Loading [MathJax]/jax/output/SVG/jax.js
Research article Special Issues

A reliable analytic technique and physical interpretation for the two-dimensional nonlinear Schrödinger equations

  • Nonlinear Schrödinger equations are a key paradigm in nonlinear research, attracting both mathematical and physical attention. This work was primarily concerned with the usage of a reliable analytic technique in order to solve two models of (2+1)-dimensional nonlinear Schrödinger equations. By applying a comprehensible wave transformation, every nonlinear model was simplified to an ordinary differential equation. A number of critical solutions were observed that correlated to various parameters. The provided approach has various advantages, including reducing difficult computations and succinctly presenting key results. Some 2D and 3D graphical representations regarding presented solitons were considered for the appropriate values of the parameters. We also showed the effect of the physical parameters on the dynamical behavior of the presented solutions. Finally, the proposed approach may be expanded to tackle increasingly complicated problems in applied science.

    Citation: Mahmoud A. E. Abdelrahman, H. S. Alayachi. A reliable analytic technique and physical interpretation for the two-dimensional nonlinear Schrödinger equations[J]. AIMS Mathematics, 2024, 9(9): 24359-24371. doi: 10.3934/math.20241185

    Related Papers:

    [1] Yanping Yang, Muhammad Shoaib Saleem, Waqas Nazeer, Ahsan Fareed Shah . New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus via exponentially convex fuzzy interval-valued function. AIMS Mathematics, 2021, 6(11): 12260-12278. doi: 10.3934/math.2021710
    [2] Muhammad Bilal Khan, Pshtiwan Othman Mohammed, Muhammad Aslam Noor, Abdullah M. Alsharif, Khalida Inayat Noor . New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation. AIMS Mathematics, 2021, 6(10): 10964-10988. doi: 10.3934/math.2021637
    [3] Hongling Zhou, Muhammad Shoaib Saleem, Waqas Nazeer, Ahsan Fareed Shah . Hermite-Hadamard type inequalities for interval-valued exponential type pre-invex functions via Riemann-Liouville fractional integrals. AIMS Mathematics, 2022, 7(2): 2602-2617. doi: 10.3934/math.2022146
    [4] Fangfang Shi, Guoju Ye, Dafang Zhao, Wei Liu . Some integral inequalities for coordinated log-h-convex interval-valued functions. AIMS Mathematics, 2022, 7(1): 156-170. doi: 10.3934/math.2022009
    [5] Muhammad Bilal Khan, Gustavo Santos-García, Hüseyin Budak, Savin Treanțǎ, Mohamed S. Soliman . Some new versions of Jensen, Schur and Hermite-Hadamard type inequalities for (p,J)-convex fuzzy-interval-valued functions. AIMS Mathematics, 2023, 8(3): 7437-7470. doi: 10.3934/math.2023374
    [6] Manar A. Alqudah, Artion Kashuri, Pshtiwan Othman Mohammed, Muhammad Raees, Thabet Abdeljawad, Matloob Anwar, Y. S. Hamed . On modified convex interval valued functions and related inclusions via the interval valued generalized fractional integrals in extended interval space. AIMS Mathematics, 2021, 6(5): 4638-4663. doi: 10.3934/math.2021273
    [7] Muhammad Bilal Khan, Muhammad Aslam Noor, Thabet Abdeljawad, Bahaaeldin Abdalla, Ali Althobaiti . Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions. AIMS Mathematics, 2022, 7(1): 349-370. doi: 10.3934/math.2022024
    [8] Muhammad Bilal Khan, Savin Treanțǎ, Hleil Alrweili, Tareq Saeed, Mohamed S. Soliman . Some new Riemann-Liouville fractional integral inequalities for interval-valued mappings. AIMS Mathematics, 2022, 7(8): 15659-15679. doi: 10.3934/math.2022857
    [9] Zehao Sha, Guoju Ye, Dafang Zhao, Wei Liu . On interval-valued K-Riemann integral and Hermite-Hadamard type inequalities. AIMS Mathematics, 2021, 6(2): 1276-1295. doi: 10.3934/math.2021079
    [10] Muhammad Bilal Khan, Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Kamsing Nonlaopon, Y. S. Hamed . Some new Jensen, Schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions. AIMS Mathematics, 2022, 7(3): 4338-4358. doi: 10.3934/math.2022241
  • Nonlinear Schrödinger equations are a key paradigm in nonlinear research, attracting both mathematical and physical attention. This work was primarily concerned with the usage of a reliable analytic technique in order to solve two models of (2+1)-dimensional nonlinear Schrödinger equations. By applying a comprehensible wave transformation, every nonlinear model was simplified to an ordinary differential equation. A number of critical solutions were observed that correlated to various parameters. The provided approach has various advantages, including reducing difficult computations and succinctly presenting key results. Some 2D and 3D graphical representations regarding presented solitons were considered for the appropriate values of the parameters. We also showed the effect of the physical parameters on the dynamical behavior of the presented solutions. Finally, the proposed approach may be expanded to tackle increasingly complicated problems in applied science.



    In convex function theory, the classical Hermite-Hadamard inequality is one of the most well-known inequalities with geometrical interpretation, and it has a wide range of applications, see [1,2].

    Let S:KR+ be a convex function on a convex set K and ρ,ςK with ρς. Then,

    S(ρ+ς2)1ςρςρS(ϖ)dϖS(ρ)+S(ς)2. (1)

    In [3], Fejér looked at the key extensions of HH-inequality which is known as Hermite-Hadamard-Fejér inequality (HH-Fejér inequality).

    Let S:KR+ be a convex function on a convex set K and ρ,ς K with ρς. Then,

    S(ρ+ς2)1ςρD(ϖ)dϖςρS(ϖ)D(ϖ)dϖS(ρ)+S(ς)2ςρD(ϖ)dϖ. (2)

    If D(ϖ)=1, then we obtain (1) from (2). We should remark that Hermite-Hadamard inequality is a refinement of the idea of convexity, and it can be simply deduced from Jensen's inequality. In recent years, the Hermite-Hadamard inequality for convex functions has gotten a lot of attention, and there have been a lot of improvements and generalizations examined. Sarikaya [4] proved the Hadamard type inequality for coordinated convex functions such that

    Let G:ΔR+ be a coordinate convex function on Δ=[ς,ρ]×[μ,ν]. If G is double fractional integrable, then following inequalities hold:

    G(μ+ν2,ς+ρ2)Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]+Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (3)

    If α=1, then we obtain the following Dragomir inequality [5] on coordinates:

    G(μ+ν2,ς+ρ2)
    12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]1(νμ)(ρς)νμρςG(x,y)dydx14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (4)

    For more details related to inequalities, see [6,7,8,9] and reference therein.

    Interval analysis, on the other hand, is a well-known example of set-valued analysis, which is the study of sets in the context of mathematical analysis and general topology. It was created as a way of dealing with the interval uncertainty that can be found in many mathematical or computer models of deterministic real-world phenomena. Archimede's method, which is used to calculate the circumference of a circle, is an old example of an interval enclosure. Moore [10], who is credited with being the first user of intervals in computational mathematics, published the first book on interval analysis in 1966. Following the publication of his book, a number of scientists began to research the theory and applications of interval arithmetic. Interval analysis is now a helpful technique in a variety of fields that are interested in ambiguous data because of its applicability. Computer graphics, experimental and computational physics, error analysis, robotics, and many more fields have applications.

    Furthermore, in recent years, numerous major inequalities (Hermite-Hadamard, Ostrowski and others) have been addressed for interval-valued functions. Chalco-Cano et al. used the Hukuhara derivative for interval-valued functions to construct Ostrowski type inequalities for interval-valued functions in [11,12,13,14]. For interval-valued functions, Román-Flores et al. developed Minkowski and Beckenbach's inequality in [15]. For fuzzy interval-valued function, Khan et al. [16,17,18] derived some new versions of Hermite-Hadamard type inequalities and proved their validity with the help of non-trivial examples. Moreover, Khan et al. [19,20] discussed some novel types of Hermite-Hadamard type inequalities in fuzzy-interval fractional calculus and proved that many classical versions are special cases of these inequalities. Recently, Khan et al. [21] introduced the new class of convexity in fuzzy-interval calculus which is known as coordinated convex fuzzy-interval-valued functions and with the support of these classes, some Hermite-Hadamard type inequalities are obtained via newly defined fuzzy-interval double integrals. We encourage readers to [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54] for other related results.

    The following is an overview of the paper's structure. Section 2 recalls some preliminary notions and definitions. Moreover, some properties of introduced coordinated LR-convex IVF are also discussed. Section 3 presents some Hermite-Hadamard type inequalities for coordinated LR-convex IVF. With the help of this class, some fractional integral inequalities are also derived for the coordinated LR-convex IVF and for the product of two coordinated LR-convex IVFs. The fourth section, Conclusions and Future Work, brings us to a close.

    Let R be the set of real numbers and RI be the space of all closed and bounded intervals of R, such that URI is defined by

    U=[U,U]={yR|UyU},(U,UR). (5)

    If U=U, then U is said to be degenerate. If U0, then [U,U] is called positive interval. The set of all positive interval is denoted by R+I and defined as R+I={[U,U]:[U,U]RIandU0}.

    Let ϱR and ϱU be defined by

    ϱ.U={[ϱU,ϱU]ifϱ>0,{0}ifϱ=0,[ϱU,ϱU]ifϱ<0. (6)

    Then, the Minkowski difference DU, addition U+D and U×D for U,DRI are defined by

    [D,D][U,U]=[DU,DU],[D,D]+[U,U]=[D+U,D+U], (7)

    and

    [D,D]×[U,U]=[min{DU,DU,DU,DU},max{DU,DU,DU,DU}].

    The inclusion "⊇" means that

    UD if and only if, [U,U][D,D], and if and only if

    UD,DU. (8)

    Remark 1. [36] (ⅰ) The relation "≤p" is defined on RI by

    [D,D]p[U,U]ifandonlyifDU,DU, (9)

    for all [D,D],[U,U]RI, and it is a pseudo order relation. The relation [D,D]p[U,U] coincident to [D,D][U,U] on RI when it is "≤p"

    (ⅱ) It can be easily seen that "p" looks like "left and right" on the real line R, so we call "p" is "left and right" (or "LR" order, in short).

    For [D,D],[U,U]RI, the Hausdorff-Pompeiu distance between intervals [D,D] and [U,U] is defined by

    d([D,D],[U,U])=max{|DU|,|DU|}. (10)

    It is familiar fact that (RI,d) is a complete metric space.

    Theorem 1. [10] If G:[μ,ν]RRI is an I-V-F given by (x) [G(x),G(x)], then G is Riemann integrable over [μ,ν] if and only if, G and G both are Riemann integrable over [μ,ν] such that

    (IR)νμG(x)dx=[(R)νμG(x)dx,(R)νμG(x)dx]. (11)

    The collection of all Riemann integrable real valued functions and Riemann integrable I-V-F is denoted by R[μ,ν] and TR[μ,ν], respectively.

    Definition 1. [31,33] Let G:[μ,ν]RI be interval-valued function and GTR[μ,ν]. Then interval Riemann-Liouville-type integrals of G are defined as

    Iαμ+G(y)=1Γ(α)yμ(yt)α1G(t)dt(y>μ), (12)
    IανG(y)=1Γ(α)νy(ty)α1G(t)dt(y<ν), (13)

    where α>0 and Γ is the gamma function.

    Theorem 2. [20] Let G:[ς,ρ]RI+ be a LR-convex I-V.F such that G(y)=[G(y),G(y)] for all y[ς,ρ]. If GL([ς,ρ],R+I), then

    G(ς+ρ2)pΓ(α+1)2(ρς)α[Iας+G(ρ)+IαρG(ς)]pG(ς)+G(ρ)2. (14)

    Theorem 3. [20] Let G,S:[ς,ρ]R+I be two LR-convex I-V.Fs such that G(x)=[G(x),G(x)] and S(x)=[S(x),S(x)] for all x[ς,ρ]. If G×SL([ς,ρ],R+I) is fuzzy Riemann integrable, then

    Γ(α+1)2(ρς)α[Iας+G(ρ)×S(ρ)+IαρG(ς)×S(ς)]
    p(12α(α+1)(α+2))M(ς,ρ)+(α(α+1)(α+2))N(ς,ρ), (15)

    and

    G(ς+ρ2)×S(ς+ρ2)
    pΓ(α+1)4(ρς)α[Iας+G(ρ)×S(ρ)+IαρG(ς)×S(ς)]
    +12(12α(α+1)(α+2))M(ς,ρ)+12(α(α+1)(α+2))N(ς,ρ), (16)

    where M(ς,ρ)=G(ς)×S(ς)+G(ρ)×S(ρ), N(ς,ρ)=G(ς)×S(ρ)+G(ρ)×S(ς),

    and M(ς,ρ)=[M(ς,ρ),M(ς,ρ)] and N(ς,ρ)=[N(ς,ρ),N(ς,ρ)].

    Note that, the Theorem 1 is also true for interval double integrals. The collection of all double integrable I-V-F is denoted TOΔ, respectively.

    Theorem 4. [35] Let Δ=[ς,ρ]×[μ,ν]. If G:ΔRI is interval-valued doubl integrable (ID-integrable) on Δ. Then, we have

    (ID)ρςνμG(x,y)dydx=(IR)ρς(IR)νμG(x,y)dydx.

    Definition 2. [36] Let G:ΔR+I and GTOΔ. The interval Riemann-Liouville-type integrals Iα,βμ+,ς+,Iα,βμ+,ρ, Iα,βν,ς+,Iα,βν,ρ of G order α,β>0 are defined by

    Iα,βμ+,ς+G(x,y)=1Γ(α)Γ(β)xμyς(xt)α1(ys)β1G(t,s)dsdt(x>μ,y>ς), (17)
    Iα,βμ+,ρG(x,y)=1Γ(α)Γ(β)xμρy(xt)α1(sy)β1G(t,s)dsdt(x>μ,y<ρ), (18)
    Iα,βν,ς+G(x,y)=1Γ(α)Γ(β)νxyς(tx)α1(ys)β1G(t,s)dsdt(x<ν,y>ς), (19)
    Iα,βν,ρG(x,y)=1Γ(α)Γ(β)νxρy(tx)α1(sy)β1G(t,s)dsdt(x<ν,y<ρ). (20)

    Definition 3. [38] The I-V.F G:ΔR+I is said to be coordinated LR-convex I-V.F on Δ if

    G(τμ+(1τ)ν,sς+(1s)ρ)
    pτsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ), (21)

    for all (μ,ν),(ς,ρ)Δ, and τ,s[0,1]. If inequality (21) is reversed, then G is called coordinate LR-concave I-V.F on Δ.

    Lemma 1. [38] Let G:ΔR+I be an coordinated I-V.F on Δ. Then, G is coordinated LR-convex I-V.F on Δ, if and only if there exist two coordinated LR-convex I-V.Fs Gx:[ς,ρ]R+I, Gx(w)=G(x,w) and Gy:[μ,ν]R+I, Gy(z)=G(z,y).

    Theorem 5. [38] Let G:ΔR+I be a I-V.F on Δ such that

    G(x,ϖ)=[G(x,ϖ),G(x,ϖ)], (22)

    for all (x,ϖ)Δ. Then, G is coordinated LR-convex I-V.F on Δ, if and only if, G(x,ϖ) and G(x,ϖ) are coordinated convex functions.

    Example 1. We consider the I-V.Fs G:[0,1]×[0,1]R+I defined by,

    G(x)(σ)={σ2(6+ex)(6+eϖ),σ[0,2(6+ex)(6+eϖ)]4(6+ex)(6+eϖ)σ2(6+ex)(6+eϖ),σ(2(6+ex)(6+eϖ),4(6+ex)(6+eϖ)]0,otherwise,

    Then, for each θ[0,1], we have G(x)=[2θ(6+ex)(6+eϖ),(4+2θ)(6+ex)(6+eϖ)]. Since end point functions G((x,ϖ),θ), G((x,ϖ),θ) are coordinate concave functions for each θ[0,1]. Hence S(x,ϖ) is coordinate LR-concave I-V.F.

    From Lemma 1, we can easily note that each LR-convex I-V.F is coordinated LR-convex I-V.F. But the converse is not true.

    Remark 2. If one takes G(x,ϖ)=G(x,ϖ), then G is known as coordinated function if G satisfies the coming inequality

    G(τμ+(1τ)ν,sς+(1s)ρ)
    τsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    is valid which is defined by Dragomir [5]

    Let one takes G(x,ϖ)G(x,ϖ), where G(x,ϖ) is affine function and G(x,ϖ) is a concave function. If coming inequality,

    G(τμ+(1τ)ν,sς+(1s)ρ)
    τsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    is valid, then G is named as coordinated IVF which is defined by Zhao et al. [37, Definition 2 and Example 2]

    In this section, we shall continue with the following fractional HH-inequality for coordinated LR-convex I-V.Fs, and we also give fractional HH-Fejér inequality for coordinated LR-convex I-V.F through fuzzy order relation.

    Theorem 6. Let G:ΔR+I be a coordinate LR-convex I-V.F on Δ such that G(x,y)=[G(x,y),G(x,y)] for all (x,y)Δ. If GTOΔ, then following inequalities holds:

    G(μ+ν2,ς+ρ2)pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)8(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (23)

    If G(x) coordinated LR-concave I-V.F, then

    G(μ+ν2,ς+ρ2)pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)8(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (24)

    Proof. Let G:[μ,ν]R+I be a coordinated LR-convex I-V.F. Then, by hypothesis, we have

    4G(μ+ν2,ς+ρ2)pG(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ).

    By using Theorem 5, we have

    4G(μ+ν2,ς+ρ2)G(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ),4G(μ+ν2,ς+ρ2)G(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ).

    By using Lemma 1, we have

    2G(x,ς+ρ2)G(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ),2G(x,ς+ρ2)G(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ), (25)

    and

    2G(μ+ν2,y)G(τμ+(1τ)ν,y)+G((1τ)μ+tν,y),2G(μ+ν2,y)G(τμ+(1τ)ν,y)+G((1τ)μ+tν,y). (26)

    From (25) and (26), we have

    2[G(x,ς+ρ2),G(x,ς+ρ2)]
    p[G(x,τς+(1τ)ρ),G(x,τς+(1τ)ρ)]
    +[G(x,(1τ)ς+τρ),G(x,(1τ)ς+τρ)],

    and

    2[G(μ+ν2,y),G(μ+ν2,y)]
    p[G(τμ+(1τ)ν,y),G(τμ+(1τ)ν,y)]
    +[G(τμ+(1τ)ν,y),G(τμ+(1τ)ν,y)],

    It follows that

    G(x,ς+ρ2)pG(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ), (27)

    and

    G(μ+ν2,y)pG(τμ+(1τ)ν,y)+G(τμ+(1τ)ν,y). (28)

    Since G(x,.) and G(.,y), both are coordinated LR-convex-IVFs, then from inequality (14), inequalities (27) and (28) we have

    Gx(ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+Gx(ρ)+IβρGx(ς)]pGx(ς)+Gx(ρ)2. (29)

    and

    Gy(μ+ν2)pΓ(α+1)2(νμ)α[Iαμ+Gy(ν)+IανGy(μ)]pGy(μ)+Gy(ν)2 (30)

    Since Gx(w)=G(x,w), the inequality (29) can be written as

    G(x,ς+ρ2)pΓ(β+1)2(ρς)β[Iας+G(x,ρ)+IαρG(x,ς)]pG(x,ς)+G(x,ρ)2. (31)

    That is

    G(x,ς+ρ2)pβ2(ρς)β[ρς(ρs)β1G(x,s)ds+ρς(sς)β1G(x,s)ds]pG(x,ς)+G(x,ρ)2.

    Multiplying double inequality (31) by α(νx)α12(νμ)α and integrating with respect to x over [μ,ν], we have

    α2(νμ)ανμG(x,ς+ρ2)(νx)α1dx
    pνμρς(νx)α1(ρs)β1G(x,s)dsdx+νμρς(νx)α1(sς)β1G(x,s)dsdx
    pα4(νμ)α[νμ(νx)α1G(x,ς)dx+νμ(νx)α1G(x,ρ)dx]. (32)

    Again multiplying double inequality (31) by α(xμ)α12(νμ)α and integrating with respect to x over [μ,ν], we have

    α2(νμ)ανμG(x,ς+ρ2)(νx)α1dx
    pαβ4(νμ)α(ρς)βνμρς(xμ)α1(ρs)β1G(x,s)dsdx
    +αβ4(νμ)α(ρς)βνμρς(xμ)α1(sς)β1G(x,s)dsdx
    pα4(νμ)α[νμ(xμ)α1G(x,ς)dx+νμ(xμ)α1G(x,d)dx]. (33)

    From (32), we have

    Γ(α+1)2(νμ)α[Iαμ+G(ν,ς+ρ2)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βν,ς+G(ν,ς)]
    pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)]. (34)

    From (33), we have

    Γ(α+1)2(νμ)α[IανG(μ,ς+ρ2)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)4(νμ)α[IανG(μ,ς)+IανG(μ,ρ)]. (35)

    Similarly, since Gy(z)=G(z,y) then, from (34) and (35), (30) we have

    Γ(β+1)2(ρς)β[Iβς+G(μ+ν2,ρ)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βν,ς+G(μ,ρ)]
    pΓ(β+1)4(ρς)β[Iβς+G(μ,ρ)+Iβς+G(ν,ρ)], (36)

    and

    Γ(β+1)2(ρς)α[IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ρG(ν,ς)+Iα,βν,ρG(μ,ς)]
    pΓ(β+1)4(ρς)β[IβρG(μ,ς)+IβρG(ν,ς)]. (37)

    After adding the inequalities (46), (35), (36) and (37), we will obtain as resultant second, third and fourth inequalities of (23).

    Now, from left part of inequality (14), we have

    G(μ+ν2,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)], (38)

    and

    G(μ+ν2,ς+ρ2)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]. (39)

    Summing the inequalities (38) and (39), we obtain the following inequality:

    G(μ+ν2,ς+ρ2)
    pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)], (40)

    this is the first inequality of (23).

    Now, from right part of inequality (14), we have

    Γ(β+1)2(ρς)β[Iβς+G(μ,ρ)+IβρG(μ,ς)]pG(μ,ς)+G(μ,ρ)2, (41)
    Γ(β+1)2(ρς)β[Iβς+G(ν,ρ)+IβρG(ν,ς)]pG(ν,ς)+G(ν,ρ)2, (42)
    Γ(α+1)2(νμ)α[Iαμ+G(ν,ς)+IανG(μ,ς)]pG(μ,ς)+G(ν,ς)2, (43)
    Γ(α+1)2(νμ)α[Iαμ+G(ν,ρ)+IανG(μ,ρ)]pG(μ,ρ)+G(ν,ρ)2. (44)

    Summing inequalities (41), (42), (43) and (44), and then taking multiplication of the resultant with 14, we have

    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)+IανG(μ,ς)+Iαμ+G(ν,ρ)+IανG(μ,ρ)]
    +Γ(β+1)2(ρς)β[Iβς+G(μ,ρ)+IβρG(μ,ς)+Iβς+G(ν,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(μ,ρ)+G(ν,ς)+G(ν,ρ)4. (45)

    This is last inequality of (23) and the result has been proven.

    Remark 3. If one to take α=1 and β=1, then from (23), we achieve the coming inequality, see [38]:

    G(μ+ν2,ς+ρ2)
    p12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]p1(νμ)(ρς)νμρςG(x,y)dydxp14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (46)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then from Remark 2 and (24), we acquire the coming inequality, see [31]:

    G(μ+ν2,ς+ρ2)Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (47)

    Let one takes α=1 and β=1, G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then Remark 2 and from (24), we acquire the coming inequality, see [37]:

    G(μ+ν2,ς+ρ2)
    12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]1(νμ)(ρς)νμρςG(x,y)dydx
    14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (48)

    Example 2. We consider the I-V-Fs G:[0,1]×[0,1]R+I defined by,

    G(x)=[2,6](6+ex)(6+ey).

    Since end point functions G(x,y), G(x,y) are convex functions on coordinate, then G(x,y) is convex I-V-F on coordinate. Then for α=1 and β=1, we have

    G(μ+ν2,ς+ρ2)=[2(5+e12)2,6(6+e12)2],
    Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    =[4(6+e12)(5+e),12(6+e12)(5+e)],
    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    =[2(5+e)2,6(5+e)2],
    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    =[(5+e)(13+e),3(5+e)(13+e)]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4=[(6+e)(20+e)+492,6((6+e)(20+e)+49)2].

    That is

    [2(5+e12)2,6(6+e12)2]p[4(6+e12)(5+e),12(6+e12)(5+e)]
    p[2(5+e)2,6(5+e)2]
    p[(5+e)(13+e),3(5+e)(13+e)]
    p[(6+e)(20+e)+492,3((6+e)(20+e)+49)].

    Hence, Theorem 3.1 has been verified

    Next both results obtain Hermite-Hadamard type inequalities for the product of two coordinate LR-convex I-V.Fs

    Theorem 7. Let G,S:ΔR+I be a coordinate LR-convex I-V.Fs on Δ such that G(x,y)=[G(x,y),G(x,y)] and S(x,y)=[S(x,y),S(x,y)] for all (x,y)Δ. If G×STOΔ, then following inequalities holds:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (49)

    If G and S both are coordinate LR-concave I-V.Fs on Δ, then above inequality can be written as

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (50)

    Where

    K(μ,ν,ς,ρ)=G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς)+G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ),
    L(μ,ν,ς,ρ)=G(μ,ς)×S(ν,ς)˜+G(ν,ρ)×S(μ,ρ)+G(ν,ς)×S(μ,ς)+G(μ,ρ)×S(ν,ρ),
    M(μ,ν,ς,ρ)=G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ)+G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς),
    N(μ,ν,ς,ρ)=G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)+G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς).

    and K(μ,ν,ς,ρ), ˜L(μ,ν,ς,ρ), M(μ,ν,ς,ρ) and N(μ,ν,ς,ρ) are defined as follows:

    K(μ,ν,ς,ρ)=[K(μ,ν,ς,ρ),K(μ,ν,ς,ρ)],
    L(μ,ν,ς,ρ)=[L(μ,ν,ς,ρ),L(μ,ν,ς,ρ)],
    M(μ,ν,ς,ρ)=[M(μ,ν,ς,ρ),M(μ,ν,ς,ρ)],
    N(μ,ν,ς,ρ)=[N(μ,ν,ς,ρ),N(μ,ν,ς,ρ)].

    Proof. Let G and S both are coordinated LR-convex I-V.Fs on [μ,ν]×[ς,ρ]. Then

    G(τμ+(1τ)ν,sς+(1s)ρ)
    pτsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    and

    S(τμ+(1τ)ν,sς+(1s)ρ)
    pτsS(μ,ς)+τ(1s)S(μ,ρ)+(1τ)sS(ν,ς)+(1τ)(1s)S(ν,ρ).

    Since G and S both are coordinated LR-convex I-V.Fs, then by Lemma 1, there exist

    Gx:[ς,ρ]R+I,Gx(y)=G(x,y),Sx:[ς,ρ]R+I,Sx(y)=S(x,y),

    Since Gx, and Sx are I-V.Fs, then by inequality (15), we have

    Γ(β+1)2(ρς)β[Iβς+Gx(ρ)×Sx(ρ)+IβρGx(ς)×Sx(ς)]
    p(12β(β+1)(β+2))(Gx(ς)×Sx(ς)+Gx(ρ)×Sx(ρ))
    +(β(β+1)(β+2))(Gx(ς)×Sx(ρ)+Gx(ρ)×Sx(ς)).

    That is

    β2(ρς)β[ρς(ρy)β1G(x,y)×S(x,y)ρy+ρς(yς)β1G(x,y)×S(x,y)ρy]
    p(12β(β+1)(β+2))(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))
    +(β(β+1)(β+2))(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς)). (51)

    Multiplying double inequality (51) by α(νx)α12(νμ)α and integrating with respect to x over [μ,ν], we get

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς)). (52)

    Again, multiplying double inequality (51) by α(xμ)α12(νμ)α and integrating with respect to x over [μ,ν], we gain

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς)). (53)

    Summing (52) and (53), we have

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς))
    +Γ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς)). (54)

    Now, again with the help of integral inequality (15) for first two integrals on the right-hand side of (54), we have the following relation

    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς))
    p(12α(α+1)(α+2))(G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς))
    +(α(α+1)(α+2))(G(μ,ς)×S(ν,ς)+G(ν,ς)×S(μ,ς)). (55)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ))
    p(12α(α+1)(α+2))(G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ))
    +(α(α+1)(α+2))(G(μ,ρ)×S(ν,ρ)+G(ν,ρ)×S(μ,ρ)). (56)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ))
    p(12α(α+1)(α+2))(G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ))
    +(α(α+1)(α+2))(G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)). (57)

    And

    Γ(α+1)2(νμ)α(Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς))
    p(12α(α+1)(α+2))(G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς))
    +(α(α+1)(α+2))(G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς)). (58)

    From (55)–(58), inequality (54) we have

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ).

    Hence, the result has been proven.

    Remark 4. If one to take α=1 and β=1, then from (49), we achieve the coming inequality, see [38]:

    1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx
    p19K(μ,ν,ς,ρ)+118[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+136N(μ,ν,ς,ρ). (59)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then by Remark 2 and (50), we acquire the coming inequality, see [36]:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    (12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (60)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then by Remark 2 and (50), we acquire the coming inequality, see [37]:

    1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx
    19K(μ,ν,ς,ρ)+118[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+136N(μ,ν,ς,ρ). (61)

    If G(x,y)=G(x,y) and S(x,y)=S(x,y), then from (49), we acquire the coming inequality, see [39]:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    (12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (62)

    Theorem 8. Let G,S:ΔR+I be a coordinate LR-convex I-V.F on Δ such that G(x,y)=[G(x,y),G(x,y)] and S(x,y)=[S(x,y),S(x,y)] for all (x,y)Δ. If G×STOΔ, then following inequalities holds:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)
    +[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)
    +[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)
    +[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (63)

    If G and S both are coordinate LR-concave I-V.Fs on Δ, then above inequality can be written as

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)
    +[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (64)

    Where K(μ,ν,ς,ρ), L(μ,ν,ς,ρ), M(μ,ν,ς,ρ) and N(μ,ν,ς,ρ) are given in Theorem 7.

    Proof. Since G,S:ΔR+I be two LR-convex I-V.Fs, then from inequality (16), we have

    2G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pα2(νμ)α[νμ(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx+νμ(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx]+(α(α+1)(α+2))(G(μ,ς+ρ2)×S(μ,ς+ρ2)+G(ν,ς+ρ2)×S(ν,ς+ρ2))+(12α(α+1)(α+2))(G(μ,ς+ρ2)×S(ν,ς+ρ2)+G(ν,ς+ρ2)×S(μ,ς+ρ2)), (65)

    and

    2G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pβ2(ρς)β[ρς(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy+ρς(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]+(β(β+1)(β+2))(G(μ+ν2,ς)×S(μ+ν2,ς)+G(μ+ν2,ρ)×S(μ+ν2,ρ))+(12β(β+1)(β+2))(G(μ+ν2,ς)×S(μ+ν2,ρ)+G(μ+ν2,ρ)×S(μ+ν2,ς)), (66)

    Adding (73) and (74), and then taking the multiplication of the resultant one by 2, we obtain

    8G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pα2(νμ)α[νμ2(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx+νμ2(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx]+β2(ρς)β[ρς2(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy+ρς2(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]+(α(α+1)(α+2))(2G(μ,ς+ρ2)×S(μ,ς+ρ2)+2G(ν,ς+ρ2)×S(ν,ς+ρ2))+(12α(α+1)(α+2))(2G(μ,ς+ρ2)×S(ν,ς+ρ2)+2G(ν,ς+ρ2)×S(μ,ς+ρ2))+(β(β+1)(β+2))(2G(μ+ν2,ς)×S(μ+ν2,ς)+2G(μ+ν2,ρ)×S(μ+ν2,ρ))+(12β(β+1)(β+2))(2G(μ+ν2,ς)×S(μ+ν2,ρ)+2G(μ+ν2,ρ)×S(μ+ν2,ς)). (67)

    Again, with the help of integral inequality (16) and Lemma 1 for each integral on the right-hand side of (67), we have

    α2(νμ)ανμ2(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dxpαβ4(νμ)α(ρς)β[νμρς(νx)α1(ρy)β1G(x,y)dydx+νμρς(νx)α1(yς)β1G(x,y)dydx]+β(β+1)(β+2)α2(νμ)ανμ(νx)α1(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))dx+(12β(β+1)(β+2))α2(νμ)ανμ(νx)α1(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς))dx,=Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)]+Γ(α+1)2(νμ)α(β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς)). (68)
    α2(νμ)ανμ2(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dxpαβ4(νμ)α(ρς)β[νμρς(xμ)α1(ρy)β1G(x,y)dydx+νμρς(xμ)α1(yς)β1G(x,y)dydx]+β(β+1)(β+2)α2(νμ)ανμ(xμ)α1(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))dx+(12β(β+1)(β+2))α2(νμ)ανμ(xμ)α1(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς))dx,=Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+Γ(α+1)2(νμ)α(β(β+1)(β+2))(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))+Γ(α+1)2(νμ)α(12β(β+1)(β+2))(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς)). (69)
    β2(ρς)β[ρς2(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)]+Γ(β+1)2(ρς)β(α(α+1)(α+2))(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(12α(α+1)(α+2))(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(ν,ρ)). (70)
    β2(ρς)β[ρς2(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ρG(ν,ς)×S(ν,ς)]+Γ(β+1)2(ρς)β(α(α+1)(α+2))(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))+Γ(β+1)2(ρς)β(12α(α+1)(α+2))(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(ν,ς)). (71)

    And

    2G(μ+ν2,ς)×S(μ+ν2,ς)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς)]+α(α+1)(α+2)(G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς))+(12α(α+1)(α+2))(G(μ,ς)×S(ν,ς)+G(ν,ς)×S(μ,ς)), (72)
    2G(μ+ν2,ρ)×S(μ+ν2,ρ)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ)]+α(α+1)(α+2)(G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ))+(12α(α+1)(α+2))(G(μ,ρ)×S(ν,ρ)+G(ν,ρ)×S(μ,ρ)), (73)
    2G(μ+ν2,ς)×S(μ+ν2,ρ)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ)]+α(α+1)(α+2)(G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ))+(12α(α+1)(α+2))(G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)), (74)
    2G(μ+ν2,ρ)×S(μ+ν2,ς)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς)]
    +α(α+1)(α+2)(G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς))+(12α(α+1)(α+2))(G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς)), (75)
    2G(μ,ς+ρ2)×S(μ,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ,ρ)×S(μ,ρ)+IβρG(μ,ρ)×S(μ,ς)]+β(β+1)(β+2)(G(μ,ς)×S(μ,ς)+G(μ,ρ)×S(μ,ρ))+(12β(β+1)(β+2))(G(μ,ς)×S(μ,ρ)+G(μ,ρ)×S(μ,ς)), (76)
    2G(ν,ς+ρ2)×Sϕ(ν,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(ν,ρ)×S(ν,ρ)+IβρG(ν,ρ)×S(ν,ς)]+β(β+1)(β+2)(G(ν,ς)×S(ν,ς)+G(ν,ρ)×S(ν,ρ))+(12β(β+1)(β+2))(G(ν,ς)×S(ν,ρ)+G(ν,ρ)×S(ν,ς)), (77)
    2G(μ,ς+ρ2)×S(ν,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ,ρ)×S(ν,ρ)+IβρG(μ,ρ)×S(ν,ς)]+β(β+1)(β+2)(G(μ,ς)×S(ν,ς)+G(μ,ρ)×S(ν,ρ))+(12β(β+1)(β+2))(G(μ,ς)×S(ν,ρ)+G(μ,ρ)×S(ν,ς)), (78)

    and

    2G(ν,ς+ρ2)×S(μ,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(ν,ρ)×S(μ,ρ)+IβρG(ν,ρ)×S(μ,ς)]+β(β+1)(β+2)(G(ν,ς)×S(μ,ς)+G(ν,ρ)×S(μ,ρ))+(12β(β+1)(β+2))(G(ν,ς)×S(μ,ρ)+G(ν,ρ)×S(μ,ς)), (79)

    From inequalities (68) to (79), inequality (67) we have

    8G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)2(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+(2α(α+1)(α+2))[Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))]+2(12α(α+1)(α+2))[Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(μ,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(μ,ς))]+2(β(β+1)(β+2))[Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))]+2(12β(β+1)(β+2))[Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))]
    +2α(α+1)(α+2)β(β+1)(β+2)K(μ,ν,ς,ρ)++(12α(α+1)(α+2))2β(β+1)(β+2)L(μ,ν,ς,ρ)
    +2α(α+1)(α+2)(12β(β+1)(β+2))M(μ,ν,ς,ρ)+2(12α(α+1)(α+2))(12β(β+1)(β+2))N(μ,ν,ς,ρ). (80)

    Again, with the help of integral inequality (15) and Lemma 1, for each integral on the right-hand side of (80), we have

    Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))p(12β(β+1)(β+2))K(μ,ν,ς,ρ)+β(β+1)(β+2)M(μ,ν,ς,ρ). (81)
    Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(μ,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(μ,ς))p(12β(β+1)(β+2))L(μ,ν,ς,ρ)+β(β+1)(β+2)N(μ,ν,ς,ρ). (82)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))p(12α(α+1)(α+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)L(μ,ν,ς,ρ). (83)
    Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))p(12α(α+1)(α+2))M(μ,ν,ς,ρ)+α(α+1)(α+2)N(μ,ν,ς,ρ). (84)

    From (77) to (84), (80) we have

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (85)

    This concludes the proof of Theorem 8 result has been proven.

    Remark 5. If we take α=1 and β=1, then from (63), we achieve the coming inequality, see [38]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)p1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx+536K(μ,ν,ς,ρ)+736[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+29N(μ,ν,ς,ρ). (86)

    Let one takes G(x,y) is an affine function and G(x,y) is convex function. If G(x,y)G(x,y), then from Remark 2 and (64), we acquire the coming inequality, see [37]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx+536K(μ,ν,ς,ρ)+736[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+29N(μ,ν,ς,ρ). (87)

    Let one takes G(x,y) is an affine function and G(x,y) is convex function. If G(x,y)G(x,y), then from Remark 2 and (64) we acquire the coming inequality, see [36]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (88)

    If we take G(x,y)=G(x,y) and S(x,y)=S(x,y), then from (63), we acquire the coming inequality, see [39]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (89)

    In this study, with the help of coordinated LR-convexity for interval-valued functions, several novel Hermite-Hadamard type inequalities are presented. It is also demonstrated that the conclusions reached in this study represent a possible extension of previously published equivalent results. Similar inequalities may be discovered in the future using various forms of convexities. This is a novel and intriguing topic, and future study will be able to find equivalent inequalities for various types of convexity and coordinated m-convexity by using different fractional integral operators.

    The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research. All authors read and approved the final manuscript. This work was funded by Taif University Researchers Supporting Project number (TURSP-2020/345), Taif University, Taif, Saudi Arabia.

    The authors declare that they have no competing interests.



    [1] H. Zhang, X. Yang, Q. Tang, D. Xu, A robust error analysis of the OSC method for a multi-term fourth-order sub-diffusion equation, Comput. Math. Appl., 109 (2022), 180–190. https://doi.org/10.1016/j.camwa.2022.01.007 doi: 10.1016/j.camwa.2022.01.007
    [2] X. Yang, H. Zhang, J. Tang, The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions, Comput. Math. Appl., 82 (2021), 1–12. https://doi.org/10.1016/j.camwa.2020.11.015 doi: 10.1016/j.camwa.2020.11.015
    [3] H. Zhang, X. Yang, D. Xu, Unconditional convergence of linearized orthogonal spline collocation algorithm for semilinear subdiffusion equation with nonsmooth solution, Numer. Meth. Part. Differ. Equ., 37 (2021), 1361–1373. https://doi.org/10.1002/num.22583 doi: 10.1002/num.22583
    [4] A. F. Daghistani, A. M. T. Abd El-Bar, A. M. Gemeay, M. A. E. Abdelrahman, S. Z. Hassan, A hyperbolic secant-squared distribution via the nonlinear evolution equation and its application, Mathematics, 11 (2023), 4270. https://doi.org/10.3390/math11204270 doi: 10.3390/math11204270
    [5] M. A. E. Abdelrahman, G. Alshreef, Closed-form solutions to the new coupled Konno–Oono equation and the Kaup-Newell model equation in magnetic field with novel statistic application, Eur. Phys. J. Plus, 136 (2021), 455. https://doi.org/10.1140/epjp/s13360-021-01472-2 doi: 10.1140/epjp/s13360-021-01472-2
    [6] Y. Cheng, A. Chertock, M. Herty, A. Kurganov, T. Wu, A new approach for designing moving-water equilibria preserving schemes for the shallow water equations, J. Sci. Comput., 80 (2019), 538–554. https://doi.org/10.1007/s10915-019-00947-w doi: 10.1007/s10915-019-00947-w
    [7] P. Ripa, Conservation laws for primitive equations models with inhomogeneous layers, Geophys. Astrophys. Fluid Dynam., 70 (1993), 85–111. https://doi.org/10.1080/03091929308203588 doi: 10.1080/03091929308203588
    [8] G. Laibe, D. J. Price, Dusty gas with one fluid, Mon. Not. R. Astron. Soc., 440 (2014), 2136–2146. https://doi.org/10.1093/mnras/stu355 doi: 10.1093/mnras/stu355
    [9] Y. Shi, X. Yang, A time two-grid difference method for nonlinear generalized viscous Burgers' equation, J. Math. Chem., 62 (2024), 1323–1356. https://doi.org/10.1007/s10910-024-01592-x doi: 10.1007/s10910-024-01592-x
    [10] C. Li, H. Zhang, X. Yang, A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel, J. Appl. Math. Comput., 70 (2024), 2045–2077. https://doi.org/10.1007/s12190-024-02039-x doi: 10.1007/s12190-024-02039-x
    [11] H. Zhang, X. Yang, Y. Liu, Y. Liu, An extrapolated CN-WSGD OSC method for a nonlinear time fractional reaction-diffusion equation, Appl. Numer. Math., 157 (2020), 619–633. https://doi.org/10.1016/j.apnum.2020.07.017 doi: 10.1016/j.apnum.2020.07.017
    [12] H. Zhang, X. Yang, D. Xu, An efficient spline collocation method for a nonlinear fourth-order reaction subdiffusion equation, J. Sci. Comput., 85 (2020), 7. https://doi.org/10.1007/s10915-020-01308-8 doi: 10.1007/s10915-020-01308-8
    [13] X. Yang, H. Zhang, Q. Tang, A spline collocation method for a fractional mobile–immobile equation with variable coefficients, Comput. Appl. Math., 39 (2020), 34. https://doi.org/10.1007/s40314-019-1013-3 doi: 10.1007/s40314-019-1013-3
    [14] H.S. Alayachi, The modulations of higher order solitonic pressure and energy of fluid filled elastic tubes, AIP Adv., 13 (2023), 115214. https://doi.org/10.1063/5.0179155 doi: 10.1063/5.0179155
    [15] X. Yang, W. Qiu, H. Zhang, L. Tang, An efficient alternating direction implicit finite difference scheme for the three-dimensional time-fractional telegraph equation, Comput. Math. Appl., 102 (2021), 233–247. https://doi.org/10.1016/j.camwa.2021.10.021 doi: 10.1016/j.camwa.2021.10.021
    [16] H. Zhang, Y. Liu, X. Yang, An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space, J. Appl. Math. Comput., 69 (2023), 651–674. https://doi.org/10.1007/s12190-022-01760-9 doi: 10.1007/s12190-022-01760-9
    [17] X. Yang, W. Qiu, H. Chen, H. Zhang, Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space, Appl. Numer. Math., 172 (2022), 497–513. https://doi.org/10.1016/j.apnum.2021.11.004 doi: 10.1016/j.apnum.2021.11.004
    [18] H. Zhang, X. Jiang, F. Wang, X. Yang, The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation, J. Appl. Math. Comput., 70 (2024), 1127–1151. https://doi.org/10.1007/s12190-024-02000-y doi: 10.1007/s12190-024-02000-y
    [19] H. G. Abdelwahed, M. A. E. Abdelrahman, M. Inc, R. Sabry, New soliton applications in earth's magnetotail plasma at critical densities, Front. Phys., 8 (2020), 181. https://doi.org/10.3389/fphy.2020.00181 doi: 10.3389/fphy.2020.00181
    [20] S. Zhang, C. Tian, W. Y. Qian, Bilinearization and new multi-soliton solutions for the (4+1)-dimensional Fokas equation, Pramana-J. Phys., 86 (2016), 1259–1267. https://doi.org/10.1007/s12043-015-1173-7 doi: 10.1007/s12043-015-1173-7
    [21] L. Akinyemi, M. Şenol, U. Akpan, K. Oluwasegun, The optical soliton solutions of generalized coupled nonlinear Schrödinger-Korteweg-de Vries equations, Opt. Quant. Electron., 53 (2021), 394. https://doi.org/10.1007/s11082-021-03030-7 doi: 10.1007/s11082-021-03030-7
    [22] F. Mirzaee, S. Rezaei, N. Samadyar, Numerical solution of two-dimensional stochastic time-fractional sine-Gordon equation on non-rectangular domains using finite difference and meshfree methods, Eng. Anal. Bound. Elem., 127 (2021), 53–63. https://doi.org/10.1016/j.enganabound.2021.03.009 doi: 10.1016/j.enganabound.2021.03.009
    [23] M. A. E. Abdelrahman, H. AlKhidhr, A robust and accurate solver for some nonlinear partial differential equations and tow applications, Phys. Scr., 95 (2020), 065212. https://doi.org/10.1088/1402-4896/ab80e7 doi: 10.1088/1402-4896/ab80e7
    [24] Z. Zhou, H. Zhang, X. Yang, CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics, Appl. Math. Comput., 474 (2024), 128680. https://doi.org/10.1016/j.amc.2024.128680 doi: 10.1016/j.amc.2024.128680
    [25] X. F. Yang, Z. C. Deng, Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Differ. Equ., 2015 (2015), 117. https://doi.org/10.1186/s13662-015-0452-4 doi: 10.1186/s13662-015-0452-4
    [26] W. Wang, H. Zhang, Z. Zhou, X. Yang, A fast compact finite difference scheme for the fourth-order diffusion-wave equation, Int. J. Comput. Math., 101 (2024), 170–193. https://doi.org/10.1080/00207160.2024.2323985 doi: 10.1080/00207160.2024.2323985
    [27] B. Q. Li, Y. L. Ma, Interaction properties between rogue wave and breathers to the manakov system arising from stationary self-focusing electromagnetic systems, Chaos Soliton. Fract., 156 (2022), 111832. https://doi.org/10.1016/j.chaos.2022.111832 doi: 10.1016/j.chaos.2022.111832
    [28] X. Jin, J. Jiang, J. Chi, X. Wu, Adaptive finite-time pinned and regulation synchronization of disturbed complex networks, Commun. Nonlinear Sci., 124 (2023), 107319. https://doi.org/10.1016/j.cnsns.2023.107319 doi: 10.1016/j.cnsns.2023.107319
    [29] Z. J. Yang, S. M. Zhang, X. L. Li, Z. G. Pang, H. X. Bu, High-order revivable complex-valued hyperbolic-sine-Gaussian solitons and breathers in nonlinear media with a spatial nonlocality, Nonlinear Dyn., 94 (2018), 2563–2573. https://doi.org/10.1007/s11071-018-4510-9 doi: 10.1007/s11071-018-4510-9
    [30] Z. Sun, J. Li, R. Bian, D. Deng, Z. Yang, Transmission mode transformation of rotating controllable beams induced by the cross phase, Opt. Express, 32 (2024), 9201–9212. https://doi.org/10.1364/OE.520342 doi: 10.1364/OE.520342
    [31] M. A. E. Abdelrahman, N. F. Abdo, On the nonlinear new wave solutions in unstable dispersive environments, Phys. Scr., 95 (2020), 045220. https://doi.org/10.1088/1402-4896/ab62d7 doi: 10.1088/1402-4896/ab62d7
    [32] H. G. Abdelwahed, M. A. E. Abdelrahman, S. Alghanim, N. F. Abdo, Higher-order Kerr nonlinear and dispersion effects on fiber optics, Results Phys., 26 (2021), 104268. https://doi.org/10.1016/j.rinp.2021.104268 doi: 10.1016/j.rinp.2021.104268
    [33] J. L. Lebowitz, H. A. Rose, E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Stat. Phys., 50 (1988), 657–687. https://doi.org/10.1007/BF01026495 doi: 10.1007/BF01026495
    [34] G. D. McDonald, C. C. N. Kuhn, K. S. Hardman, S. Bennetts, P. J. Everitt, P. A. Altin, et al., Bright solitonic matter-wave interferometer, Phys. Rev. Lett., 113 (2014), 013002. https://doi.org/10.1103/PhysRevLett.113.013002 doi: 10.1103/PhysRevLett.113.013002
    [35] Y. L. Ma, Nth-order rogue wave solutions for a variable coefficient Schrödinger equation in inhomogeneous optical fibers, Optik, 251 (2022), 168103. https://doi.org/10.1016/j.ijleo.2021.168103 doi: 10.1016/j.ijleo.2021.168103
    [36] B. Q. Li, Y. L. Ma, Interaction properties between rogue wave and breathers to the manakov system arising from stationary self-focusing electromagnetic systems, Chaos Soliton. Fract., 156 (2022), 111832. https://doi.org/10.1016/j.chaos.2022.111832 doi: 10.1016/j.chaos.2022.111832
    [37] O. V. Marchukov, B. A. Malomed, V. A. Yurovsky, M. Olshanii, V. Dunjko, R. G. Hulet, Splitting of nonlinear-Schrödinger-equation breathers by linear and nonlinear localized potentials, Phys. Rev. A, 99 (2019), 063623. https://doi.org/10.1103/PhysRevA.99.063623 doi: 10.1103/PhysRevA.99.063623
    [38] S. Shen, Z. Yang, X. Li, S. Zhang, Periodic propagation of complex-valued hyperbolic-cosine-Gaussian solitons and breathers with complicated light field structure in strongly nonlocal nonlinear media, Commun. Nonlinear Sci., 103 (2021), 106005. https://doi.org/10.1016/j.cnsns.2021.106005 doi: 10.1016/j.cnsns.2021.106005
    [39] Z. Y. Sun, D. Deng, Z. G. Pang, Z. J. Yang, Nonlinear transmission dynamics of mutual transformation between array modes and hollow modes in elliptical sine-Gaussian cross-phase beams, Chaos Soliton. Fract., 178 (2024), 114398. https://doi.org/10.1016/j.chaos.2023.114398 doi: 10.1016/j.chaos.2023.114398
    [40] S. Shen, Z. J. Yang, Z. G. Pang, Y. R. Ge, The complex-valued astigmatic cosine-Gaussian soliton solution of the nonlocal nonlinear Schrödinger equation and its transmission characteristics, Appl. Math. Lett., 125 (2022), 107755. https://doi.org/10.1016/j.aml.2021.107755 doi: 10.1016/j.aml.2021.107755
    [41] L. M. Song, Z. J. Yang, X. L. Li, S. M. Zhang, Coherent superposition propagation of Laguerre-Gaussian and Hermite-Gaussian solitons, Appl. Math. Lett., 102 (2020), 106114. https://doi.org/10.1016/j.aml.2019.106114 doi: 10.1016/j.aml.2019.106114
    [42] M. Najafi, S. Arbabi, Traveling wave solutions for nonlinear Schrödinger equations, Optik, 126 (2015), 3992–3997. https://doi.org/10.1016/j.ijleo.2015.07.165 doi: 10.1016/j.ijleo.2015.07.165
    [43] M. Dehghan, A. Shokri, A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions, Comput. Math. Appl., 54 (2007), 136–146. https://doi.org/10.1016/j.camwa.2007.01.038 doi: 10.1016/j.camwa.2007.01.038
    [44] S. V. Mousavi, S. Miret-Artés, On non-linear Schrödinger equations for open quantum systems, Eur. Phys. J. Plus, 134 (2019), 431. https://doi.org/10.1140/epjp/i2019-12965-6 doi: 10.1140/epjp/i2019-12965-6
    [45] W. Huang, C. Xu, S. T. Chu, S. K. Chaudhuri, The finite-difference vector beam propagation method: Analysis and assessment, J. Lightwave Technol., 10 (1992), 295–305. https://doi.org/10.1109/50.124490 doi: 10.1109/50.124490
    [46] A. I. Aliyu, M. Inc, A. Yusuf, D. Baleanu, Optical solitary waves and conservation laws to the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation, Mod. Phys. Lett. B, 32 (2018), 1850373. https://doi.org/10.1142/S0217984918503736 doi: 10.1142/S0217984918503736
    [47] H. Durur, E. Ilhan, H. Bulut, Novel complex wave solutions of the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation, Fractal Fract., 4 (2020), 41. https://doi.org/10.3390/fractalfract4030041 doi: 10.3390/fractalfract4030041
    [48] D. Baleanu, K. Hosseini, S. Salahshour, K. Sadri, M. Mirzazadeh, C. Park, A. Ahmadian, The (2+1)-dimensional hyperbolic nonlinear Schrödinger equation and its optical solitons, AIMS Mathematics, 6 (2021), 9568–9581. https://doi.org/10.3934/math.2021556 doi: 10.3934/math.2021556
    [49] G. Ai-Lin, L. Ji, Exact solutions of (2+1)-dimensional HNLS equation, Commun. Theor. Phys., 54 (2010), 401. https://doi.org/10.1088/0253-6102/54/3/04 doi: 10.1088/0253-6102/54/3/04
    [50] X. Yang, H. Zhang, The uniform l1 long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data, Appl. Math. Lett., 124 (2022), 107644. https://doi.org/10.1016/j.aml.2021.107644 doi: 10.1016/j.aml.2021.107644
    [51] X. Yang, H. Zhang, Q. Zhang, G. Yuan, Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes, Nonlinear Dyn., 108 (2022), 3859–3886. https://doi.org/10.1007/s11071-022-07399-2 doi: 10.1007/s11071-022-07399-2
    [52] X. Yang, Z. Zhang, Analysis of a new NFV scheme preserving DMP for two-dimensional sub-diffusion equation on distorted meshes, J. Sci. Comput., 99 (2024), 80. https://doi.org/10.1007/s10915-024-02511-7 doi: 10.1007/s10915-024-02511-7
    [53] X. Yang, Z. Zhang, On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations, Appl. Math. Lett., 150 (2024), 108972. https://doi.org/10.1016/j.aml.2023.108972 doi: 10.1016/j.aml.2023.108972
  • This article has been cited by:

    1. Waqar Afzal, Khurram Shabbir, Thongchai Botmart, Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued (h1,h2)-Godunova-Levin functions, 2022, 7, 2473-6988, 19372, 10.3934/math.20221064
    2. Muhammad Bilal Khan, Omar Mutab Alsalami, Savin Treanțǎ, Tareq Saeed, Kamsing Nonlaopon, New class of convex interval-valued functions and Riemann Liouville fractional integral inequalities, 2022, 7, 2473-6988, 15497, 10.3934/math.2022849
    3. Miguel J. Vivas-Cortez, Hasan Kara, Hüseyin Budak, Muhammad Aamir Ali, Saowaluck Chasreechai, Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions, 2022, 20, 2391-5455, 1887, 10.1515/math-2022-0477
    4. Tareq Saeed, Eze R. Nwaeze, Muhammad Bilal Khan, Khalil Hadi Hakami, New Version of Fractional Pachpatte-Type Integral Inequalities via Coordinated ℏ-Convexity via Left and Right Order Relation, 2024, 8, 2504-3110, 125, 10.3390/fractalfract8030125
    5. HAIYANG CHENG, DAFANG ZHAO, GUOHUI ZHAO, FRACTIONAL QUANTUM HERMITE–HADAMARD-TYPE INEQUALITIES FOR INTERVAL-VALUED FUNCTIONS, 2023, 31, 0218-348X, 10.1142/S0218348X23501049
    6. Haiyang Cheng, Dafang Zhao, Guohui Zhao, Delfim F. M. Torres, New quantum integral inequalities for left and right log-ℏ-convex interval-valued functions, 2024, 31, 1072-947X, 381, 10.1515/gmj-2023-2088
  • Reader Comments
  • © 2024 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(858) PDF downloads(47) Cited by(0)

Figures and Tables

Figures(7)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog