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

Stochastic comparisons of extreme order statistic from dependent and heterogeneous lower-truncated Weibull variables under Archimedean copula

  • Received: 13 October 2021 Revised: 31 December 2021 Accepted: 09 January 2022 Published: 26 January 2022
  • MSC : Primary 90B25; Secondary 60E15, 60K10

  • This article studies the stochastic comparisons of order statistics with dependent and heterogeneous lower-truncated Weibull samples under Archimedean copula. To begin, we obtain the usual stochastic and hazard rate orders of the largest and smallest order statistics from heterogeneous and dependent lower-truncated Weibull samples under Archimedean copula. Second, under Archimedean copula, we get the convex transform and the dispersive orders of the largest and smallest order statistics from dependent and heterogeneous lower-truncated Weibull samples. Finally, several numerical examples are given to demonstrate the theoretical conclusions.

    Citation: Xiao Zhang, Rongfang Yan. Stochastic comparisons of extreme order statistic from dependent and heterogeneous lower-truncated Weibull variables under Archimedean copula[J]. AIMS Mathematics, 2022, 7(4): 6852-6875. doi: 10.3934/math.2022381

    Related Papers:

    [1] Shakir Ali, Amal S. Alali, Atif Ahmad Khan, Indah Emilia Wijayanti, Kok Bin Wong . XOR count and block circulant MDS matrices over finite commutative rings. AIMS Mathematics, 2024, 9(11): 30529-30547. doi: 10.3934/math.20241474
    [2] Weitao Xie, Jiayu Zhang, Wei Cao . On the number of the irreducible factors of xn1 over finite fields. AIMS Mathematics, 2024, 9(9): 23468-23488. doi: 10.3934/math.20241141
    [3] Xiaoer Qin, Li Yan . Some specific classes of permutation polynomials over Fq3. AIMS Mathematics, 2022, 7(10): 17815-17828. doi: 10.3934/math.2022981
    [4] Qian Liu, Jianrui Xie, Ximeng Liu, Jian Zou . Further results on permutation polynomials and complete permutation polynomials over finite fields. AIMS Mathematics, 2021, 6(12): 13503-13514. doi: 10.3934/math.2021783
    [5] Phitthayathon Phetnun, Narakorn R. Kanasri . Further irreducibility criteria for polynomials associated with the complete residue systems in any imaginary quadratic field. AIMS Mathematics, 2022, 7(10): 18925-18947. doi: 10.3934/math.20221042
    [6] Taekyun Kim, Dae San Kim, Dmitry V. Dolgy, Hye Kyung Kim, Hyunseok Lee . A new approach to Bell and poly-Bell numbers and polynomials. AIMS Mathematics, 2022, 7(3): 4004-4016. doi: 10.3934/math.2022221
    [7] Jovanny Ibarguen, Daniel S. Moran, Carlos E. Valencia, Rafael H. Villarreal . The signature of a monomial ideal. AIMS Mathematics, 2024, 9(10): 27955-27978. doi: 10.3934/math.20241357
    [8] Kaimin Cheng . Permutational behavior of reversed Dickson polynomials over finite fields. AIMS Mathematics, 2017, 2(2): 244-259. doi: 10.3934/Math.2017.2.244
    [9] Ian Marquette . Representations of quadratic Heisenberg-Weyl algebras and polynomials in the fourth Painlevé transcendent. AIMS Mathematics, 2024, 9(10): 26836-26853. doi: 10.3934/math.20241306
    [10] Varsha Jarali, Prasanna Poojary, G. R. Vadiraja Bhatta . A recent survey of permutation trinomials over finite fields. AIMS Mathematics, 2023, 8(12): 29182-29220. doi: 10.3934/math.20231495
  • This article studies the stochastic comparisons of order statistics with dependent and heterogeneous lower-truncated Weibull samples under Archimedean copula. To begin, we obtain the usual stochastic and hazard rate orders of the largest and smallest order statistics from heterogeneous and dependent lower-truncated Weibull samples under Archimedean copula. Second, under Archimedean copula, we get the convex transform and the dispersive orders of the largest and smallest order statistics from dependent and heterogeneous lower-truncated Weibull samples. Finally, several numerical examples are given to demonstrate the theoretical conclusions.



    Partial differential equations and fractional partial differential equations have developed in a variety of scientific disciplines [1,2,3,4]. The popularity of FPDEs, on the other hand, has recently increased as a result of their extraordinary capacity to offer more accurate simulations for a variety of physical phenomena. The integration of non-local memory effects and long-range interactions through fractional derivatives, which take into account the complexity present in many complex systems, is the cause of this increased accuracy [5,6,7,8,9]. FPDEs are a strong tool for modeling and analyzing a wide range of phenomena, including heat conduction, wave propagation, image processing and data analysis. They have a wide range of applications in physics, biology, economics and engineering. As a result, the study of FPDEs has developed into a booming research area, stimulating the identification of conceptual foundations and novel methods for resolving these equations, enabling new applications in a variety of real-world situations [10,11,12,13].

    The solution of FPDEs may be handled in two ways: Numerical and analytical techniques. The finite difference approach [14,15], finite element method [16], this Monte Carlo method [17], shooting method [18] and the adaptive moving mesh and uniform mesh methods [19,20,21,22,23] are examples of numerical methodologies that rely on numerical algorithms and computational simulations to approximate the solution. Analytical methods, on the other hand, use algebraic and calculus techniques to solve issues and produce accurate solutions. Analytical procedures are frequently preferred over numerical methods because they provide this more thorough knowledge of the problem and its underlying behavior. Analytical approaches are very useful when dealing with issues that have simple geometries or equations that can be solved in closed form, increasing efficiency in such situations [24,25,26].

    As a result, several researchers have explored distinct FPDEs with diverse analytical methodologies. The Laplace Adomian decomposition technique [27], (G'/G)-expansion technique [28,29,30], perturbation methods [31], direct algebraic methods [32], variational iteration method [33], exp-function approach [34,35], auxiliary equation method [36], Jacobian elliptic function method [37], Riccati mapping method [38], Darboux transformation method [39], Hirota bilinear method [40] and modified extended DAM (mEDAM) [41] are some widely used analytical methods. For example, in groundwater modeling, Al-Mdallal et al. employed the Laplace transform approach to solve a FPDE [42]. Jiang et al. used a variable separation approach to solve the multi-term time-fractional diffusion-wave equation in a finite domain in another work [43]. Similarly, Zheng solved the nonlinear fractional Sharma-Tasso-Olver problem by using the exp-function technique [44]. Finally, Khan et al. used the (G'/G)-expansion approach to achieve accurate solutions for FPDEs [45]. Overall, these analytical strategies have been shown to be helpful in addressing various types of FPDEs in a variety of scientific and technical domains.

    For its accuracy and effectiveness in solving FPDEs, the DAM stands out as a very strong and effective analytical strategy. The DAM differs from previous transformation-based systems in that it can convert FPDEs directly into a system of nonlinear equations without the need for a linearization phase. Using a recommended series-based solution derived from an ordinary differential equation (ODE) solution, the FPDE is first transformed into a nonlinear ODE (NODE), which is then transformed into a system of algebraic equations. The DAM is distinguished by three variants: simple DAM [46], extended DAM (EDAM) [47] and mEDAM [48,49,50]. An enhanced version of the DAM known as the mEDAM, has shown to be extremely efficient in handling a variety of FPDE forms. As a result, the DAM offers a simple, effective, and precise solution to FPDEs, emphasizing its potential for significant contributions in a range of fields of science and technology [51,52,53].

    The generalized modified Caamassa-Holm (CH) and Degasperis-Procesi (DP) equations were initially presented and studied by Wazwaz [54]. These modifications to the DP equation were created by Wazwaz [54] as a tool for identifying discrepancies in the physical makeup of the generated solution. It is important to remember that the standard DP equation has multi-peakon solutions. The characteristics of these peakon solutions are altered to bell-shaped solutions in the updated DP equation, though. When examining shallow water dynamics, this equation is applicable, practical and integrable. The fractional modified DP (MDP) equation is a nonlinear FPDE that is created by fusing the MDP equation with fractional derivatives. A versatile equation with uses in fluid dynamics, oceanography and image processing is the fractional MDP equation. The equation has been applied to fluid dynamics to study interactions with submerged objects, characterize wave packet propagation and mimic wave behavior in shallow water. The equation has been applied to oceanography to analyze wave dynamics, interactions between waves and currents and soliton formation. The equation has been applied to image processing to create algorithms for object recognition, image segmentation and edge detection. Because standard derivatives are unable to simulate events with non-local repercussions and long-range dependencies, the inclusion of fractional derivatives in the equation is crucial. Particularly fractional derivatives capture the non-local interactions and memory effects that are prevalent in many real-world systems. The fractional generalized MDP and CH equation has the following mathematical form:

    DαtuDαt(Dβx(Dβxu))+(μ+1)u2Dβxu=μDβx(u)Dβx(Dβxu)+uDβx(Dβx(Dβxu)), (1.1)

    where u=u(x,t), 0<α,β1, μ is a constant and t0. In this study we have solved (1.1) with μ=3 thus (1.1) becomes the fractional MDP equation given by

    DαtuDαt(Dβx(Dβxu))+4u2Dβxu=3Dβx(u)Dβx(Dβxu)+uDβx(Dβx(Dβxu)). (1.2)

    To model compressible fluid flow in a medium, the fractional gas dynamics equation is utilized. Aeronautical engineering, combustion research, and materials science are all fields that utilize the equation. Its solutions have been applied in the simulation of shock waves, turbulent flows, and other complex fluid phenomena. The use of fractional derivatives in the equation results in a more accurate representation of the system's dynamics, making it a useful tool for understanding the behavior of compressible fluid flow under a variety of conditions [55,56,57]. The fractional gas dynamics equation has the following mathematical form [45]:

    Dαtu(x,t)+12Dβx(Dβxu2(x,t))u(x,t)+u2(x,t)=0.0<α,β1,t0. (1.3)

    Both the fractional MDP and fractional gas dynamics equations have been treated analytically and numerically in the literature. For example, Dubey et al. in [58] investigated the time-fractional MDP equation with a Caputo fractional derivative by using the Sumudu transform and q-homotopy analysis approach. In [59], Zhang et al. offered a detailed comparison of two strong analytical approaches for generating series solutions to fractional DP equations, namely the homotopy perturbation transform method and the Aboodh Adomian decomposition method. Similarly, Das and Kumar successfully used the differential transform approach to derive approximate analytical solutions to the nonlinear fractional gas dynamics equation [60]. Finally, Khan et al. used the (G'/G)-expansion approach to achieve accurate solutions for fractional gas dynamics equations [45].

    However, the fundamental goal and novelty of this research is to enhance the area of nonlinear science by introducing the revolutionary mEDAM approach, which results in the discovery of a slew of new solitary wave solution families for both the fractional MDP and fractional gas dynamics equations. This accomplishment not only broadens current knowledge, it also goes deeper into the complexities of these mathematical models. Furthermore, our research attempted to completely examine the wave behavior of these solitary waves in both models and develop relevant linkages between the wave dynamics and the underlying mathematical formulations, providing insight into the fundamental relationships driving these systems. These combined objectives have our research to make major contributions to the understanding and practical uses of soliton waves in a variety of scientific disciplines.

    Because of its widespread acceptance and well-established mathematical qualities, the authors elected to employ Caputo's fractional derivative in this study. Caputo's fractional derivative is well known for retaining the physical meaning of standard derivatives, making it more appropriate for modeling real-world events and systems. It has also improved behavior for non-smooth functions and produces consistent results across a wide range of applications. Here, the authors establish consistency with current literature and increase the credibility and comparability of their findings within the scientific community by employing Caputo's fractional derivative. This operator is defined as follows [61]:

    Dαtf(x,t)={1Γ(1α)t0τf(x,τ)(τt)αdτ,α(0,1),f(x,t)t,α=1, (1.4)

    where f(x,t) is a sufficiently smooth function. The following two properties of this derivative will be utilized while transforming targeted FPDEs into NODEs:

    Dαxxj=Γ(1+j)Γ(1+jα)xjα, (1.5)
    Dαxf[g(x)]=fg(g(x))Dαxg(x)=Dαgf(g(x))[g(x)]α, (1.6)

    where f(x) and g(x) are differentiable functions and j is a real number.

    In this section, we present the methodology of the mEDAM. Consider the FPDE of the following form [49,50,62]:

    P(y,αty,βv1y,γv2y,yβv1y,)=0,  0<α,β,γ1, (2.1)

    where y is a function of v1,v2,v3,,vn and t.

    To solve Eq (2.1), we follow the following steps:

    (1) First we perform variable transformation y(t,v1,v2,v3,,vn)=Y(ξ), ξ=ξ(t,v1,v2,v3,,vn), where ξ can be defined in various ways. This transformation converts Eq (2.1) into a nonlinear ODE of the form

    R(Y,Y,)=0, (2.2)

    where the derivatives of Y in Eq (2.2) are with respect to ξ. The constants of integration can then be obtained by integrating Eq (2.2) one or more times.

    (2) We then suppose the following solution to Eq (2.2):

    Y(ξ)=m2ρ=m1aρ(Q(ξ))ρ, (2.3)

    where aρ(ρ=m1,,0,1,2,,m2) are constants to be calculated, and Q(ξ) satisfies the following ODE:

    Q(ξ)=Ln(A)(a+bQ(ξ)+c(Q(ξ))2), (2.4)

    where A0 or 1 and a,b and c are constants.

    (3) The positive integers m1 and m2 given in Eq (2.3) are calculated by finding the homogeneous balance between the highest order derivative and the largest nonlinear term in Eq (2.2).

    (4) After that, we plug Eq (2.3) into Eq (2.4) or the equation generated by integrating Eq (2.4) and gather all terms of (Q(ξ)) in the same order. We then set each coefficient of the following polynomial as equal to zero, yielding a system of algebraic equations for aρ(ρ=m1,...,0,1,2,...,m2) and other parameters in the system of algebraic equations.

    (5) We use Maple to solve this system of algebraic equations.

    (6) Finally, we retrieve the unknown values and plug them into Eq (2.3) along with Q(ξ) (i.e., the solution of Eq (2.4)), which gives us the analytical solutions to Eq (2.1). We can generate the following families of solutions by using the generic solution of Eq (2.4).

    Family 1. When Z<0  and  c0,

    Q1(ξ)=b2c+ZtanA(1/2Zξ)2c, (2.5)
    Q2(ξ)=b2cZcotA(1/2Zξ)2c, (2.6)
    Q3(ξ)=b2c+Z(tanA(Zξ)±(pqsecA(Zξ)))2c, (2.7)
    Q4(ξ)=b2cZ(cotA(Zξ)±(pqcscA(Zξ)))2c (2.8)

    and

    Q5(ξ)=b2c+Z(tanA(14Zξ)cotA(14Zξ))4c. (2.9)

    Family 2. When Z>0  and  c0,

    Q6(ξ)=b2cZtanhA(1/2Zξ)2c, (2.10)
    Q7(ξ)=b2cZcothA(1/2Zξ)2c, (2.11)
    Q8(ξ)=b2cZ(tanhA(Zξ)±(pqsechA(Zξ)))2c, (2.12)
    Q9(ξ)=b2cZ(cothA(Zξ)±(pqcschA(Zξ)))2c (2.13)

    and

    Q10(ξ)=b2cZ(tanhA(14Zξ)cothA(14Zξ))4c. (2.14)

    Family 3. When ac>0 and b=0,

    Q11(ξ)=actanA(acξ), (2.15)
    Q12(ξ)=accotA(acξ), (2.16)
    Q13(ξ)=ac(tanA(2acξ)±(pqsecA(2acξ))), (2.17)
    Q14(ξ)=ac(cotA(2acξ)±(pqcscA(2acξ))) (2.18)

    and

    Q15(ξ)=12ac(tanA(1/2acξ)cotA(1/2acξ)). (2.19)

    Family 4. When ac>0 and b=0,

    Q16(ξ)=actanhA(acξ), (2.20)
    Q17(ξ)=accothA(acξ), (2.21)
    Q18(ξ)=ac(tanhA(2acξ)±(ipqsechA(2acξ))), (2.22)
    Q19(ξ)=ac(cothA(2acξ)±(pqcschA(2acξ))) (2.23)

    and

    Q20(ξ)=12ac(tanhA(1/2acξ)+cothA(1/2acξ)). (2.24)

    Family 5. When c=a and b=0,

    Q21(ξ)=tanA(aξ), (2.25)
    Q22(ξ)=cotA(aξ), (2.26)
    Q23(ξ)=tanA(2aξ)±(pqsecA(2aξ)), (2.27)
    Q24(ξ)=cotA(2aξ)±(pqcscA(2aξ)) (2.28)

    and

    Q25(ξ)=12tanA(1/2aξ)1/2cotA(1/2aξ). (2.29)

    Family 6. When c=a and b=0,

    Q26(ξ)=tanhA(aξ), (2.30)
    Q27(ξ)=cothA(aξ), (2.31)
    Q28(ξ)=tanhA(2aξ)±(ipqsechA(2aξ)), (2.32)
    Q29(ξ)=cothA(2aξ)±(pqcschA(2aξ)) (2.33)

    and

    Q30(ξ)=12tanhA(1/2aξ)1/2cothA(1/2aξ). (2.34)

    Family 7. When Z=0,

    Q31(ξ)=2a(bξLnA+2)b2ξLnA. (2.35)

    Family 8. When b=λ, a=nλ(n0) and c=0,

    Q32(ξ)=Aλξn. (2.36)

    Family 9. When b=c=0,

    Q33(ξ)=aξLnA. (2.37)

    Family 10. When b=a=0,

    Q34(ξ)=1cξLnA. (2.38)

    Family 11. When a=0, b0 and c0,

    Q35(ξ)=pbc(coshA(bξ)sinhA(bξ)+p) (2.39)

    and

    Q36(ξ)=b(coshA(bξ)+sinhA(bξ))c(coshA(bξ)+sinhA(bξ)+q). (2.40)

    Family 12. When b=λ, c=nλ(n0) and a=0,

    Q37(ξ)=pAλξpnqAλξ. (2.41)

    Here, p,q>0, and they are called the deformation parameters while Z=b24ac. The generalized trigonometric and hyperbolic functions are defined as follow:

    sinA(ξ)=pAiξqAiξ2i,cosA(ξ)=pAiξ+qAiξ2,tanA(ξ)=sinA(ξ)cosA(ξ),cotA(ξ)=cosA(ξ)sinA(ξ),secA(ξ)=1cosA(ξ),cscA(ξ)=1sinA(ξ). (2.42)

    Similarly,

    sinhA(ξ)=pAξqAξ2,coshA(ξ)=pAξ+qAξ2,tanhA(ξ)=sinhA(ξ)coshA(ξ),cothA(ξ)=coshA(ξ)sinhA(ξ),sechA(ξ)=1coshA(ξ),cschA(ξ)=1sinhA(ξ). (2.43)

    In this section, we utilize our suggested this improved mEDAM approach to address the targeted problems.

    First, the equation for the fractional MDP stated in Eq (1.2) is taken into account. In order to transform Eq (1.2) into a nonlinear ODE, we utilize the following complex transformation:

    u(x,t)=U(ξ),ξ=k1xβΓ(β+1)k2tαΓ(α+1), (3.1)

    which results in the following

    k2U+k2k21U+4k1U2U3k31UUk31UU=0; (3.2)

    integrating Eq (3.2) with respect to the wave variable ξ and the constant of integration to zero, yields:

    k2(k21UU)k31UU+4k1U33k31(U)2=0. (3.3)

    We balance the linear and nonlinear terms of the greatest order, which we may do by putting m1=m2. When we attempt m1=m2=1, however, the system of algebraic equations generated via Eq (3.3) only has trivial solutions. As a result, we choose this m1=m2=2 instead. By replacing m1=m2=2 into Eq (2.3), we get the following series solution for Eq (3.3):

    U(ξ)=2ρ=2dρ(G(ξ))ρ=d2(G(ξ))2+d1(G(ξ))1+d0+d1G(ξ)+d2(G(ξ))2. (3.4)

    By substituting Eq (3.3) into Eq (3.2) and equating the coefficients of (G(ξ))i to zero for i=6,5,...,0,1,...,6, we obtain a system of nonlinear algebraic equations. We can solve this system for the unknowns d2, d1, d0, d1, d2, k1 and k2 by using Maple. The solution produces four sets of answers:

    Case 1.

    d1=0,d2=0,d1=15cb8ac2b2,d2=15c28ac2b2,k1=(4acb2)1(ln(A))1,k2=5/2(4acb2)1(ln(A))1,d0=15ac8ac2b2. (3.5)

    Case 2.

    d1=15ab8ac2b2,d2=15a28ac2b2,d1=0,d2=0,k1=(4acb2)1(ln(A))1,k2=5/2(4acb2)1(ln(A))1,d0=15ac8ac2b2. (3.6)

    Case 3.

    d1=0,d2=0,d1=38i(25i±(15))cb4acb2,d2=38i(25i±(15))c24acb2,k1=110(25i±(53))5i(4acb2)(25i±(53))ln(A),k2=140(25i±(53))5(739+27i±(53))(73+15i±(53))i(4acb2)(25i±(53))ln(A),d0=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2. (3.7)

    Case 4.

    d1=0,d2=0,d1=38i(25i±(15))ab4acb2,d2=38i(25i±(15))a24acb2,k1=110(25i±(53))5i(4acb2)(25i±(53))ln(A),k2=140(25i±(53))5(739+27i±(53))(73+15i±(53))i(4acb2)(25i±(53))ln(A),d0=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2. (3.8)

    If we consider Case 1, we obtain the following sets of traveling wave solutions:

    Family 1. When Z<0 and a, b, c are nonzero, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u1(x,t)=158ac2b2(ac+cb(b2c+ZtanA(1/2Zξ)2c)+c2(b2c+ZtanA(1/2Zξ)2c)2), (3.9)
    u2(x,t)=158ac2b2(ac+cb(b2cZcotA(1/2Zξ)2c)+c2(b2cZcotA(1/2Zξ)2c)2), (3.10)
    u3(x,t)=158ac2b2(ac+cb(b2c+Z(tanA(Zξ)±(pqsecA(Zξ)))2c)+c2(b2c+Z(tanA(Zξ)±(pqsecA(Zξ)))2c)2), (3.11)
    u4(x,t)=158ac2b2(ac+cb(b2cZ(cotA(Zξ)±(pqcscA(Zξ)))2c)+c2(b2cZ(cotA(Zξ)±(pqcscA(Zξ)))2c)2) (3.12)

    and

    u5(x,t)=158ac2b2(ac+cb(b2c+Z(tanA(14Zξ)cotA(14Zξ))4c)+c2(b2c+Z(tanA(14Zξ)cotA(14Zξ))2c)2). (3.13)

    Family 2. When Z>0 and a, b, c are nonzero and the corresponding family of solitary wave solutions for Eq (1.2) is given as follow:

    u6(x,t)=158ac2b2(ac+cb(b2cZtanhA(1/2Zξ)2c)+c2(b2cZtanhA(1/2Zξ)2c)2), (3.14)
    u7(x,t)=158ac2b2(ac+cb(b2cZcothA(1/2Zξ)2c)+c2(b2cZcothA(1/2Zξ)2c)2), (3.15)
    u8(x,t)=158ac2b2(ac+cb(b2cZ(tanhA(Zξ)±(pqsechA(Zξ)))2c)+c2(b2cZ(tanhA(Zξ)±(pqsechA(Zξ)))2c)2), (3.16)
    u9(x,t)=158ac2b2(ac+cb(b2cZ(cothA(Zξ)±(pqcschA(Zξ)))2c)+c2(b2cZ(cothA(Zξ)±(pqcschA(Zξ)))2c)2) (3.17)

    and

    u10(x,t)=158ac2b2(ac+cb(b2cZ(tanhA(14Zξ)cothA(14Zξ))2c)+c2(b2cZ(tanhA(14Zξ)cothA(14Zξ))2c)2). (3.18)

    Family 3. When ac>0 and b=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follow:

    u11(x,t)=158(1+(tanA(acξ))2), (3.19)
    u12(x,t)=158(1+(cotA(acξ))2), (3.20)
    u13(x,t)=158(1+(tanA(2acξ)±(pqsecA(2acξ)))2), (3.21)
    u14(x,t)=158(1+(cotA(2acξ)±(pqcscA(2acξ)))2) (3.22)

    and

    u15(x,t)=158(1+14(tanA(1/2acξ)cotA(1/2acξ))2). (3.23)

    Family 4. When ac>0 and b=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follow:

    u16(x,t)=158(1(tanhA(acξ))2), (3.24)
    u17(x,t)=158(1(cothA(acξ))2), (3.25)
    u18(x,t)=158(1(tanhA(2acξ)±(ipqsechA(2acξ)))2), (3.26)
    u19(x,t)=158(1(cothA(2acξ)±(pqcschA(2acξ)))2) (3.27)

    and

    u20(x,t)=158(114(tanhA(1/2acξ)+cothA(1/2acξ))2). (3.28)

    Family 5. If c=a and b=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u21(x,t)=158(1+(tanA(aξ))2), (3.29)
    u22(x,t)=158(1+(cotA(aξ))2), (3.30)
    u23(x,t)=158(1+(tanA(2aξ)±(pqsecA(2aξ)))2), (3.31)
    u24(x,t)=158(1+(cotA(2aξ)±(pqcscA(2aξ)))2) (3.32)

    and

    u25(x,t)=158(1+(1/2tanA(1/2aξ)1/2cotA(1/2aξ))2). (3.33)

    Family 6. When c=a and b=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u26(x,t)=158(1+(tanhA(aξ))2), (3.34)
    u27(x,t)=158(1+(cothA(aξ))2), (3.35)
    u28(x,t)=158(1+(tanhA(2aξ)±(ipqsechA(2aξ)))2, (3.36)
    u29(x,t)=158(1+(cothA(2aξ)±(pqcschA(2aξ)))2) (3.37)

    and

    u30(x,t)=158(1+(1/2tanhA(1/2aξ)1/2cothA(1/2aξ))2). (3.38)

    Family 7. If a=0, b0 and c0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u31(x,t)=15p2(coshA(bξ)sinhA(bξ)+p)+15p22(coshA(bξ)sinhA(bξ)+p)2 (3.39)

    and

    u32(x,t)=15(coshA(bξ)+sinhA(bξ))2(coshA(bξ)+sinhA(bξ)+q)+15(coshA(bξ)+sinhA(bξ))22(coshA(bξ)+sinhA(bξ)+q)2. (3.40)

    Family 8. If b=λ, c=nλ (where n0) and a=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u33(x,t)=15npAλξ2(pnqAλξ)+15n2p2(Aλξ)22(pnqAλξ)2, (3.41)

    where ξ=(4acb2)1xβln(A)Γ(β+1)5(4acb2)1tα2ln(A)Γ(α+1).

    Now by assuming Case 2, we obtain the following families of solutions:

    Family 9. If Z<0 and a, b, c are nonzero then the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u34(x,t)=158ac2b2(ac+a2(b2c+ZtanA(1/2Zξ)2c)2+ab(b2c+ZtanA(1/2Zξ)2c)1), (3.42)
    u35(x,t)=158ac2b2(ac+a2(b2cZcotA(1/2Zξ)2c)2+ab(b2cZcotA(1/2Zξ)2c)1), (3.43)
    u36(x,t)=158ac2b2(ac+a2(b2c+Z(tanA(Zξ)±(pqsecA(Zξ)))2c)2+ab(b2c+Z(tanA(Zξ)±(pqsecA(Zξ)))2c)1), (3.44)
    u37(x,t)=158ac2b2(ac+a2(b2cZ(cotA(Zξ)±(pqcscA(Zξ)))2c)2+ab(b2cZ(cotA(Zξ)±(pqcscA(Zξ)))2c)1) (3.45)

    and

    u38(x,t)=158ac2b2(ac+a2(b2c+Z(tanA(14Zξ)cotA(14Zξ))2c)2+ab(b2c+Z(tanA(14Zξ)cotA(14Zξ))2c)1). (3.46)

    Family 10. If Z>0 and a, b, c are nonzero then the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u39(x,t)=158ac2b2(ac+a2(b2cZtanhA(1/2Zξ)2c)2+ab(b2cZtanhA(1/2Zξ)2c)1), (3.47)
    u40(x,t)=158ac2b2(ac+a2(b2cZcothA(1/2Zξ)2c)2+ab(b2cZcothA(1/2Zξ)2c)1), (3.48)
    u41(x,t)=158ac2b2(ac+a2(b2cZ(tanhA(Zξ)±(pqsechA(Zξ)))2c)2+ab(b2cZ(tanhA(Zξ)±(pqsechA(Zξ)))2c)1), (3.49)
    u42(x,t)=158ac2b2(ac+a2(b2cZ(cothA(Zξ)±(pqcschA(Zξ)))2c)2+ab(b2cZ(cothA(Zξ)±(pqcschA(Zξ)))2c)1) (3.50)

    and

    u43(x,t)=158ac2b2(ac+a2(b2cZ(tanhA(14Zξ)cothA(14Zξ))4c)2+ab(b2cZ(tanhA(14Zξ)cothA(14Zξ))4c)1). (3.51)

    Family 11. When ac>0 and b=0, this the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u44(x,t)=158(1+1(tanA(acξ))2), (3.52)
    u45(x,t)=158(1+1(cotA(acξ))2), (3.53)
    u46(x,t)=158(1+1(tanA(2acξ)±(pqsecA(2acξ)))2), (3.54)
    u47(x,t)=158(1+1(cotA(2acξ)±(pqcscA(2acξ)))2) (3.55)

    and

    u48(x,t)=158(1+41(tanA(1/2acξ)cotA(1/2acξ))2). (3.56)

    Family 12. When ac>0 and b=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u49(x,t)=158(11(tanhA(acξ))2), (3.57)
    u50(x,t)=158(11(cothA(acξ))2), (3.58)
    u51(x,t)=158(11(tanhA(2acξ)±(ipqsechA(2acξ)))2), (3.59)
    u52(x,t)=158(11(cothA(2acξ)±(pqcschA(2acξ)))2) (3.60)

    and

    u53(x,t)=158(141(tanhA(1/2acξ)+cothA(1/2acξ))2). (3.61)

    Family 13. If c=a and b=0, this the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u54(x,t)=158(1+1(tanA(aξ))2), (3.62)
    u55(x,t)=158(1+1(cotA(aξ))2), (3.63)
    u56(x,t)=158(1+1(tanA(2aξ)±(pqsecA(2aξ)))2), (3.64)
    u57(x,t)=158(1+1(cotA(2aξ)±(pqcscA(2aξ)))2) (3.65)

    and

    u58(x,t)=158(1+1(1/2tanA(1/2aξ)1/2cotA(1/2aξ))2). (3.66)

    Family 14. If c=a and b=0, then the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u59(x,t)=158(1+1(tanhA(aξ))2), (3.67)
    u60(x,t)=158(1+1(cothA(aξ))2), (3.68)
    u61(x,t)=158(1+1+(tanhA(2aξ)±(ipqsechA(2aξ)))2), (3.69)
    u62(x,t)=158(1+1(cothA(2aξ)±(pqcschA(2aξ)))2) (3.70)

    and

    u63(x,t)=158(1+1(1/2tanhA(1/2aξ)1/2cothA(1/2aξ))2). (3.71)

    Family 15. If b=λ, a=nλ (where n0) and c=0, then the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u64(x,t)=152(n2(Aλξn)2+n(Aλξn)) (3.72)

    where ξ=(4acb2)1xβln(A)Γ(β+1)5(4acb2)1tα2ln(A)Γ(α+1).

    If we consider Case 3, we obtain the following sets of traveling wave solutions:

    Family 16. If Z<0 and a, b, c are nonzero then the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u65(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(cb(b2c+ZtanA(1/2Zξ)2c)+c2(b2c+ZtanA(1/2Zξ)2c)2), (3.73)
    u66(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(cb(b2cZcotA(1/2Zξ)2c)+c2(b2cZcotA(1/2Zξ)2c)2), (3.74)
    u67(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(cb(b2c+Z(tanA(Zξ)±(pqsecA(Zξ)))2c)+c2(b2c+Z(tanA(Zξ)±(pqsecA(Zξ)))2c)2), (3.75)
    u68(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(cb(b2cZ(cotA(Zξ)±(pqcscA(Zξ)))2c)+c2(b2cZ(cotA(Zξ)±(pqcscA(Zξ)))2c)2) (3.76)

    and

    u69(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(cb(b2c+Z(tanA(14Zξ)cotA(14Zξ))4c)+c2(b2c+Z(tanA(14Zξ)cotA(14Zξ))2c)2). (3.77)

    Family 17. If Z>0 and a, b, c are nonzero then the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u70(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(cb(b2cZtanhA(1/2Zξ)2c)+c2(b2cZtanhA(1/2Zξ)2c)2), (3.78)
    u71(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(cb(b2cZcothA(1/2Zξ)2c)+c2(b2cZcothA(1/2Zξ)2c)2), (3.79)
    u72(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(cb(b2cZ(tanhA(Zξ)±(pqsechA(Zξ)))2c)+c2(b2cZ(tanhA(Zξ)±(pqsechA(Zξ)))2c)2), (3.80)
    u73(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(cb(b2cZ(cothA(Zξ)±(pqcschA(Zξ)))2c)+c2(b2cZ(cothA(Zξ)±(pqcschA(Zξ)))2c)2) (3.81)

    and

    u74(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(cb(b2cZ(tanhA(14Zξ)cothA(14Zξ))2c)+c2(b2cZ(tanhA(14Zξ)cothA(14Zξ))2c)2). (3.82)

    Family 18. When ac>0 and b=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u75(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8((tanA(acξ))2), (3.83)
    u76(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8((cotA(acξ))2), (3.84)
    u77(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8((tanA(2acξ)±(pqsecA(2acξ)))2), (3.85)
    u78(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8((cotA(2acξ)±(pqcscA(2acξ)))2) (3.86)

    and

    u79(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(14(tanA(1/2acξ)cotA(1/2acξ))2). (3.87)

    Family 19. When ac>0 and b=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u80(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8((tanhA(acξ))2), (3.88)
    u81(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8((cothA(acξ))2), (3.89)
    u82(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8((tanhA(2acξ)±(ipqsechA(2acξ)))2), (3.90)
    u83(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8((cothA(2acξ)±(pqcschA(2acξ)))2) (3.91)

    and

    u84(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(14(tanhA(1/2acξ)+cothA(1/2acξ))2). (3.92)

    Family 20. If c=a and b=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u85(x,t)=141311a2+329ia2±2(15)292a2+60ia2±2(15)+3/4i(25i±(15))8((tanA(aξ))2), (3.93)
    u86(x,t)=141311a2+329ia2±2(15)292a2+60ia2±2(15)+3/4i(25i±(15))8((cotA(aξ))2), (3.94)
    u87(x,t)=141311a2+329ia2±2(15)292a2+60ia2±2(15)+3/4i(25i±(15))8((tanA(2aξ)±(pqsecA(2aξ)))2), (3.95)
    u88(x,t)=141311a2+329ia2±2(15)292a2+60ia2±2(15)+3/4i(25i±(15))8((cotA(2aξ)±(pqcscA(2aξ)))2) (3.96)

    and

    u89(x,t)=141311a2+329ia2±2(15)292a2+60ia2±2(15)+3/4i(25i±(15))8((1/2tanA(1/2aξ)1/2cotA(1/2aξ))2). (3.97)

    Family 21. When c=a and b=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u90(x,t)=141311a2329ia2±2(15)292a260ia2±2(15)+3/4i(25i±(15))8((tanhA(aξ))2), (3.98)
    u91(x,t)=141311a2329ia2±2(15)292a260ia2±2(15)+3/4i(25i±(15))8((cothA(aξ))2), (3.99)
    u92(x,t)=141311a2329ia2±2(15)292a260ia2±2(15)+3/4i(25i±(15))8((tanhA(2aξ)±(ipqsechA(2aξ)))2, (3.100)
    u93(x,t)=141311a2329ia2±2(15)292a260ia2±2(15)+3/4i(25i±(15))8((cothA(2aξ)±(pqcschA(2aξ)))2) (3.101)

    and

    u94(x,t)=141311a2329ia2±2(15)292a260ia2±2(15)+3/4i(25i±(15))8((1/2tanhA(1/2aξ)1/2cothA(1/2aξ))2). (3.102)

    Family 22. If a=0, b0 and c0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u95(x,t)=14±2(15)+441b2+31ib2±2(15)73b215ib2+3/4i(25i±(15))p2(coshA(bξ)sinhA(bξ)+p)+3/4i(25i±(15))p22(coshA(bξ)sinhA(bξ)+p)2 (3.103)

    and

    u96(x,t)=14±2(15)+441b2+31ib2±2(15)73b215ib2+3/4i(25i±(15))(coshA(bξ)+sinhA(bξ))2(coshA(bξ)+sinhA(bξ)+q)+3/4i(25i±(15))(coshA(bξ)+sinhA(bξ))22(coshA(bξ)+sinhA(bξ)+q)2. (3.104)

    Family 23. If b=λ, c=nλ (where n0) and a=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u97(x,t)=14±2(15)+441λ2+31iλ2±2(15)73λ215iλ2+3/4i(25i±(15))npAλξ2(pnqAλξ)+3/4i(25i±(15))n2p2(Aλξ)22(pnqAλξ)2, (3.105)

    where ξ=(25i±(53))5xβ10i(4acb2)(25i±(53))ln(A)Γ(β+1)

    +(25i±(53))5(739+27i±(53))tα40(73+15i±(53))i(4acb2)(25i±(53))ln(A)Γ(α+1).

    Assuming Case 4, we obtain the following families of solutions:

    Family 24. If Z<0 and a, b, c are nonzero then the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u98(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(a2(b2c+ZtanA(1/2Zξ)2c)2+ab(b2c+ZtanA(1/2Zξ)2c)1), (3.106)
    u99(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(a2(b2cZcotA(1/2Zξ)2c)2+ab(b2cZcotA(1/2Zξ)2c)1), (3.107)
    u100(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(a2(b2c+Z(tanA(Zξ)±(pqsecA(Zξ)))2c)2+ab(b2c+Z(tanA(Zξ)±(pqsecA(Zξ)))2c)1), (3.108)
    u101(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(a2(b2cZ(cotA(Zξ)±(pqcscA(Zξ)))2c)2+ab(b2cZ(cotA(Zξ)±(pqcscA(Zξ)))2c)1) (3.109)

    and

    u102(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(a2(b2c+Z(tanA(14Zξ)cotA(14Zξ))2c)2+ab(b2c+Z(tanA(14Zξ)cotA(14Zξ))2c)1). (3.110)

    Family 25. If Z>0 and a, b, c are nonzero then the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u103(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(a2(b2cZtanhA(1/2Zξ)2c)2+ab(b2cZtanhA(1/2Zξ)2c)1), (3.111)
    u104(x,t)=141311ac+329ac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(a2(b2cZcothA(1/2Zξ)2c)2+ab(b2cZcothA(1/2Zξ)2c)1), (3.112)
    u105(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(a2(b2cZ(tanhA(Zξ)±(pqsechA(Zξ)))2c)2+ab(b2cZ(tanhA(Zξ)±(pqsechA(Zξ)))2c)1), (3.113)
    u106(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(a2(b2cZ(cothA(Zξ)±(pqcschA(Zξ)))2c)2+ab(b2cZ(cothA(Zξ)±(pqcschA(Zξ)))2c)1) (3.114)

    and

    u107(x,t)=141311ac+329iac±2(15)+441b2+31ib2292ac+60iac±2(15)73b215ib2+3/4i(25i±(15))8ac2b2(a2(b2cZ(tanhA(14Zξ)cothA(14Zξ))4c)2+ab(b2cZ(tanhA(14Zξ)cothA(14Zξ))4c)1). (3.115)

    Family 26. When ac>0 and b=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u108(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(1(tanA(acξ))2), (3.116)
    u109(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(1(cotA(acξ))2), (3.117)
    u110(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(1(tanA(2acξ)±(pqsecA(2acξ)))2), (3.118)
    u111(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(1(cotA(2acξ)±(pqcscA(2acξ)))2) (3.119)

    and

    u112(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(41(tanA(1/2acξ)cotA(1/2acξ))2). (3.120)

    Family 27. When ac>0 and b=0, the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u113(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(1(tanhA(acξ))2), (3.121)
    u114(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(1(cothA(acξ))2), (3.122)
    u115(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(1(tanhA(2acξ)±(ipqsechA(2acξ)))2), (3.123)
    u116(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(1(cothA(2acξ)±(pqcschA(2acξ)))2) (3.124)

    and

    u117(x,t)=141311ac+329iac±2(15)292ac+60iac±2(15)+3/4i(25i±(15))8(41(tanhA(1/2acξ)+cothA(1/2acξ))2). (3.125)

    Family 28. If c=a and b=0, then the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u118(x,t)=141311a2+329ia2±2(15)292a2+60ia2±2(15)+3/4i(25i±(15))8(1(tan(aξ))2), (3.126)
    u119(x,t)=141311a2+329ia2±2(15)292a2+60ia2±2(15)+3/4i(25i±(15))8(1(cotA(aξ))2), (3.127)
    u120(x,t)=141311a2+329ia2±2(15)292a2+60ia2±2(15)+3/4i(25i±(15))8(1(tanA(2aξ)±(pqsecA(2aξ)))2), (3.128)
    u121(x,t)=141311a2+329ia2±2(15)292a2+60ia2±2(15)+3/4i(25i±(15))8(1(cotA(2aξ)±(pqcscA(2aξ)))2) (3.129)

    and

    u122(x,t)=141311a2+329ia2±2(15)292a2+60ia2±2(15)+3/4i(25i±(15))8(1(1/2tanA(1/2aξ)1/2cotA(1/2aξ))2). (3.130)

    Family 29. If c=a and b=0, then the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u123(x,t)=141311a2329ia2±2(15)292a260ia2±2(15)+3/4i(25i±(15))8(1(tanhA(aξ))2), (3.131)
    u124(x,t)=141311a2329ia2±2(15)292a260ia2±2(15)+3/4i(25i±(15))8(1(cothA(aξ))2), (3.132)
    u125(x,t)=141311a2329ia2±2(15)292a260ia2±2(15)+3/4i(25i±(15))8(1(tanhA(2aξ)±(ipqsechA(2aξ)))2), (3.133)
    u126(x,t)=141311a2329ia2±2(15)292a260ia2±2(15)+3/4i(25i±(15))8(1(cothA(2aξ)±(pqcschA(2aξ)))2) (3.134)

    and

    u127(x,t)=141311a2329ia2±2(15)292a260ia2±2(15)+3/4i(25i±(15))8(1(1/2tanhA(1/2aξ)1/2cothA(1/2aξ))2). (3.135)

    Family 30. If b=λ, a=nλ (where n0) and c=0, then the corresponding family of solitary wave solutions for Eq (1.2) is given as follows:

    u128(x,t)=14±2(15)+441λ2+31iλ2±2(15)73λ215iλ2+3/4i(25i±(15))2(n2(Aλξn)2+n(Aλξn)), (3.136)

    where ξ=(25i±(53))5xβ10i(4acb2)(25i±(53))ln(A)Γ(β+1)

    +(25i±(53))5(739+27i±(53))tα40(73+15i±(53))i(4acb2)(25i±(53))ln(A)Γ(α+1).

    Consider the fractional gas dynamics equation given by Eq (1.3). In order to convert Eq (1.3) into a NODE, we apply the following complex transformation:

    u(x,t)=U(ξ),ξ=k1tαΓ(α+1)+k2xβΓ(β+1). (3.137)

    This yields

    k1U+k2UUU+U2=0. (3.138)

    If we balance the highest order linear term U with the nonlinear term U2, we obtain that m1=m2=1. Substituting m1=m2=1 into Eq (2.3), we can obtain a series form solution for Eq (3.138) as

    U(ξ)=1ρ=1dρ(G(ξ))ρ=d1(G(ξ))1+d0+d1G(ξ). (3.139)

    By substituting Eq (3.139) into Eq (3.138), we can obtain a system of nonlinear algebraic equations by equating the coefficients of (G(ξ))i for i=3,...,0,...,3 to zero. Solving this system for the unknown d1, d0, d1, k1 and k2 by using Maple, we obtain the following two sets of solutions:

    Case 1.

    d1=(b2+4ac)1a,d1=0,d0=12((b2+4ac)1b+1),k1=(b2+4ac)1(ln(A))1,k2=0. (3.140)

    Case 2.

    d1=0,d1=(b2+4ac)1c,d0=12((b2+4ac)1b+1),k1=(b2+4ac)1(ln(A))1,k2=0. (3.141)

    Assuming Case 1, we can obtain the following families of solutions

    Family 1. If Z<0 and a, b, c are nonzero then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u1(x,t)=(b2+4ac)1a(b2c+ZtanA(1/2Zξ)2c)1+12((b2+4ac)1b+1), (3.142)
    u2(x,t)=(b2+4ac)1a(b2cZcotA(1/2Zξ)2c)1+12((b2+4ac)1b+1), (3.143)
    u3(x,t)=(b2+4ac)1a×(b2c+Z(tanA(Zξ)±(pqsecA(Zξ)))2c)1+12((b2+4ac)1b+1), (3.144)
    u4(x,t)=(b2+4ac)1a×(b2cZ(cotA(Zξ)±(pqcscA(Zξ)))2c)1+12((b2+4ac)1b+1) (3.145)

    and

    u5(x,t)=(b2+4ac)1a×(b2c+Z(tanA(14Zξ)cotA(14Zξ))4c)1+12((b2+4ac)1b+1). (3.146)

    Family 2. If Z>0 and a, b, c are nonzero then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u6(x,t)=(b2+4ac)1a(b2cZtanhA(1/2Zξ)2c)1+12((b2+4ac)1b+1), (3.147)
    u7(x,t)=(b2+4ac)1a(b2cZcothA(1/2Zξ)2c)1+12((b2+4ac)1b+1), (3.148)
    u8(x,t)=(b2+4ac)1a×(b2cZ(tanhA(Zξ)±(pqsechA(Zξ)))2c)1+12((b2+4ac)1b+1), (3.149)
    u9(x,t)=(b2+4ac)1a×(b2cZ(cothA(Zξ)±(pqcschA(Zξ)))2c)1+12((b2+4ac)1b+1) (3.150)

    and

    u10(x,t)=(b2+4ac)1a×(b2cZ(tanhA(14Zξ)cothA(14Zξ))4c)1+12((b2+4ac)1b+1). (3.151)

    Family 3. If ac>0 and b=0, then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u11(x,t)=i2(tanA(acξ))1+12, (3.152)
    u12(x,t)=i2(cotA(acξ))1+12, (3.153)
    u13(x,t)=i2(tanA(2acξ)±(pqsecA(2acξ)))1+12, (3.154)
    u14(x,t)=i2(cotA(2acξ)±(pqcscA(2acξ)))1+12 (3.155)

    and

    u15(x,t)=i(tanA(1/2acξ)cotA(1/2acξ))1+12. (3.156)

    Family 4. If ac<0 and b=0, then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u16(x,t)=12(tanhA(acξ))1+12, (3.157)
    u17(x,t)=12(cothA(acξ))1+12, (3.158)
    u18(x,t)=12(tanhA(2acξ)±(ipqsechA(2acξ)))1+12, (3.159)
    u19(x,t)=12(cothA(2acξ)±(pqcschA(2acξ)))1+12 (3.160)

    and

    u20(x,t)=(tanhA(1/2acξ)+cothA(1/2acξ))1+12. (3.161)

    Family 5. If c=a and b=0, then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u21(x,t)=i2(tanA(aξ))1+12, (3.162)
    u22(x,t)=i2(cotA(aξ))1+12, (3.163)
    u23(x,t)=i2(tanA(2aξ)±(pqsecA(2aξ)))1+12, (3.164)
    u24(x,t)=i2(cotA(2aξ)±(pqcscA(2aξ)))1+12 (3.165)

    and

    u25(x,t)=i2(1/2tanA(1/2aξ)1/2cotA(1/2aξ))1+12. (3.166)

    Family 6. If c=a and b=0, then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u26(x,t)=12(tanhA(aξ))1+12, (3.167)
    u27(x,t)=12(cothA(aξ))1+12, (3.168)
    u28(x,t)=12(tanhA(2aξ)±(ipqsechA(2aξ)))1+12, (3.169)
    u29(x,t)=12(cothA(2aξ)±(pqcschA(2aξ)))1+12 (3.170)

    and

    u30(x,t)=12(1/2tanhA(1/2aξ)1/2cothA(1/2aξ))1+12. (3.171)

    Family 7. If b=λ, a=nλ(n0), and c=0, then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u31(x,t)=n(Aλξn)1+12(λ2+4ac)1λ+12, (3.172)

    where ξ=(b2+4ac)1(ln(A))1tαΓ(α+1).

    By assuming Case 2, we can derive the following families of solutions:

    Family 8. When Z<0 and a, b, c are nonzero then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u32(x,t)=12(1(b2+4ac)1b)(b2+4ac)1c(b2c+ZtanA(1/2Zξ)2c), (3.173)
    u33(x,t)=12(1(b2+4ac)1b)(b2+4ac)1c(b2cZcotA(1/2Zξ)2c), (3.174)
    u34(x,t)=12(1(b2+4ac)1b)(b2+4ac)1c×(b2c+Z(tanA(Zξ)±(pqsecA(Zξ)))2c), (3.175)
    u35(x,t)=12(1(b2+4ac)1b)(b2+4ac)1c×(b2cZ(cotA(Zξ)±(pqcscA(Zξ)))2c) (3.176)

    and

    u36(x,t)=12(1(b2+4ac)1b)(b2+4ac)1c×(b2c+Z(tanA(14Zξ)cotA(14Zξ))4c). (3.177)

    Family 9. When Z>0 and a, b, c are nonzero then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u37(x,t)=12(1(b2+4ac)1b)(b2+4ac)1c(b2cZtanhA(1/2Zξ)4c), (3.178)
    u38(x,t)=12(1(b2+4ac)1b)(b2+4ac)1c(b2cZcothA(1/2Zξ)2c), (3.179)
    u39(x,t)=12(1(b2+4ac)1b)(b2+4ac)1c×(b2cZ(tanhA(Zξ)±(pqsechA(Zξ)))2c), (3.180)
    u40(x,t)=12(1(b2+4ac)1b)(b2+4ac)1c×(b2cZ(cothA(Zξ)±(pqcschA(Zξ)))2c) (3.181)

    and

    u41(x,t)=12(1(b2+4ac)1b)(b2+4ac)1c×(b2cZ(tanhA(14Zξ)cothA(14Zξ))4c). (3.182)

    Family 10. When ac>0 and b=0 then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u42(x,t)=12i2tanA(acξ), (3.183)
    u43(x,t)=12+i2cotA(acξ), (3.184)
    u44(x,t)=12i2(tanA(2acξ)±(pqsecA(2acξ))), (3.185)
    u45(x,t)=12+i2(cotA(2acξ)±(pqcscA(2acξ))) (3.186)

    and

    u46(x,t)=12i2(tanA(1/2acξ)cotA(1/2acξ)). (3.187)

    Family 11. When ac<0 and b=0 then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u47(x,t)=12+12tanhA(acξ), (3.188)
    u48(x,t)=12+12cothA(acξ), (3.189)
    u49(x,t)=12+12(tanhA(2acξ)±(ipqsechA(2acξ))), (3.190)
    u50(x,t)=12+(cothA(2acξ)±(pqcschA(2acξ))) (3.191)

    and

    u51(x,t)=12+(tanhA(1/2acξ)+cothA(1/2acξ)). (3.192)

    Family 12. When c=a and b=0, the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u52(x,t)=12i2tanA(aξ), (3.193)
    u53(x,t)=12+i2cotA(aξ), (3.194)
    u54(x,t)=12i2(tanA(2aξ)±(pqsecA(2aξ))), (3.195)
    u55(x,t)=12i2(cotA(2aξ)±(pqcscA(2aξ))) (3.196)

    and

    u56(x,t)=12i2(1/2tanA(1/2aξ)1/2cotA(1/2aξ)). (3.197)

    Family 13. When c=a and b=0, the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u57(x,t)=12+12tanhA(aξ), (3.198)
    u58(x,t)=12+12cothA(aξ), (3.199)
    u59(x,t)=1212(tanhA(2aξ)±(ipqsechA(2aξ))), (3.200)
    u60(x,t)=1212(cothA(2aξ)±(pqcschA(2aξ))) (3.201)

    and

    u61(x,t)=1212(12tanhA(1/2aξ)12cothA(1/2aξ)). (3.202)

    Family 14. If a=0, b0 and c0, then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u62(x,t)=p(coshA(bξ)sinhA(bξ)+p)1 (3.203)

    and

    u63(x,t)=(coshA(bξ)+sinhA(bξ))(coshA(bξ)+sinhA(bξ)+q)1. (3.204)

    Family 15. If b=λ, c=nλ  (n0) and a=0, then the corresponding family of solitary wave solutions for Eq (1.3) is given as follows:

    u64(x,t)=npAλξ(pnqAλξ)1, (3.205)

    where ξ=(b2+4ac)1(ln(A))1tαΓ(α+1).

    We discovered solitary wave solutions for the fractional MDP and fractional gas dynamics equations by using a unique mEDAM approach in this work. Our findings contain a variety of essential features, such as periodic waves, hyperbolic waves, singular waves, singular kink waves, shock waves and solitons, among others. Periodic waves are distinguished by their consistent amplitude and wavelength oscillations that are continuous and regular. Hyperbolic waves, on the other hand, are more complicated in shape and this distinguished by steep, concave or convex profiles. A singular wave is a wave that has a singularity or a concentrated energy distribution. Kink waves, on the other hand, are distinguished by abrupt discontinuities in the wave profile. Solitons, on the other hand, are self-reinforcing solitary waves that keep their shape and speed as they travel across a medium without dispersing or losing energy.

    The fundamental goal of our research was to enhance nonlinear science by introducing the revolutionary mEDAM approach, which resulted in the discovery of a slew of new solitary wave solution families for both the fractional MDP and fractional gas dynamics equations. This accomplishment not only broadens current knowledge, also goes deeper into the complexities of these mathematical models. Furthermore, our research to examine the wave behavior of these solitary waves in both models in depth and to develop significant linkages between the wave dynamics and the underlying mathematical formulations, giving insight into the fundamental interconnections that drive these systems. These combined goals have allowed our research to make substantial contributions to the understanding and practical uses of soliton waves in a variety of scientific disciplines. The relationship between these waves and the solved FPDEs is fascinating. The fractional MDP equation is a nonlinear dispersive wave equation that models complex wave propagation. The equation's fractional structure allows it to replicate waves with nonlocal interactions, making it an effective tool for modeling complex wave phenomena. The fractional gas dynamics equation, on the other hand, is a model that depicts the mobility of gas in a fluid medium.

    It is critical to recognize that the fractional gas dynamics equation is time dependent but not space-dependant. As a result, while the wave profile does not change in space, it does change over time. This is due to the fractional order derivative of the equation, which creates a memory effect and allows the wave to recall information about its prior behavior. Thus, the equation may be utilized to forecast wave occurrences including long-range interactions and memory effects.

    Remark 1. Figure 1 illustrates a singular kink wave profile. The fractional MDP equation is known for backing singular kink wave solutions, which are fascinating wave dynamics phenomena. These isolated kinks are caused by localized wave structures with abrupt, non-smooth characteristics. They appear in the context of the fractional MDP equation due to the interaction of nonlinearities and fractional derivatives, resulting in the development of these separate solitary waves. In this model, studying singular kink waves can provide valuable insights 1) into how fractional calculus influences wave behavior and 2) the emergence of complex localized structures in various physical systems, providing a deeper understanding of the equation's behaviour in applications such as fluid dynamics and oceanography.

    Figure 1.  The 3D graph of (3.104) is plotted for a=2,b=0,c=2,p=3,q=4,A=2,α=β=1. The 3D depiction is plotted with t=0 and for the same values of parameters involved.

    Remark 2. In Figure 2, (a) depicts a singular wave (which is formed by the combination of two shock waves that propagate in opposite directions with a common asymptote) while (b) shows a singular kink wave profile. Singular waves, particularly shock waves, in gas dynamics equations provide critical insights into the behavior of compressible fluids and the propagation of disturbances. These waves, which are distinguished by sudden changes in fluid characteristics, shed light on the phenomena of gas compression and the rarefaction found in barriers or flow shifts. Their importance lies in understanding high-speed flow physics, particularly in supersonic and hypersonic contexts, through aspects such as shock wave formation, wave propagation governed by Rankine-Hugoniot relations, strength and speed determination influenced by multiple factors, energy dissipation and heat transfer mechanisms. Furthermore, precise solitary wave modelling is critical in engineering and aerospace applications to optimize designs and assure safety in high-speed transportation systems.

    Figure 2.  The 2D graphs of (3.200) and its squared norm are depicted for a=2,b=10,c=2,k2=0,A=e,α=β=1 in (a) and (b) respectively.

    Similarly, within the fractional gas dynamics equation, singular kink wave solutions reflect highly localized, abrupt changes in the density or pressure profiles of compressible flows. The complicated interplay between the nonlinear components and fractional derivatives in the equation causes these peculiar bends. In conventional gas dynamics, they are akin to shock waves, but with fractional influences controlling their generation and behavior. Investigating singular kink waves in the fractional gas dynamics equation yields valuable insights 1) into how fractional calculus affects the dynamics of compressible flows and 2) the formation of sharp, non-smooth wave structures, providing a deeper understanding of wave phenomena in the context of gas dynamics, particularly in scenarios involving rarefaction waves and other complex wave interactions.

    The research resulted in the creation of the mEDAM, a ground breaking approach for generating solitary wave solutions for both the fractional MDP and fractional gas dynamics equations with Caputo's derivatives. This method uses complex transformations and series-based solutions, as well as generalized hyperbolic and trigonometric functions, to build families of solitary wave solutions. The study makes an important addition to nonlinear science, with applications ranging from fluid dynamics to plasma physics to nonlinear optics. These answers give essential insights into the behavior of soliton waves in these systems. In terms of future work, we want to adapt and apply the mEDAM approach to different FPDEs that use varied and modern derivative operators. This extension intends to improve our understanding of wave behavior across FPDEs and open up new paths for practical applications.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, the Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. 4271).

    Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, the Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. 4271).

    The authors declare that they have no competing interests.



    [1] R. Dykstra, S. Kochar, J. Rojo, Stochastic comparisons of parallel systems of heterogeneous exponential components, J. Stat. Plan. Infer., 65 (1997), 203–211. https://doi.org/10.1016/S0378-3758(97)00058-X doi: 10.1016/S0378-3758(97)00058-X
    [2] P. Zhao, N. Balakrishnan, Some characterization results for parallel systems with two heterogeneous exponential components, Statistics, 45 (2011), 593–604. https://doi.org/10.1080/02331888.2010.485276 doi: 10.1080/02331888.2010.485276
    [3] C. Li, X. Li, Likelihood ratio order of sample minimum from heterogeneous Weibull random variables, Stat. Probabil. Lett., 97 (2015), 46–53. https://doi.org/10.1016/j.spl.2014.10.019 doi: 10.1016/j.spl.2014.10.019
    [4] N. Torrado, S. C. Kochar, Stochastic order relations among parallel systems from Weibull distributions, J. Appl. Probab., 52 (2015), 102–116. https://doi.org/10.1239/jap/1429282609 doi: 10.1239/jap/1429282609
    [5] B. E. Khaledi, S. Kochar, Weibull distribution: Some stochastic comparisons results, J. Stat. Plan. Infer., 136 (2006), 3121–3129. https://doi.org/10.1016/j.jspi.2004.12.013 doi: 10.1016/j.jspi.2004.12.013
    [6] S. Kochar, M. Xu, Stochastic comparisons of parallel systems when components have proportional hazard rates, Probab. Eng. Inform. Sci., 21 (2007), 597–609. https://doi.org/10.1017/S0269964807000344 doi: 10.1017/S0269964807000344
    [7] N. Torrado, Comparisons of smallest order statistics from Weibull distributions with different scale and shape parameters, J. Korean Stat. Soc., 44 (2015), 68–76. https://doi.org/10.1016/j.jkss.2014.05.004 doi: 10.1016/j.jkss.2014.05.004
    [8] S. Kochar, M. Xu, On the skewness of order statistics with applications, Ann. Oper. Res., 212 (2012), 127–138. https://doi.org/10.1007/s10479-012-1212-4 doi: 10.1007/s10479-012-1212-4
    [9] N. Balakrishnan, A. Haidari, K. Masoumifard, Stochastic comparisons of series and parallel systems with generalized exponential components, IEEE T. Reliab., 64 (2015), 333–348. https://doi.org/10.1109/TR.2014.2354192 doi: 10.1109/TR.2014.2354192
    [10] L. Fang, X. Zhang, Stochastic comparisons of parallel systems with exponentiated Weibull components, Stat. Probabil. Lett., 97 (2015), 25–31. https://doi.org/10.1016/j.spl.2014.10.017 doi: 10.1016/j.spl.2014.10.017
    [11] A. Kundu, S. Chowdhury, Ordering properties of order statistics from heterogeneous exponentiated Weibull models, Stat. Probabil. Lett., 114 (2016), 119–127. https://doi.org/10.1016/j.spl.2016.03.017 doi: 10.1016/j.spl.2016.03.017
    [12] N. K. Hazra, M. R. Kuiti, M. Finkelstein, A. K. Nanda, On stochastic comparisons of minimum order statistics from the location-scale family of distributions, Metrika, 81 (2017), 105–123. https://doi.org/10.1007/s00184-017-0636-x doi: 10.1007/s00184-017-0636-x
    [13] N. Balakrishnan, G. Barmalzan, A. Haidari, Modified proportional hazard rates and proportional reversed hazard rates models via marshall olkin distribution and some stochastic comparisons, J. Korean Stat. Soc., 47 (2018), 127–138. https://doi.org/10.1016/j.jkss.2017.10.003 doi: 10.1016/j.jkss.2017.10.003
    [14] G. Barmalzan, S. Kosari, N. Balakrishnan, Usual stochastic and reversed hazard orders of parallel systems with independent heterogeneous components, Commun. Stat.-Theory Methods, 2020, 1–26. https://doi.org/10.1080/03610926.2020.1823415
    [15] G. Barmalzan, S. Ayat, N. Balakrishnan, Stochastic comparisons of series and parallel systems with dependent burr type XII components, Commun. Stat.-Theory Methods, 2020, 1–22. https://doi.org/10.1080/03610926.2020.1772307
    [16] S. Naqvi, W. Ding, P. Zhao, Stochastic comparison of parallel systems with pareto components, Probab. Eng. Inform. Sci., 2021, 1–13. https://doi.org/10.1017/S0269964821000176
    [17] A. Arriaza, A. Di Crescenzo, M. A. Sordo, A. Suárez-Llorens, Shape measures based on the convex transform order, Metrika, 82 (2019), 99–124. https://doi.org/10.1007/s00184-018-0667-y doi: 10.1007/s00184-018-0667-y
    [18] J. Zhang, R. Yan, Stochastic comparison at component level and system level series system with two proportional hazards rate components, J. Quant. Econ., 35 (2018), 91–95. https://doi.org/10.16339/j.cnki.hdjjsx.2018.04.035 doi: 10.16339/j.cnki.hdjjsx.2018.04.035
    [19] G. S. Mudholkar, D. K. Srivastava, Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE T. Reliab., 42 (1993), 299–302. https://doi.org/10.1109/24.229504 doi: 10.1109/24.229504
    [20] P. Zhao, N. Balakrishnan, New results on comparisons of parallel systems with heterogeneous gamma components, Stat. Probabil. Lett., 81 (2011), 36–44. https://doi.org/10.1016/j.spl.2010.09.016 doi: 10.1016/j.spl.2010.09.016
    [21] B. E. Khaledi, S. Farsinezhad, S. C. Kochar, Stochastic comparisons of order statistics in the scale model, J. Stat. Plan. Infer., 141 (2011), 276–286. https://doi.org/10.1016/j.jspi.2010.06.006 doi: 10.1016/j.jspi.2010.06.006
    [22] N. Misra, A. K. Misra, New results on stochastic comparisons of two-component series and parallel systems, Stat. Probabil. Lett., 82 (2012), 283–290. https://doi.org/10.1016/j.spl.2011.10.010 doi: 10.1016/j.spl.2011.10.010
    [23] N. Balakrishnan, P. Zhao, Hazard rate comparison of parallel systems with heterogeneous gamma components, J. Multivariate Anal., 113 (2013), 153–160. https://doi.org/10.1016/j.jmva.2011.05.001 doi: 10.1016/j.jmva.2011.05.001
    [24] P. Zhao, N. Balakrishnan, Comparisons of largest order statistics from multiple-outlier gamma models, Methodol. Comput. Appl. Probab., 17 (2015), 617–645. https://doi.org/10.1007/s11009-013-9377-0 doi: 10.1007/s11009-013-9377-0
    [25] W. Ding, Y. Zhang, P. Zhao, Comparisons of k-out-of-n systems with heterogenous components, Stat. Probabil. Lett., 83 (2013), 493–502. https://doi.org/10.1016/j.spl.2012.10.012 doi: 10.1016/j.spl.2012.10.012
    [26] Z. Guo, J. Zhang, R. Yan, The residual lifetime of surviving components of coherent system under periodical inspections, Mathematics, 8 (2020), 2181. https://doi.org/10.3390/math8122181 doi: 10.3390/math8122181
    [27] S. C. Kochar, N. Torrado, On stochastic comparisons of largest order statistics in the scale model, Commun. Stat.-Theory Methods, 44 (2015), 4132–4143. https://doi.org/10.1080/03610926.2014.985839 doi: 10.1080/03610926.2014.985839
    [28] A. Panja, P. Kundu, B. Pradhan, Variability and skewness ordering of sample extremes from dependent random variables following the proportional odds model, arXiv, 2020. Available from: https://arXiv.org/abs/2006.04454.
    [29] Z. Guo, J. Zhang, R. Yan, On inactivity times of failed components of coherent system under double monitoring, Probab. Eng. Inform. Sci., 2021, 1–18. https://doi.org/10.1017/S0269964821000152
    [30] A. Kundu, S. Chowdhury, A. K. Nanda, N. K. Hazra, Some results on majorization and their applications, J. Comput. Appl. Math., 301 (2016), 161–177. https://doi.org/10.1016/j.cam.2016.01.015 doi: 10.1016/j.cam.2016.01.015
    [31] R. Yan, B. Lu, X. Li, On redundancy allocation to series and parallel systems of two components, Commun. Stat.-Theory Methods, 48 (2019), 4690–4701. https://doi.org/10.1080/03610926.2018.1500603 doi: 10.1080/03610926.2018.1500603
    [32] X. Li, R. Fang, Ordering properties of order statistics from random variables of Archimedean copulas with applications, J. Multivariate Anal., 133 (2015), 304–320. https://doi.org/10.1016/j.jmva.2014.09.016 doi: 10.1016/j.jmva.2014.09.016
    [33] C. Li, R. Fang, X. Li, Stochastic somparisons of order statistics from scaled and interdependent random variables, Metrika, 79 (2016), 553–578. https://doi.org/10.1007/s00184-015-0567-3 doi: 10.1007/s00184-015-0567-3
    [34] R. Fang, C. Li, X. Li, Stochastic comparisons on sample extremes of dependent and heterogenous observations, Statistics, 50 (2016), 930–955. https://doi.org/10.1080/02331888.2015.1119151 doi: 10.1080/02331888.2015.1119151
    [35] Y. Zhang, X. Cai, P. Zhao, H. Wang, Stochastic comparisons of parallel and series systems with heterogeneous resilience-scaled components, Statistics, 53 (2019), 126–147. https://doi.org/10.1080/02331888.2018.1546705 doi: 10.1080/02331888.2018.1546705
    [36] C. Li, X. Li, Hazard rate and reversed hazard rate orders on extremes of heterogeneous and dependent random variables, Stat. Probabil. Lett., 146 (2019), 104–111. https://doi.org/10.1016/j.spl.2018.11.005 doi: 10.1016/j.spl.2018.11.005
    [37] Y. Zhang, X. Cai, P. Zhao, Ordering properties of extreme claim amounts from heterogeneous portfolios, ASTIN Bull., 49 (2019), 525–554. https://doi.org/10.1017/asb.2019.7 doi: 10.1017/asb.2019.7
    [38] G. Barmalzan, S. M. Ayat, N. Balakrishnan, R. Roozegar, Stochastic comparisons of series and parallel systems with dependent heterogeneous extended exponential components under archimedean copula, J. Comput. Appl. Math., 380 (2020), 112965. https://doi.org/10.1016/j.cam.2020.112965 doi: 10.1016/j.cam.2020.112965
    [39] L. Zhang, R. Yan, Stochastic comparisons of series and parallel systems with dependent and heterogeneous topp leone generated components, AIMS Math., 6 (2021), 2031–2047. https://doi.org/10.3934/math.2021124 doi: 10.3934/math.2021124
    [40] M. Zhang, B. Lu, R. Yan, Ordering results of extreme order statistics from dependent and heterogeneous modified proportional (reversed) hazard variables, AIMS Math., 6 (2021), 584–606. https://doi.org/10.3934/math.2021036 doi: 10.3934/math.2021036
    [41] N. Torrado, Comparing the reliability of coherent systems with heterogeneous, dependent and distribution-free components, Qual. Technol. Quant. M., 18 (2021), 740–770. https://doi.org/10.1080/16843703.2021.1963033 doi: 10.1080/16843703.2021.1963033
    [42] E. Amini-Seresht, J. Qiao, Y. Zhang, P. Zhao, On the skewness of order statistics in multiple-outlier PHR models, Metrika, 79 (2016), 817–836. https://doi.org/10.1007/s00184-016-0579-7 doi: 10.1007/s00184-016-0579-7
    [43] B. E. Khaledi, S. Kochar, Dispersive ordering among linear combinations of uniform random variables, J. Stat. Plan. Infer., 100 (2002), 13–21. https://doi.org/10.1016/S0378-3758(01)00091-X doi: 10.1016/S0378-3758(01)00091-X
    [44] J. Jeon, S. Kochar, C. G. Park, Dispersive ordering some applications and examples, Stat. Pap., 47 (2006), 227–247. https://doi.org/10.1007/s00362-005-0285-4 doi: 10.1007/s00362-005-0285-4
    [45] S. Kochar, Stochastic comparisons of order statistics and spacings: A review, Probab. Stat., 2012 (2012), 839473. https://doi.org/10.5402/2012/839473 doi: 10.5402/2012/839473
    [46] W. Ding, J. Yang, X. Ling, On the skewness of extreme order statistics from heterogeneous samples, Commun. Stat.-Theory Methods, 46 (2016), 2315–2331. https://doi.org/10.1080/03610926.2015.1041984 doi: 10.1080/03610926.2015.1041984
    [47] J. Wu, M. Wang, X. Li, Convex transform order of the maximum of independent Weibull random variables, Stat. Probabil. Lett., 156 (2020), 108597. https://doi.org/10.1016/j.spl.2019.108597 doi: 10.1016/j.spl.2019.108597
    [48] R. Yan, J. Wang, Component level versus system level at active redundancies for coherent systems with dependent heterogeneous components, Commun. Stat.-Theory Methods, 2020, 1–21. https://doi.org/10.1080/03610926.2020.1767140
    [49] R. Yan, J. Zhang, Y. Zhang, Optimal allocation of relevations in coherent systems, J. Appl. Probab., 58 (2021), 1152–1169. https://doi.org/10.1017/jpr.2021.23 doi: 10.1017/jpr.2021.23
    [50] T. Lando, I. Arab, P. E. Oliveira, Second-order stochastic comparisons of order statistics, Statistics, 55 (2021), 561–579. https://doi.org/10.1080/02331888.2021.1960527 doi: 10.1080/02331888.2021.1960527
    [51] T. Lando, L. Bertoli-Barsotti, Second-order stochastic dominance for decomposable multiparametric families with applications to order statistics, Stat. Probabil. Lett., 159 (2020), 108691. https://doi.org/10.1016/j.spl.2019.108691 doi: 10.1016/j.spl.2019.108691
    [52] J. Zhang, R. Yan, J. Wang, Reliability optimization of parallel-series and series-parallel systems with statistically dependent components, Appl. Math. Model., 102 (2022), 618–639. https://doi.org/10.1016/j.apm.2021.10.003 doi: 10.1016/j.apm.2021.10.003
    [53] L. Jiao, R. Yan, Stochastic comparisons of lifetimes of series and parallel systems with dependent heterogeneous MOTL-G components under random shocks, Symmetry, 13 (2021), 2248. https://doi.org/10.3390/sym13122248 doi: 10.3390/sym13122248
    [54] B. Lu, J. Zhang, R. Yan, Optimal allocation of a coherent system with statistical dependent subsystems, Probab. Eng. Inform. Sci., 2021, 1–20. https://doi.org/10.1017/S0269964821000437
    [55] S. Blumenthal, A survey of estimating distributional parameters and sample sizes from truncated samples, In: Statistical distributions in scientific work, Springer Netherlands: Dordrecht, 79 (1981), 75–86. https://doi.org/10.1007/978-94-009-8552-0_6
    [56] S. Blumenthal, R. Marcus, Estimating population size with exponential failure, J. Am. Stat. Assoc., 70 (1975), 913–922. https://doi.org/10.2307/2285457 doi: 10.2307/2285457
    [57] L. Fang, J. Ling, N. Balakrishnan, Stochastic comparisons of series and parallel systems with independent heterogeneous lower-truncated Weibull components, Commun. Stat.-Theory Methods, 45 (2015), 540–551. https://doi.org/10.1080/03610926.2015.1099671 doi: 10.1080/03610926.2015.1099671
    [58] F. Famoye, Continuous univariate distributions, Technometrics, 37 (1995), 466. https://doi.org/10.1080/00401706.1995.10484392 doi: 10.1080/00401706.1995.10484392
    [59] D. N. P. Murthy, M. Xie, R. Jiang, Weibull models, Wiley, 2004.
    [60] W. R. Van Zwet, Convex transformations of random variables, Mathematisch Centrum Amsterdam, 1964.
    [61] M. Shaked, J. G. Shanthikumar, Stochastic orders, New York: Springer, 2007. https://doi.org/10.1007/978-0-387-34675-5
    [62] H. Li, X. Li, Stochastic orders in reliability and risk, New York: Springer, 2013. https://doi.org/10.1007/978-1-4614-6892-9
    [63] A. W. Marshall, I. Olkin, B. C. Arnold, Inequalities: Theory of majorization and its applications, Springer, 1979.
    [64] R. B. Nelsen, An introduction to copulas, New York: Springer, 2007. https://doi.org/10.1007/0-387-28678-0
    [65] A. J. McNeil, J. Neslehova, Multivariate Archimedean copulas, d-monotone functions and 1-norm symmetric distributions, Ann. Stat., 37 (2009), 3059–3097. https://doi.org/10.1214/07-AOS556 doi: 10.1214/07-AOS556
  • Reader Comments
  • © 2022 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(2522) PDF downloads(87) Cited by(1)

Figures and Tables

Figures(7)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog