Processing math: 100%
Research article Topical Sections

Magnetic fraction from phosphate mining tailings as heterogeneous catalyst for biodiesel production through transesterification reaction of triacylglycerols in bio-oil

  • Received: 18 July 2017 Accepted: 13 October 2017 Published: 19 September 2017
  • Biodiesel is an interesting alternative fuel for complementing or even completely replacing the mineral diesel. It is industrially obtained through the transesterification reaction of triacylglycerol in bio-oils with alcohol of short molecular chain, to produce the corresponding mixture of esters of fatty acids (biodiesel). Mineral rejects from mining usually constitute a major environmental and economic problem. Herein, it is described the study devoted to evaluate the chemical efficiency of the transesterification reaction of triacylglycerols in soybean oil with methanol, catalyzed by a material based on the magnetic fraction from rejects of a phosphate ore being commercially exploited in Tapira, Minas Gerais, Brazil. The magnetite-containing material from those mining tailings was used to form a new heterogeneous catalyst, by first mixing it with a commercial synthetic calcium oxide. The mixture was then heated at 200 ℃ for 4 h. The magnetic, crystallographic and 57Fe hyperfine structures of the resulting catalyst were assessed by VSM magnetometer, X-ray diffraction and 57Fe Mössbauer spectroscopy, respectively. The transesterification reaction was performed at 65 ℃, at a molar ratio methanol:oil 30:1. The chemical yields in esters for this heterogeneously catalyzed transesterification was 99 ± 1 mass%, through a reaction completed in 135 ± 30 min. The reaction catalyzed by the sole magnetic fraction, without CaO, did not produce any esters even after 24 h reaction; the pure CaO catalyst yielded 84 ± 10 mass% esters, after reaction completion, which took 128 ± 16 min. The magnetic fraction with CaO was found to act synergically on the transesterification reaction. From the technological, economic and environmental points of view, these results strongly evidence the real viability of using this magnetic fraction-CaO catalyst, to produce biodiesel.

    Citation: Bárbara Gonçalves Rocha, Alice Lopes Macedo, Bárbara Rodrigues Freitas, Priscylla Caires de Almeida, Vany P. Ferraz, Luis Carlos Duarte Cavalcante, José Domingos Fabris, José Domingos Ardisson. Magnetic fraction from phosphate mining tailings as heterogeneous catalyst for biodiesel production through transesterification reaction of triacylglycerols in bio-oil[J]. AIMS Energy, 2017, 5(5): 864-872. doi: 10.3934/energy.2017.5.864

    Related Papers:

    [1] K. M. S. Y. Konara, M. L. Kolhe, Arvind Sharma . Power dispatching techniques as a finite state machine for a standalone photovoltaic system with a hybrid energy storage. AIMS Energy, 2020, 8(2): 214-230. doi: 10.3934/energy.2020.2.214
    [2] Syed Sabir Hussain Rizvi, Krishna Teerth Chaturvedi, Mohan Lal Kolhe . A review on peak shaving techniques for smart grids. AIMS Energy, 2023, 11(4): 723-752. doi: 10.3934/energy.2023036
    [3] Sulabh Sachan . Integration of electric vehicles with optimum sized storage for grid connected photo-voltaic system. AIMS Energy, 2017, 5(6): 997-1012. doi: 10.3934/energy.2017.6.997
    [4] Dalong Guo, Chi Zhou . Potential performance analysis and future trend prediction of electric vehicle with V2G/V2H/V2B capability. AIMS Energy, 2016, 4(2): 331-346. doi: 10.3934/energy.2016.2.331
    [5] Habibullah Fedayi, Mikaeel Ahmadi, Abdul Basir Faiq, Naomitsu Urasaki, Tomonobu Senjyu . BESS based voltage stability improvement enhancing the optimal control of real and reactive power compensation. AIMS Energy, 2022, 10(3): 535-552. doi: 10.3934/energy.2022027
    [6] Hassan Shirzeh, Fazel Naghdy, Philip Ciufo, Montserrat Ros . Stochastic energy balancing in substation energy management. AIMS Energy, 2015, 3(4): 810-837. doi: 10.3934/energy.2015.4.810
    [7] Shota Tobaru, Ryuto Shigenobu, Foday Conteh, Naomitsu Urasaki, Abdul Motin Howlader, Tomonobu Senjyu, Toshihisa Funabashi . Optimal operation method coping with uncertainty in multi-area small power systems. AIMS Energy, 2017, 5(4): 718-734. doi: 10.3934/energy.2017.4.718
    [8] Aaron St. Leger . Demand response impacts on off-grid hybrid photovoltaic-diesel generator microgrids. AIMS Energy, 2015, 3(3): 360-376. doi: 10.3934/energy.2015.3.360
    [9] Mohamed Elweddad, Muhammet Güneşer, Ziyodulla Yusupov . Designing an energy management system for household consumptions with an off-grid hybrid power system. AIMS Energy, 2022, 10(4): 801-830. doi: 10.3934/energy.2022036
    [10] HVV Priyadarshana, MA Kalhan Sandaru, KTMU Hemapala, WDAS Wijayapala . A review on Multi-Agent system based energy management systems for micro grids. AIMS Energy, 2019, 7(6): 924-943. doi: 10.3934/energy.2019.6.924
  • Biodiesel is an interesting alternative fuel for complementing or even completely replacing the mineral diesel. It is industrially obtained through the transesterification reaction of triacylglycerol in bio-oils with alcohol of short molecular chain, to produce the corresponding mixture of esters of fatty acids (biodiesel). Mineral rejects from mining usually constitute a major environmental and economic problem. Herein, it is described the study devoted to evaluate the chemical efficiency of the transesterification reaction of triacylglycerols in soybean oil with methanol, catalyzed by a material based on the magnetic fraction from rejects of a phosphate ore being commercially exploited in Tapira, Minas Gerais, Brazil. The magnetite-containing material from those mining tailings was used to form a new heterogeneous catalyst, by first mixing it with a commercial synthetic calcium oxide. The mixture was then heated at 200 ℃ for 4 h. The magnetic, crystallographic and 57Fe hyperfine structures of the resulting catalyst were assessed by VSM magnetometer, X-ray diffraction and 57Fe Mössbauer spectroscopy, respectively. The transesterification reaction was performed at 65 ℃, at a molar ratio methanol:oil 30:1. The chemical yields in esters for this heterogeneously catalyzed transesterification was 99 ± 1 mass%, through a reaction completed in 135 ± 30 min. The reaction catalyzed by the sole magnetic fraction, without CaO, did not produce any esters even after 24 h reaction; the pure CaO catalyst yielded 84 ± 10 mass% esters, after reaction completion, which took 128 ± 16 min. The magnetic fraction with CaO was found to act synergically on the transesterification reaction. From the technological, economic and environmental points of view, these results strongly evidence the real viability of using this magnetic fraction-CaO catalyst, to produce biodiesel.


    DFC has captivated a lot of consideration across various analysis and engineering disciplines, particularly in modelling [1], neural networks [2] and image encryption [3]. The developing approach portraying real-world problems have been exhibited to be helpful in numerical devices to analyze, comprehend and predict the nature within humankind live [4,5,6,7,8,9,10]. In 1974, Daiz et al. [11] introduced the idea of DFC and composed it with an infinite sum. Later on, in 1988, Gray et al. [12] extended this concept and implemented it on the finite sum. This concept is known as the nabla difference operator in the literature. Atici and Eloe [13] proposed the theory of fractional difference equations, although the practical implementation is presented in [14]. Yilmazer [15] proposed discrete fractional solution of a nonhomogeneous non-Fuchsian differential equations. Yilmazer and Ali [16] derived the discrete fractional solutions of the Hydrogen atom type equations. Many researchers' focus is directed towards modeling and analysis of various problems in bio-mathematical sciences. This field demonstrates several distinguished kernels depending on discrete power law, discrete exponential-law and discrete Mittag-Leffler law kernels which correspond to the Liouville-Caputo, Caputo-Fabrizio and the Atangana-Baleanu nabla(delta) difference operators generalized Z time scale [17,18,19].

    Numerous utilities have been developed via DFC such as the solution of fractional difference equations and discrete boundary value problems are proposed in terms of new mathematical techniques [20,21,22,23]. Therefore, the conventional methodology of DFC have some intriguing and less-acknowledged opportunities for modelling. DFC is proposed to depict the customary practice of time scale analysis, with discussing its numerical approximations in ˇZ. Furthermore, we observe that ˇ-discrete fractional calculus is tremendously momentous in applied sciences and can also address the requirements of synchronous operation of various mechanisms, see [24,25,26].

    Among the computational models formulated in fractional calculus, discrete AB-fractional operators, which is a universal operator of fractional calculus that has been traditionally employed to develop modern operators and their characterizations have been proposed in research article [27,28]. Moreover, DFC has been theoretically presented more by introducing and analyzing discrete forms of these fractional operators [13]. Here, we intend to find the discrete fractional inequalities analogous to fractional operators having -discrete Mittag-Leffler kernels, encompassing and simplifying these operators in such a manner as to recuperate certain appropriate traits such as discrete inequalities for AB-fractional sums.

    Mathematical inequalities [29,30,31,32,33,34,35,36,37,38] initially alluded to adjust, harmony, and coordination. Until modern times, refinements of inequalities were characterized as invariance to change [39,40,41,42,43]. Physics comprehends fractional inequalities as predictability, while Psychology accentuates that inequality is the trait of magnificence and art [44].

    Numerous investigations have been directed on fractional inequalities in the natural science [45], engineering sciences, see [41,46,47,48] and the references cited therein. Landscapes, structures, and mechanical equipment all demonstrate inequalities attributes. Therefore, we intend to find the discrete version of the Grüss type and some further connected modifications by the -discrete AB-fractional sums depending on -discrete generalized Mittag-Leffler kernel. This stands as an inspiration for the current paper. The intensively investigated Grüss inequality can be presented as follows:

    Theorem 1.1. (See [49]) Let F,G:[c,d]R be two positive functions such that αF(x)A and βG(x)B for all x[c,d] and α,β,A,BR. Then

    |1dcdcF(x)G(x)dx1(dc)2dcF(x)dxdcG(x)dx| (1.1)
    14(Aα)(Bβ),

    where the constant 1/4 can not be improved.

    The Grüss inequality Eq (1.1) has been broadly and intensely investigated in engineering and applied analysis, and various developed consequences have been acquired so far. Nevertheless, the prevalent existence of Grüss inequality in scientific fields is not in direct proportion to the consideration it has acknowledged. In application viewpoint, practically all mechanical structures are found to have inequality Eq (1.1), and the vast majority of them have the qualities of discrete and continuous fractional operators [50,51,52,53,54,55,56,57,58,59,60,61,62,63].

    Inspired by the excellent dynamical properties of -discrete AB-fractional sums differences formulation [64], the limitations of fractional calculus can be ameliorated via discrete and continuous state-of-the-art techniques for effective information chaotic map applications, that can be inferred as a generalization of nonlocal/nonsingular type kernels. These investigations promote further sum/difference operators and related inequalities. It is our aim in this investigation to explore the discrete version of the Grüss type and certain other associated variants with some traditional and forthright inequalities in the frame of -discrete AB-fractional sums. We also would like to mention that besides these variants, several other intriguing generalizations are derived. The comparison of Grüss type with other discrete fractional calculus frameworks is currently under investigation. Finally, two examples are presented that correlate with some well-known inequalities in the relative literature and with the proposed strategy.

    In this section, we evoke some basic ideas related to fractional operator, discrete generalized Mittag Leffler functions and the time scale calculus, see the detailed information in [13]. For the sake of simplicity, we use the notation, for c,dR and >0, Nc,={c,c+,c+2,...} and Nd,={d,d+,d+2,...}.

    Definition 2.1. ([65])The backward difference operator of a function F on Z is stated as

    ˆF(t)=F(t)F(ρ(t)), (2.1)

    where ρ(t)=t denotes the backward jump operator. Also, the forward difference operator of a function F on Z is stated as

    ˆΔF(t)=F(ρ(t))F(t), (2.2)

    where σ(t)=t+ denotes the forward jump operator.

    Definition 2.2. ([65]) (ⅰ) For any t,αR and >0, the delta -factorial function is stated as

    t(α)=αΓ(t+1)Γ(t+1α), (2.3)

    where Γ denotes the Euler gamma function. For =1, then t(α)=Γ(t+1)Γ(t+1α). Also, the division by a pole leads to zero.

    (ⅱ) For any t,αR and >0, the nabla -factorial function is stated as

    t(α)=αΓ(t+α)Γ(t). (2.4)

    For =1, we observe that t(α)=Γ(t+α)Γ(t).

    Lemma 2.3. ([64]) Let tT=Nc,, then for all tTι, we obtain

    ˆx,{(xt)ι+1(ι+1)!}=(xt)ιι!. (2.5)

    Lemma 2.4. ([66]) For the time scale T=Nc, then the nabla Taylor polynomial

    ˆBι(x,t)=(xt)ιι!,ιN0. (2.6)

    Now we present the concept of nabla -discrete Mittag-Leffler function which is introduced by [6].

    Definition 2.5. ([6]) Let α,ϱ,ΩC having (α)>0 such that λR with |λα|<1, then the nabla discrete Mittag-leffler function is defined

    ˇEα,ϱ(λ,Ω)=ι=0λιΩια+ϱ1Γ(αι+ϱ),|λα|<1. (2.7)

    For ϱ=1, we have

    ˇEα(λ,y)ˇEα,1(λ,y)=ι=0λιyιαΓ(αι+1),|λα|<1. (2.8)

    The following remark illustrates the strengthening properties why Z is important.

    Remark 1. In view of Z:

    Ⅰ. letting =1, we attain the nabla discrete Mittag-Leffler function stated in [67,68].

    Ⅱ. letting 0<<1, the interval of convergence to which λ lies. Observe that, when 0, then α(0,1). Moreover, when 1 guarantee convergence for λ=α1α,α(0,12).

    For further investigation of the discrete Mittag-Leffler function we refer the reader to [4].

    Definition 2.6. ([26]) For some ιN,α>0 and let d=c+ι. Assume that a function F be defined on T=Nc,Nd,. Then the delta -fractional sums in the left and right case are defined as follows

    (cˆΔαF)(t)=1Γ(α)x/αι=c/(xσ(ι))(α1)F(ι),x{x+α:xT}

    and

    (ˆΔαdF)(t)=1Γ(α)d/αι=x/+α(ισ(x))(α1)F(ι),x{xα:xT},

    respectively.

    Definition 2.7. ([6,66]) Assume that >0 and the backward jump operator is ρ(x)=x. A function F:Nc,R is said to be nabla -fractional sum of order α, if

    (cˆαF)(t)=1Γ(α)x/αι=c/+1(xρ(ι))(α1)F(ι),xNc+,.

    Also, the nabla right -fractional sum of order α>0(ending at d) for F:Nd,R is described as follows

    (ˆαdF)(t)=1Γ(α)d/1ι=x/(ιρ(x))(α1)F(ι).

    Now, we demonstrate the some new concepts which we will be utilized for proving coming results of this paper, see [4]. Also, we use the notation, λ=α1α and ρ(x)=x.

    Definition 2.8. ([64]) For α[0,1],>0 with |λα|<1 and let F be a function defined on Nc,d,N with c<d such that cd(mod), then the left nabla ABC-fractional difference (in the sense of Atangana and Baleanu) is described as

    (ABCcˆαF)(x)=B(α,)1α+α1αx/ι=c/+1ˆF(ι)ˇEα(λ,xρ(ι)) (2.9)

    and in the left Riemann sense by

    (ABRcˆαF)(x)=B(α,)1α+α1αˆx/ι=c/+1F(ι)ˇEα(λ,xρ(ι)). (2.10)

    Definition 2.9. ([64]) For 0<α<1 and let the left -fractional sum concern to (ABRcˆαF)(x) defined on Nc, is stated as follows

    (ABcˆαF)(x)=1αB(α,)(1α+α)F(x)+αB(α,)(1α+α)Γ(α)x/ι=c/+1(xρ(ι))α1F(ι). (2.11)

    The right -fractional sum is defined on d,N by

    (ABˆαdF)(x)=1αB(α,)(1α+α)F(x)+αB(α,)(1α+α)Γ(α)d/1ι=x/(ιρ(x))α1F(ι). (2.12)

    In this section, we present a different concept of Grüss type inequalities, which consolidates the ideas of -discrete AB-fractional sums.

    Theorem 3.1. Let α(0,1) and let F be a positive function on Nc,. Suppose that there exist two positive functions ϕ1,ϕ2 on Nc, such that

    ϕ1(x)F(x)ϕ2(x),xNc,. (3.1)

    Then, for x{c,c+,c+2,...}, one has

    ABcˆβ[ϕ2(x)]ABcˆα[F(x)]+ABcˆβ[F(x)]ABcˆα[ϕ1(x)]ABcˆβ[ϕ2(x)]ABcˆα[ϕ1(x)]+ABcˆα[F(x)]ABcˆβ[F(x)]. (3.2)

    Proof. From Eq (3.1), for θ,λNc,, we have

    (ϕ2(θ)F(θ))(F(λ)ϕ1(λ))0. (3.3)

    Therefore,

    ϕ2(θ)F(λ)+ϕ1(λ)F(θ)ϕ1(λ)ϕ2(θ)+F(θ)F(λ). (3.4)

    Taking product both sides of Eq (3.4) by 1αB(α,)(1α+α), we get

    (1α)ϕ2(θ)F(λ)B(α,)(1α+α)+(1α)ϕ1(λ)F(θ)B(α,)(1α+α)(1α)ϕ1(λ)ϕ2(θ)B(α,)(1α+α)+(1α)F(θ)F(λ)B(α,)(1α+α). (3.5)

    Replacing λ by t in Eq (3.5) and conducting product both sides by α(xρ(t))α1B(α,)Γ(α), we have

    α(xρ(t))α1B(α,)Γ(α)ϕ2(θ)F(t)+α(xρ(t))α1B(α,)Γ(α)ϕ1(t)F(θ)α(xρ(t))α1B(α,)Γ(α)ϕ1(t)ϕ2(θ)+α(xρ(t))α1B(α,)Γ(α)F(θ)F(t).

    Summing both sides for t{c,c+,c+2,...}, we get

    x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)ϕ2(θ)F(ι)+x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)ϕ1(ι)F(θ)α(xρ(t))α1B(α,)Γ(α)ϕ1(ι)ϕ2(θ)+x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)F(θ)F(ι). (3.6)

    Adding Eqs (3.5) and (3.6), we have

    (1α)ϕ2(θ)F(λ)B(α,)(1α+α)+x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)ϕ2(θ)F(ι)+(1α)ϕ1(λ)F(θ)B(α,)(1α+α)+x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)ϕ1(ι)F(θ)(1α)ϕ1(λ)ϕ2(θ)B(α,)(1α+α)+x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)ϕ1(ι)ϕ2(θ)+(1α)F(θ)F(λ)B(α,)(1α+α)+x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)F(θ)F(ι),

    arrives at

    ϕ2(θ)ABcˆα[F(x)]+F(θ)ABcˆα[ϕ1(x)]ϕ2(θ)ABcˆα[ϕ1(x)]+F(θ)ABcˆα[F(x)]. (3.7)

    Taking product both sides of Eq (3.7) by 1βB(β,)(1β+β), we have

    (1β)ϕ2(θ)B(β,)(1β+β)ABcˆα[F(x)]+(1β)F(θ)B(β,)(1β+β)ABcˆα[ϕ1(x)](1β)ϕ2(θ)B(β,)(1β+β)ABcˆα[ϕ1(x)]+(1β)F(θ)B(β,)(1β+β)ABcˆα[F(x)]. (3.8)

    Also, replacing θ by ˉt in Eq (3.8) and conducting product both sides by β(xρ(ˉt))β1B(β,)Γ(β), we have

    β(xρ(ˉt))β1B(β,)Γ(β)ϕ2(θ)ABcˆα[F(x)]+β(xρ(ˉt))β1B(β,)Γ(β)F(θ)ABcˆα[ϕ1(x)]β(xρ(ˉt))β1B(β,)Γ(β)ϕ2(θ)ABcˆα[ϕ1(x)]+β(xρ(ˉt))β1B(β,)Γ(β)F(θ)ABcˆα[F(x)].

    Summing both sides for ˉt{c,c+,c+2,...}, we get

    x/j=c/+1β(xρ(j))β1B(β,)Γ(β)ϕ2(j)ABcˆα[F(x)]+x/j=c/+1β(xρ(j))β1B(β,)Γ(β)F(j)ABcˆα[ϕ1(x)]x/j=c/+1β(xρ(j))β1B(β,)Γ(β)ϕ2(j)ABcˆα[ϕ1(x)]+x/j=c/+1β(xρ(j))β1B(β,)Γ(β)F(j)ABcˆα[F(x)]. (3.9)

    Adding Eqs (3.8) and (3.9), then in view of Definition 2.9, yields the inequality Eq (3.2). This completes the proof.

    Some special cases which can be derived immediately from Theorem 3.1.

    Choosing =1, then we attain a new result for discrete AB-fractional sum.

    Corollary 1. Let α(0,1) and let F be a positive function on Nc. Suppose that there exist two positive functions ϕ1,ϕ2 on Nc such that

    ϕ1(x)F(x)ϕ2(x),xNc. (3.10)

    Then, for x{c,c+1,c+2,...}, one has

    ABcˆβ[ϕ2(x)]ABcˆα[F(x)]+ABcˆβ[F(x)]ABcˆα[ϕ1(x)]ABcˆβ[ϕ2(x)]ABcˆα[ϕ1(x)]+ABcˆα[F(x)]ABcˆβ[F(x)]. (3.11)

    Theorem 3.2. Let α,β(0,1) and let F and G be two positive functions on Nc,. Suppose that Eq (3.1) satisfies and also one assumes that there exist two positive functions Ω1,Ω2 on Nc, such that

    Ω1(x)G(x)Ω2(x),xNc,. (3.12)

    Then, for x{c,c+,c+2,...}, one has

    (M1)ABcˆβ[ϕ2(x)]ABcˆα[G(x)]+ABcˆβ[F(x)]ABcˆα[Ω1(x)]ABcˆβ[ϕ2(x)]ABcˆα[Ω1(x)]+ABcˆβ[F(x)]ABcˆα[G(x)],(M2)ABcˆα[ϕ1(x)]ABcˆβ[G(x)]+ABcˆα[Ω2(x)]ABcˆα[F(x)]ABcˆα[ϕ1(x)]ABcˆβ[Ω2(x)]+ABcˆα[F(x)]ABcˆβ[G(x)],(M3)ABcˆα[Ω2(x)]ABcˆβ[ϕ2(x)]+ABcˆβ[F(x)]ABcˆα[G(x)]ABcˆβ[ϕ2(x)]ABcˆα[G(x)]+ABcˆβ[F(x)]ABcˆα[Ω2(x)],(M4)ABcˆβ[ϕ1(x)]ABcˆα[Ω1(x)]+ABcˆβ[F(x)]ABcˆα[G(x)]ABcˆβ[ϕ1(x)]ABcˆα[G(x)]+ABcˆα[Ω1(x)]ABcˆβ[F(x)]. (3.13)

    Proof. To prove Eq (M1), from Eqs (3.1) and (3.12), we have for λ,θNc, that

    (ϕ2(θ)F(θ))(G(λ)Ω1(λ))0. (3.14)

    Therefore,

    ϕ2(θ)G(λ)+Ω1(λ)F(θ)Ω1(λ)ϕ2(θ)+G(λ)F(θ). (3.15)

    Taking product both sides of Eq (3.17) by 1αB(α,)(1α+α), we get

    (1α)ϕ2(θ)G(λ)B(α,)(1α+α)+(1α)Ω1(λ)F(θ)B(α,)(1α+α)(1α)Ω1(λ)ϕ2(θ)B(α,)(1α+α)+(1α)G(λ)F(θ)B(α,)(1α+α). (3.16)

    Moreover, replacing λ by t in Eq (3.17) and conducting product both sides by α(xρ(t))α1B(α,)Γ(α), we have

    α(xρ(t))α1B(α,)Γ(α)ϕ2(θ)G(λ)+α(xρ(t))α1B(α,)Γ(α)Ω1(λ)F(θ)α(xρ(t))α1B(α,)Γ(α)Ω1(λ)ϕ2(θ)+α(xρ(t))α1B(α,)Γ(α)G(λ)F(θ). (3.17)

    Summing both sides for t{c,c+,c+2,...}, we get

    x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)ϕ2(θ)G(ι)+x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)Ω1(ι)F(θ)x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)Ω1(ι)ϕ2(θ)+x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)G(ι)F(θ).

    Then, we have

    ABcˆα[G(x)]ϕ2(θ)+ABcˆα[Ω1(x)]F(θ)ABcˆα[Ω1(x)]ϕ2(θ)+ABcˆα[G(x)]F(θ). (3.18)

    Taking product both sides of Eq (3.16) by 1βB(β,)(1β+β), we have

    1βB(β,)(1β+β)ABcˆβ[G(x)]ϕ2(θ)+1βB(β,)(1β+β)ABcˆβ[Ω1(x)]F(θ)1βB(β,)(1β+β)ABcˆβ[Ω1(x)]ϕ2(θ)+1βB(β,)(1β+β)ABcˆβ[G(x)]F(θ). (3.19)

    Further, replacing θ by ˉt in Eq (3.19) and conducting product both sides by β(xρ(ˉt))β1B(β,)Γ(β), we have

    ABcˆα[G(x)]β(xρ(ˉt))β1B(β,)Γ(β)ϕ2(θ)+ABcˆα[Ω1(x)]β(xρ(ˉt))β1B(β,)Γ(β)F(θ)ABcˆα[Ω1(x)]β(xρ(ˉt))β1B(β,)Γ(β)ϕ2(θ)+ABcˆα[G(x)]β(xρ(ˉt))β1B(β,)Γ(β)F(θ). (3.20)

    Summing both sides for ˉt{c,c+,c+2,...}, we get

    ABcˆα[G(x)]x/j=c/+1β(xρ(j))β1B(β,)Γ(β)ϕ2(j)+ABcˆα[Ω1(x)]x/j=c/+1β(xρ(j))β1B(β,)Γ(β)F(j)ABcˆα[Ω1(x)]x/j=c/+1β(xρ(j))β1B(β,)Γ(β)ϕ2(j)+ABcˆα[G(x)]x/j=c/+1β(xρ(j))β1B(β,)Γ(β)F(j). (3.21)

    Adding Eqs (3.19) and (3.21), we conclude the desired inequality Eq (M1).

    To prove Eqs (M2)(M4), we utilize the following inequalities:

    (M2)(Ω2(θ)G(θ))(F(λ)ϕ1(λ))0,
    (M3)(ϕ2(θ)F(θ))(G(λ)Ω2(λ))0,
    (M4)(ϕ1(θ)F(θ))(G(λ)Ω1(λ))0.

    Some special cases which can be derived immediately from Theorem 3.2.

    Choosing =1, then we attain a new result for discrete AB-fractional sums.

    Corollary 2. Let α,β(0,1) and let F and G be two positive functions on Nc. Suppose that Eq (3.1) satisfies and also one assumes that there exist two positive functions Ω1,Ω2 on Nc such that

    Ω1(x)G(x)Ω2(x),xNc.

    Then, for x{c,c+1,c+2,...}, one has

    (M5)ABcˆβ[ϕ2(x)]ABcˆα[G(x)]+ABcˆβ[F(x)]ABcˆα[Ω1(x)]ABcˆβ[ϕ2(x)]ABcˆα[Ω1(x)]+ABcˆβ[F(x)]ABcˆα[G(x)],(M6)ABcˆα[ϕ1(x)]ABcˆβ[G(x)]+ABcˆα[Ω2(x)]ABcˆα[F(x)]ABcˆα[ϕ1(x)]ABcˆβ[Ω2(x)]+ABcˆα[F(x)]ABcˆβ[G(x)],(M7)ABcˆα[Ω2(x)]ABcˆβ[ϕ2(x)]+ABcˆβ[F(x)]ABcˆα[G(x)]ABcˆβ[ϕ2(x)]ABcˆα[G(x)]+ABcˆβ[F(x)]ABcˆα[Ω2(x)],(M8)ABcˆβ[ϕ1(x)]ABcˆα[Ω1(x)]+ABcˆβ[F(x)]ABcˆα[G(x)]ABcˆβ[ϕ1(x)]ABcˆα[G(x)]+ABcˆα[Ω1(x)]ABcˆβ[F(x)].

    Theorem 3.3. Let α,β(0,1) and let F and G be two positive functions on Nc, with p,q>0 satisfying 1p+1q=1. Then, for x{c,c+,c+2,...}, one has

    (M9)1pABcˆβ[Fp(x)]ABcˆα[Gp(x)]+1qABcˆβ[Gq(x)]ABcˆα[Fq(x)]ABcˆβ[FG(x)]ABcˆα[GF(x)],(M10)1pABcˆα[Gq(x)]ABcˆβ[Fp(x)]+1qABcˆα[Fp(x)]ABcˆβ[Gq(x)]ABcˆα[Gq1Fp1(x)]ABcˆβ[FG(x)],(M11)1pABcˆα[G2(x)]ABcˆβ[Fp(x)]+1qABcˆα[F2(x)]ABcˆβ[Gq(x)]ABcˆα[F2qG2p(x)]ABcˆβ[FG(x)],(M12)1pABcˆα[Gq(x)]ABcˆβ[F2(x)]+1qABcˆα[Fp(x)]ABcˆβ[G2(x)]ABcˆα[Fp1Gq1(x)]ABcˆβ[F2pG2q(x)]. (3.22)

    Proof. According to the well-known Young's inequality:

    1pap+1qbqab,a,b0,p,q>0,1p+1q=1, (3.23)

    setting a=F(θ)G(λ) and b=F(λ)G(θ),θ,λ>0, we have

    1p(F(θ)G(λ))p+1q(F(λ)G(θ))q(F(θ)G(λ))(F(λ)G(θ)). (3.24)

    Taking product both sides of Eq (3.24) by 1αB(α,)(1α+α), we have

    1p(1α)Fp(θ)Gp(λ)B(α,)(1α+α)+1q(1α)Fq(λ)Gq(θ)B(α,)(1α+α)(1α)F(θ)G(λ))(F(λ)G(θ)B(α,)(1α+α). (3.25)

    Moreover, replacing λ by t in Eq (3.25) and conducting product both sides by α(xρ(t))α1B(α,)Γ(α), we have

    Fp(θ)pα(xρ(t))α1B(α,)Γ(α)Gp(t)+Gq(θ)qα(xρ(t))α1B(α,)Γ(α)vq(t)F(θ)G(θ)α(xρ(t))α1B(α,)Γ(α)F(t)G(t). (3.26)

    Summing both sides for t{c,c+,c+2,...}, we get

    Fp(θ)px/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)Gp(ι)+Gq(θ)qx/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)Fq(ι)F(θ)G(θ)x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)F(ι)G(ι). (3.27)

    Adding Eqs (3.24) and (3.27), we get

    1p(1α)Fp(θ)Gp(λ)B(α,)(1α+α)+Fp(θ)px/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)Gp(ι)+1q(1α)Fq(λ)Gq(θ)B(α,)(1α+α)+Gq(θ)qx/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)Fq(ι)(1α)F(θ)G(λ)F(λ)G(θ)B(α,)(1α+α)+F(θ)G(θ)x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)F(ι)G(ι). (3.28)

    In view of Definition 2.9, yields

    Fp(θ)pABcˆα[Gp(x)]+Gq(θ)qABcˆα[Fp(x)]F(θ)G(θ)ABcˆα[F(x)G(x)]. (3.29)

    Again, taking product both sides of Eq (3.29) by 1βB(β,)(1β+β), we have

    Fp(θ)p(1β)ABcˆα[Gp(x)]B(β,)(1β+β)+Gq(θ)q(1β)ABcˆα[Fp(x)]B(β,)(1β+β)(1β)ABcˆα[F(x)G(x)]B(β,)(1β+β)F(θ)G(θ). (3.30)

    Further, replacing θ by ˉt in Eq (3.29) and conducting product both sides by β(xρ(ˉt))β1B(β,)Γ(β), we have

    1pABcˆα[Gp(x)]β(xρ(ˉt))β1B(β,)Γ(β)Fp(ˉt)+1qABcˆα[Fp(x)]β(xρ(ˉt))β1B(β,)Γ(β)Gq(ˉt)ABcˆα[F(x)G(x)]β(xρ(ˉt))β1B(β,)Γ(β)F(ˉt)G(ˉt). (3.31)

    After summing the above inequality Eq (3.31) both sides for ˉt{c,c+,c+2,...}, yields the desired assertion Eq (M9).

    The remaining variants can be derived by adopting the same technique and accompanying the selection of parameters in Young inequality.

    (M10)a=F(θ)F(λ),b=G(θ)G(λ),F(λ),G(λ)0,(M11)a=F(θ)G2p(λ),b=F2q(λ)G(θ),(M12)a=F2p(θ)F(λ),b=G2q(θ)G(λ),F(λ),G(λ)0.

    Repeating the foregoing argument, we obtain Eqs (M10)(M12).

    (I) Letting =1, then we attain a result for discrete AB-fractional sums.

    Corollary 3. Let α,β(0,1) and let F and G be two positive functions on Nc with p,q>0 satisfying 1p+1q=1. Then, for x{c,c+1,c+2,...}, one has

    (M13)1pABcˆβ[Fp(x)]ABcˆα[Gp(x)]+1qABcˆβ[Gq(x)]ABcˆα[Fq(x)]ABcˆβ[FG(x)]ABcˆα[GF(x)],(M14)1pABcˆα[Gq(x)]ABcˆβ[Fp(x)]+1qABcˆα[Fp(x)]ABcˆβ[Gq(x)]ABcˆα[Gq1Fp1(x)]ABcˆβ[FG(x)],(M15)1pABcˆα[G2(x)]ABcˆβ[Fp(x)]+1qABcˆα[F2(x)]ABcˆβ[Gq(x)]ABcˆα[F2qG2p(x)]ABcˆβ[FG(x)],(M16)1pABcˆα[Gq(x)]ABcˆβ[F2(x)]+1qABcˆα[Fp(x)]ABcˆβ[G2(x)]ABcˆα[Fp1Gq1(x)]ABcˆβ[F2pG2q(x)]. (3.32)

    Example 3.4. Let α,β(0,1) and let F and G be two positive functions on Nc, with p,q>0 satisfying p+q=1. Then, for x{c,c+,c+2,...}, one has

    (M17)pABcˆβ[F(x)]ABcˆα[G(x)]+qABcˆα[F(x)]ABcˆβ[G(η)ABcˆβ[FpGq(x)]ABcˆα[FqGp(x)],(M18)pABcˆβ[Fp1(x)]ABcˆα[(F(x)]Gq(x)])+qABcˆα[Gq1(x)]ABcˆβ[FqG(x)]ABcˆβ[Gq(x)]ABcˆα[Fp(x)],(M19)pABcˆβ[F(x)]ABcˆα[G2p(x)]+qABcˆβ[G(x)]ABcˆα[F2q(x)]ABcˆβ[FpG(x)]ABcˆα[GqF2(x)],(M20)pABcˆβ[F2pGq(x)]ABcˆα[Gp1(x)]+qABcˆβ[Gq1(x)]ABcˆα[F2qGp(x)]ABcˆβ[F2(x)]ABcˆα[G2(x)]. (3.33)

    Proof. The example can be proved with the aid of the weighted AM–GM inequality with the same technique as we did in Theorem 3.3 and utilizing the following assumptions:

    (M17)a=F(θ)G(λ),b=F(λ)G(θ).(M18)a=F(λ)F(θ),b=G(θ)G(λ),F(θ),G(λ)0.(M19)a=F(θ)G2p(λ),b=F2q(λ)G(θ).(M20)a=F2p(θ)G(λ),b=F2q(λ)G(θ),G(θ),G(θ)0.

    Example 3.5. Let α(0,1) and let F and G be two positive functions on Nc, with p,q>1 satisfying 1p+1q=1. Let

    γ=minθNc,F(θ)G(θ)andΥ=maxθNc,F(θ)G(θ). (3.34)

    Then, for x{c,c+,c+2,...}, one has

    (i)0ABcˆα[F2(x)]ABcˆα[G2(x)]γ+Υ4γΥ(ABcˆα[FG(x)])2,(ii)0ABcˆα[F2(x)]ABcˆα[G2(x)](ABcˆα[FG(x)])Υγ2Υγ(ABcˆα[FG(x)]),(iii)0ABcˆα[F2(x)]ABcˆα[G2(x)](ABcˆα[FG(x)])2Υγ4γΥ(ABcˆα[FG(x)])2.

    Proof. From Eq (3.34) and the inequality

    (F(θ)G(θ)γ)(ΥF(θ)G(θ))G2(θ)0,θNc, (3.35)

    then we can write as,

    F2(θ)+γΥG2(θ)(γ+Υ)F(θ)G(θ). (3.36)

    Taking product both sides of Eq (3.36) by 1αB(α,)(1α+α), we have

    (1α)F2(θ)B(α,)(1α+α)+(1α)G2(θ)B(α,)(1α+α)γx1αB(α,)(1α+α)(γ+x)F(θ)G(θ). (3.37)

    Replacing θ by t in Eq (3.36) and conducting product both sides by α(xρ(t))α1B(α,)Γ(α), we have

    α(xρ(t))α1B(α,)Γ(α)F2(t)+γΥα(xρ(t))α1B(α,)Γ(α)G2(θ)(γ+Υ)α(xρ(t))α1B(α,)Γ(α)F(t)G(t). (3.38)

    Summing both sides for t{c,c+,c+2,...}, we get

    x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)F2(ι)+γΥx/ι=c/+1α(xρ(t))α1B(α,)Γ(α)G2(ι)(γ+Υ)x/ι=c/+1α(xρ(ι))α1B(α,)Γ(α)F(ι)G(ι). (3.39)

    Adding Eqs (3.37) and (3.39), yields

    ABcˆα[F2(x)]+γΥABcˆα[G2(x)](γ+Υ)ABcˆα[FG(x)], (3.40)

    on the other hand, it follows from γΥ>0 and

    (ABcˆα[F2(x)]γΥABcˆα[G2(x)])20, (3.41)

    that

    2ABcˆα[F2(x)]γΥABcˆα[G2(x)]ABcˆα[F2(x)]+γΥABcˆα[G2(x)] (3.42)

    then from Eqs (3.40) and (3.42), we obtain,

    4γΥABcˆα[F2(x)]ABcˆα[G2(x)](γ+Υ)2(ABcˆα[FG(x)]). (3.43)

    Which implies (i). By some change of (i), analogously, we get (ii) and (iii).

    Unlike some known and established inequalities in the literature, the Grüss type inequalities have been presented via the -discrete AB-fractional sums with different values of parameters on the domain Z that can be implemented to solve the qualitative properties of difference equations. Our consequences can be applied to overcome the obstacle of obtaining estimation on the explicit bounds of unknown functions and also to extend and unify continuous inequalities by using the simple technique. Several novel consequences have been derived by the use of discrete -fractional sums. The noted consequences can also be extended to the weighted function case. Certainly, the case 1 recaptures the outcomes of the discrete AB-fractional sums. For indicating the strength of the offered fallouts, we employ them to investigate numerous initial value problems of fractional difference equations.

    Authors are grateful to the referees for their valuable suggestions and comments.

    The authors declare no conflict of interest.

    [1] Galvão LPFC, Barbosa MN, Araujo AS, et al. (2012) Iodeto de potássio suportado em peneiras moleculares mesoporosas (SBA-15 e MCM-41) como catalisador básico para síntese de biodiesel. Quim Nova 35: 41–44. doi: 10.1590/S0100-40422012000100008
    [2] Schuchardt U, Sercheli R, Vargas RM (1998) Transesterification of vegetable oils: a review. J Brazil Chem Soc 9: 199–210.
    [3] Saifuddin N, Samiuddin A, Kumaran P (2015) A review on processing technology for biodiesel production. Trends Appl Sci Res 10: 1–37. doi: 10.3923/tasr.2015.1.37
    [4] Ruhul AM, Kalam MA, Masjuki HH, et al. (2015) State of the art of biodiesel production processes: a review of the heterogeneous catalyst. RSC Adv 5: 101023–101044. doi: 10.1039/C5RA09862A
    [5] Abdullah SHYS, Hanapi NHM, Azid A, et al. (2017) A review of biomass-derived heterogeneous catalyst for a sustainable biodiesel production. Renew Sust Energ Rev 70: 1040–1051. doi: 10.1016/j.rser.2016.12.008
    [6] Mardhiah HH, Ong HC, Masjuki HH, et al. (2017) A review on latest developments and future prospects of heterogeneous catalyst in biodiesel production from non-edible oils. Renew Sust Energ Rev 67: 1225–1236. doi: 10.1016/j.rser.2016.09.036
    [7] Mansir N, Taufiq-Yap YH, Rashid U, et al. (2017) Investigation of heterogeneous solid acid catalyst performance on low grade feedstocks for biodiesel production: a review. Energy Convers Manage 141: 171–182. doi: 10.1016/j.enconman.2016.07.037
    [8] Macedo AL, Fabris JD, Pires MJM, et al. (2016) A mesoporous SiO2/γ-Fe2O3/KI heterogeneous magnetic catalyst for the green synthesis of biodiesel. J Braz Chem Soc 27: 2290–2299.
    [9] Wang H, Covarrubias J, Prock H, et al. (2015) Acid-functionalized magnetic nanoparticle as heterogeneous catalysts for biodiesel synthesis. J Phys Chem C 119: 26020–26028. doi: 10.1021/acs.jpcc.5b08743
    [10] JCPDS-Joint Committee on Powder Diffraction Standards (1980) Mineral Powder diffraction files data book. Swarthmore, Pennsylvania.
    [11] Holland TJB, Redfern SAT (1997) Unit cell refinement from powder diffraction data: the use of regression diagnostics. Mineral Mag 61: 65–77. doi: 10.1180/minmag.1997.061.404.07
    [12] Cornell RM, Schwertmann U (2003) The iron oxides: structure, properties, reactions, occurrences and uses. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.
  • This article has been cited by:

    1. Abdul Conteh, Mohammed Elsayed Lotfy, Kiptoo Mark Kipngetich, Tomonobu Senjyu, Paras Mandal, Shantanu Chakraborty, An Economic Analysis of Demand Side Management Considering Interruptible Load and Renewable Energy Integration: A Case Study of Freetown Sierra Leone, 2019, 11, 2071-1050, 2828, 10.3390/su11102828
    2. Mark Kipngetich Kiptoo, Mohammed Elsayed Lotfy, Oludamilare Bode Adewuyi, Abdul Conteh, Abdul Motin Howlader, Tomonobu Senjyu, Integrated approach for optimal techno-economic planning for high renewable energy-based isolated microgrid considering cost of energy storage and demand response strategies, 2020, 215, 01968904, 112917, 10.1016/j.enconman.2020.112917
    3. Ryuto Shigenobu, Mitsunaga Kinjo, Paras Mandal, Abdul Howlader, Tomonobu Senjyu, Optimal Operation Method for Distribution Systems Considering Distributed Generators Imparted with Reactive Power Incentive, 2018, 8, 2076-3417, 1411, 10.3390/app8081411
    4. P. Sokolnikova, P. Lombardi, B. Arendarski, K. Suslov, A.M. Pantaleo, M. Kranhold, P. Komarnicki, Net-zero multi-energy systems for Siberian rural communities: A methodology to size thermal and electric storage units, 2020, 155, 09601481, 979, 10.1016/j.renene.2020.03.011
    5. Abdan Hanifan Dharmasakya, Lesnanto Multa Putranto, Roni Irnawan, 2021, A Review of Reactive Power Management in Distribution Network with PV System, 978-1-6654-4878-9, 1, 10.1109/APPEEC50844.2021.9687706
    6. Sandip Chanda, Suparna Maity, Abhinandan De, A differential evolution modified quantum PSO algorithm for social welfare maximisation in smart grids considering demand response and renewable generation, 2022, 0946-7076, 10.1007/s00542-022-05399-1
    7. Xin Huang, Kai Wang, Mintong Zhao, Jiajia Huan, Yundong Yu, Kai Jiang, Xiaohe Yan, Nian Liu, Optimal Dispatch and Control Strategy of Integrated Energy System Considering Multiple P2H to Provide Integrated Demand Response, 2022, 9, 2296-598X, 10.3389/fenrg.2021.824255
    8. Ryuto Shigenobu, Masakazu Ito, Kosuke Uchida, Harun Or Rashid Howlader, Tomonobu Senjyu, 2022, Chapter 9, 978-981-16-3127-6, 157, 10.1007/978-981-16-3128-3_9
    9. Xiang Huo, Mingxi Liu, Two-Facet Scalable Cooperative Optimization of Multi-Agent Systems in the Networked Environment, 2022, 30, 1063-6536, 2317, 10.1109/TCST.2022.3143115
    10. Neyre Tekbıyık‐Ersoy, 2023, 9781119899426, 93, 10.1002/9781119899457.ch4
    11. Oludamilare Bode Adewuyi, Komla A. Folly, David T. O. Oyedokun, Yanxia Sun, 2023, Chapter 4, 978-3-031-26495-5, 55, 10.1007/978-3-031-26496-2_4
    12. Gbubemi Kevin Akporhonor, Pegah Mirzania, 2025, Chapter 4, 978-3-031-83164-5, 47, 10.1007/978-3-031-83165-2_4
  • Reader Comments
  • © 2017 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(4717) PDF downloads(977) Cited by(2)

Figures and Tables

Figures(3)  /  Tables(3)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog