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 |
Antimicrobial resistance (AMR) is becoming a major problem for animal and human health. Reports of resistance to colistin, an antibiotic that is considered a last resort drug against resistant Gram-negative bacteria, have been increasing over the last years. Among the different mechanisms that cause AMR to colistin, the mobilized colistin resistance (mcr) gene has been reported as responsible for the increased incidence in animals and humans since 2015. There are ten recognized distinct variants of this gene in bacteria isolated from animals, humans, food, and the environment. Companion animals could have a role in human infection by pathogenic and resistant E. coli strains as they share the same environment and are in close contact with humans. Considering this, our aim was to investigate antimicrobial resistance in companion domestic and stray dogs in Western Macedonia, Greece. Our results revealed that of the 43 individual fecal samples examined, 16% of them hosted the mcr-1 gene, all of which were isolated from stray dogs. Our results suggested that companion dogs and stray dogs can serve as reservoirs for colistin-resistant E. coli strains.
Citation: Ioannis Tsakmakidis, Anastasia Parisi, Dimitrios K. Papadopoulos, Maria V. Alvanou, Konstantinos V. Papageorgiou, Evanthia Petridou, Ioannis A. Giantsis. Stray dogs as carriers of E. coli resistant strains for the retracted and re-emerged antibiotic colistin, based on the mcr-1 gene presence[J]. AIMS Molecular Science, 2024, 11(4): 367-378. doi: 10.3934/molsci.2024022
[1] | Mubashir Qayyum, Efaza Ahmad, Hijaz Ahmad, Bandar Almohsen . New solutions of time-space fractional coupled Schrödinger systems. AIMS Mathematics, 2023, 8(11): 27033-27051. doi: 10.3934/math.20231383 |
[2] | Aslı Alkan, Halil Anaç . The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method. AIMS Mathematics, 2024, 9(9): 25333-25359. doi: 10.3934/math.20241237 |
[3] | Rasool Shah, Abd-Allah Hyder, Naveed Iqbal, Thongchai Botmart . Fractional view evaluation system of Schrödinger-KdV equation by a comparative analysis. AIMS Mathematics, 2022, 7(11): 19846-19864. doi: 10.3934/math.20221087 |
[4] | Sunil Kumar, Amit Kumar , Zaid Odibat, Mujahed Aldhaifallah, Kottakkaran Sooppy Nisar . A comparison study of two modified analytical approach for the solution of nonlinear fractional shallow water equations in fluid flow. AIMS Mathematics, 2020, 5(4): 3035-3055. doi: 10.3934/math.2020197 |
[5] | Mohammed Al-Refai, Dumitru Baleanu . Comparison principles of fractional differential equations with non-local derivative and their applications. AIMS Mathematics, 2021, 6(2): 1443-1451. doi: 10.3934/math.2021088 |
[6] | Thongchai Botmart, Ravi P. Agarwal, Muhammed Naeem, Adnan Khan, Rasool Shah . On the solution of fractional modified Boussinesq and approximate long wave equations with non-singular kernel operators. AIMS Mathematics, 2022, 7(7): 12483-12513. doi: 10.3934/math.2022693 |
[7] | Alessandra Jannelli, Maria Paola Speciale . On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations. AIMS Mathematics, 2021, 6(8): 9109-9125. doi: 10.3934/math.2021529 |
[8] | K. Pavani, K. Raghavendar . A novel method to study time fractional coupled systems of shallow water equations arising in ocean engineering. AIMS Mathematics, 2024, 9(1): 542-564. doi: 10.3934/math.2024029 |
[9] | Changdev P. Jadhav, Tanisha B. Dale, Vaijanath L. Chinchane, Asha B. Nale, Sabri T. M. Thabet, Imed Kedim, Miguel Vivas-Cortez . On solutions of fractional differential equations for the mechanical oscillations by using the Laplace transform. AIMS Mathematics, 2024, 9(11): 32629-32645. doi: 10.3934/math.20241562 |
[10] | Naveed Iqbal, Muhammad Tajammal Chughtai, Nehad Ali Shah . Numerical simulation of fractional-order two-dimensional Helmholtz equations. AIMS Mathematics, 2023, 8(6): 13205-13218. doi: 10.3934/math.2023667 |
Antimicrobial resistance (AMR) is becoming a major problem for animal and human health. Reports of resistance to colistin, an antibiotic that is considered a last resort drug against resistant Gram-negative bacteria, have been increasing over the last years. Among the different mechanisms that cause AMR to colistin, the mobilized colistin resistance (mcr) gene has been reported as responsible for the increased incidence in animals and humans since 2015. There are ten recognized distinct variants of this gene in bacteria isolated from animals, humans, food, and the environment. Companion animals could have a role in human infection by pathogenic and resistant E. coli strains as they share the same environment and are in close contact with humans. Considering this, our aim was to investigate antimicrobial resistance in companion domestic and stray dogs in Western Macedonia, Greece. Our results revealed that of the 43 individual fecal samples examined, 16% of them hosted the mcr-1 gene, all of which were isolated from stray dogs. Our results suggested that companion dogs and stray dogs can serve as reservoirs for colistin-resistant E. coli strains.
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∗|.∴(1−ρ)|ϑ−ϑ∗|≤0, | (4.6) |
where ρ=(1+ℏ)+ℏ(λ+ξ)ς. (4.7)
Since 0<ρ<1, therefore |ϑ−ϑ∗|=0, which provides ϑ=ϑ∗. This ensures the aspect of uniqueness of the solution of Eq (3.1).
Theorem 4.2. (Convergence theorem). Suppose Ξ is a Banach space and Θ:Ξ→Ξ is a nonlinear mapping. Assume that
‖Θ(ω)−Θ(ϑ)‖⩽−λ‖ω−ϑ‖,∀ϑ,ω∈Ξ. | (4.8) |
Then Banach's fixed point theory [58,59] suggests the existence of a fixed point forΘ. Moreover, the sequence constructed by LFNHAM converges to the fixed points of Θ with arbitrary choices of ω0,ϑ0∈Ξ and
‖ϑl1−ϑl2‖⩽−λl21−−λ‖ϑ1−ϑ0‖,∀ϑ,ω∈Ξ. | (4.9) |
Proof. It is presumed that (Π[Ω],‖.‖), where Π[Ω] signifies the Banach space of continuous functions on real line interval Ω holding the sup norm. Now, it is sufficient to prove that {ϑl2} is a Cauchy sequence in the Banach space Ξ.
Now, consider
‖ϑl1−ϑl2‖=maxτ∈Ω|ϑl1−ϑl2|=maxτ∈Ω|(1+ℏ)(ϑl1−1−ϑl2−1)+ℏLFN−1σ(uqσsqσLFNσ[Pσ(ϑl1−1−ϑl2−1)+Qσ(ϑl1−1−ϑl2−1)])|⩽maxτ∈Ω{(1+ℏ)|ϑl1−1−ϑl2−1|+ℏLFN−1σ(uqσsqσLFNσ[|Pσ(ϑl1−1−ϑl2−1)|+|Qσ(ϑl1−1−ϑl2−1)|])}. | (4.10) |
Employing the local fractional convolution theorem for LFNT in Eq (4.10), we have
‖ϑl1−ϑl2‖⩽maxτ∈Ω{(1+ℏ)|ϑl1−1−ϑl2−1|+ℏΓ(1+σ)∫τ0[|Pσ(ϑl1−1−ϑl2−1)|+|Qσ(ϑl1−1−ϑl2−1)|](τ−η)(q−1)σΓ(1+(q−1)σ)(dη)σ}⩽maxτ∈Ω{(1+ℏ)|ϑl1−1−ϑl2−1|+ℏΓ(1+σ)∫τ0(λ+ξ)|ϑl1−1−ϑl2−1|(τ−η)(q−1)σΓ(1+(q−1)σ)(dη)σ}. | (4.11) |
Now, application of MVT of LFI calculus [56,57] reduces the inequality (4.11) in the following form
‖ϑl1−ϑl2‖⩽maxτ∈Ω{(1+ℏ)|ϑl1−1−ϑl2−1|+ℏ(λ+ξ)|ϑl1−1−ϑl2−1|ς}=maxt∈Γ{(1+ℏ)+ℏ(λ+ξ)ς}|ϑl1−1−ϑl2−1|=−λ‖ϑl1−1−ϑl2−1‖.∴‖ϑl1−ϑl2‖⩽−λ‖ϑl1−1−ϑl2−1‖, | (4.12) |
where−λ=(1+ℏ)+ℏ(λ+ξ)ς | (4.13) |
Assume l1=l2+1, then it produces
‖ϑl2+1−ϑl2‖⩽−λ‖ϑl2−ϑl2−1‖⩽−λ2‖ϑl2−1−ϑl2−2‖⋯⩽−λl2‖ϑ1−ϑ0‖. | (4.14) |
Utilizing the triangular inequality, we have
‖ϑl1−ϑl2‖⩽‖ϑl2+1−ϑl2‖+‖ϑl2+2−ϑl2+1‖+⋯+‖ϑl1−ϑl1−1‖⩽(−λl2+−λl2+1+−λl2+2+⋯+−λl1−1)‖ϑ1−ϑ0‖=−λl2(1+−λ+−λ2+⋯+−λl1−l2−1)‖ϑ1−ϑ0‖=−λl2[1−−λl1−l2−11−−λ]‖ϑ1−ϑ0‖. | (4.15) |
Since 0<−λ<1, thus 1−−λl1−l2−1<1, then
‖ϑl1−ϑl2‖⩽−λl21−−λ‖ϑ1−ϑ0‖. | (4.16) |
But ‖ϑ1−ϑ0‖<∞, thus ‖ϑl1−ϑl2‖→0 as n→∞, hence {ϑl2} is a Cauchy sequence in Π[Ω] and so {ϑl2} is convergent. This ensures the convergence of the solution ϑ(γ,τ) 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
∂2σϑ1(γ,τ)∂τ2σ+∂2σϑ2(γ,τ)∂γ2σ−ϑ1(γ,τ)=0,0<σ⩽1, | (5.1) |
∂2σϑ2(γ,τ)∂τ2σ+∂2σϑ1(γ,τ)∂γ2σ−ϑ2(γ,τ)=0,0<σ⩽1, | (5.2) |
subject to the fractal initial conditions:
ϑ1(γ,0)=0,∂σϑ1(γ,0)∂τσ=Eσ(γσ), |
ϑ2(γ,0)=0,∂σϑ2(γ,0)∂τσ=−Eσ(γσ), | (5.3) |
where ϑ1(γ,τ) and ϑ2(γ,τ) represent the local fractional continuous functions.
On account of the initial conditions (5.3) and algorithm of LFNHAM, the initial guess are written as
ϑ1,0(γ,τ)=ϑ1(γ,0)+τσΓ(1+σ)ϑ1,0(σ)(γ,0)=Eσ(γσ)τσΓ(1+σ), |
ϑ2,0(γ,τ)=ϑ2(γ,0)+τσΓ(1+σ)ϑ2,0(σ)(γ,0)=−Eσ(γσ)τσΓ(1+σ). | (5.4) |
Employing the LFNT operator LFNσ on Eqs (5.1) and (5.2), we get
LFNσ[∂2σϑ1(γ,τ)∂τ2σ]+LFNσ[∂2σϑ2(γ,τ)∂γ2σ]−LFNσ(ϑ1(γ,τ))=0, | (5.5) |
LFNσ[∂2σϑ2(γ,τ)∂τ2σ]+LFNσ[∂2σϑ1(γ,τ)∂γ2σ]−LFNσ(ϑ2(γ,τ))=0. | (5.6) |
Now, the implementation of formula of LFNT for local fractional derivatives yields
s2σu2σˉϑ1(γ,s,u)−sσu2σϑ1(γ,0)−1uσϑ1(σ)(γ,0)+LFNσ[∂2σϑ2(γ,τ)∂γ2σ]−LFNσ(ϑ1(γ,τ))=0, | (5.7) |
s2σu2σˉϑ2(γ,s,u)−sσu2σϑ2(γ,0)−1uσϑ2(σ)(γ,0)+LFNσ[∂2σϑ1(γ,τ)∂γ2σ]−LFNσ(ϑ2(γ,τ))=0. | (5.8) |
After rearranging the terms, we get
ˉϑ1(γ,s,u)=1sσϑ1(γ,0)+uσs2σϑ1(σ)(γ,0)−u2σs2σLFNσ[∂2σϑ2(γ,τ)∂γ2σ]+u2σs2σLFNσ(ϑ1(γ,τ)), | (5.9) |
ˉϑ2(γ,s,u)=1sσϑ2(γ,0)+uσs2σϑ2(σ)(γ,0)−u2σs2σLFNσ[∂2σϑ1(γ,τ)∂γ2σ]+u2σs2σLFNσ(ϑ2(γ,τ)). | (5.10) |
Now, further simplification in view of initial condition (5.3) reduces Eqs (5.9) and (5.10) in the following way
ˉϑ1(γ,s,u)−uσs2σEσ(γσ)+u2σs2σLFNσ[∂2σϑ2(γ,τ)∂γ2σ]−u2σs2σLFNσ(ϑ1)=0, | (5.11) |
ˉϑ2(γ,s,u)+uσs2σEσ(γσ)+u2σs2σLFNσ[∂2σϑ1(γ,τ)∂γ2σ]−u2σs2σLFNσ(ϑ2)=0. | (5.12) |
Now in view of Eqs (5.11) and (5.12), the nonlinear operators are formed as:
Φ1[φ1(γ,τ;κ)]=LFNσ(φ1(γ,τ;κ))−uσs2σEσ(γσ)+u2σs2σLFNσ[∂2σφ2(γ,τ;κ)∂γ2σ]−u2σs2σLFNσ(φ1(γ,τ;κ)), | (5.13) |
Φ2[φ2(γ,τ;κ)]=LFNσ(φ1(γ,τ;κ))+uσs2σEσ(γσ)+u2σs2σLFNσ[∂2σφ1(γ,τ;κ)∂γ2σ]−u2σs2σLFNσ(φ2(γ,τ;κ)), | (5.14) |
where κ is an embedding parameter and φ1(γ,τ;κ) & φ2(γ,τ;κ) are real valued functions of γ,τ and κ.
Now using the steps of the LFHAM [43,46] and basic methodology of HAM [52,53,54], μth-order deformation equations are constructed as follows:
ϑ1,μ(γ,τ)=χμϑ1,(μ−1)(γ,τ)+ℏLFN−1σ(ℜμ(ϑ1,(μ−1)(γ,τ))), | (5.15) |
ϑ2,μ(γ,τ)=χμϑ2,(μ−1)(γ,τ)+ℏLFN−1σ(ℜμ(ϑ2,(μ−1)(γ,τ))). | (5.16) |
In Eqs (5.15) and (5.16), the terms ℜμ(ϑ1,(μ−1)(γ,τ)) and ℜμ(ϑ2,(μ−1)(γ,τ)) are expressed as
ℜμ(ϑ1,(μ−1)(γ,τ))=LFNσ[ϑ1,(μ−1)(γ,τ)]−(1−χμ)uσs2σEσ(γσ)+u2σs2σLFNσ[∂2σϑ2,(μ−1)(γ,τ)∂γ2σ]−u2σs2σLFNσ(ϑ1,(μ−1)(γ,τ)), | (5.17) |
ℜμ(ϑ2,(μ−1)(γ,τ))=LFNσ[ϑ2,(μ−1)(γ,τ)]+(1−χμ)uσs2σEσ(γσ)+u2σs2σLFNσ[∂2σϑ1,(μ−1)(γ,τ)∂γ2σ]−u2σs2σLFNσ(ϑ2,(μ−1)(γ,τ)). | (5.18) |
Now implementing the LFNHAM and using Eqs (5.15) to (5.18), we have
ϑ1,μ(γ,τ)=(χμ+ℏ)ϑ1,(μ−1)(γ,τ)−ℏ(1−χμ)Eσ(γσ)τσΓ(1+σ)+ℏLFN−1σ(u2σs2σLFNσ[∂2σϑ2,(μ−1)(γ,τ)∂γ2σ]−u2σs2σLFNσ(ϑ1,(μ−1)(γ,τ))),μ⩾1, | (5.19) |
ϑ2,μ(γ,τ)=(χμ+ℏ)ϑ2,(μ−1)(γ,τ)+ℏ(1−χμ)Eσ(γσ)τσΓ(1+σ)+ℏLFN−1σ(u2σs2σLFNσ[∂2σϑ1,(μ−1)(γ,τ)∂γ2σ]−u2σs2σLFNσ(ϑ2,(μ−1)(γ,τ))),μ⩾1. | (5.20) |
On account of Eqs (5.19) and (5.20) for μ=1, we have
ϑ1,1(γ,τ)=ℏϑ1,0(γ,τ)−ℏEσ(γσ)τσΓ(1+σ)+ℏLFN−1σ(u2σs2σLFNσ[∂2σϑ2,0(γ,τ)∂γ2σ]−u2σs2σLFNσ(ϑ1,0(γ,τ))), | (5.21) |
ϑ2,1(γ,τ)=ℏϑ2,0(γ,τ)+ℏEσ(γσ)τσΓ(1+σ)+ℏLFN−1σ(u2σs2σLFNσ[∂2σϑ1,0(γ,τ)∂γ2σ]−u2σs2σLFNσ(ϑ2,0(γ,τ))). | (5.22) |
Using initial guess values (5.4) and further simplification reduces Eqs (5.21) and (5.22) in this way
ϑ1,1(γ,τ)=−2ℏEσ(γσ)LFN−1σ(u2σs2σuσs2σ)=−2ℏEσ(γσ)τ3σΓ(1+3σ), | (5.23) |
ϑ2,1(γ,τ)=2ℏEσ(γσ)LFN−1σ(u2σs2σuσs2σ)=2ℏEσ(γσ)τ3σΓ(1+3σ). | (5.24) |
By means of Eqs (5.19) and (5.20) for μ=2, we have
ϑ1,2(γ,τ)=(1+ℏ)ϑ1,1(γ,τ)+ℏLFN−1σ(u2σs2σLFNσ[∂2σϑ2,1(γ,τ)∂γ2σ]−u2σs2σLFNσ(ϑ1,1(γ,τ))), | (5.25) |
ϑ2,2(γ,τ)=(1+ℏ)ϑ2,1(γ,τ)+ℏLFN−1σ(u2σs2σLFNσ[∂2σϑ1,1(γ,τ)∂γ2σ]−u2σs2σLFNσ(ϑ2,1(γ,τ))). | (5.26) |
Utilizing the values provided by Eqs (5.23) and (5.24) in Eqs (5.25) and (5.26), we obtain
ϑ1,2(γ,τ)=−2ℏ(1+ℏ)Eσ(γσ)τ3σΓ(1+3σ)+4ℏ2Eσ(γσ)τ5σΓ(1+5σ), | (5.27) |
ϑ2,2(γ,τ)=2ℏ(1+ℏ)Eσ(γσ)τ3σΓ(1+3σ)−4ℏ2Eσ(γσ)τ5σΓ(1+5σ). | (5.28) |
Following the similar procedure, we obtain the rest of the values for ϑ1,μ(γ,τ) and ϑ2,μ(γ,τ) for μ⩾3.
Setting the convergence-control parameter ℏ=−1, we attain the following values
ϑ1,1(γ,τ)=2Eσ(γσ)τ3σΓ(1+3σ), |
ϑ2,1(γ,τ)=−2Eσ(γσ)τ3σΓ(1+3σ), |
ϑ1,2(γ,τ)=4Eσ(γσ)τ5σΓ(1+5σ), |
ϑ2,2(γ,τ)=−4Eσ(γσ)τ5σΓ(1+5σ), | (5.29) |
⋮ |
and so on.
Hence, the solutions of Eqs (5.1) and (5.2) are obtained as
ϑ1(γ,τ)=∞∑n=0ϑ1,n(γ,τ)=Eσ(γσ)(τσΓ(1+σ)+2τ3σΓ(1+3σ)+4τ5σΓ(1+5σ)+8τ7σΓ(1+7σ)+⋯)=Eσ(γσ)(∞∑n=02nτ(2n+1)σΓ(1+(2n+1)σ))=1√2Eσ(γσ)(∞∑n=02n+12τ(2n+1)σΓ(1+(2n+1)σ))=1√2Eσ(γσ)sinhσ(√2τσ). | (5.30) |
Similarly, we have
ϑ2(γ,τ)=∞∑n=0ϑ2,n(γ,τ)=Eσ(γσ)(−τσΓ(1+σ)−2τ3σΓ(1+3σ)−4τ5σΓ(1+5σ)−8τ7σΓ(1+7σ)−⋯)=−Eσ(γσ)(∞∑n=02nτ(2n+1)σΓ(1+(2n+1)σ))=−1√2Eσ(γσ)sinhσ(√2τσ). | (5.31) |
Finally, the solutions of the coupled Helmholtz Eqs (5.1) & (5.2) are expressed as
ϑ1(γ,τ)=Eσ(γσ)sinhσ(√2τσ)√2, |
ϑ2(γ,τ)=−Eσ(γσ)sinhσ(√2τσ)√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.
∂σϑ1∂τσ+∂2σϑ1∂γ2σ−2∂σϑ1∂γσϑ1+∂σ[ϑ1ϑ2]∂γσ=0,ϑ1=ϑ1(γ,τ),0<σ⩽1, | (6.1) |
∂σϑ2∂τσ+∂2σϑ2∂γ2σ−2∂σϑ2∂γσϑ2+∂σ[ϑ1ϑ2]∂γσ=0,ϑ2=ϑ2(γ,τ),0<σ⩽1, | (6.2) |
subject to the fractal initial conditions:
ϑ1(γ,0)=ϑ2(γ,0)=Eσ(γσ), | (6.3) |
where ϑ1(γ,τ) and ϑ2(γ,τ) are local fractional continuous functions.
In view of initial conditions (6.3) and LFNHAM, the initial guess are expressed as
ϑ1,0(γ,τ)=ϑ2,0(γ,τ)=Eσ(γσ). | (6.4) |
Employing the LFNT operator LFNσ on Eqs (6.1) and (6.2), we get
LFNσ(∂σϑ1∂τσ)+LFNσ(∂2σϑ1∂γ2σ)−2LFNσ(∂σϑ1∂γσϑ1)+LFNσ(∂σ[ϑ1ϑ2]∂γσ)=0, | (6.5) |
LFNσ(∂σϑ2∂τσ)+LFNσ(∂2σϑ2∂γ2σ)−2LFNσ(∂σϑ2∂γσϑ2)+LFNσ(∂σ[ϑ1ϑ2]∂γσ)=0. | (6.6) |
Now, employing the formula of LFNT for LFDs yields
sσuσˉϑ1(γ,s,u)−1uσϑ1(γ,0)+LFNσ(∂2σϑ1∂γ2σ)−2LFNσ(∂σϑ1∂γσϑ1)+LFNσ(∂σ[ϑ1ϑ2]∂γσ)=0, | (6.7) |
sσuσˉϑ2(γ,s,u)−1uσϑ2(γ,0)+LFNσ(∂2σϑ2∂γ2σ)−2LFNσ(∂σϑ2∂γσϑ2)+LFNσ(∂σ[ϑ1ϑ2]∂γσ)=0. | (6.8) |
After rearranging the terms, we get
ˉϑ1(γ,s,u)=1sσϑ1(γ,0)−uσsσLFNσ(∂2σϑ1∂γ2σ)+2uσsσLFNσ(∂σϑ1∂γσϑ1)−uσsσLFNσ(∂σ[ϑ1ϑ2]∂γσ), | (6.9) |
ˉϑ2(γ,s,u)=1sσϑ2(γ,0)−uσsσLFNσ(∂2σϑ2∂γ2σ)+2uσsσLFNσ(∂σϑ2∂γσϑ2)−uσsσLFNσ(∂σ[ϑ1ϑ2]∂γσ). | (6.10) |
Now, further simplification in view of initial conditions (6.3) reduces Eqs (6.9) and (6.10) as follows:
ˉϑ1(γ,s,u)−1sσEσ(γσ)+uσsσLFNσ(∂2σϑ1∂γ2σ)−2uσsσLFNσ(∂σϑ1∂γσϑ1)+uσsσLFNσ(∂σ[ϑ1ϑ2]∂γσ)=0 | (6.11) |
ˉϑ2(γ,s,u)−1sσEσ(γσ)+uσsσLFNσ(∂2σϑ2∂γ2σ)−2uσsσLFNσ(∂σϑ2∂γσϑ2)+uσsσLFNσ(∂σ[ϑ1ϑ2]∂γσ)=0. | (6.12) |
Now in view of Eqs (6.11) and (6.12), the nonlinear operators are formed as
Φ1[φ1(γ,τ;κ)]=LFNσ(φ1(γ,τ;κ))−1sσEσ(γσ)+uσsσLFNσ[∂2σφ1(γ,τ;κ)∂γ2σ]−2uσsσLFNσ(∂σφ1∂γσφ1)+uσsσLFNσ(∂σ[φ1(γ,τ;κ)φ2(γ,τ;κ)]∂γσ), | (6.13) |
Φ2[φ2(γ,τ;κ)]=LFNσ(φ2(γ,τ;κ))−1sσEσ(γσ)+uσsσLFNσ[∂2σφ2(γ,τ;κ)∂γ2σ]−2uσsσLFNσ(∂σφ2∂γσφ2)+uσsσLFNσ(∂σ[φ1(γ,τ;κ)φ2(γ,τ;κ)]∂γσ), | (6.14) |
where κ∈[0,1] is an embedding parameter and φ1(γ,τ;κ) & φ2(γ,τ;κ) are real valued functions of γ,τ, and κ.
Performing the steps of the LFNHAM, μth-order deformation equations are formed as
ϑ1,μ(γ,τ)=χμϑ1,(μ−1)(γ,τ)+ℏLFN−1σ(ℜμ(ϑ1,(μ−1)(γ,τ))), | (6.15) |
ϑ2,μ(γ,τ)=χμϑ2,(μ−1)(γ,τ)+ℏLFN−1σ(ℜμ(ϑ2,(μ−1)(γ,τ))). | (6.16) |
In Eqs (6.15) and (6.16), the terms ℜμ(ϑ1,(μ−1)(γ,τ)) and ℜμ(ϑ2,(μ−1)(γ,τ)) are computed as
ℜμ(ϑ1,(μ−1)(γ,τ))=LFNσ[ϑ1,(μ−1)(γ,τ)]−(1−χμ)1sσEσ(γσ)+uσsσLFNσ[∂2σϑ1,(μ−1)(γ,τ)∂γ2σ]−2uσsσLFNσ(Pμ−1)+uσsσLFNσ(∂σP′μ−1∂γσ), | (6.17) |
ℜμ(ϑ2,(μ−1)(γ,τ))=LFNσ[ϑ2,(μ−1)(γ,τ)]−(1−χμ)1sσEσ(γσ)+uσsσLFNσ[∂2σϑ2,(μ−1)(γ,τ)∂γ2σ]−2uσsσLFNσ(P″μ−1)+uσsσLFNσ(∂σP′μ−1∂γσ), | (6.18) |
where Pμ−1, P′μ−1, P″μ−1 denote the homotopy polynomials [55] and are formulated as
Pμ−1=1Γ(μ)[∂μ∂κμ(∂σφ1(γ,τ;κ)∂γσφ1(γ,τ;κ))], | (6.19) |
P′μ−1=1Γ(μ)[∂μ∂κμ(φ1(γ,τ;κ)φ2(γ,τ;κ))], | (6.20) |
P″μ−1=1Γ(μ)[∂μ∂κμ(∂σφ2(γ,τ;κ)∂γσφ2(γ,τ;κ))], | (6.21) |
and
φ1(γ,t,κ)=φ1,0+κφ1,1+κ2φ1,2+⋯, | (6.22) |
φ2(γ,t,κ)=φ2,0+κφ2,1+κ2φ2,2+⋯. | (6.23) |
On account of LFNHAM and Eqs (6.15)–(6.18), we have
ϑ1,μ(γ,τ)=(χμ+ℏ)ϑ1,(μ−1)(γ,τ)−ℏ(1−χμ)Eσ(γσ)+ℏLFN−1σ(uσsσLFNσ[∂2σϑ1,(μ−1)(γ,τ)∂γ2σ]−2uσsσLFNσ(Pμ−1)+uσsσLFNσ(∂σP′μ−1∂γσ)),μ⩾1. | (6.24) |
ϑ2,μ(γ,τ)=(χμ+ℏ)ϑ2,(μ−1)(γ,τ)−ℏ(1−χμ)Eσ(γσ)+ℏLFN−1σ(uσsσLFNσ[∂2σϑ2,(μ−1)(γ,τ)∂γ2σ]−2uσsσLFNσ(P″μ−1)+uσsσLFNσ(∂σP′μ−1∂γσ)),μ⩾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 μ are computed as follows:
ϑ1,1(γ,τ)=ℏEσ(γσ)τσΓ(1+σ),ϑ2,1(γ,τ)=ℏEσ(γσ)τσΓ(1+σ),ϑ1,2(γ,τ)=ℏ(1+ℏ)Eσ(γσ)τσΓ(1+σ)+ℏ2Eσ(γσ)τ2σΓ(1+2σ),ϑ2,2(γ,τ)=ℏ(1+ℏ)Eσ(γσ)τσΓ(1+σ)+ℏ2Eσ(γσ)τ2σΓ(1+2σ),ϑ1,3(γ,τ)=ℏ(1+ℏ)Eσ(γσ)((1+ℏ)τσΓ(1+σ)+ℏτ2σΓ(1+2σ))+ℏ2Eσ(γσ)((1+ℏ)τ2σΓ(1+2σ)+ℏτ3σΓ(1+3σ)),ϑ2,3(γ,τ)=ℏ(1+ℏ)Eσ(γσ)((1+ℏ)τσΓ(1+σ)+ℏτ2σΓ(1+2σ))+ℏ2Eσ(γσ)((1+ℏ)τ2σΓ(1+2σ)+ℏτ3σΓ(1+3σ)). | (6.26) |
Proceeding in the similar way, we get the remaining values for ϑ1,μ(γ,τ) and ϑ2,μ(γ,τ) for μ⩾4.
Setting the convergence-control parameter ℏ=−1, we attain the following values
ϑ1,1(γ,τ)=−Eσ(γσ)τσΓ(1+σ), |
ϑ2,1(γ,τ)=−Eσ(γσ)τσΓ(1+σ), |
ϑ1,2(γ,τ)=Eσ(γσ)τ2σΓ(1+2σ), |
ϑ2,2(γ,τ)=Eσ(γσ)τ2σΓ(1+2σ), |
ϑ1,3(γ,τ)=−Eσ(γσ)(τ3σΓ(1+3σ)), |
ϑ2,3(γ,τ)=−Eσ(γσ)(τ3σΓ(1+3σ)), | (6.27) |
⋮ |
and so on.
Proceeding in the same way, the rest of the terms of ϑ1,μ(γ,τ) and ϑ2,μ(γ,τ) for μ⩾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
ϑ1(γ,τ)=∞∑n=0ϑ1,n(γ,τ)=Eσ(γσ)(1−τσΓ(1+σ)+τ2σΓ(1+2σ)−τ3σΓ(1+3σ)+τ4σΓ(1+4σ)−⋯)=Eσ(γσ)Eσ(−τσ). | (6.28) |
Similarly,
ϑ2(γ,τ)=∞∑n=0ϑ2,n(γ,τ)=Eσ(γσ)Eσ(−τσ). | (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 ℏ=−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 σ=ln2/ln3. 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 ϑ1(γ,τ) and ϑ2(γ,τ) for the LFCHEs. Similarly, Figures 3 and 4 show the 3D surface graphics of coupled solutions ϑ1(γ,τ) and ϑ2(γ,τ) for LFCBEs. Here, γ and τ have been taken in the closed interval of 0 to 1. The nature of ϑ1(γ,τ) and ϑ2(γ,τ) have been explored with respect to γ and τ. The fractal solutions of the LFCHEs and LFCBEs show interesting characteristics for σ=ln2/ln3. The graphical presentation demonstrates that the computed solutions for the LFCHEs and LFCBEs are consistently dependent on the fractal order σ. Furthermore, the 3D figures drawn on Cantor sets indicate that the coupled solutions ϑ1(γ,τ) and ϑ2(γ,τ) 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 ϑ1(γ,τ) and ϑ2(γ,τ) 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] |
Brink AJ, Richards GA, Colombo G, et al. (2014) Multicomponent antibiotic substances produced by fermentation: Implications for regulatory authorities, critically ill patients and generics. Int J Antimicrob Ag 43: 1-6. https://doi.org/10.1016/j.ijantimicag.2013.06.013 ![]() |
[2] |
Poirel L, Jayol A, Nordmann P (2017) Polymyxins: Antibacterial activity, susceptibility testing, and resistance mechanisms encoded by plasmids or chromosomes. Clin Microbiol Rev 30: 557-596. https://doi.org/10.1128/cmr.00064-16 ![]() |
[3] |
Kaye KSK, Pogue JMP, Tran TB, et al. (2016) Agents of last resort: Polymyxin resistance. Infect Dis Clin N Am 30: 391-414. https://doi.org/10.1016/j.idc.2016.02.005 ![]() |
[4] |
Mohapatra SS, Dwibedy SK, Padhy I (2021) Polymyxins, the last-resort antibiotics: Mode of action, resistance emergence, and potential solutions. J Biosci 46: 85. https://doi.org/10.1007/s12038-021-00209-8 ![]() |
[5] |
Andrade FF, Silva D, Rodrigues A, et al. (2020) Colistin update on its mechanism of action and resistance, present and future challenges. Microorganisms 8: 1716. https://doi.org/10.3390/microorganisms8111716 ![]() |
[6] |
Li B, Yin F, Zhao X, et al. (2020) Colistin resistance gene mcr-1 mediates cell permeability and resistance to hydrophobic antibiotics. Front Microbiol 10: 3015. https://doi.org/10.3389/fmicb.2019.03015 ![]() |
[7] |
Rhouma M, Beaudry F, Letellier A (2016) Resistance to colistin: What is the fate for this antibiotic in pig production?. Int J Antimicrob Agents 48: 119-126. https://doi.org/10.1016/j.ijantimicag.2016.04.008 ![]() |
[8] |
Gharaibeh MH, Shatnawi SQ (2019) An overview of colistin resistance, mobilized colistin resistance genes dissemination, global responses, and the alternatives to colistin: A review. Vet World 12: 1735-1746. https://doi.org/10.14202/vetworld.2019.1735-1746 ![]() |
[9] |
El-Sayed Ahmed MAE, Zhong LL, Shen C, et al. (2020) Colistin and its role in the Era of antibiotic resistance: An extended review (2000–2019). Emerg Microbes Infect 9: 868-885. https://doi.org/10.1080/22221751.2020.1754133 ![]() |
[10] |
Gurjar M (2015) Colistin for lung infection: An update. J Intensive Care 3: 3. https://doi.org/10.1186/s40560-015-0072-9 ![]() |
[11] |
Abd El-Baky RM, Masoud SM, Mohamed DS, et al. (2020) Prevalence and some possible mechanisms of colistin resistance among multidrug-resistant and extensively drug-resistant Pseudomonas aeruginosa. Infect Drug Resist 2020: 323-332. https://doi.org/10.2147/IDR.S238811 ![]() |
[12] |
Falagas ME, Rafailidis PI, Matthaiou DK (2010) Resistance to polymyxins: Mechanisms, frequency and treatment options. Drug Resist Update 13: 132-138. https://doi.org/10.1016/j.drup.2010.05.002 ![]() |
[13] |
Kempf I, Jouy E, Chauvin C (2016) Colistin use and colistin resistance in bacteria from animals. Int J Antimicrob Ag 48: 598-606. https://doi.org/10.1016/j.ijantimicag.2016.09.016 ![]() |
[14] |
Osei Sekyere J, Govinden U, Bester LA, et al. (2016) Colistin and tigecycline resistance in carbapenemase-producing Gram-negative bacteria: Emerging resistance mechanisms and detection methods. J Appl Microbiol 121: 601-617. https://doi.org/10.1111/jam.13169 ![]() |
[15] |
Son SJ, Huang R, Squire CJ, et al. (2019) MCR-1: A promising target for structure-based design of inhibitors to tackle polymyxin resistance. Drug Discov Today 24: 206-216. https://doi.org/10.1016/j.drudis.2018.07.004 ![]() |
[16] |
MacNair CR, Stokes JM, Carfrae LA, et al. (2018) Overcoming mcr-1 mediated colistin resistance with colistin in combination with other antibiotics. Nat Commun 9: 458. https://doi.org/10.1038/s41467-018-02875-z ![]() |
[17] |
Liu YY, Wang Y, Walsh TR, et al. (2016) Emergence of plasmid-mediated colistin resistance mechanism MCR-1 in animals and human beings in China: A microbiological and molecular biological study. Lancet Infect Dis 16: 161-168. https://doi.org/10.1016/S1473-3099(15)00424-7 ![]() |
[18] |
Hamame A, Davoust B, Hasnaoui B, et al. (2022) Screening of colistin-resistant bacteria in livestock animals from France. Vet Res 53: 96. https://doi.org/10.1186/s13567-022-01113-1 ![]() |
[19] |
Hussein NH, Al-Kadmy IMS, Taha BM, et al. (2021) Mobilized colistin resistance (mcr) genes from 1 to 10: A comprehensive review. Mol Biol Rep 48: 2897-2907. https://doi.org/10.1007/s11033-021-06307-y ![]() |
[20] |
Ling Z, Yin W, Shen Z, et al. (2020) Epidemiology of mobile colistin resistance genes mcr-1 to mcr-9. J Antimicrob Chemoth 75: 3087-3095. https://doi.org/10.1093/jac/dkaa205 ![]() |
[21] |
Liu JH, Liu YY, Shen YB, et al. (2024) Plasmid-mediated colistin-resistance genes: mcr. Trends Microbiol 32: 365-378. https://doi.org/10.1016/j.tim.2023.10.006 ![]() |
[22] |
Schwarz S, Johnson AP (2016) Transferable resistance to colistin: A new but old threat. J Antimicrob Chemoth 71: 2066-2070. https://doi.org/10.1093/jac/dkw274 ![]() |
[23] |
El Ouazzani ZEB, Benaicha H, Reklaoui L, et al. (2024) First detection of colistin resistance encoding gene mcr-1 in clinical enterobacteriaceae isolates in Morocco. Iran J Med Microbiol 18: 33-40. https://doi.org/10.30699/ijmm.18.1.33 ![]() |
[24] |
Loayza-Villa F, Salinas L, Tijet N, et al. (2020) Diverse Escherichia coli lineages from domestic animals carrying colistin resistance gene mcr-1 in an Ecuadorian household. J Glob Antimicrob Re 22: 63-67. https://doi.org/10.1016/j.jgar.2019.12.002 ![]() |
[25] |
Pormohammad A, Nasiri MJ, Azimi T (2019) Prevalence of antibiotic resistance in Escherichia coli strains simultaneously isolated from humans, animals, food, and the environment: A systematic review and meta-analysis. Infect Drug Resist 12: 1181-1197. https://doi.org/10.2147/IDR.S201324 ![]() |
[26] |
Khairy RM, Mohamed ES, Abdel Ghany HM, et al. (2019) Phylogenic classification and virulence genes profiles of uropathogenic E. coli and diarrhegenic E. coli strains isolated from community acquired infections. Plos One 14: e0222441. https://doi.org/10.1371/journal.pone.0222441 ![]() |
[27] |
Aguirre-Sánchez JR, Valdez-Torres JB, Del Campo NC, et al. (2022) Phylogenetic group and virulence profile classification in Escherichia coli from distinct isolation sources in Mexico. Infect Genet Evol 106: 105380. https://doi.org/10.1016/j.meegid.2022.105380 ![]() |
[28] |
Kaper JB, Nataro JP, Mobley HLT (2004) Pathogenic Escherichia coli. Nat Rev Microbiol 2: 123-140. https://doi.org/10.1038/nrmicro818 ![]() |
[29] |
Chakraborty A, Saralaya V, Adhikari P, et al. (2015) Characterization of Escherichia coli phylogenetic groups associated with extraintestinal infections in south indian oopulation. Ann Med Health Sci Res 5: 241-246. https://doi.org/10.4103/2141-9248.160192 ![]() |
[30] |
Lemlem M, Aklilu E, Mohamed M, et al. (2023) Phenotypic and genotypic characterization of colistin-resistant Escherichia Coli with mcr-4, mcr-5, mcr-6, and mcr-9 genes from broiler chicken and farm environment. BMC Microbiol 23: 392. https://doi.org/10.1186/s12866-023-03118-y ![]() |
[31] |
Lim JY, Yoon JW, Hovde CJ (2010) A brief overview of Escherichia coli O157:H7 and its plasmid O157. (2010). J Microbiol Biotechnol 20: 5-14. https://doi.org/10.4014/jmb.0908.08007 ![]() |
[32] |
Xexaki A, Papadopoulos DK, Alvanou MV, et al. (2023) Prevalence of antibiotic resistant E. coli strains isolated from farmed broilers and hens in Greece, based on phenotypic and molecular analyses. Sustainability 15: 9421. https://doi.org/10.3390/su15129421 ![]() |
[33] |
Bélanger L, Garenaux A, Harel J, et al. (2011) Escherichia coli from animal reservoirs as a potential source of human extraintestinal pathogenic E. coli. FEMS Immunol Med Mic 62: 1-10. https://doi.org/10.1111/j.1574-695X.2011.00797.x ![]() |
[34] |
Persad AK, LeJeune JT (2014) Animal reservoirs of Shiga toxin-producing Escherichia coli. Microbiol Spectr 2: EHEC-0027-2014. https://doi.org/10.1128/microbiolspec.EHEC-0027-2014 ![]() |
[35] |
Mulder AC, van de Kassteele J, Heederik D, et al. (2020) Spatial effects of livestock farming on human infections with Shiga toxin-producing Escherichia coli O157 in small but densely populated regions: The case of the Netherlands. GeoHealth 4: e2020GH000276. https://doi.org/10.1029/2020GH000276 ![]() |
[36] |
Marchetti L, Buldain D, Castillo LC, et al. (2021) Pet and stray dogs as reservoirs of antimicrobial-resistant Escherichia coli. Int J Microbiol 2021: 6664557. https://doi.org/10.1155/2021/6664557 ![]() |
[37] |
Hata A, Fujitani N, Ono F, et al. (2022) Surveillance of antimicrobial-resistant Escherichia coli in Sheltered dogs in the Kanto Region of Japan. Sci Rep 12: 773. https://doi.org/10.1038/s41598-021-04435-w ![]() |
[38] |
Sun L, Meng N, Wang Z, et al. (2022) Genomic characterization of ESBL/AmpC-producing Escherichia coli in stray dogs sheltered in Yangzhou, China. Infect Drug Resist 15: 7741-7750. https://doi.org/10.2147/IDR.S397872 ![]() |
[39] |
Powell L, Reinhard C, Satriale D, et al. (2021) Characterizing unsuccessful animal adoptions: Age and breed predict the likelihood of return, reasons for return and post-return outcomes. Sci Rep 11: 8018. https://doi.org/10.1038/s41598-021-87649-2 ![]() |
[40] |
Guenther KM (2023) Understanding the durable myth of the “Irresponsible Pet Owner”. Contexts 22: 32-37. ![]() |
[41] | FEDIAFAnnual-report: Facts and figures 2022 (2023). https://europeanpetfood.comingsoon.site/wp-content/uploads/2023/06/FEDIAF_Annual-Report_2023_Facts-Figures.pdf |
[42] |
Albán MV, Núñez EJ, Zurita J, et al. (2020) Canines with different pathologies as carriers of diverse lineages of Escherichia coli harbouring mcr-1 and clinically relevant β-lactamases in central Ecuador. J Glob Antimicrob Re 22: 182-183. https://doi.org/10.1016/j.jgar.2020.05.017 ![]() |
[43] |
Zheng HH, Yu C, Tang XY, et al. (2023) Isolation, identification and antimicrobial resistance analysis of canine oral and intestinal Escherichia coli resistant to colistin. Int J Mol. Sci 24: 13428. https://doi.org/10.3390/ijms241713428 ![]() |
[44] |
Dalla Villa P, Kahn S, Stuardo L, et al. (2010) Free-roaming dog control among OIE-member countries. Prev Vet Med 97: 58-63. https://doi.org/10.1016/j.prevetmed.2010.07.001 ![]() |
[45] | Zorgios.The problem of the increase of stray dogs in Marathonas (in Greek). Official Government Gazette 5732/Β/28-12-2020/KYA2654/356295 (2020) . |
[46] | Giantsis IA, Beleri S, Balatsos G, et al. (2021) Sand fly (Diptera: Psychodidae: Phlebotominae) population dynamics and natural Leishmania infections in Attica region, Greece. J Med Entomol 58: 480-485. https://doi.org/10.1093/jme/tjaa158 |
[47] |
Tong YC, Zhang YN, Li PC, et al. (2023) Detection of antibiotic-resistant canine origin Escherichia coli and the synergistic effect of magnolol in reducing the resistance of multidrug-resistant Escherichia coli. Front Vet Sci 10: 1104812. https://doi.org/10.3389/fvets.2023.1104812 ![]() |
[48] |
Kumar S, Stecher G, Tamura K (2016) MEGA7: Molecular evolutionary genetics analysis version 7.0 for bigger datasets. Mol Biol Evol 33: 1870-1874. https://doi.org/10.1093/molbev/msw054 ![]() |
[49] | O'Neill J (2016) Tackling drug-resistant infections globally: Final report and recommendations. The review on antimicrobial resistance . Available from: https://amr-review.org/sites/default/files/160518_Final%20paper_with%20cover.pdf |
[50] |
Lencina FA, Bertona M, Stegmayer MA, et al. (2024) Prevalence of colistin-resistant Escherichia coli in foods and food-producing animals through the food chain: A worldwide systematic review and meta-analysis. Heliyon 10: e26579. https://doi.org/10.1016/j.heliyon.2024.e26579 ![]() |
[51] |
Tiseo K, Huber L, Gilbert M, et al. (2020) Global trends in antimicrobial use in food animals from 2017 to 2030. Antibiotics 9: 918. https://doi.org/10.3390/antibiotics9120918 ![]() |
[52] |
Zhao C, Wang Y, Mulchandani R, et al. (2024) Global surveillance of antimicrobial resistance in food animals using priority drugs maps. Nat Commun 15: 763. https://doi.org/10.1038/s41467-024-45111-7 ![]() |
[53] |
Sterneberg-van der Maaten T, Turner D, Van Tilburg J, et al. (2016) Benefits and risks for people and livestock of keeping companion animals: Searching for a healthy balance. J Comp Pathol 155: S8-S17. https://doi.org/10.1016/j.jcpa.2015.06.007 ![]() |
[54] |
Zhang XF, Doi Y, Huang X, et al. (2016) Possible transmission of mcr-1-Harboring Escherichia coli between companion animals and human. Emerg Infect Dis 22: 1679-1681. https://doi.org/10.3201/eid2209.160464 ![]() |
[55] |
Bhat AH (2021) Bacterial zoonoses transmitted by household pets and as reservoirs of antimicrobial resistant bacteria. Microb Pathogenesis 155: 104891. https://doi.org/10.1016/j.micpath.2021.104891 ![]() |
[56] |
Humphrey M, Larrouy-Maumus GJ, Furniss RCD, et al. (2021) Colistin resistance in Escherichia coli confers protection of the cytoplasmic but not outer membrane from the polymyxin antibiotic. Microbiology 167: 001104. https://doi.org/10.1099/mic.0.001104 ![]() |
[57] |
Poirel L, Madec J, Lupo A, et al. (2018) Antimicrobial resistance in Escherichia coli. Microbiol Spectrum 6: ARBA-0026-2017. https://doi.org/10.1128/microbiolspec.arba-0026-2017 ![]() |
[58] |
Puvača N, de Llanos Frutos R (2021) Antimicrobial resistance in Escherichia coli strains isolated from humans and pet animals. Antibiotics 10: 69. https://doi.org/10.3390/antibiotics10010069 ![]() |
[59] |
Hamame A, Davoust B, Cherak Z, et al. (2022) Mobile colistin resistance (mcr) genes in cats and dogs and their zoonotic transmission risks. Pathogens 11: 698. https://doi.org/10.3390/pathogens11060698 ![]() |
[60] |
Menezes J, Moreira da Silva J, Frosini SM, et al. (2022) mcr-1 colistin resistance gene sharing between Escherichia coli from cohabiting dogs and humans, Lisbon, Portugal, 2018 to 2020. Euro Surveill 27: 2101144. https://doi.org/10.2807/1560-7917.ES.2022.27.44.2101144 ![]() |
[61] |
Caneschi A, Bardhi A, Barbarossa A, et al. (2023) The use of antibiotics and antimicrobial resistance in veterinary medicine, a complex phenomenon: A narrative review. Antibiotics 12: 487. https://doi.org/10.3390/antibiotics12030487 ![]() |
[62] | Pomba C, Rantala M, Greko C, et al. (2017) Public health risk of antimicrobial resistance transfer from companion animals. J Antimicrob Chemoth 72: 957-968. https://doi.org/10.1093/jac/dkw481 |
[63] |
Joosten P, Ceccarelli D, Odent E, et al. (2020) Antimicrobial usage and resistance in companion animals: A cross-sectional study in Three European Countries. Antibiotics 9: 87. https://doi.org/10.3390/antibiotics9020087 ![]() |
[64] |
Lei L, Wang Y, Schwarz S, et al. (2017) mcr-1 in Enterobacteriaceae from Companion Animals, Beijing, China, 2012-2016. Emerg Infect Dis 23: 710-711. https://doi.org/10.3201/eid2304.161732 ![]() |
[65] |
Kobs VC, Valdez RE, de Medeiros F, et al. (2020) mcr-1-carrying Enterobacteriaceae isolated from companion animals in Brazil. Pesq Vet Bras 40: 690-695. https://doi.org/10.1590/1678-5150-pvb-6635 ![]() |
[66] |
Smith LM, Quinnell RJ, Goold C, et al. (2022) Assessing the impact of free-roaming dog population management through systems modelling. Sci Rep 12: 11452. https://doi.org/10.1038/s41598-022-15049-1 ![]() |
[67] |
Dănilă G, Simioniuc V, Duduman ML (2023) Research on the Ethology and diet of the stray dog population in the areas bordering the municipality of Suceava, Romania. Vet Sci 10: 188. https://doi.org/10.3390/vetsci10030188 ![]() |
[68] |
Abdulkarim A, Bin Goriman Khan MAK, Aklilu E (2021) Stray animal population control: methods, public health concern, ethics, and animal welfare issues. World's Vet J 11: 319-326. https://doi.org/10.54203/scil.2021.wvj44 ![]() |
[69] |
Lei L, Wang Y, He J, et al. (2021) Prevalence and risk analysis of mobile colistin resistance and extended-spectrum β-lactamase genes carriage in pet dogs and their owners: a population based cross-sectional study. Emerg Microbes Infect 10: 242-251. https://doi.org/10.1080/22221751.2021.1882884 ![]() |
[70] |
Lima T, Loureiro D, Henriques A, et al. (2022) Occurrence and biological cost of mcr-1-carrying plasmids co-harbouring beta-lactamase resistance genes in zoonotic pathogens from intensive animal production. Antibiotics 11: 1356. https://doi.org/10.3390/antibiotics11101356 ![]() |
[71] |
Ortega-Paredes D, Haro M, Leoro-Garzón P, et al. (2019) Multidrug-resistant Escherichia coli isolated from canine faeces in a public park in Quito, Ecuador. J Glob Antimicrob Re 18: 263-268. https://doi.org/10.1016/j.jgar.2019.04.002 ![]() |
[72] |
Buranasinsup S, Wiratsudakul A, Chantong B, et al. (2023) Prevalence and characterization of antimicrobial-resistant Escherichia coli isolated from veterinary staff, pets, and pet owners in Thailand. J Infect Public Health 16: 194-202. https://doi.org/10.1016/j.jiph.2023.11.006 ![]() |
[73] |
Gill GS, Singh BB, Dhand NK, et al. (2022) Stray dogs and public health: Population estimation in Punjab, India. Vet Sci 9: 75. https://doi.org/10.3390/vetsci9020075 ![]() |
1. | Raimund Bürger, Elvis Gavilán, Daniel Inzunza, Pep Mulet, Luis Miguel Villada, Implicit-Explicit Methods for a Convection-Diffusion-Reaction Model of the Propagation of Forest Fires, 2020, 8, 2227-7390, 1034, 10.3390/math8061034 | |
2. | Josephine Davies, Kamalini Lokuge, Kathryn Glass, Routine and pulse vaccination for Lassa virus could reduce high levels of endemic disease: A mathematical modelling study, 2019, 37, 0264410X, 3451, 10.1016/j.vaccine.2019.05.010 | |
3. | Alberto d’Onofrio, Malay Banerjee, Piero Manfredi, Spatial behavioural responses to the spread of an infectious disease can suppress Turing and Turing–Hopf patterning of the disease, 2020, 545, 03784371, 123773, 10.1016/j.physa.2019.123773 | |
4. | Raimund Bürger, Elvis Gavilán, Daniel Inzunza, Pep Mulet, Luis Miguel Villada, Exploring a Convection–Diffusion–Reaction Model of the Propagation of Forest Fires: Computation of Risk Maps for Heterogeneous Environments, 2020, 8, 2227-7390, 1674, 10.3390/math8101674 | |
5. | Yanni Xiao, Yunhu Zhang, Min Gao, Modeling hantavirus infections in mainland China, 2019, 360, 00963003, 28, 10.1016/j.amc.2019.05.009 | |
6. | Rinaldo M. Colombo, Elena Rossi, Well‐posedness and control in a hyperbolic–parabolic parasitoid–parasite system, 2021, 147, 0022-2526, 839, 10.1111/sapm.12402 | |
7. | Robert R. Parmenter, Gregory E. Glass, Hantavirus outbreaks in the American Southwest: Propagation and retraction of rodent and virus diffusion waves from sky-island refugia, 2022, 36, 0217-9792, 10.1142/S021797922140052X | |
8. | Juan Pablo Gutiérrez Jaraa, María Teresa Quezada, Modeling of hantavirus cardiopulmonary syndrome, 2022, 22, 07176384, e002526, 10.5867/medwave.2022.03.002526 | |
9. | Asep K. Supriatna, Herlina Napitupulu, Meksianis Z. Ndii, Bapan Ghosh, Ryusuke Kon, Chung-Min Liao, A Mathematical Model for Transmission of Hantavirus among Rodents and Its Effect on the Number of Infected Humans, 2023, 2023, 1748-6718, 1, 10.1155/2023/9578283 |
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 |