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Research article Special Issues

Enhanced structural, thermal, mechanical and electrical properties of nano ZTA/epoxy composites

  • Epoxy composites were prepared by doping nano Zirconia Toughened Alumina (ZTA) which were synthesized by solution combustion method into epoxy resin and hardener. Initially ZTA nanopowder was characterized to check its purity, morphology and to confirm its metal-oxide bonding using XRD, SEM and FTIR respectively. The thermal properties such as TGA and DTG were also analysed. The polymer composites were obtained by uniformly dispersing ZTA nanopowder into epoxy using an ultrasonicator. Polymer composites of various concentrations viz, 0.5, 1, 1.5, 2 and 2.5 wt% were synthesized, all concentrations were prepared on weight basis. All the polymer composites were tested for compression properties, flexural properties and tensile properties. Best results for all the mechanical properties were obtained for epoxy with 1.5 wt% ZTA composites. Electrical properties such as breakdown voltage and breakdown strength were analysed and outstanding results were observed for epoxy with 2.5 wt% ZTA composite.

    Citation: Chaitra Srikanth, G.M. Madhu, Shreyas J. Kashyap. Enhanced structural, thermal, mechanical and electrical properties of nano ZTA/epoxy composites[J]. AIMS Materials Science, 2022, 9(2): 214-235. doi: 10.3934/matersci.2022013

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  • Epoxy composites were prepared by doping nano Zirconia Toughened Alumina (ZTA) which were synthesized by solution combustion method into epoxy resin and hardener. Initially ZTA nanopowder was characterized to check its purity, morphology and to confirm its metal-oxide bonding using XRD, SEM and FTIR respectively. The thermal properties such as TGA and DTG were also analysed. The polymer composites were obtained by uniformly dispersing ZTA nanopowder into epoxy using an ultrasonicator. Polymer composites of various concentrations viz, 0.5, 1, 1.5, 2 and 2.5 wt% were synthesized, all concentrations were prepared on weight basis. All the polymer composites were tested for compression properties, flexural properties and tensile properties. Best results for all the mechanical properties were obtained for epoxy with 1.5 wt% ZTA composites. Electrical properties such as breakdown voltage and breakdown strength were analysed and outstanding results were observed for epoxy with 2.5 wt% ZTA composite.



    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)γ2σω12σϑ1(γ,τ)=1(γ,τ),0<σ1, (1.1)
    2σϑ2(γ,τ)τ2σ+2σϑ1(γ,τ)γ2σω22σϑ2(γ,τ)=2(γ,τ),0<σ1, (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 1(γ,τ) and 2(γ,τ) are the nondifferentiable source terms.

    The system of nonlinear coupled Burger's equations with local fractional derivatives can be described as:

    σϑ1τσ+ξ12σϑ1γ2σ+ξ2σϑ1γσϑ1+ρσ[ϑ1ϑ2]γσ=0,ϑ1=ϑ1(γ,τ),0<σ1, (1.4)
    σϑ2τσ+μ12σϑ2γ2σ+μ2σϑ2γσϑ2+ησ[ϑ1ϑ2]γσ=0,ϑ2=ϑ2(γ,τ),0<σ1, (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 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,...,N1, t0=l1, tN=l2 with Δtj=tj+1tj. Now, the LFI of ϑ(γ) is formulated as

    l1Iσl2ϑ(γ)=1Γ(1+σ)l2l1ϑ(t)(dt)σ=1Γ(1+σ)limΔt0N1j=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<σ1. (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<σ1, (2.3)
    Cosσ(γσ)=q=0(1)qγ2qσΓ(1+2qσ),0<σ1, (2.4)
    Sinhσ(γσ)=q=0γ(2q+1)σΓ(1+(2q+1)σ),0<σ1, (2.5)
    Coshσ(γσ)=q=0γ2qσΓ(1+2qσ),0<σ1. (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))(γγ0)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:

    στσϑ(γ,τ)=Δσ(ϑ(γ,τ)ϑ(γ,τ0))(ττ0)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(q1)σΓ(1+(q1)σ),qN,
    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),qN,

    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<σ1, (2.9)

    and the corresponding inverse LFNT LFN1σ is formulated as

    LFN1σ[ˉϑσ(γ,s,u)]=ϑ(γ,τ)=1(2πi)σρ+iρiEσ(sστσuσ)ˉϑσ(γ,s,u)(ds)σ,0<σ1, (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)q1k=0s(qk1)σu(qk)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,τη)(dη)σ. (2.15)

    Some useful formulae for LFNT are listed in Table 1 [47].

    Table 1.  Formulae of LFNT.
    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))=sσs2σu2σ

     | Show Table
    DownLoad: CSV

    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 Lσqστqσ denotes the linear local fractional differential operator (LFDO) of order qσ i.e., a number δ>0 such that Lσϑδϑ, 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(ϑϑ)|λ|ϑϑ|. Qσ denotes the nonlinear differential operator which is Lipschitz continuous with ξ>0 fulfilling the criteria |Qσ(ϑϑ)|ξ|ϑϑ|, 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)=uqσsqσq1k=0s(qk1)σu(qk)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)uqσsqσq1k=0s(qk1)σu(qk)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)]uqσsqσq1k=0s(qk1)σu(qk)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:

    (1κ)LFNσ[φ(γ,τ;κ)ϑ0(γ,τ)]=κΦ[φ(γ,τ;κ)], (3.7)

    where 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 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)χμϑμ1(γ,τ)]=μ[ϑμ1(γ,τ)]. (3.13)

    Operating the inverse of LFNT on Eq (3.13), we get

    ϑμ(γ,τ)=χμϑμ1(γ,τ)+LFN1σ(μ(ϑμ1(γ,τ))). (3.14)

    In Eq (3.14), the value of μ(ϑμ1(γ,τ)) can be written in a new look as

    μ(ϑμ1(γ,τ))=LFNσ[ϑμ1(γ,τ)](1χμ)(uqσsqσq1k=0s(qk1)σu(qk)σϑ(kσ)(γ,0)+uqσsqσ(LFNσ[ω(γ,τ)]))+uqσsqσLFNσ(Pσϑμ1(γ,τ)+Pμ1), (3.15)

    where the value of χμ is presented as

    χμ={0,μ11,μ>1. (3.16)

    In Eq (3.15), Pμ denotes homotopy polynomial suggested in [55] in functioning of LFHAM [43,46], and is formulated as

    Pμ=1Γ(μ)[μκμQσ(φ(γ,τ;κ))]κ=0, (3.17)

    where

    φ=φ0+κφ1+κ2φ2+κ3φ3. (3.18)

    Putting the value of μ(ϑμ1) from Eq (3.15) in Eq (3.14) transforms the Eq (3.14) as follows:

    ϑμ(γ,τ)=(χμ+)ϑμ1(γ,τ)(1χμ)LFN1σ(uqσsqσq1k=0s(qk1)σu(qk)σϑ(kσ)(γ,0)+uqσsqσ(LFNσ[ω(γ,τ)]))+LFN1σ(uqσsqσLFNσ(Pσϑμ1(γ,τ)+Pμ1)). (3.19)

    From Eq (3.19), the components ϑμ(γ,τ) can be evaluated for μ1 and the LFNHAM solution is presented in the following way:

    ϑ(γ,τ)=limNNμ=0ϑμ(γ,τ). (3.20)

    The significant aspect of the LFNHAM is the auxiliary parameter 0 which guarantees the convergence of the series solution of Eq (3.1).

    Theorem 3.1. If a constant 0<<1 can be estimated such that ϑμ+1(γ,τ)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)N+1(1)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+1ϑμ(γ,τ)μ=N+1μϑ0(γ,τ)()N+1[1++2+]ϑ0(γ,τ)N+1(1)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+)+(λ+ξ)ς. (4.1)

    Proof. The solution of nonlinear LFPDE (3.1) is obtained as

    ϑ(γ,τ)=limNNμ=0ϑμ(γ,τ), (4.2)

    where

    ϑμ(γ,τ)=(χμ+)ϑμ1(γ,τ)(1χμ)LFN1σ(uqσsqσq1k=0s(qk1)σu(qk)σϑ(kσ)(γ,0)+uqσsqσ(LFNσ[ω(γ,τ)]))+LFN1σ(uqσsqσLFNσ(Pσϑμ1(γ,τ)+Pμ1)). (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+)(ϑϑ)+LFN1σ(uqσsqσLFNσ[Pσ(ϑϑ)+Qσ(ϑϑ)])|(1+)|ϑϑ|+LFN1σ(uqσsqσLFNσ|Pσ(ϑϑ)+Qσ(ϑϑ)|). (4.4)

    Utilization of the local fractional convolution theorem for LFNT in Eq (4.4) gives

    |ϑϑ|(1+)|ϑϑ|+Γ(1+σ)τ0|Pσ(ϑϑ)+Qσ(ϑϑ)|(τη)(q1)σΓ(1+(q1)σ)(dη)σ(1+)|ϑϑ|+Γ(1+σ)τ0(λ+ξ)|ϑϑ|(τη)(q1)σΓ(1+(q1)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+)|ϑϑ|+(λ+ξ)|ϑϑ|ς=[(1+)+(λ+ξ)ς]|ϑϑ|=ρ|ϑϑ|. (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.

    Figure 1.  3D plot of the solution {\vartheta _1}\, \left({\gamma, \tau } \right) with respect to \gamma and \tau in case of LFCHEs for \sigma = \ln 2/\ln 3 .
    Figure 2.  3D plot of the solution {\vartheta _2}\, \left({\gamma, \tau } \right) with respect to \gamma and \tau in case of LFCHEs for \sigma = \ln 2/\ln 3 .
    Figure 3.  3D nature of the solution {\vartheta _1}\, \left({\gamma, \tau } \right) with respect to \gamma and \tau in case of LFCBEs for \sigma = \ln 2/\ln 3 .
    Figure 4.  3D behavior of the solution {\vartheta _2}\, \left({\gamma, \tau } \right) with respect to \gamma and \tau in case of LFCBEs for \sigma = \ln 2/\ln 3 .

    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.



    [1] Wetzel B, Haupert F, Zhang MQ (2003) Epoxy nanocomposites with high mechanical and tribological performance. Compos Sci Technol 63: 2055-2067. https://doi.org/10.1016/S0266-3538(03)00115-5 doi: 10.1016/S0266-3538(03)00115-5
    [2] Carolan D, Ivankovic A, Kinloch AJ, et al. (2017) Toughened carbon fibre-reinforced polymer composites with nanoparticle-modified epoxy matrices. J Mater Sci 52: 1767-1788. https://doi.org/10.1007/s10853-016-0468-5 doi: 10.1007/s10853-016-0468-5
    [3] Dorigato A, Pegoretti A, Bondioli F, et al. (2010) Improving epoxy adhesives with zirconia nanoparticles. Compos Interface 17: 873-892. https://doi.org/10.1163/092764410X539253 doi: 10.1163/092764410X539253
    [4] Bondioli F, Cannillo V, Fabbri E, et al. (2006) Preparation and characterization of epoxy resins filled with submicron spherical zirconia particles. Polimery 51: 794-798. https://doi.org/10.14314/polimery.2006.794 doi: 10.14314/polimery.2006.794
    [5] Dorigato A, Pegoretti A (2011) The role of alumina nanoparticles in epoxy adhesives. J Nanopart Res 13: 2429-2441. https://doi.org/10.1007/s11051-010-0130-0 doi: 10.1007/s11051-010-0130-0
    [6] Yu ZQ, You SL, Yang ZG, et al. (2011) Effect of surface functional modification of nano-alumina particles on thermal and mechanical properties of epoxy nanocomposites. Adv Compos Mater 20: 487-502. https://doi.org/10.1163/092430411X579104 doi: 10.1163/092430411X579104
    [7] Reyes-Rojas A, Dominguez-Rios C, Garcia-Reyes A, et al. (2018) Sintering of carbon nanotube-reinforced zirconia-toughened alumina composites prepared by uniaxial pressing and cold isostatic pressing. Mater Res Express 5: 105602. https://doi.org/10.1088/2053-1591/aada35 doi: 10.1088/2053-1591/aada35
    [8] Chuankrerkkul N, Somton K, Wonglom T, et al. (2016) Physical and mechanical properties of zirconia toughened alumina (ZTA) composites fabricated by powder injection moulding. Chiang Mai J Sci 43: 375-380.
    [9] Ponnilavan V, Kannan S (2019) Structural, optical tuning, and mechanical behavior of zirconia toughened alumina through europium substitutions. J Biomed Mater Res Part B 107: 1170-1179. https://doi.org/10.1002/jbm.b.34210 doi: 10.1002/jbm.b.34210
    [10] Srikanth C, Madhu GM (2020) Effect of ZTA concentration on structural, thermal, mechanical and dielectric behavior of novel ZTA-PVA nanocomposite films. SN Appl Sci 2: 1-12. https://doi.org/10.1007/s42452-020-2232-3 doi: 10.1007/s42452-020-2232-3
    [11] Zhang J, Ge L, Chen ZG, et al. (2019) Cracking behavior and mechanism of gibbsite crystallites during calcination. Cryst Res Technol 54: 1800201. https://doi.org/10.1002/crat.201800201 doi: 10.1002/crat.201800201
    [12] Bhaduri S, Bhaduri SB, Zhou E (1998) Auto ignition synthesis and consolidation of Al2O3-ZrO2 nano/nano composite powders. J Mater Res 13: 156-165. https://doi.org/10.1557/JMR.1998.0021 doi: 10.1557/JMR.1998.0021
    [13] Vasylkiv O, Sakka Y, Skorokhod VV (2003) Low-temperature processing and mechanical properties of zirconia and zirconia-alumina nanoceramics. J Am Ceram Soc 86: 299-304. https://doi.org/10.1111/j.1151-2916.2003.tb00015.x doi: 10.1111/j.1151-2916.2003.tb00015.x
    [14] Sagar JS, Kashyap SJ, Madhu GM, et al. (2020) Investigation of mechanical, thermal and electrical parameters of gel combustion-derived cubic zirconia/epoxy resin composites for high-voltage insulation. Cerâmica 66: 186-196. https://doi.org/10.1590/0366-69132020663782887 doi: 10.1590/0366-69132020663782887
    [15] Ho MW, Lam CK, Lau K, et al. (2006) Mechanical properties of epoxy-based composites using nanoclays. Compos Struct 75: 415-421. https://doi.org/10.1016/j.compstruct.2006.04.051 doi: 10.1016/j.compstruct.2006.04.051
    [16] Uhl FM, Davuluri SP, Wong SC, et al. (2004) Organically modified montmorillonites in UV curable urethane acrylate films. Polymer 45: 6175-6187. https://doi.org/10.1016/j.polymer.2004.07.001 doi: 10.1016/j.polymer.2004.07.001
    [17] Nguyen TA, Nguyen TV, Thai H, et al. (2016) Effect of nanoparticles on the thermal and mechanical properties of epoxy coatings. J Nanosci Nanotechnol 16: 9874-9881. https://doi.org/10.1166/jnn.2016.12162 doi: 10.1166/jnn.2016.12162
    [18] Baiquni M, Soegijono B, Hakim AN (2019) Thermal and mechanical properties of hybrid organoclay/rockwool fiber reinforced epoxy composites. J Phys Conf Ser 1191: 012056. https://doi.org/10.1088/1742-6596/1191/1/012056 doi: 10.1088/1742-6596/1191/1/012056
    [19] Zhang X, Alloul O, He Q, et al. (2013) Strengthened magnetic epoxy nanocomposites with protruding nanoparticles on the graphene nanosheets. Polymer 54: 3594-3604. https://doi.org/10.1016/j.polymer.2013.04.062 doi: 10.1016/j.polymer.2013.04.062
    [20] Nazarenko OB, Melnikova TV, Visakh PM (2016) Thermal and mechanical characteristics of polymer composites based on epoxy resin, aluminium nanopowders and boric acid. J Phys Conf Ser 671: 012040. https://doi.org/10.1088/1742-6596/671/1/012040 doi: 10.1088/1742-6596/671/1/012040
    [21] Sand Chee S, Jawaid M (2019) The effect of Bi-functionalized MMT on morphology, thermal stability, dynamic mechanical, and tensile properties of epoxy/organoclay nanocomposites. Polymers 11: 2012. https://doi.org/10.3390/polym11122012 doi: 10.3390/polym11122012
    [22] Bikiaris D (2011) Can nanoparticles really enhance thermal stability of polymers? Part Ⅱ: An overview on thermal decomposition of polycondensation polymers. Thermochim Acta 523: 25-45. https://doi.org/10.1016/j.tca.2011.06.012 doi: 10.1016/j.tca.2011.06.012
    [23] Xue Y, Shen M, Zeng S, et al. (2019) A novel strategy for enhancing the flame resistance, dynamic mechanical and the thermal degradation properties of epoxy nanocomposites. Mater Res Express 6: 125003. https://doi.org/10.1088/2053-1591/ab537f doi: 10.1088/2053-1591/ab537f
    [24] Colomban P (1989) Structure of oxide gels and glasses by infrared and Raman scattering. J Mater Sci 24: 3011-3020. https://doi.org/10.1007/BF02385660 doi: 10.1007/BF02385660
    [25] Taavoni-Gilan A, Taheri-Nassaj E, Naghizadeh R, et al. (2010) Properties of sol-gel derived Al2O3-15 wt% ZrO2 (3 mol% Y2O3) nanopowders using two different precursors. Ceram Int 36: 1147-1153. https://doi.org/10.1016/j.ceramint.2009.11.011 doi: 10.1016/j.ceramint.2009.11.011
    [26] Noma T, Sawaoka A (1984) Fracture toughness of high pressure sintered Al2O3-ZrO2 ceramics. J Mater Sci Lett 3: 533-535. https://doi.org/10.1007/BF00720992 doi: 10.1007/BF00720992
    [27] Shukla DK, Kasisomayajula SV, Parameswaran V (2008) Epoxy composites using functionalized alumina platelets as reinforcements. Compos Sci Technol 68: 3055-3063. https://doi.org/10.1016/j.compscitech.2008.06.025 doi: 10.1016/j.compscitech.2008.06.025
    [28] Abbate M, Martuscelli E, Musto P, et al. (1994) Toughening of a highly cross-linked epoxy resin by reactive blending with bisphenol A polycarbonate. I. FTIR spectroscopy. J Polym Sci Pol Phys 32: 395-408. https://doi.org/10.1002/polb.1994.090320301 doi: 10.1002/polb.1994.090320301
    [29] Katon JE, Bentley FF (1963) New spectra-structure correlations of ketones in the 700-750 cm-1 region. Spectrochim Acta 19: 639-653. https://doi.org/10.1016/0371-1951(63)80127-7 doi: 10.1016/0371-1951(63)80127-7
    [30] Magnani G, Brillante A (2005) Effect of the composition and sintering process on mechanical properties and residual stresses in zirconia-alumina composites. J Eur Ceram Soc 25: 3383-3392. https://doi.org/10.1016/j.jeurceramsoc.2004.09.025 doi: 10.1016/j.jeurceramsoc.2004.09.025
    [31] Ashamol A, Priyambika VS, Avadhani GS, et al. (2013) Nanocomposites of crosslinked starch phthalate and silane modified nanoclay: Study of mechanical, thermal, morphological, and biodegradable characteristics. Starch-Stärke 65: 443-452. https://doi.org/10.1002/star.201200145 doi: 10.1002/star.201200145
    [32] Jumahat A, Soutis C, Mahmud J, et al. (2012) Compressive properties of nanoclay/epoxy nanocomposites. Procedia Eng 41: 1607-1613. https://doi.org/10.1016/j.proeng.2012.07.361 doi: 10.1016/j.proeng.2012.07.361
    [33] Abbass A, Abid S, Özakça M (2019) Experimental investigation on the effect of steel fibers on the flexural behavior and ductility of high-strength concrete hollow beams. Adv Civ Eng 2019: 8390345. https://doi.org/10.1155/2019/8390345 doi: 10.1155/2019/8390345
    [34] Konnola R, Deeraj BDS, Sampath S, et al. (2019) Fabrication and characterization of toughened nanocomposites based on TiO2 nanowire-epoxy system. Polym Compos 40: 2629-2638. https://doi.org/10.1002/pc.25058 doi: 10.1002/pc.25058
    [35] Zhao S, Schadler LS, Duncan R, et al. (2008) Mechanisms leading to improved mechanical performance in nanoscale alumina filled epoxy. Compos Sci Technol 68: 2965-2975. https://doi.org/10.1016/j.compscitech.2008.01.009 doi: 10.1016/j.compscitech.2008.01.009
    [36] Goyat MS, Rana S, Halder S, et al. (2018) Facile fabrication of epoxy-TiO2 nanocomposites: a critical analysis of TiO2 impact on mechanical properties and toughening mechanisms. Ultrason Sonochem 40: 861-873. https://doi.org/10.1016/j.ultsonch.2017.07.040 doi: 10.1016/j.ultsonch.2017.07.040
    [37] Johnsen BB, Kinloch AJ, Mohammed RD, et al. (2007) Toughening mechanisms of nanoparticle-modified epoxy polymers. Polymer 48: 530-541. https://doi.org/10.1016/j.polymer.2006.11.038 doi: 10.1016/j.polymer.2006.11.038
    [38] Johnsen BB, Kinloch AJ, Taylor AC (2005) Toughness of syndiotactic polystyrene/epoxy polymer blends: microstructure and toughening mechanisms. Polymer 46: 7352-7369. https://doi.org/10.1016/j.polymer.2005.05.151 doi: 10.1016/j.polymer.2005.05.151
    [39] Mohanty A, Srivastava VK (2013) Dielectric breakdown performance of alumina/epoxy resin nanocomposites under high voltage application. Mater Design 47: 711-716. https://doi.org/10.1016/j.matdes.2012.12.052 doi: 10.1016/j.matdes.2012.12.052
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