartheta _1}}}{{\partial {\gamma ^\sigma }}}{\vartheta _1}} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }\left[ {{\vartheta _1}{\vartheta _2}} \right]}}{{\partial {\gamma ^\sigma }}}} \right) = 0 | (6.11) |
{\bar \vartheta _2}\left( {\gamma , s, u} \right) - \frac{1}{{{s^\sigma }}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^{2\sigma }}{\vartheta _2}}}{{\partial {\gamma ^{2\sigma }}}}} \right) \\ \;\;\;\;\;\;\;\;\;\;- 2\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\vartheta _2}}}{{\partial {\gamma ^\sigma }}}{\vartheta _2}} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }\left[ {{\vartheta _1}{\vartheta _2}} \right]}}{{\partial {\gamma ^\sigma }}}} \right) = 0 . | (6.12) |
Now in view of Eqs (6.11) and (6.12), the nonlinear operators are formed as
{\Phi _1}\left[ {{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)} \right] = {}^{LF}{N_\sigma }\left( {{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)} \right) - \frac{1}{{{s^\sigma }}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] \\ \;\;\;\;\;\;\;\;\;\; - 2\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\varphi _1}}}{{\partial {\gamma ^\sigma }}}{\varphi _1}} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }\left[ {{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)\, {\varphi _2}\left( {\gamma , \tau ;\kappa } \right)} \right]}}{{\partial {\gamma ^\sigma }}}} \right) , | (6.13) |
{\Phi _2}\left[ {{\varphi _2}\left( {\gamma , \tau ;\kappa } \right)} \right] = {}^{LF}{N_\sigma }\left( {{\varphi _2}\left( {\gamma , \tau ;\kappa } \right)} \right) - \frac{1}{{{s^\sigma }}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\varphi _2}\left( {\gamma , \tau ;\kappa } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] \\ \;\;\;\;\;\;\;\;\;\; - 2\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{\varphi _2}}}{{\partial {\gamma ^\sigma }}}{\varphi _2}} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }\left[ {{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)\, {\varphi _2}\left( {\gamma , \tau ;\kappa } \right)} \right]}}{{\partial {\gamma ^\sigma }}}} \right) , | (6.14) |
where \kappa \in \left[{0, \, 1} \right] is an embedding parameter and {\varphi _1}\left({\gamma, \tau; \kappa } \right) & {\varphi _2}\left({\gamma, \tau; \kappa } \right) are real valued functions of \gamma, \tau , and \kappa .
Performing the steps of the LFNHAM, \mu \, th -order deformation equations are formed as
{\vartheta _{1, \mu }}\left( {\gamma , \, \tau } \right) = {\chi _\mu }{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right) + \hbar \, {}^{LF}N_\sigma ^{ - 1}\left( {{\Re _\mu }\left( {{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right)} \right)\, , | (6.15) |
{\vartheta _{2, \mu }}\left( {\gamma , \, \tau } \right) = {\chi _\mu }{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right) + \hbar \, {}^{LF}N_\sigma ^{ - 1}\left( {{\Re _\mu }\left( {{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right)} \right)\, . | (6.16) |
In Eqs (6.15) and (6.16), the terms {\Re _\mu }\left({{\vartheta _{1, \, (\mu - 1)}}\left({\gamma, \, \tau } \right)} \right) and {\Re _\mu }\left({{\vartheta _{2, \, (\mu - 1)}}\left({\gamma, \, \tau } \right)} \right) are computed as
{\Re _\mu }\left( {{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right) = {}^{LF}{N_\sigma }\left[ {{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right] - \left( {1 - {\chi _\mu }} \right)\, \frac{1}{{{s^\sigma }}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] \\ \;\;\;\;\;\;\;\;\;\;- 2\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {{P_{\mu - 1}}} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{{P'}_{\mu - 1}}}}{{\partial {\gamma ^\sigma }}}} \right) , | (6.17) |
{\Re _\mu }\left( {{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right) = {}^{LF}{N_\sigma }\left[ {{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)} \right] - \left( {1 - {\chi _\mu }} \right)\, \frac{1}{{{s^\sigma }}}{E_\sigma }\left( {{\gamma ^\sigma }} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] \\ \;\;\;\;\;\;\;\;\;\; - 2\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {{{P''}_{\mu - 1}}} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{{P'}_{\mu - 1}}}}{{\partial {\gamma ^\sigma }}}} \right) , | (6.18) |
where {P_{\mu - 1}} , {P'_{\mu - 1}} , {P''_{\mu - 1}} denote the homotopy polynomials [55] and are formulated as
{P_{\mu - 1}} = \frac{1}{{\Gamma \left( \mu \right)}}\left[ {\frac{{{\partial ^\mu }}}{{\partial {\kappa ^\mu }}}\left( {\frac{{{\partial ^\sigma }{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)}}{{\partial {\gamma ^\sigma }}}{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)} \right)} \right] , | (6.19) |
{P'_{\mu - 1}} = \frac{1}{{\Gamma \left( \mu \right)}}\left[ {\frac{{{\partial ^\mu }}}{{\partial {\kappa ^\mu }}}\left( {{\varphi _1}\left( {\gamma , \tau ;\kappa } \right)\, {\varphi _2}\left( {\gamma , \tau ;\kappa } \right)} \right)} \right] , | (6.20) |
{P''_{\mu - 1}} = \frac{1}{{\Gamma \left( \mu \right)}}\left[ {\frac{{{\partial ^\mu }}}{{\partial {\kappa ^\mu }}}\left( {\frac{{{\partial ^\sigma }{\varphi _2}\left( {\gamma , \tau ;\kappa } \right)}}{{\partial {\gamma ^\sigma }}}{\varphi _2}\left( {\gamma , \tau ;\kappa } \right)} \right)} \right] , | (6.21) |
and
{\varphi _1}\left( {\gamma , t, \kappa } \right) = {\varphi _{1, 0}} + \kappa \, {\varphi _{1, 1}} + {\kappa ^2}{\varphi _{1, 2}} + \cdots , | (6.22) |
{\varphi _2}\left( {\gamma , t, \kappa } \right) = {\varphi _{2, 0}} + \kappa \, {\varphi _{2, 1}} + {\kappa ^2}{\varphi _{2, 2}} + \cdots . | (6.23) |
On account of LFNHAM and Eqs (6.15)–(6.18), we have
{\vartheta _{1, \mu }}\left( {\gamma , \, \tau } \right) = \left( {{\chi _\mu } + \hbar } \right){\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right) - \hbar \, \left( {1 - {\chi _\mu }} \right)\, {E_\sigma }\left( {{\gamma ^\sigma }} \right) \\ \;\;\;\;\;\;\;\;\;\;+ \hbar {}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{1, \, (\mu - 1)}}\left( {\gamma , \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - 2\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {{P_{\mu - 1}}} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{{P'}_{\mu - 1}}}}{{\partial {\gamma ^\sigma }}}} \right)} \right) , \mu \geqslant 1 . | (6.24) |
{\vartheta _{2, \mu }}\left( {\gamma , \, \tau } \right) = \left( {{\chi _\mu } + \hbar } \right){\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \, \tau } \right) - \hbar \, \left( {1 - {\chi _\mu }} \right)\, {E_\sigma }\left( {{\gamma ^\sigma }} \right) \\ \;\;\;\;\;\;\;\;\;\; + \hbar {}^{LF}N_\sigma ^{ - 1}\left( {\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left[ {\frac{{{\partial ^{2\sigma }}{\vartheta _{2, \, (\mu - 1)}}\left( {\gamma , \tau } \right)}}{{\partial {\gamma ^{2\sigma }}}}} \right] - 2\frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {{{P''}_{\mu - 1}}} \right) + \frac{{{u^\sigma }}}{{{s^\sigma }}}{}^{LF}{N_\sigma }\left( {\frac{{{\partial ^\sigma }{{P'}_{\mu - 1}}}}{{\partial {\gamma ^\sigma }}}} \right)} \right) , \mu \geqslant 1 . | (6.25) |
Taking into account the set of iterative schemes (6.24) & (6.25) and initial conditions (6.3), the iterative terms for various values of \mu are computed as follows:
{\vartheta _{1, 1}}\left( {\gamma , \, \tau } \right) = \hbar \, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} , \\ {\vartheta _{2, 1}}\left( {\gamma , \, \tau } \right) = \hbar \, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} , \\ {\vartheta _{1, 2}}\left( {\gamma , \, \tau } \right) = \hbar \left( {1 + \hbar } \right)\, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} + {\hbar ^2}{E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{2\sigma }}}}{{\Gamma \left( {1 + 2\sigma } \right)}} , \\ {\vartheta _{2, 2}}\left( {\gamma , \, \tau } \right) = \hbar \left( {1 + \hbar } \right)\, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} + {\hbar ^2}{E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{2\sigma }}}}{{\Gamma \left( {1 + 2\sigma } \right)}} , \\{\vartheta _{1, 3}}\left( {\gamma , \, \tau } \right) = \hbar \left( {1 + \hbar } \right)\, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( {\left( {1 + \hbar } \right)\frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} + \hbar \frac{{{\tau ^{2\sigma }}}}{{\Gamma \left( {1 + 2\sigma } \right)}}} \right)\\ \;\;\;\;\;\;\;\;\;\;+ {\hbar ^2}{E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( {\left( {1 + \hbar } \right)\frac{{{\tau ^{2\sigma }}}}{{\Gamma \left( {1 + 2\sigma } \right)}} + \hbar \frac{{{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}}} \right) , \\ {\vartheta _{2, 3}}\left( {\gamma , \, \tau } \right) = \hbar \left( {1 + \hbar } \right)\, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( {\left( {1 + \hbar } \right)\frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} + \hbar \frac{{{\tau ^{2\sigma }}}}{{\Gamma \left( {1 + 2\sigma } \right)}}} \right)\\ \;\;\;\;\;\;\;\;\;\;+ {\hbar ^2}{E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( {\left( {1 + \hbar } \right)\frac{{{\tau ^{2\sigma }}}}{{\Gamma \left( {1 + 2\sigma } \right)}} + \hbar \frac{{{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}}} \right) . | (6.26) |
Proceeding in the similar way, we get the remaining values for {\vartheta _{1, \mu }}\left({\gamma, \, \tau } \right) and {\vartheta _{2, \mu }}\left({\gamma, \, \tau } \right) for \mu \geqslant 4 .
Setting the convergence-control parameter \hbar = - 1 , we attain the following values
{\vartheta _{1, 1}}\left( {\gamma , \, \tau } \right) = - {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} , |
{\vartheta _{2, 1}}\left( {\gamma , \, \tau } \right) = - \, {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} , |
{\vartheta _{1, 2}}\left( {\gamma , \, \tau } \right) = {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{2\sigma }}}}{{\Gamma \left( {1 + 2\sigma } \right)}} , |
{\vartheta _{2, 2}}\left( {\gamma , \, \tau } \right) = {E_\sigma }\left( {{\gamma ^\sigma }} \right)\frac{{{\tau ^{2\sigma }}}}{{\Gamma \left( {1 + 2\sigma } \right)}} , |
{\vartheta _{1, 3}}\left( {\gamma , \, \tau } \right) = - {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( {\frac{{{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}}} \right) , |
{\vartheta _{2, 3}}\left( {\gamma , \, \tau } \right) = - {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( {\frac{{{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}}} \right) , | (6.27) |
\vdots |
and so on.
Proceeding in the same way, the rest of the terms of {\vartheta _{1, \mu }}\left({\gamma, \, \tau } \right) and {\vartheta _{2, \mu }}\left({\gamma, \, \tau } \right) for \mu \geqslant 4 are evaluated in a smooth manner, and finally the local fractional series solutions are obtained.
Hence, the solutions of Eqs (6.1) and (6.2) are obtained as
{\vartheta _1}\, \left( {\gamma , \tau } \right) = \sum\limits_{n = 0}^\infty {{\vartheta _{1, n}}\, \left( {\gamma , \tau } \right)}\\ \;\;\;\;\;\;\;\;\;\; = {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, \left( {1 - \frac{{{\tau ^\sigma }}}{{\Gamma \left( {1 + \sigma } \right)}} + \frac{{{\tau ^{2\sigma }}}}{{\Gamma \left( {1 + 2\sigma } \right)}} - \frac{{{\tau ^{3\sigma }}}}{{\Gamma \left( {1 + 3\sigma } \right)}} + \frac{{\, {\tau ^{4\sigma }}}}{{\Gamma \left( {1 + 4\sigma } \right)}} - \cdots } \right) \\ \;\;\;\;\;\;\;\;\;\; = {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, {E_\sigma }\left( { - {\tau ^\sigma }} \right) . | (6.28) |
Similarly,
{\vartheta _2}\, \left( {\gamma , \tau } \right) = \sum\limits_{n = 0}^\infty {{\vartheta _{2, n}}\, \left( {\gamma , \tau } \right) = \, } {E_\sigma }\left( {{\gamma ^\sigma }} \right)\, {E_\sigma }\left( { - {\tau ^\sigma }} \right) . | (6.29) |
The solutions of LFCHEs and LFCBEs acquired by implementing the LFNHAM are the general form of solutions as compared to the LFADM, LFHPM, and LFNHPM. It is noteworthy that the LFNHAM solution transforms to the LFNHPM solution for \hbar = - 1 . The computational results validate the reliability and accuracy of the proposed method to achieve solutions for LFCHEs and LFCBEs. Moreover, the solutions of LFCHEs are in excellent match with the solutions obtained by Yang and Hua [60]. These facts authenticate the reliability of the solutions obtained by LFNHAM. Conclusively, the suggested hybrid framework can be employed to a variety of local fractional models occurring in a fractal medium.
In this segment, the numerical simulations are presented for the solutions of the LFCHEs and LFCBEs under fractal initial conditions obtained via LFNHAM. The 3D plots for solutions of LFCHEs and LFCBEs have been generated for the fractal order \sigma = \ln 2/\ln 3 . Here, all the 3D plots on the Cantor sets have been prepared with the aid of MATLAB software. Figures 1 and 2 depict the 3D surface graphics of coupled solutions {\vartheta _1}\left({\gamma, \, \tau } \right) and {\vartheta _2}\left({\gamma, \, \tau } \right) for the LFCHEs. Similarly, Figures 3 and 4 show the 3D surface graphics of coupled solutions {\vartheta _1}\left({\gamma, \, \tau } \right) and {\vartheta _2}\left({\gamma, \, \tau } \right) for LFCBEs. Here, \gamma and \tau have been taken in the closed interval of 0 to 1. The nature of {\vartheta _1}\left({\gamma, \, \tau } \right) and {\vartheta _2}\left({\gamma, \, \tau } \right) have been explored with respect to \gamma and \tau . The fractal solutions of the LFCHEs and LFCBEs show interesting characteristics for \sigma = \ln 2/\ln 3 . The graphical presentation demonstrates that the computed solutions for the LFCHEs and LFCBEs are consistently dependent on the fractal order \sigma . Furthermore, the 3D figures drawn on Cantor sets indicate that the coupled solutions {\vartheta _1}\left({\gamma, \, \tau } \right) and {\vartheta _2}\left({\gamma, \, \tau } \right) are of fractal nature.
In this paper, the LFNHAM is proposed for computation of solutions for LFCHEs and LFCBEs on Cantor sets. The local fractional series solutions for LFCHEs and LFCBEs have been depicted in terms of Mittag-Leffler function. The 3D plots are presented for solutions of LFCHEs and LFCBEs by using the MATLAB software. It is clearly observed from the surface graphics of the solutions that the figures plotted on the Cantor set for the functions {\vartheta _1}\left({\gamma, \, \tau } \right) and {\vartheta _2}\left({\gamma, \, \tau } \right) are of fractal nature. The computational results authenticate the reliability and accuracy of the implemented method to obtain solutions for LFCHEs and LFCBEs. The combination of LFHAM and LFNT performs faster calculations than LFHAM. The convergence and uniqueness of the LFNHAM solution for a general LFPDE is also discussed in view of Banach's fixed point theory. In a nutshell, the suggested hybrid approach in connection with LFNT can be employed to such types of local fractional models appearing in a fractal media.
The authors are very grateful to the referees for constructive comments and suggestions towards the improvement of this paper. This research received no external funding.
The authors declare no conflict of interest.
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{}^{LF}{N_\sigma }\left(1 \right) = \frac{1}{{{s^\sigma }}} | {}^{LF}{N_\sigma }\left({Co{s_\sigma }\left({{\tau ^\sigma }} \right)} \right) = \frac{{{s^\sigma }}}{{{s^{2\sigma }} + {u^{2\sigma }}}} |
{}^{LF}{N_\sigma }\left({\frac{{{\tau ^\sigma }}}{{\Gamma \left({1 + \sigma } \right)}}} \right) = \frac{{{u^\sigma }}}{{{s^{2\sigma }}}} | {}^{LF}{N_\sigma }\left({Si{n_\sigma }\left({{\tau ^\sigma }} \right)} \right) = \frac{{{u^\sigma }}}{{{s^{2\sigma }} + {u^{2\sigma }}}} |
{}^{LF}{N_\sigma }\left({\frac{{{\tau ^{q\sigma }}}}{{\Gamma \left({1 + q\sigma } \right)}}} \right) = \frac{{{u^{q\sigma }}}}{{{s^{(q + 1)\sigma }}}} | {}^{LF}{N_\sigma }\left({Cos{h_\sigma }\left({{\tau ^\sigma }} \right)} \right) = \frac{{{s^\sigma }}}{{{s^{2\sigma }} - {u^{2\sigma }}}} |
{}^{LF}{N_\sigma }\left(1 \right) = \frac{1}{{{s^\sigma }}} | {}^{LF}{N_\sigma }\left({Co{s_\sigma }\left({{\tau ^\sigma }} \right)} \right) = \frac{{{s^\sigma }}}{{{s^{2\sigma }} + {u^{2\sigma }}}} |
{}^{LF}{N_\sigma }\left({\frac{{{\tau ^\sigma }}}{{\Gamma \left({1 + \sigma } \right)}}} \right) = \frac{{{u^\sigma }}}{{{s^{2\sigma }}}} | {}^{LF}{N_\sigma }\left({Si{n_\sigma }\left({{\tau ^\sigma }} \right)} \right) = \frac{{{u^\sigma }}}{{{s^{2\sigma }} + {u^{2\sigma }}}} |
{}^{LF}{N_\sigma }\left({\frac{{{\tau ^{q\sigma }}}}{{\Gamma \left({1 + q\sigma } \right)}}} \right) = \frac{{{u^{q\sigma }}}}{{{s^{(q + 1)\sigma }}}} | {}^{LF}{N_\sigma }\left({Cos{h_\sigma }\left({{\tau ^\sigma }} \right)} \right) = \frac{{{s^\sigma }}}{{{s^{2\sigma }} - {u^{2\sigma }}}} |