Research article

Optimal operation of flexible IES based on energy hub and demand-side response

  • Received: 25 March 2025 Revised: 24 May 2025 Accepted: 05 June 2025 Published: 16 June 2025
  • In this paper, we explored optimal operation strategies for integrated energy systems (IES) in electrolytic aluminum industrial parks, highlighting energy hubs (EH) as key to improving energy efficiency and operational flexibility. A review of IES research covers system modeling, optimization algorithms, and demand-side response (DR) strategies. By integrating EH into an energy coupling model with a DR framework, we analyzed the system's economic viability, constraints, and optimization approaches. A case study was performed to validate the model's effectiveness across scenarios. Our results demonstrated that the proposed framework significantly improves IES performance: When compared with single-energy-conversion equipment configurations, the EH-integrated model reduces total operating costs by 21.05%–38.02%. By incorporating a DR model tailored to electrolytic aluminum's rigid load characteristics, including load shifting and multi–energy substitution, the system achieves a 53.6%–62.1% reduction in wind/solar curtailment rates and shortens energy storage payback periods by 18.7%. Further analysis indicated that the DR-IES model can dynamically balance electrical and thermal loads, incentivize user participation, and enhance environmental benefits. Empirical results showed that this approach reduces total operating costs by 5.56% and narrows the peak-valley differences of electrical and thermal loads by 24.78% and 17.11%, respectively. This study provides a new paradigm for high-energy-consuming industries to achieve low–carbon transformation through the collaborative optimization of EH and DR, offering theoretical and practical guidance for energy management in industrial parks.

    Citation: Xin Jin, Ruoli Tang, Tingzhe Pan, Xin Li, Zongyi Wang, Chao Jiang, Rui Zhang. Optimal operation of flexible IES based on energy hub and demand-side response[J]. AIMS Energy, 2025, 13(3): 569-589. doi: 10.3934/energy.2025022

    Related Papers:

    [1] Nadiyah Hussain Alharthi, Abdon Atangana, Badr S. Alkahtani . Numerical analysis of some partial differential equations with fractal-fractional derivative. AIMS Mathematics, 2023, 8(1): 2240-2256. doi: 10.3934/math.2023116
    [2] Hasib Khan, Jehad Alzabut, Anwar Shah, Sina Etemad, Shahram Rezapour, Choonkil Park . A study on the fractal-fractional tobacco smoking model. AIMS Mathematics, 2022, 7(8): 13887-13909. doi: 10.3934/math.2022767
    [3] Abdon Atangana, Seda İğret Araz . Extension of Chaplygin's existence and uniqueness method for fractal-fractional nonlinear differential equations. AIMS Mathematics, 2024, 9(3): 5763-5793. doi: 10.3934/math.2024280
    [4] Manal Alqhtani, Khaled M. Saad . Numerical solutions of space-fractional diffusion equations via the exponential decay kernel. AIMS Mathematics, 2022, 7(4): 6535-6549. doi: 10.3934/math.2022364
    [5] Abdon Atangana, Ali Akgül . Analysis of a derivative with two variable orders. AIMS Mathematics, 2022, 7(5): 7274-7293. doi: 10.3934/math.2022406
    [6] Emile Franc Doungmo Goufo, Abdon Atangana . On three dimensional fractal dynamics with fractional inputs and applications. AIMS Mathematics, 2022, 7(2): 1982-2000. doi: 10.3934/math.2022114
    [7] Rahat Zarin, Amir Khan, Pushpendra Kumar, Usa Wannasingha Humphries . Fractional-order dynamics of Chagas-HIV epidemic model with different fractional operators. AIMS Mathematics, 2022, 7(10): 18897-18924. doi: 10.3934/math.20221041
    [8] Muhammad Farman, Ali Akgül, Kottakkaran Sooppy Nisar, Dilshad Ahmad, Aqeel Ahmad, Sarfaraz Kamangar, C Ahamed Saleel . Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel. AIMS Mathematics, 2022, 7(1): 756-783. doi: 10.3934/math.2022046
    [9] Amir Ali, Abid Ullah Khan, Obaid Algahtani, Sayed Saifullah . Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels. AIMS Mathematics, 2022, 7(8): 14975-14990. doi: 10.3934/math.2022820
    [10] Muhammad Aslam, Muhammad Farman, Hijaz Ahmad, Tuan Nguyen Gia, Aqeel Ahmad, Sameh Askar . Fractal fractional derivative on chemistry kinetics hires problem. AIMS Mathematics, 2022, 7(1): 1155-1184. doi: 10.3934/math.2022068
  • In this paper, we explored optimal operation strategies for integrated energy systems (IES) in electrolytic aluminum industrial parks, highlighting energy hubs (EH) as key to improving energy efficiency and operational flexibility. A review of IES research covers system modeling, optimization algorithms, and demand-side response (DR) strategies. By integrating EH into an energy coupling model with a DR framework, we analyzed the system's economic viability, constraints, and optimization approaches. A case study was performed to validate the model's effectiveness across scenarios. Our results demonstrated that the proposed framework significantly improves IES performance: When compared with single-energy-conversion equipment configurations, the EH-integrated model reduces total operating costs by 21.05%–38.02%. By incorporating a DR model tailored to electrolytic aluminum's rigid load characteristics, including load shifting and multi–energy substitution, the system achieves a 53.6%–62.1% reduction in wind/solar curtailment rates and shortens energy storage payback periods by 18.7%. Further analysis indicated that the DR-IES model can dynamically balance electrical and thermal loads, incentivize user participation, and enhance environmental benefits. Empirical results showed that this approach reduces total operating costs by 5.56% and narrows the peak-valley differences of electrical and thermal loads by 24.78% and 17.11%, respectively. This study provides a new paradigm for high-energy-consuming industries to achieve low–carbon transformation through the collaborative optimization of EH and DR, offering theoretical and practical guidance for energy management in industrial parks.



    Fractional calculus is a generalization of classical calculus and many researchers have paid attention to this science as they encounter many of these issues in the real world. Most of these issues do not have analytical exact solution. Which made many researchers interest and search in numerical and approximate methods to obtain solutions using these methods. There are many of these methods, such as the homotopy analysis [1,2,3,4], He's variational iteration method [5,6], Adomians decomposition method [7,8,9], Fourier spectral methods [10], finite difference schemes [11], collocation methods [12,13,14]. To find out more about the fractal calculus, refer to the following references [15,16]. More recently, a new concept was introduced for the fractional operator, as this operator has two orders, the first representing the fractional order, and the second representing the fractal dimension. In our work we aim to applied the idea of fractal-fractional derivative of orders β,k to a reaction-diffusion equation with q-th nonlinear. To this end [17], we replace the derivative with respect to t by the fractal-fractional derivatives power (FFP) law, the fractal-fractional exponential(FFE) law and the fractal-fractional Mittag-Leffler (FFM) law kernels which corresponds to the [18], Caputo-Fabrizio (CF) [19] and the Atangana-Baleanu (AB) [20] fractional derivatives, respectively. This topic has attracted many researchers and has been applied to research related to the real world, such as [21,22,23,24,25,26]. Some recent developments in the area of numerical techniques can be found in [27,28,29,30,31].

    Merkin and Needham [32] considered the reaction-diffusion travelling waves that can develop in a coupled system involving simple isothermal autocatalysis kinetics. They assumed that reactions took place in two separate and parallel regions, with, in I, the reaction being given by quadratic autocatalysis

    F+G2G(ratek1fg), (1.1)

    together with a linear decay step

    GH(ratek2g) (1.2)

    where f and g are the concentrations of reactant F and autocatalyst H, the ki(i=1,2) are the rate constants and H is some inert product of reaction. The reaction in region II was the quadratic autocatalytic step (1.1) only. The two regions were assumed to be coupled via a linear diffusive interchange of the autocatalytic species G. We shall consider a similar system as I, but with cubic autocatalysis

    F+2G3G(ratek3fg2) (1.3)

    together with a linear decay step

    GH(ratek4g). (1.4)

    For q-th autocatalytic, we have

    F+qG(q+1)G(ratek3fgq),1q2, (1.5)

    together with a linear decay step

    GH(rate k4g). (1.6)

    This yields to the following system

    η1t=2η1ξ2+ν(η2η1)η1ζq1, (1.7)
    ζ1t=2ζ1ξ2κζ1+η1ζq1, (1.8)
    η2t=2η2ξ2+ν(η1η2)η2ζq2, (1.9)
    ζ2t=2ζ2ξ2+η2ζq2 (1.10)

    where ν represents the couple between (I) and (II) and κ represents the strength of the auto-catalyst decay. For more details see [32]. Omitting the diffusion terms in the system (1.7)-(1.10), one has the following ordinary differential equations

    η1t=ν(η2η1)η1ζq1, (1.11)
    ζ1t=κζ1+η1ζq1, (1.12)
    η2t=ν(η1η2)η2ζq2, (1.13)
    ζ2t=η2ζq2. (1.14)

    Now we provide some basic definitions that be needed in this work. As for the theorems and proofs related to the three fractal-fractional operators, they are found in details in [17]. Thus we suffice in this work by constructing the algorithms and making the numerical simulations of the set of Eqs (1.7)-(1.10) with the three fractal-fractional operators.

    Definition 1. If η(t) is continuous and fractal differentiable on (a,b) of order k, then the fractal-fractional derivative of η(t) of order β in Riemann Liouville sense with the power law is given by [17]:

    {FFP}0Dβ,ktη(t)=1Γ(1β)ddtkt0(tτ)βη(τ)dτ,(0<β,k1), (1.15)

    and the fractal-fractional integral of η(t) is given by

    FFP0Iβ,ktη(t)=kΓ(β)t0τk1(tτ)β1η(τ)dτ. (1.16)

    Definition 2. If η(t) is continuous in the (a,b) and fractal differentiable on (a,b) with order k, then the fractal-fractional derivative of η(t) of order β in Riemann Liouville sense with the exponential decay kernel is given by [17]:

    FFE0Dβ,ktη(t)=M(β)1βddtkt0eβ1β(tτ)η(τ)dτ,(0<β,k1), (1.17)

    and the fractal-fractional integral of η(t) is given by

    FFE0Iβ,ktη(t)=(1β)ktk1M(β)η(t)+βkM(β)t0τk1η(τ)dτ (1.18)

    where M(β) is the normalization function such that M(0)=M(1)=1.

    Definition 3. If η(t) is continuous in the (a,b) and fractal differentiable on (a,b) with order k, then the fractal-fractional derivative of η(t) of order β in Riemann Liouville sense with the Mittag-Leffler type kernel is given by [17]:

    FFE0Dβ,ktη(t)=A(β)1βddtkt0Eβ(β1β(tτ))η(τ)dτ,(0<β,k1), (1.19)

    and the fractal-fractional integral of η(t) is given by

    FFE0Iβ,ktη(t)=(1β)ktk1A(β)η(t)+βkA(β)Γ(β)t0τk1(tτ)β1η(τ)dτ, (1.20)
    dη(t)dtk=limτtη(τ)η(t)τktk (1.21)

    where where A(β)=1β+βΓ(β) is a normalization function such that A(0)=A(1)=1.

    Our contribution to this paper is to construct the successive approximations and evaluate the numerical solutions of the FFRDE. These successive approximations allow us to study the behavior of numerical solutions based on power, exponential, and the Mittag-Leffler kernels. Also we can study the behavior of approximate solutions in the case of nonlinearity of the FFRDE in general. To our best knowledge, this is the first study of the FFRDE using fractal-fractional with these kernels. The importance of these results lies in the fact that they highlight the possibility of using these results for the benefit of chemical and physical researchers, by trying to link the numerical results of these mathematical models with the laboratory results. These results also contribute to the reliance on numerical results in the case of many models related to the real world, which often cannot find an analytical solution. The structure of this paper is summarized as follows: In sections, two, three and four, the FFRDE is presented with the three kernels that proposed in this work and construct the successive approximations. In section Five, numerical solutions for the FFRDE are discussed with a study of their behavior. Section Six the conclusion is presented.

    The new model is obtained by replacing the ordinary derivative with the the fractal-fractional derivative the power law kernel as [17]

    FFP0Dβtη1(t)=ν(η2(t)η1(t))η1(t)ζq1(t), (2.1)
    FFP0Dβtζ1(t)=κζ1(t)+η1(t)ζq1(t), (2.2)
    FFP0Dβtη2(t)=ν(η1(t)η2(t))η2(t)ζq2(t), (2.3)
    FFP0Dβtζ2(t)=η2(t)ζq2(t). (2.4)

    By following the procedure in [17], we can obtain the following successive approximations:

    η1(t)η1(0)=kΓ(β)t0τk1(tτ)β1φ1(η1,ζ1,η2,ζ2,τ)dτ, (2.5)
    ζ1(t)ζ2(0)=kΓ(β)t0τk1(tτ)β1φ2(η1,ζ1,η2,ζ2,τ)dτ, (2.6)
    η2(t)η3(0)=kΓ(β)t0τk1(tτ)β1φ3(η1,ζ1,η2,ζ2,τ)dτ, (2.7)
    ζ2(t)ζ2(0)=kΓ(β)t0τk1(tτ)β1φ4(η1,ζ1,η2,ζ2,τ)dτ (2.8)

    where

    φ1(η1,ζ1,η2,ζ2,τ)=(ν(η2(τ)η1(τ))η1(τ)ζq1(τ)), (2.9)
    φ2(η1,ζ1,η2,ζ2,τ)=(κζ1(τ)+η1(τ)ζq1(τ)), (2.10)
    φ3(η1,ζ1,η2,ζ2,τ)=(ν(η1(τ)η2(τ))η2(τ)ζq2(τ)), (2.11)
    φ4(η1,ζ1,η2,ζ2,τ)=η2(τ)ζq2(τ). (2.12)

    Equation (2.5)-(2.8) can be reformulated as

    η1(t)η1(0)=kΓ(β)nm=0tm+1tmτk1(tn+1τ)β1φ1(η1,ζ1,η2,ζ2,τ)dτ, (2.13)
    ζ1(t)ζ1(0)=kΓ(β)nm=0tm+1tmτk1(tn+1τ)β1φ2(η1(τ),ζ1(τ),η2(τ),ζ2(τ),τ)dτ, (2.14)
    η2(t)η2(0)=kΓ(β)nm=0tm+1tmτk1(tn+1τ)β1φ3(η1,ζ1,η2,ζ2,τ)dτ, (2.15)
    ζ2(t)ζ2(0)=kΓ(β)nm=0tm+1tmτk1(tn+1τ)β1φ4(η1,ζ1,η2,ζ2,τ)dτ. (2.16)

    Using the two-step Lagrange polynomial interpolation, we obtain

    η1(t)η1(0)=kΓ(β)nm=0tm+1tm(tn+1τ)β1Q1,m(τ)dτ, (2.17)
    ζ1(t)ζ1(0)=kΓ(β)nm=0tm+1tm(tn+1τ)β1Q2,m(τ)dτ, (2.18)
    η2(t)η2(0)=kΓ(β)nm=0tm+1tm(tn+1τ)β1Q3,m(τ)dτ, (2.19)
    ζ2(t)ζ2(0)=kΓ(β)nm=0tm+1tm(tn+1τ)β1Q4,m(τ)dτ, (2.20)

    where,

    Q1,m(τ)=τtm1tmtm1tk1mφ1(η1(τm),ζ1(τm),η2(τm),ζ2(τm),τm)τtmtmtm1×tk1m1φ1(η1(τm1),ζ1(τm1),η2(τm1),ζ2(τm1),τm1), (2.21)
    Q2,m(τ)=τtm1tmtm1tk1mφ2(η1(τm),ζ1(τm),η2(τm),ζ2(τm),τm)τtmtmtm1×tk1m1φ2(η1(τm1),ζ1(τm1),η2(τm1),ζ2(τm1),τm1), (2.22)
    Q3,m(τ)=τtm1tmtm1tk1mφ3(η1(τm),ζ1(τm),η2(τm),ζ2(τm),τm)τtmtmtm1×tk1m1φ3(η1(τm1),ζ1(τm1),η2(τm1),ζ2(τm1),τm1), (2.23)
    Q4,m(τ)=τtm1tmtm1tk1mφ4(η4(τm),ζ1(τm),η2(τm),ζ2(τm),τm)τtmtmtm1×tk1m1φ4(η1(τm1),ζ1(τm1),η2(τm1),ζ2(τm1),τm1). (2.24)

    These integrals are evaluated directly and the numerical solutions of (2.1)-(2.4) involving the FFP derivative are given by

    η1(tn+1)=η1(0)+khβΓ(β+2)nm=0tk1mφ1(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)tk1m1φ1(η1(τm1),ζ1(tm1),η2(tm1),ζ2(tm1),tm1)Ξ2(n,m)), (2.25)
    ζ1(tn+1)=ζ1(0)+khβΓ(β+2)nm=0tk1mφ2(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)tk1m1φ2(η1(τm1),ζ1(tm1),η2(tm1),ζ2(tm1),tm1)Ξ2(n,m)), (2.26)
    η2(tn+1)=η2(0)+khβΓ(β+2)nm=0tk1mφ3(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)tk1m1φ4(η1(τm1),ζ1(tm1),η2(tm1),ζ2(tm1),tm1)Ξ2(n,m)), (2.27)
    ζ2(tn+1)=ζ2(0)+khβΓ(β+2)nm=0tk1mφ4(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)tk1m1φ4(η1(τm1),ζ1(tm1),η2(tm1),ζ2(tm1),tm1)Ξ2(n,m)), (2.28)
    Ξ1(n,m)=((n+1m)β(nm+2+β)(nm)β×(nm+2+2β)), (2.29)
    Ξ2(n,m)=((n+1m)β+1(nm)β(nm+1+β)). (2.30)

    Considering the FFE derivative, we have from [17]

    FFE0Dβtη1(t)=ν(η2(t)η1(t))η1(t)ζq1(t), (3.1)
    FFE0Dβtζ1(t)=κζ1(t)+η1(t)ζq1(t), (3.2)
    FFE0Dβtη2(t)=ν(η1(t)η2(t))η2(t)ζq2(t), (3.3)
    FFE0Dβtζ2(t)=η2(t)ζq2(t). (3.4)

    For the successive approximations of the system (3.1)-(3.4), we follow the same procedures as in [17], we obtain

    η1(t)η1(0)=ktk1(1β)M(β)φ1(η1,ζ1,η2,ζ2,t)+βM(β)t0kτk1φ1(η1,ζ1,η2,ζ2,τ)dτ, (3.5)
    ζ1(t)ζ1(0)=ktk1(1β)M(β)φ2(η1,ζ1,η2,ζ2,t)+βM(β)t0kτk1φ2(η1,ζ1,η2,ζ2,τ)dτ, (3.6)
    η2(t)η2(0)=ktk1(1β)M(β)φ3(η1,ζ1,η2,ζ2,t)+βM(β)t0kτk1φ3(η1,ζ1,η2,ζ2,τ)dτ, (3.7)
    ζ2(t)ζ2(0)=ktk1(1β)M(β)φ4(η1,ζ1,η2,ζ2,t)+βM(β)t0kτk1φ4(η1,ζ1,η2,ζ2,τ)dτ. (3.8)

    Using t=tn+1 the following is established

    η1(tn+1)η1(0)=ktk1(1β)M(β)φ1(η1,ζ1,η2,ζ2,tn)+βM(β)tn+10kτk1φ1(η1,ζ1,η2,ζ2,τ)dτ, (3.9)
    ζ1(tn+1)ζ1(0)=ktk1(1β)M(β)φ2(η1,ζ1,η2,ζ2,tn)+βM(β)tn+10kτk1φ2(η1,ζ1,η2,ζ2,τ)dτ, (3.10)
    η2(tn+1)η2(0)=ktk1(1β)M(β)φ3(η1,ζ1,η2,ζ2,tn)+βM(β)tn+10kτk1φ3(η1,ζ1,η2,ζ2,τ)dτ, (3.11)
    ζ2(tn+1)ζ2(0)=ktk1(1β)M(β)φ4(η1,ζ1,η2,ζ2,tn)+βM(β)tn+10kτk1φ4(η1,ζ1,η2,ζ2,τ)dτ. (3.12)

    Further, we have the following:

    η1(tn+1)η1(tn)=ktk1n(1β)M(β)φ1(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ1(η1,ζ1,η2,ζ2,tn1)+βM(β)tn+1tnkτk1φ1(η1,ζ1,η2,ζ2,τ)dτ, (3.13)
    ζ1(tn+1)ζ1(tn)=ktk1n(1β)M(β)φ2(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ2(η1,ζ1,η2,ζ2,tn1)+βM(β)tn+1tnkτk1φ2(η1,ζ1,η2,ζ2,τ)dτ, (3.14)
    η2(tn+1)η2(tn)=ktk1n(1β)M(β)φ3(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ3(η1,ζ1,η2,ζ2,tn1)+βM(β)tn+1tnkτk1φ3(η1,ζ1,η2,ζ2,τ)dτ, (3.15)
    ζ2(tn+1)ζ2(tn)=ktk1n(1β)M(β)φ4(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ4(η1,ζ1,η2,ζ2,tn1)+βM(β)tn+1tnkτk1φ4(η1,ζ1,η2,ζ2,τ)dτ. (3.16)

    It follows from the Lagrange polynomial interpolation and integrating the following expressions:

    η1(tn+1)η1(tn)=ktk1n(1β)M(β)φ1(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ1(η1,ζ1,η2,ζ2,tn1)+khβ2M(β)×(3tk1nφ1(η1,ζ1,η2,ζ2,tn)tk1n1φ1(η1,ζ1,η2,ζ2,tn1), (3.17)
    ζ1(tn+1)ζ1(tn)=ktk1n(1β)M(β)φ2(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ2(η1,ζ1,η2,ζ2,tn1)+khβ2M(β)×(3tk1nφ2(η1,ζ1,η2,ζ2,tn)tk1n1φ2(η1,ζ1,η2,ζ2,tn1), (3.18)
    η2(tn+1)η2(tn)=ktk1n(1β)M(β)φ3(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ3(η1,ζ1,η2,ζ2,tn1)+khβ2M(β)×(3tk1nφ3(η1,ζ1,η2,ζ2,tn)tk1n1φ3(η1,ζ1,η2,ζ2,tn1), (3.19)
    ζ2(tn+1)ζ2(tn)=ktk1n(1β)M(β)φ4(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ4(η1,ζ1,η2,ζ2,tn1)+khβ2M(β)×(3tk1nφ4(η1,ζ1,η2,ζ2,tn)tk1n1φ4(η1,ζ1,η2,ζ2,tn1). (3.20)

    Finally, it is appropriate to write the successive approximations of the system (3.1)-(3.4) as follows:

    η1(tn+1)η1(tn)=ktk1n((1β)M(β)+3hβ2M(β))φ1(η1,ζ1,η2,ζ2,tn)ktk1n1((1β)M(β)+hβ2M(β))φ1(η1,ζ1,η2,ζ2,tn1), (3.21)
    ζ1(tn+1)ζ1(tn)=ktk1n((1β)M(β)+3hβ2M(β))φ2(η1,ζ1,η2,ζ2,tn)ktk1n1((1β)M(β)+hβ2M(β))φ2(η1,ζ1,η2,ζ2,tn1), (3.22)
    η2(tn+1)η2(tn)=ktk1n((1β)M(β)+3hβ2M(β))φ3(η1,ζ1,η2,ζ2,tn)ktk1n1((1β)M(β)+hβ2M(β))φ3(η1,ζ1,η2,ζ2,tn1), (3.23)
    ζ2(tn+1)ζ2(tn)=ktk1n((1β)M(β)+3hβ2M(β))φ4(η1,ζ1,η2,ζ2,tn)ktk1n1((1β)M(β)+hβ2M(β))φ4(η1,ζ1,η2,ζ2,tn1). (3.24)

    Considering the FFM derivative, we have [18]

    FFM0Dβtη1(t)=ν(η2(t)η1(t))η1(t)ζq1(t), (4.1)
    FFM0Dβtζ1(t)=κζ1(t)+η1(t)ζq1(t), (4.2)
    FFM0Dβtη2(t)=ν(η1(t)η2(t))η2(t)ζq2(t), (4.3)
    FFM0Dβtζ2(t)=η2(t)ζq2(t). (4.4)

    Also, for this system (4.1)-(4.4), we follow the same treatment that was done in [17] to obtain the successive approximate solutions as follows:

    η1(t)η1(0)=ktk1(1β)A(β)φ1(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)t0kτk1(tτ)β1φ1(η1,ζ1,η2,ζ2,τ)dτ, (4.5)
    ζ1(t)ζ1(0)=ktk1(1β)A(β)φ2(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)t0kτk1(tτ)β1φ2(η1,ζ1,η2,ζ2,τ)dτ, (4.6)
    η2(t)η2(0)=ktk1(1β)A(β)φ3(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)t0kτk1(tτ)β1φ3(η1,ζ1,η2,ζ2,τ)dτ, (4.7)
    ζ2(t)ζ2(0)=ktk1(1β)A(β)φ4(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)t0kτk1(tτ)β1φ4(η1,ζ1,η2,ζ2,τ)dτ. (4.8)

    At tn+1 we obtain the following

    η1(tn+1)η1(0)=ktk1n(1β)A(β)φ1(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)tn+10kτk1(tn+1τ)β1φ1(η1,ζ1,η2,ζ2,τ)dτ, (4.9)
    ζ1(tn+1)ζ1(0)=ktk1n(1β)A(β)φ2(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)tn+10kτk1(tn+1τ)β1φ2(η1,ζ1,η2,ζ2,τ)dτ, (4.10)
    η2(tn+1)η2(0)=ktk1n(1β)A(β)φ3(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)tn+10kτk1(tn+1τ)β1φ3(η1,ζ1,η2,ζ2,τ)dτ, (4.11)
    ζ2(tn+1)ζ2(0)=ktk1n(1β)A(β)φ4(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)tn+10kτk1(tn+1τ)β1φ4(η1,ζ1,η2,ζ2,τ)dτ, (4.12)

    The integrals involving in (4.9)-(4.12) can be approximated as:

    η1(tn+1)η1(0)=ktk1n(1β)A(β)φ1(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)nm=0tm+1tmkτk1(tn+1τ)β1φ1(η1,ζ1,η2,ζ2,τ)dτ, (4.13)
    ζ1(tn+1)ζ1(0)=ktk1n(1β)A(β)φ2(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)nm=0tm+1tmkτk1(tn+1τ)β1φ2(η1,ζ1,η2,ζ2,τ)dτ, (4.14)
    η2(tn+1)η2(0)=ktk1n(1β)A(β)φ3(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)nm=0tm+1tmkτk1(tn+1τ)β1φ3(η1,ζ1,η2,ζ2,τ)dτ, (4.15)
    ζ2(tn+1)ζ2(0)=ktk1n(1β)A(β)φ4(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)nm=0tm+1tmkτk1(tn+1τ)β1φ4(η1,ζ1,η2,ζ2,τ)dτ. (4.16)

    The following numerical schemes after approximating the expressions τk1φi(η1,ζ1,η2,ζ2,τ),i=1,2,3,4 in the interval [tm,tm+1] in (4.13)-(4.16) are given by

    η1(tn+1)η1(0)=ktk1n(1β)A(β)φ1(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+khβA(β)Γ(β+2)nm=0[tk1mφ1(η1(tm),ζ1(tm),η2(tm),ζ2(tm),(tm))Ξ1(n,m)tk1m1φ1(η1(tm1),ζ1(tm1),η2(tm1),ζ2(tm1),(tm1))Ξ2(n,m)], (4.17)
    ζ1(tn+1)ζ1(0)=ktk1n(1β)A(β)φ2(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+khβA(β)Γ(β+2)nm=0[tk1mφ2(η1(tm),ζ1(tm),η2(tm),ζ2(tm),(tm))Ξ1(n,m)tk1m1φ2(η1(tm1),ζ1(tm1),η2(tm1),ζ2(tm1),(tm1))Ξ2(n,m)], (4.18)
    η2(tn+1)η2(0)=ktk1n(1β)A(β)φ3(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+khβA(β)Γ(β+2)nm=0[tk1mφ3(η1(tm),ζ1(tm),η2(tm),ζ2(tm),(tm))Ξ1(n,m)tk1m1φ3(η1(tm1),ζ1(tm1),η2(tm1),ζ2(tm1),(tm1))Ξ2(n,m)], (4.19)
    ζ2(tn+1)ζ2(0)=ktk1n(1β)A(β)φ4(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+khβA(β)Γ(α+2)nm=0[tk1mφ4(η1(tm),ζ1(tm),η2(tm),ζ2(tm),(tm))Ξ1(n,m)tk1m1φ4(η1(tm1),ζ1(tm1),η2(tm1),ζ2(tm1),(tm1))Ξ2(n,m)]. (4.20)

    In this section, we study in detail the effect of the non-linear term in general, as well as the effect of the fractal-fractional order on the numerical solutions that we obtained by using successive approximations in the above sections. First we begin by satisfying the effective of the numerical solutions of the proposed system when β=1 and k=1.

    We compare only for the power kernel with a known numerical method which is the finite differences method. This is because all numerical solutions based on the three fractal-fractional operators that presented in this paper are very close each other when β=1 and k=1. Figure 1 illustrates the comparison between numerical solutions (2.25)-(2.28) and numerical solutions computed by using the finite differences method with k and β. The parameters that used are γ=0.4,κ=0.004,h=0.02. From this figure we note that an excellent agreement. And the accurate is increasing as we take small h. From, Figure 1(a) and 1(c), we can see, that the profiles for η1 and η2 are very similar, but the profiles of ζ1 and ζ2 are more distinct with ζ2>ζ2. For Figure 1(b), the profiles of ζ1 and ζ2 are very close than in Figure 1(a) and 1(c), also for ζ1 and ζ2. Figures 2 and 3 show that the behavior of the approximate solutions based on FFP, FFE and FFM, when the degree of the non-linear term is cubic and for different values of k and β. For the parameters γ and κ, we fixed them in all computations. The remain parameters are the same as in Figure 1. Similarly, in Figures 4 and 5, the approximate solutions are plotted in the case of a non-linear with quadratic degree and for different values of k and β. Finally in Figures 6 and 7, the approximate solutions are shown in the case of non-linear with fractional order and for different values for k and β. For the Figures 2 and 3 which the nonlinear is cubic, all the profiles are distinct. Similarly with Figures 6 and 7 when the nonlinear is quadratic. From Figures 4 and 5, we can see in the case of fraction non-linear, the profiles of η1 and η2 are very close to each other than the profiles of ζ1 and ζ2.

    Figure 1.  Comparison between the numerical solutions (2.25)-(2.28) and numerical based on finite difference methods for β=1,k=1,γ=0.4,κ=0.001,h=0.01.(a)q=2;(b)q=1;(c)q=1.8; (Green solid color: Numerical solutions (2.25)-(2.28); Red dashed color: FDM).
    Figure 2.  Graph of the numerical solutions with q=2 for β=0.8,k=1,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).
    Figure 3.  Graph of the numerical solutions with q=2 for β=0.7,k=0.8,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).
    Figure 4.  Graph of the numerical solutions with q=1 for β=0.8,k=1,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).
    Figure 5.  Graph of the numerical solutions with q=1 for β=0.7,k=0.8,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).
    Figure 6.  Graph of the numerical solutions with q=1.8 for β=0.8,k=1,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).
    Figure 7.  Graph of the numerical solutions with q=1.8 for β=0.7,k=0.8,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).

    In this paper, numerical solutions of the of the fractal-fractional reaction diffusion equations with general nonlinear have been studied. We introduced the FFRDE in three instances of fractional derivatives based on power, exponential, and Mittag-Leffler kernels. After that, we used the fundamental fractional calculus with the help of Lagrange polynomial functions. We obtained the iterative and approximate formulas in the three cases. We studied the effect of the non-linear term order, in the case of cubic, quadratic, and fractional for different values of the fractal-fractional derivative order. The accuracy of the numerical solutions in the classic case of the FFRDE was tested in the case of power kernel, where all the numerical solutions in the classic case of integer order coincide to each other, and the comparison result has excellent agreement. In all calculations was used the Mathematica Program Package.

    The authors would like to express their Gratitudes to the ministry of education and the deanship of scientific research-Najran University-Kingdom of Saudi Arabia for their financial and Technical support under code number (NU/ESCI/17/025).

    The authors declare that there is no conflict of interests regarding the publication of this paper.



    [1] Li YZ, Wang D, Jia HJ, et al. (2023) Research on the diversity modeling and typical applicability of energy hubs in integrated energy systems. Integr Intell Energy 45: 22–29. https://doi.org/10.3969/j.issn.2097-0706.2023.07.003 doi: 10.3969/j.issn.2097-0706.2023.07.003
    [2] Gao X, Lin H, Jing D, et al. (2025) A novel framework for optimal design of solar-powered integrated energy system considering long timescale characteristics. Energy 325: 136137. https://doi.org/10.1016/j.energy.2025.136137 doi: 10.1016/j.energy.2025.136137
    [3] Ao X, Zhang J, Yan R, et al. (2025) More flexibility and waste heat recovery of a combined heat and power system for renewable consumption and higher efficiency. Energy 315: 134392. https://doi.org/10.1016/j.energy.2025.134392 doi: 10.1016/j.energy.2025.134392
    [4] Dai ZK, Zeng G, Shi KJ, et al. (2023) A multi-energy hub load-source coordination optimization method considering the benefits of energy consumption. Sci, Technol Eng 23: 14603–14608. https://doi.org/10.12404/j.issn.1671-1815.2211286. doi: 10.12404/j.issn.1671-1815.2211286
    [5] Li HW, Jing HJ, Wu L, et al. (2023) Optimization operation of electric thermal grids based on the variable energy efficiency of energy hubs. J Zhengzhou Univ 44: 76–83. https://doi.org/10.13705/j.issn.1671-6833.2023.04.015. doi: 10.13705/j.issn.1671-6833.2023.04.015
    [6] Wang L, Xie Q, Sun L (2023) Optimization operation of energy hubs considering energy storage systems. J Zhengzhou Univ 51: 1–6. https://doi.org/10.16109/j.cnki.jldl.2023.06.012 doi: 10.16109/j.cnki.jldl.2023.06.012
    [7] Wang MY, Wang RQ, Liu JY, et al. (2022) Operation optimization for park with integrated energy system based on integrated demand response. Energy Rep 8: 249–259. https://doi.org/10.1016/j.egyr.2022.05.060 doi: 10.1016/j.egyr.2022.05.060
    [8] Guo ZH, Zhang R, Wang L, et al. (2021) Optimal operation of regional integrated energy system considering demand response. Appl Therm Eng 191: 18. https://doi.org/10.1016/j.applthermaleng.2021.116860 doi: 10.1016/j.applthermaleng.2021.116860
    [9] Li P, Zhang F, Ma XY, et al. (2021) Multi-time scale economic optimization dispatch of the park integrated energy system. Front Energy Res 9: 12. https://doi.org/10.3389/fenrg.2021.743619 doi: 10.3389/fenrg.2021.743619
    [10] Xu CS, Dong SF, Zhang SP, et al. (2021) Centralized-distributed integrated demand response method for industrial parks. Power Syst Technol 45: 489–497. https://doi.org/10.13335/j.1000-3673.pst.2020.0945 doi: 10.13335/j.1000-3673.pst.2020.0945
    [11] Chong Z, Yang L, Jiang Y, et al. (2024) Hybrid-timescale optimal dispatch strategy for electricity and heat integrated energy system considering integrated demand response. Renewable Energy 232: 121123. https://doi.org/10.1016/j.renene.2024.121123 doi: 10.1016/j.renene.2024.121123
    [12] Duan J, Tian Q, Liu F, et al. (2024) Optimal scheduling strategy with integrated demand response based on stepped incentive mechanism for integrated electricity-gas energy system. Energy 313: 133689. https://doi.org/10.1016/j.energy.2024.133689 doi: 10.1016/j.energy.2024.133689
    [13] Nourizadeh H, Nazar MS (2024) Customer-oriented scheduling of active distribution system considering integrated demand response programs and multi-carrier energy hubs. J Clean Prod 447: 141308. https://doi.org/10.1016/j.jclepro.2024.141308 doi: 10.1016/j.jclepro.2024.141308
    [14] Wang M, Zheng JH, Li ZG, et al. (2022) Multi-attribute decision analysis for optimal design of park-level integrated energy systems based on load characteristics. Energy 254: 23. https://doi.org/10.1016/j.energy.2022.124379 doi: 10.1016/j.energy.2022.124379
    [15] Liu ZF, Zhao SX, Luo XF, et al. (2025) Two-layer energy dispatching and collaborative optimization of regional integrated energy system considering stakeholders game and flexible load management. Appl Energy 379: 124918. https://doi.org/10.1016/j.apenergy.2024.124918 doi: 10.1016/j.apenergy.2024.124918
    [16] Liu Y, Zheng R, Shen R, et al. (2025) Research on capacity configuration optimization of integrated energy system by integrating energy hub and response surface methodology. Energy 136348. https://doi.org/10.1016/j.energy.2025.136348
    [17] Li S, Zhu J, Dong H, et al. (2024) Multi-time-scale energy management of renewable microgrids considering grid-friendly interaction. Appl Energy 367: 123428. https://doi.org/10.1016/j.apenergy.2024.123428 doi: 10.1016/j.apenergy.2024.123428
    [18] Fan J, Yan R, He Y, et al. (2025) Stochastic optimization of combined energy and computation task scheduling strategies of hybrid system with multi-energy storage system and data center. Renewable Energy 242: 122466. https://doi.org/10.1016/j.renene.2025.122466 doi: 10.1016/j.renene.2025.122466
    [19] Zhao Z, Xu H, Bao G (2025) Study on energy resource-project mode-load demand chain flexibility adaptation of park-level integrated energy systems. Energy 320: 135246. https://doi.org/10.1016/j.energy.2025.135246 doi: 10.1016/j.energy.2025.135246
    [20] Xie T, Ma K, Zhang G, et al. (2024) Optimal scheduling of multi-regional energy system considering demand response union and shared energy storage. Energy Strateg Rev 53: 101413. https://doi.org/10.1016/j.esr.2024.101413 doi: 10.1016/j.esr.2024.101413
    [21] Yu J, Chen L, Wang Q, et al. (2024) Towards sustainable regional energy solutions: An optimized operational model for integrated energy systems with price-responsive planning. Energy 305: 132278. https://doi.org/10.1016/j.energy.2024.132278 doi: 10.1016/j.energy.2024.132278
    [22] Huang A, Bi Q, Dai L (2025) Integrated economic and environmental optimization for industrial consumers: A dual-objective approach with multi-carrier energy systems and fuzzy decision-making. Energy 324: 135787. https://doi.org/10.1016/j.energy.2025.135787 doi: 10.1016/j.energy.2025.135787
    [23] Jordehi AR, Mansouri SA, Tostado-Véliz M, et al. (2024) Industrial energy hubs with electric, thermal and hydrogen demands for resilience enhancement of mobile storage-integrated power systems. Int J Hydrogen Energy 50: 77–91. https://doi.org/10.1016/j.ijhydene.2023.07.205 doi: 10.1016/j.ijhydene.2023.07.205
    [24] Sepehrzad R, Al-Durra A, Anvari-Moghaddam A, et al. (2025) Short-term and probability scenario-oriented energy management of integrated energy distribution systems with considering energy market interactions and end-user participation. Energy 322: 135691. https://doi.org/10.1016/j.energy.2025.135691 doi: 10.1016/j.energy.2025.135691
    [25] Zhang XP (2019) Optimal expansion planning of energy hub with multiple energy infrastructures. South Energy Constr 6: 6–12. https://doi.org/10.16516/j.gedi.issn2095-8676.2019.04.002 doi: 10.16516/j.gedi.issn2095-8676.2019.04.002
    [26] Wang R, Cheng S, Xu JY, et al. (2023) Multi-time scale optimal scheduling strategy of energy hub based on master-slave game and hybrid demand response. Electr Power Autom Equip 43: 32–40. https://doi.org/10.16081/j.epae.202204055 doi: 10.16081/j.epae.202204055
    [27] Zhu XP, Yao XY, Fu Q, et al. (2022) Optimized operation mode considering cooperation among energy hubs. Modern Electric Power 39: 397–405. https://doi.org/10.19725/j.cnki.1007-2322.2021.0168 doi: 10.19725/j.cnki.1007-2322.2021.0168
    [28] Song TL (2020) Study on integrated port energy system considering demand response. South Univ https://doi.org/10.1016/j.ijepes.2019.105654
    [29] Wang LM, Liu XM, Li Y, et al. (2024) Low-carbon optimal dispatch of integrated energy system considering demand response under the tiered carbon trading mechanism. Electr Power Constr 45: 102–114. https://doi.org/10.12204/j.issn.1000-7229.2024.02.009 doi: 10.12204/j.issn.1000-7229.2024.02.009
    [30] Chen JP, Hu ZJ, Chen JB, et al. (2021) Optimal dispatch of integrated energy system considering ladder-type carbon trading and flexible double response of supply and demand. High Voltage Eng 47: 3094–3106. https://doi.org/10.13336/j.1003-6520.hve.20211094 doi: 10.13336/j.1003-6520.hve.20211094
    [31] Yang HZ, Li ML, Jiang ZY, et al. (2020) Optimal operation of regional integrated energy system considering demand side electricity heat and natural-gas loads response. Power Syst Prot Control 48: 30–37. https://doi.org/10.19783/j.cnki.pspc.190774 doi: 10.19783/j.cnki.pspc.190774
    [32] Chen JP, Hu ZJ, Chen YG, et al. (2021) Thermoelectric optimization of integrated energy system considering ladder-type carbon trading mechanism and electric hydrogen production. Electr Power Autom Equip 41: 48–55. https://doi.org/10.16081/j.epae.202109032 doi: 10.16081/j.epae.202109032
    [33] Li ZM, Zhang F, Liang J, et al. (2015). Optimization on microgrid with combined heat and power system. Proc CSEE 35: 3569–3576. https://doi.org/10.13334/j.0258-8013.pcsee.2015.14.011 doi: 10.13334/j.0258-8013.pcsee.2015.14.011
  • This article has been cited by:

    1. Jia-Bao Liu, Saad Ihsan Butt, Jamshed Nasir, Adnan Aslam, Asfand Fahad, Jarunee Soontharanon, Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator, 2022, 7, 2473-6988, 2123, 10.3934/math.2022121
    2. Hasib Khan, Jehad Alzabut, Anwar Shah, Sina Etemad, Shahram Rezapour, Choonkil Park, A study on the fractal-fractional tobacco smoking model, 2022, 7, 2473-6988, 13887, 10.3934/math.2022767
    3. Hasib Khan, Muhammad Ibrahim, Abdel-Haleem Abdel-Aty, M. Motawi Khashan, Farhat Ali Khan, Aziz Khan, A fractional order Covid-19 epidemic model with Mittag-Leffler kernel, 2021, 148, 09600779, 111030, 10.1016/j.chaos.2021.111030
    4. Krunal B. Kachhia, Chaos in fractional order financial model with fractal–fractional derivatives, 2023, 7, 26668181, 100502, 10.1016/j.padiff.2023.100502
    5. Hari M. Srivastava, Khaled Mohammed Saad, Walid M. Hamanah, Certain New Models of the Multi-Space Fractal-Fractional Kuramoto-Sivashinsky and Korteweg-de Vries Equations, 2022, 10, 2227-7390, 1089, 10.3390/math10071089
    6. Hasnaa H Alzahrani, Marco Lucchesi, Kassem Mustapha, Olivier P Le Maître, Omar M Knio, Bayesian calibration of order and diffusivity parameters in a fractional diffusion equation, 2021, 5, 2399-6528, 085014, 10.1088/2399-6528/ac1507
    7. Jagdev Singh, Arpita Gupta, Dumitru Baleanu, On the analysis of an analytical approach for fractional Caudrey-Dodd-Gibbon equations, 2022, 61, 11100168, 5073, 10.1016/j.aej.2021.09.053
    8. Xiaojun Zhou, Yue Dai, A spectral collocation method for the coupled system of nonlinear fractional differential equations, 2022, 7, 2473-6988, 5670, 10.3934/math.2022314
    9. Nauman Ahmed, Ali Raza, Ali Akgül, Zafar Iqbal, Muhammad Rafiq, Muhammad Ozair Ahmad, Fahd Jarad, New applications related to hepatitis C model, 2022, 7, 2473-6988, 11362, 10.3934/math.2022634
    10. Esra Karatas Akgül, Wasim Jamshed, Kottakkaran Sooppy Nisar, S.K. Elagan, Nawal A. Alshehri, On solutions of gross domestic product model with different kernels, 2022, 61, 11100168, 1289, 10.1016/j.aej.2021.06.067
    11. Rubayyi T. Alqahtani, Shabir Ahmad, Ali Akgül, On Numerical Analysis of Bio-Ethanol Production Model with the Effect of Recycling and Death Rates under Fractal Fractional Operators with Three Different Kernels, 2022, 10, 2227-7390, 1102, 10.3390/math10071102
    12. A. DLAMINI, EMILE F. DOUNGMO GOUFO, M. KHUMALO, CHAOTIC BEHAVIOR OF MODIFIED STRETCH–TWIST–FOLD FLOW UNDER FRACTAL-FRACTIONAL DERIVATIVES, 2022, 30, 0218-348X, 10.1142/S0218348X22402071
    13. Anwar Zeb, Abdon Atangana, Zareen A. Khan, Salih Djillali, A robust study of a piecewise fractional order COVID-19 mathematical model, 2022, 61, 11100168, 5649, 10.1016/j.aej.2021.11.039
    14. Shabir Ahmad, Aman Ullah, Ali Akgül, Manuel De la Sen, A study of fractional order Ambartsumian equation involving exponential decay kernel, 2021, 6, 2473-6988, 9981, 10.3934/math.2021580
    15. Raheel Kamal, Gul Rahmat, Kamal Shah, Ricardo Escobar, On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations, 2021, 2021, 1563-5147, 1, 10.1155/2021/4640467
    16. Kaihong Zhao, Shuang Ma, Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses, 2022, 7, 2473-6988, 3169, 10.3934/math.2022175
    17. Kamsing Nonlaopon, Muhammad Naeem, Ahmed M. Zidan, Rasool Shah, Ahmed Alsanad, Abdu Gumaei, Muhammad Imran Asjad, Numerical Investigation of the Time-Fractional Whitham–Broer–Kaup Equation Involving without Singular Kernel Operators, 2021, 2021, 1099-0526, 1, 10.1155/2021/7979365
    18. Saima Rashid, Rehana Ashraf, Ebenezer Bonyah, Azhar Hussain, On Analytical Solution of Time-Fractional Biological Population Model by means of Generalized Integral Transform with Their Uniqueness and Convergence Analysis, 2022, 2022, 2314-8888, 1, 10.1155/2022/7021288
    19. Khadija Tul Kubra, Rooh Ali, Modeling and analysis of novel COVID-19 outbreak under fractal-fractional derivative in Caputo sense with power-law: a case study of Pakistan, 2023, 2363-6203, 10.1007/s40808-023-01747-w
    20. ZAREEN A. KHAN, KAMAL SHAH, BAHAAELDIN ABDALLA, THABET ABDELJAWAD, A NUMERICAL STUDY OF COMPLEX DYNAMICS OF A CHEMOSTAT MODEL UNDER FRACTAL-FRACTIONAL DERIVATIVE, 2023, 31, 0218-348X, 10.1142/S0218348X23401813
    21. Kamal Shah, Thabet Abdeljawad, On complex fractal-fractional order mathematical modeling of CO 2 emanations from energy sector, 2024, 99, 0031-8949, 015226, 10.1088/1402-4896/ad1286
    22. Samy A. Abdelhafeez, Anas A. M. Arafa, Yousef H. Zahran, Ibrahim S. I. Osman, Moutaz Ramadan, Adapting Laplace residual power series approach to the Caudrey Dodd Gibbon equation, 2024, 14, 2045-2322, 10.1038/s41598-024-57780-x
    23. Krunal B. Kachhia, Prit P. Parmar, A novel fractional mask for image denoising based on fractal–fractional integral, 2024, 11, 26668181, 100833, 10.1016/j.padiff.2024.100833
    24. Harpreet Kaur, Amanpreet Kaur, Palwinder Singh, Scale-3 Haar wavelet-based method of fractal-fractional differential equations with power law kernel and exponential decay kernel, 2024, 13, 2192-8029, 10.1515/nleng-2022-0380
    25. Muhammad Farman, Changjin Xu, Perwasha Abbas, Aceng Sambas, Faisal Sultan, Kottakkaran Sooppy Nisar, Stability and chemical modeling of quantifying disparities in atmospheric analysis with sustainable fractal fractional approach, 2025, 142, 10075704, 108525, 10.1016/j.cnsns.2024.108525
    26. Ashish Rayal, System of fractal-fractional differential equations and Bernstein wavelets: a comprehensive study of environmental, epidemiological, and financial applications, 2025, 100, 0031-8949, 025236, 10.1088/1402-4896/ada592
    27. Mashael M. AlBaidani, Abdul Hamid Ganie, Adnan Khan, Fahad Aljuaydi, An Approximate Analytical View of Fractional Physical Models in the Frame of the Caputo Operator, 2025, 9, 2504-3110, 199, 10.3390/fractalfract9040199
  • Reader Comments
  • © 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(260) PDF downloads(71) Cited by(0)

Figures and Tables

Figures(9)  /  Tables(4)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog