Research article

Simulation analysis, properties and applications on a new Burr XII model based on the Bell-X functionalities

  • Received: 02 December 2022 Revised: 21 December 2022 Accepted: 02 January 2023 Published: 11 January 2023
  • MSC : 60E05, 62N05, 62F10, 62D05

  • In this article, we make mathematical and practical contributions to the Bell-X family of absolutely continuous distributions. As a main member of this family, a special distribution extending the modeling perspectives of the famous Burr XII (BXII) distribution is discussed in detail. It is called the Bell-Burr XII (BBXII) distribution. It stands apart from the other extended BXII distributions because of its flexibility in terms of functional shapes. On the theoretical side, a linear representation of the probability density function and the ordinary and incomplete moments are among the key properties studied in depth. Some commonly used entropy measures, namely Rényi, Havrda and Charvat, Arimoto, and Tsallis entropy, are derived. On the practical (inferential) side, the associated parameters are estimated using seven different frequentist estimation methods, namely the methods of maximum likelihood estimation, percentile estimation, least squares estimation, weighted least squares estimation, Cramér von-Mises estimation, Anderson-Darling estimation, and right-tail Anderson-Darling estimation. A simulation study utilizing all these methods is offered to highlight their effectiveness. Subsequently, the BBXII model is successfully used in comparisons with other comparable models to analyze data on patients with acute bone cancer and arthritis pain. A group acceptance sampling plan for truncated life tests is also proposed when an item's lifetime follows a BBXII distribution. Convincing results are obtained.

    Citation: Ayed. R. A. Alanzi, Muhammad Imran, M. H. Tahir, Christophe Chesneau, Farrukh Jamal, Saima Shakoor, Waqas Sami. Simulation analysis, properties and applications on a new Burr XII model based on the Bell-X functionalities[J]. AIMS Mathematics, 2023, 8(3): 6970-7004. doi: 10.3934/math.2023352

    Related Papers:

    [1] Guowei Zhang . The lower bound on the measure of sets consisting of Julia limiting directions of solutions to some complex equations associated with Petrenko's deviation. AIMS Mathematics, 2023, 8(9): 20169-20186. doi: 10.3934/math.20231028
    [2] Hong Li, Keyu Zhang, Hongyan Xu . Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables. AIMS Mathematics, 2021, 6(11): 11796-11814. doi: 10.3934/math.2021685
    [3] Arunachalam Murali, Krishnan Muthunagai . Generation of Julia and Mandelbrot fractals for a generalized rational type mapping via viscosity approximation type iterative method extended with s-convexity. AIMS Mathematics, 2024, 9(8): 20221-20244. doi: 10.3934/math.2024985
    [4] Nan Li, Jiachuan Geng, Lianzhong Yang . Some results on transcendental entire solutions to certain nonlinear differential-difference equations. AIMS Mathematics, 2021, 6(8): 8107-8126. doi: 10.3934/math.2021470
    [5] Minghui Zhang, Jianbin Xiao, Mingliang Fang . Entire solutions for several Fermat type differential difference equations. AIMS Mathematics, 2022, 7(7): 11597-11613. doi: 10.3934/math.2022646
    [6] Wenjie Hao, Qingcai Zhang . The growth of entire solutions of certain nonlinear differential-difference equations. AIMS Mathematics, 2022, 7(9): 15904-15916. doi: 10.3934/math.2022870
    [7] Yeyang Jiang, Zhihua Liao, Di Qiu . The existence of entire solutions of some systems of the Fermat type differential-difference equations. AIMS Mathematics, 2022, 7(10): 17685-17698. doi: 10.3934/math.2022974
    [8] Zheng Wang, Zhi Gang Huang . On transcendental directions of entire solutions of linear differential equations. AIMS Mathematics, 2022, 7(1): 276-287. doi: 10.3934/math.2022018
    [9] Bin He . Developing a leap-frog meshless methods with radial basis functions for modeling of electromagnetic concentrator. AIMS Mathematics, 2022, 7(9): 17133-17149. doi: 10.3934/math.2022943
    [10] Wenju Tang, Keyu Zhang, Hongyan Xu . Results on the solutions of several second order mixed type partial differential difference equations. AIMS Mathematics, 2022, 7(2): 1907-1924. doi: 10.3934/math.2022110
  • In this article, we make mathematical and practical contributions to the Bell-X family of absolutely continuous distributions. As a main member of this family, a special distribution extending the modeling perspectives of the famous Burr XII (BXII) distribution is discussed in detail. It is called the Bell-Burr XII (BBXII) distribution. It stands apart from the other extended BXII distributions because of its flexibility in terms of functional shapes. On the theoretical side, a linear representation of the probability density function and the ordinary and incomplete moments are among the key properties studied in depth. Some commonly used entropy measures, namely Rényi, Havrda and Charvat, Arimoto, and Tsallis entropy, are derived. On the practical (inferential) side, the associated parameters are estimated using seven different frequentist estimation methods, namely the methods of maximum likelihood estimation, percentile estimation, least squares estimation, weighted least squares estimation, Cramér von-Mises estimation, Anderson-Darling estimation, and right-tail Anderson-Darling estimation. A simulation study utilizing all these methods is offered to highlight their effectiveness. Subsequently, the BBXII model is successfully used in comparisons with other comparable models to analyze data on patients with acute bone cancer and arthritis pain. A group acceptance sampling plan for truncated life tests is also proposed when an item's lifetime follows a BBXII distribution. Convincing results are obtained.



    Due to its well-established applications in various scientific and technical fields, fractional calculus has gained prominence during the last three decades. Many pioneers have shown that when adjusted by integer-order models, fractional-order models may accurately represent complex events [1,2]. The Caputo fractional derivatives are nonlocal in contrast to the integer-order derivatives, which are local in nature [1]. In other words, the integer-order derivative may be used to analyze changes in the area around a point, but the Caputo fractional derivative can be used to analyze changes in the whole interval. Senior mathematicians including Riemann [4], Caputo [5], Podlubny [6], Ross [7], Liouville [8], Miller and others, collaborated to create the fundamental foundation for fractional order integrals and derivatives. The theory of fractional-order calculus has been related to real-world projects, and it has been applied to chaos theory [9], signal processing [10], electrodynamics [11], human diseases [12,13], and other areas [14,15,16].

    Due to the numerous applications of fractional differential equations in engineering and science such as electrodynamics [17], chaos ideas [18], accounting [19], continuum and fluid mechanics [20], digital signal [21] and biological population designs [22] fractional differential equations are now more widely known. For such issues to be resolved, efficient tools are needed [23,24,25]. Because of this, we will attempt to apply an efficient analytical technique to solve nonlinear arbitrary order differential equations in this article. Many strategies in collaboration fields may be delightfully and even more accurately analyzed using fractional differential equations. Various strategies have been developed in this regard, some of them are as follows, such as the fractional Reduced differential transformation technique [26], Adomian decomposition technique [27], the fractional Variational iteration technique [28], Elzaki decomposition technique [29,30], iterative transformation technique [31], the fractional natural decomposition method (FNDM) [32], and the fractional homotopy perturbation method [33].

    The power series solution is used to solve some classes of the differential and integral equations of fractional or non-fractional order, and it is based on assuming that the solution of the equation can be expanded as a power series. RPS is an easy and fast technique for determining the coefficients of the power series solution. The Jordanian mathematician Omar Abu Arqub created the residual power series method in 2013, as a technique for quickly calculating the coefficients of the power series solutions for 1st and 2nd-order fuzzy differential equations [34]. Without perturbation, linearization, or discretization, the residual power series method provides a powerful and straightforward power series solution for highly linear and nonlinear equations [35,36,37,38]. The residual power series method has been used to solve an increasing variety of nonlinear ordinary and partial differential equations of various sorts, orders, and classes during the past several years. It has been used to make non-linear fractional dispersive partial differential equation have solitary pattern results and to predict them [39], to solve the highly nonlinear singular differential equation known as the generalized Lane-Emden equation [40], to solve higher-order ordinary differential equations numerically [41], to approximate solve the fractional nonlinear KdV-Burger equations, to predict and represent the RPSM differs from several other analytical and numerical approaches in some crucial ways [42]. First, there is no requirement for a recursion connection or for the RPSM to compare the coefficients of the related terms. Second, by reducing the associated residual error, the RPSM offers a straightforward method to guarantee the convergence of the series solution. Thirdly, the RPSM doesn't suffer from computational rounding mistakes and doesn't use a lot of time or memory. Fourth, the approach may be used immediately to the provided issue by selecting an acceptable starting guess approximation since the residual power series method does not need any converting when transitionary from low-order to higher-order and from simple linearity to complicated nonlinearity [43,44,45]. The process of solving linear differential equations using the LT method consists of three steps. The first step depends on transforming the original differential equation into a new space, called the Laplace space. In the second step, the new equation is solved algebraically in the Laplace space. In the last step, the solution in the second step is transformed back into the original space, resulting in the solution of the given problem.

    In this article, we apply the Laplace residual power series method to achieve the definitive solution of the fractional-order nonlinear partial differential equations. The Laplace transformation efficiently integrates the residual power series method for the renewability algorithmic technique. This proposed technique produces interpretive findings in the sense of a convergent series. The Caputo fractional derivative operator explains quantitative categorizations of the partial differential equations. The offered methodology is well demonstrated in modelling and enumeration investigations. The exact-analytical findings are a valuable way to analyze the problematic dynamics of systems, notably for computational fractional partial differential equations.

    Definition 2.1. The fractional Caputo derivative of a function u(ζ,t) of order α is given as [46]

    CDαtu(ζ,t)=Jmαtum(ζ,t),m1<αm,t>0, (2.1)

    where mN and Jαt is the fractional integral Riemann-Liouville (RL) of u(ζ,t) of order α is given as

    Jσtu(ζ,t)=1Γ(α)t0(tτ)α1u(φ,τ)dτ (2.2)

    Definition 2.2. The Laplace transformation (LT) of u(ζ,t) is given as [46]

    u(ζ,s)=Lt[u(ζ,t)]=0estu(ζ,t)dt,s>α, (2.3)

    where the Laplace transform inverse is defined as

    u(ζ,t)=L1t[u(ζ,s)]=l+iliestu(ζ,s)ds,l=Re(s)>l0. (2.4)

    Lemma 2.1. Suppose that u(ζ,t) is piecewise continue term and U(ζ,s)=Lt[u(ζ,t)], we get

    (1) Lt[Jαtu(ζ,t)]=U(ζ,s)sα,α>0.

    (2) Lt[Dαtu(ζ,t)]=sσU(ζ,s)m1k=0sαk1uk(ζ,0),m1<αm.

    (3) Lt[Dnαtu(ζ,t)]=snαU(ζ,s)n1k=0s(nk)α1Dkαtu(ζ,0),0<α1.

    Proof. For proof see Refs. [46].

    Theorem 2.1. Let u(ζ,t) be a piecewise continuous function on I×[0,) with exponential order ζ. Assume that the fractional expansion of the function U(ζ,s)=Lt[u(ζ,t)] is as follows:

    U(ζ,s)=n=0fn(ζ)s1+nα,0<α1,ζI,s>ζ. (2.5)

    Then, fn(ζ)=Dnσtu(ζ,0).

    Proof. For proof see Refs. [46].

    Remark 2.1. The inverse Laplace transform of the Eq (2.5) is represented as [46]

    u(ζ,t)=i=0Dαtu(ζ,0)Γ(1+iα)ti(ζ),0<ζ1,t0. (2.6)

    Consider the fractional order partial differential equation,

    DαtU(ζ,t)+3U(ζ,t)tζ24U(ζ,t)t2ζ2+4U(ζ,t)ζ4+a(2U(ζ,t)ζ2)2b(2U(ζ,t)t2)3+cU(ζ,t)=0. (3.1)

    Applying LT of Eq (3.1), we get

    U(ζ,s)+f0(ζ,s)s+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+aL(L1t(2U(ζ,s)ζ2))2bL(L1t(2U(ζ,s)t2))3+cU(ζ,s)]=0. (3.2)

    Suppose that the result of Eq (3.2), we get

    U(ζ,s)=n=0fn(ζ,s)snα+1. (3.3)

    The kth-truncated term series are

    U(ζ,s)=f0(ζ,s)s+kn=1fn(ζ,s)snα+1,k=1,2,3,4. (3.4)

    Residual Laplace function (RLF) is given as

    LtResu(ζ,s)=U(ζ,s)+f0(ζ,s)s+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+aL(L1t(2U(ζ,s)ζ2))2bL(L1t(2U(ζ,s)t2))3+cU(ζ,s)]. (3.5)

    And the kth-LRFs as

    LtResk(ζ,s)=Uk(ζ,s)+f0(ζ,s)s+1sσ[3Uk(ζ,s)tζ24Uk(ζ,s)t2ζ2+4U(ζ,s)ζ4+aL(L1t(2Uk(ζ,s)ζ2))2bL(L1t(2Uk(ζ,s)t2))3+cUk(ζ,s)]. (3.6)

    To illustrate a few facts, the following LRPSM features are provided:

    (1) LtRes(ζ,s)=0 and limjLtResk(ζ,s)=LtResu(ζ,s) for each s>0.

    (2) limssLtResu(ζ,s)=0limssLtResu,k(ζ,s)=0.

    (3) limsskα+1LtResu,k(ζ,s)=limsskα+1LtResu,k(ζ,s)=0,0<α1,k=1,2,3,.

    To calculate the coefficients using fn(ζ,s), gn(ζ,s), hn(ζ,s) and ln(ζ,s), the following system is recursively solved:

    limsskα+1LtResu,k(α,s)=0,k=1,2,. (3.7)

    In finally inverse Laplace transform to Eq (3.4), to get the kth analytical result of uk(ζ,t).

    Example 4.1. Consider the fractional partial differential equations [47],

    Dαtu(ζ,t)3u(ζ,t)tζ24u(ζ,t)t2ζ2+4u(ζ,t)ζ4+19(2u(ζ,t)ζ2)21216(2u(ζ,t)t2)3+16u(ζ,t)=0, where 2<α3, (4.1)

    with the following IC's:

    u(x,0)=ζ4,tu(ζ,0)=0,2t2u(ζ,0)=0. (4.2)

    Using Laplace transform to Eq (4.1), we get

    U(ζ,s)+ζ4s+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+19L(L1t(2U(ζ,s)ζ2))21216L(L1t(2U(ζ,s)t2))3+16U(ζ,s)]=0, (4.3)

    and so the kth-truncated term series are

    ζu(ζ,s)=ζ4s+kn=1fn(ζ,s)snα+1,k=1,2,3,4. (4.4)

    Residual Laplace function is given as

    LtResu(ζ,s)=U(ζ,s)+ζ4s+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+19L(L1t(2U(ζ,s)ζ2))21216L(L1t(2U(ζ,s)t2))3+16U(ζ,s)], (4.5)

    and the kth-LRFs as:

    LtResk(ζ,s)=Uk(ζ,s)+ζ4s+1sσ[3Uk(ζ,s)tζ24Uk(ζ,s)t2ζ2+4U(ζ,s)ζ4+19L(L1t(2Uk(ζ,s)ζ2))21216L(L1t(2Uk(ζ,s)t2))3+16Uk(ζ,s)]. (4.6)
    Table 1.  Comparison of the exact and proposed technique solution and various fractional-orders α and t=0.25 for Example 4.1.
    ζ α=2.5 α=2.7 α=2.9 α=3 HPM [47] Exact
    0 0.0222397 0.0111683 0.00553658 0.00388069 0.00388069 0.0038812
    0.2 0.0206397 0.00956834 0.00393658 0.00228069 0.00228069 0.0022812
    0.4 -0.00336028 -0.0144317 -0.0200634 -0.0217193 -0.0217193 -0.0217188
    0.6 -0.10736 -0.118432 -0.124063 -0.125719 -0.125719 -0.125719
    0.8 -0.38736 -0.398432 -0.404063 -0.405719 -0.405719 -0.405719
    1.0 -0.97776 -0.988832 -0.994463 -0.996119 -0.996119 -0.996119

     | Show Table
    DownLoad: CSV

    Now, we calculate fk(ζ,s), k=1,2,3,, substituting the kth-truncate series of Eq (4.4) into the kth residual Laplace term Eq (4.6), multiply the solution equation by skα+1, and then solve recursively the link lims(skα+1LtResu,k(ζ,s))=0, k=1,2,3,. Following are the first some term:

    f1(ζ,s)=24,f2(ζ,s)=384,f3(ζ,s)=6144. (4.7)

    Putting the value of fk(ζ,s), k=1,2,3,, in Eq (4.4), we get

    U(ζ,s)=ζ4s+24sα+1384s2α+1+6144s3α+1+. (4.8)

    Using inverse LT, we get

    u(ζ,t)=ζ4+24tαΓ(α+1)384t2αΓ(2α+1)+6144t3αΓ(3α+1)+, (4.9)

    and the exact solution are

    u=ζ4+4t3. (4.10)

    In Figure 1, the exact and LRPSM solutions for u(ζ,t) at α=3 at ζ and t=0.3 of Example 4.1. In Figure 2, analytical solution for u(ζ,t) at different value of α=2.8 and 2.6 at ζ and t=0.3. In Figure 3, analytical solution for u(ζ,t) at various value of α at t=0.3 of Example 4.1.

    Figure 1.  The actual and LRPSM results for u(ζ,t) at α=3 at ζ and t=0.3.
    Figure 2.  Analytical solution for u(ζ,t) at different value of α=2.8 and 2.6 at ζ and t=0.3.
    Figure 3.  Analytical solution for u(ζ,t) at various value of α at t=0.3.

    Example 4.2. Consider the fractional partial differential equations [47]:

    DαtU(ζ,t)3U(ζ,t)tζ24U(ζ,t)t2ζ2+4U(ζ,t)ζ4+(2U(ζ,t)ζ2)2(2U(ζ,t)t2)2+2U2(ζ,t)=0, where 2<α3, (4.11)

    with the following IC's:

    U(ζ,0)=eζ,tU(ζ,0)=eζ,2t2U(ζ,0)=eζ. (4.12)

    Using Laplace transform to Eq (4.11), we get

    U(ζ,s)eζseζs2eζs3+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+Lt(L1t(2U(ζ,s)ζ2))2Lt(L1t(2U(ζ,s)t2))2+2Lt(L1t(U(ζ,s)))2]=0. (4.13)
    Table 2.  Comparison of the exact and proposed technique solution and various fractional-orders α and t=0.099 for Example 4.2.
    ζ α=2.5 α=2.7 α=2.9 α=3 Exact
    0 1.08911 1.09052 1.09122 1.09142 1.09199
    0.2 1.32968 1.33169 1.33268 1.33297 1.33376
    0.4 1.62325 1.62612 1.62754 1.62795 1.62905
    0.6 1.98139 1.98553 1.98759 1.98818 1.98973
    0.8 2.41823 2.42422 2.42721 2.42807 2.43026
    1 2.95086 2.95959 2.96394 2.96519 2.96833

     | Show Table
    DownLoad: CSV

    Residual Laplace function is given as

    LtResu(ζ,s)=U(ζ,s)eζseζs2eζs3+1sα[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+Lt(L1t(2U(ζ,s)ζ2))2Lt(L1t(2U(ζ,s)t2))2+2Lt(L1t(U(ζ,s)))2], (4.14)

    and so the kth-truncated term series are

    u(ζ,s)=eζs+eζs2+eζs3+kn=1fn(ζ,s)snα+1,k=1,2,3,4, (4.15)

    and the kth-LRFs as:

    LtResk(ζ,s)=Uk(ζ,s)eζseζs2eζs3+1sσ[3Uk(ζ,s)tζ24Uk(ζ,s)t2ζ2+4Uk(ζ,s)ζ4+Lt(L1t(2Uk(ζ,s)ζ2))2Lt(L1t(2Uk(ζ,s)t2))2+2Lt(L1t(Uk(ζ,s)))2]. (4.16)

    Now, we calculate fk(ζ,s), k=1,2,3,, substituting the kth-truncate series of Eq (4.15) into the kth residual Laplace term Eq (4.16), multiply the solution equation by skα+1, and then solve recursively the link lims(skα+1LtResu,k(ζ,s))=0, k=1,2,3,. Following are the first some term:

    f1(ζ,s)=(eζ+3e2ζ),f2(ζ,s)=eζ+54e2ζ+36e3ζ,f3(ζ,s)=(eζ+870e2ζ+3564e3ζ+792e4ζ).   (4.17)

    Putting the value of fk(ζ,s), k=1,2,3,, in Eq (4.15), we get

    U(ζ,s)=eζs+eζs2+eζs3eζ+3e2ζsα+1eζ+54e2ζ+36e3ζs2α+1eζ+870e2ζ+3564e3ζ+792e4ζs3α+1+. (4.18)

    Using inverse LT, we get

    u(ζ,t)=eζ+eζt+eζt2(eζ+3e2ζ)tαΓ(α+1)+(eζ+54e2ζ+36e3ζ)t2αΓ(2α+1)+(eζ+870e2ζ+3564e3ζ+792e4ζ)t3αΓ(3α+1)+, (4.19)

    and the exact solution are

    u=eζ+t. (4.20)

    In Figure 4, the exact and LRPSM solutions for u(ζ,t) at α=3 at ζ and t=0.3 of Example 4.2. In Figure 5, LRPSM solutions for u(ζ,t) at α=2.5 and α=2.8 and t=0.3 of Example 4.2.

    Figure 4.  Exact and LRPSM solutions for u(ζ,t) at α=3 at ζ and t=0.3.
    Figure 5.  LRPSM solutions for u(ζ,t) at α=2.5 and α=2.8 and t=0.3.

    Example 4.3. Consider the fractional partial differential equations [47]:

    Dαtu(ζ,t)3u(ζ,t)tζ24u(ζ,t)t2x2+4u(ζ,t)ζ4(2u(ζ,t)t2)(u(ζ,t)ζ)u(ζ,t)(u(ζ,t)t)=0, where 2<α3, (4.21)

    with the following IC's:

    U(ζ,0)=cosζ,tU(ζ,0)=sinζ,2t2U(ζ,0)=cosζ. (4.22)
    Table 3.  Comparison of the exact and proposed technique solution and various fractional-orders α and t=0.22 for Example 4.3.
    ζ α=2.5 α=2.7 α=2.9 α=3 Exact
    0 0. 0.97178 0.973463 0.974025 0.975897
    0.2 -0.04370738 0.908702 0.910351 0.910903 0.913089
    0.4 -0.085672 0.809397 0.810946 0.811465 0.813878
    0.6 -0.124221 0.677823 0.679212 0.679677 0.682221
    0.8 -0.157818 0.519227 0.5204 0.520792 0.523366
    1 -0.185124 0.339931 0.34084 0.341145 0.343646

     | Show Table
    DownLoad: CSV

    Using Laplace transform to Eq (4.21), we get

    U(ζ,s)cosζs+sinζs2+cosζs3+1sσ[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+L(L1t(2u(ζ,s)t2)L1t(u(ζ,s)ζ))L(L1t(U(ζ,s))L1t(U(ζ,s)t))]=0. (4.23)

    Residual Laplace function is given as

    LtResu(ζ,s)=U(ζ,s)cosζs+sinζs2+cosζs3+1sσ[3U(ζ,s)tζ24U(ζ,s)t2ζ2+4U(ζ,s)ζ4+L(L1t(2u(ζ,s)t2)L1t(u(ζ,s)ζ))L(L1t(U(ζ,s))L1t(U(ζ,s)t))], (4.24)

    and so the kth-truncated term series are

    u(ζ,s)=cosζs+sinζs2+cosζs3+kn=1fn(ζ,s)snα+1,k=1,2,3,4, (4.25)

    and the kth-LRFs as:

    LtResk(ζ,s)=Uk(ζ,s)cosζs+sinζs2+cosζs3+1sα[3Uk(ζ,s)tζ24Uk(ζ,s)t2ζ2+4Uk(ζ,s)ζ4+Lt(L1t(2Uk(ζ,s)t2)L1t(Uk(ζ,s)ζ))Lt(L1t(Uk(ζ,s))L1t(Uk(ζ,s)t))]. (4.26)

    Now, we calculate fk(ζ,s), k=1,2,3,, substituting the kth-truncate series of Eq (4.25) into the kth residual Laplace term Eq (4.26), multiply the solution equation by skα+1, and then solve recursively the link lims(skα+1LtResu,k(ζ,s))=0, k=1,2,3,. Following are the first some term:

    f1(ζ,s)=cosζ,f2(ζ,s)=cosζ,f3(ζ,s)=cosζ. (4.27)

    Putting the value of fk(x,s), k=1,2,3,, in Eq (4.25), we get

    U(ζ,s)=cosζssinζs2cosζs3cosζsα+1+cosζs2α+1cosζs3α+1+. (4.28)

    Using inverse LT, we get

    u(ζ,t)=cosζtsinζt2cosζ2tαcosζΓ(α+1)+t2αcosζΓ(2α+1)t3αcosζΓ(3α+1)+, (4.29)

    and the exact solution are

    u=cos(ζ+t). (4.30)

    In Figure 6, exact and LRPSM solutions for u(ζ,t) at α=3 and t=0.3 of Example 4.3. Figure 7, LRPSM solutions for u(ζ,t) at α=2.5, α=2.8, and t=0.3.

    Figure 6.  Exact and LRPSM solutions for u(ζ,t) at α=3 at and t=0.3.
    Figure 7.  LRPSM solutions for u(ζ,t) at α=2.5 α=2.8, and t=0.3.

    In this article, the fractional partial differential equation has been solved analytically by employing the Laplace residual power series method in conjunction with the Caputo operator. To demonstrate the validity of the recommended method, we analyzed three distinct partial differential equation problems. The simulation results demonstrate that the outcomes of our method are in close accordance with the exact answer. The new method is highly straightforward, efficient, and suitable for getting numerical solutions to partial differential equations. The primary advantage of the proposed approach is the series form solution, which rapidly converges to the exact answer. We can therefore conclude that the suggested approach is quite methodical and efficient for a more thorough investigation of fractional-order mathematical models.

    The authors declare no conflicts of interest.



    [1] C. Lee, F. Famoye, A. Y. Alzaatreh, Methods for generating families of univariate continuous distributions in the recent decades, Wiley Interdiscip. Rev.: Comput. Stat., 5 (2013), 219–238. https://doi.org/10.1002/wics.1255 doi: 10.1002/wics.1255
    [2] S. K. Maurya, S. Nadarajah, Poisson generated family of distributions: a review, Sankhya B, 83 (2021), 484–540. https://doi.org/10.1007/s13571-020-00237-8 doi: 10.1007/s13571-020-00237-8
    [3] M. H. Tahir, G. M. Cordeiro, Compounding of distributions: a survey and new generalized classes, J. Stat. Distrib. Appl., 3 (2016), 13. https://doi.org/10.1186/s40488-016-0052-1 doi: 10.1186/s40488-016-0052-1
    [4] A. Alzaatreh, C. Lee, F. Famoye, A new method for generating families of continuous distributions, METRON, 71 (2013), 63–79. https://doi.org/10.1007/s40300-013-0007-y doi: 10.1007/s40300-013-0007-y
    [5] I. W. Burr, Cumulative frequency functions, Ann. Math. Stat., 13 (1942), 215–232.
    [6] R. N. Rodriguez, A guide to the Burr type XII distributions, Biometrika, 64 (1977), 129–134. https://doi.org/10.1093/biomet/64.1.129 doi: 10.1093/biomet/64.1.129
    [7] R. V. da Silva, F. Gomes-Silva, M. W. A. Ramos, G. M. Cordeiro, The exponentiated Burr XII Poisson distribution with application to lifetime data, Int. J. Stat. Probab., 4 (2015), 112. https://doi.org/10.5539/ijsp.v4n4p112 doi: 10.5539/ijsp.v4n4p112
    [8] P. F. Paranaíba, E. M. Ortega, G. M. Cordeiro, M. A. de Pascoa, The Kumaraswamy Burr XII distribution: theory and practice, J. Stat. Comput. Simul., 83 (2013), 2117–2143. https://doi.org/10.1080/00949655.2012.683003 doi: 10.1080/00949655.2012.683003
    [9] A. Y. Al-Saiari, L. A. Baharith, S. A. Mousa, Marshall-Olkin extended Burr type XII distribution, Int. J. Stat. Probab., 3 (2014), 78–84. https://doi.org/10.5539/ijsp.v3n1p78 doi: 10.5539/ijsp.v3n1p78
    [10] H. M. Reyad, S. A. Othman, The Topp-Leone Burr-XII distribution: properties and applications, Br. J. Math. Comput. Sci., 21 (2017), 1–15.
    [11] P. F. Paranaíba, E. M. M. Ortega, G. M. Cordeiro, R. R. Pescim, The beta Burr XII distribution with application to lifetime data, Comput. Stat. Data Anal., 55 (2011), 1118–1136. https://doi.org/10.1016/j.csda.2010.09.009 doi: 10.1016/j.csda.2010.09.009
    [12] W. J. Zimmer, J. B. Keats, F. K. Wang, The Burr XII distribution in reliability analysis, J. Qual. Technol., 30 (1998), 386–394. https://doi.org/10.1080/00224065.1998.11979874 doi: 10.1080/00224065.1998.11979874
    [13] P. R. Tadikamalla, A look at the Burr and related distributions, Int. Stat. Review/Revue Int. Stat., 48 (1980), 337–344. https://doi.org/10.2307/1402945 doi: 10.2307/1402945
    [14] C. Kleiber, S. Kotz, Statistical size distributions in economics and actuarial sciences, John Wiley & Sons, 2003. https://doi.org/10.1002/0471457175
    [15] E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258–277. https://doi.org/10.2307/1968431
    [16] F. Castellares, S. L. P. Ferrari, A. J. Lemonte, On the Bell distribution and its associated regression model for count data, Appl. Math. Model., 56 (2018), 172–185. https://doi.org/10.1016/j.apm.2017.12.014 doi: 10.1016/j.apm.2017.12.014
    [17] A. Fayomi, M. Tahir, A. Algarni, M. Imran, F. Jamal, A new useful exponential model with applications to quality control and actuarial data, Comput. Intel. Neurosci., 2022 (2022), 2489998. https://doi.org/10.1155/2022/2489998 doi: 10.1155/2022/2489998
    [18] M. H. Tahir, M. Zubair, G. M. Cordeiro, A. Alzaatreh, M. Mansoor, The Poisson-X family of distributions, J. Stat. Comput. Simul., 86 (2016), 2901–2921. https://doi.org/10.1080/00949655.2016.1138224 doi: 10.1080/00949655.2016.1138224
    [19] A. Z. Afify, O. A. Mohamed, A new three-parameter exponential distribution with variable shapes for the hazard rate: estimation and applications, Mathematics, 8 (2020), 135. https://doi.org/10.3390/math8010135 doi: 10.3390/math8010135
    [20] M. Nassar, A. Z. Afify, M. K. Shakhatreh, S. Dey, On a new extension of Weibull distribution: properties, estimation, and applications to one and two causes of failures, Qual. Reliab. Eng. Int., 36 (2020), 2019–2043. https://doi.org/10.1002/qre.2671 doi: 10.1002/qre.2671
    [21] T. Dey, D. Kundu, Two-parameter Rayleigh distribution: different methods of estimation, Amer. J. Math. Manage. Sci., 33 (2014), 55–74. https://doi.org/10.1080/01966324.2013.878676 doi: 10.1080/01966324.2013.878676
    [22] S. Dey, S. Ali, C. Park, Weighted exponential distribution: properties and different methods of estimation, J. Stat. Comput. Simul., 85 (2015), 3641–3661. https://doi.org/10.1080/00949655.2014.992346 doi: 10.1080/00949655.2014.992346
    [23] S. Dey, T. Dey, S. Ali, M. S. Mulekar, Two-parameter Maxwell distribution: properties and different methods of estimation, J. Stat. Theory Pract., 10 (2016), 291–310. https://doi.org/10.1080/15598608.2015.1135090 doi: 10.1080/15598608.2015.1135090
    [24] S. Dey, A. Alzaatreh, C. Zhang, D. Kumar, A new extension of generalized exponential distribution with application to ozone data, Ozone: Sci. Eng., 39 (2017), 273–285. https://doi.org/10.1080/01919512.2017.1308817 doi: 10.1080/01919512.2017.1308817
    [25] J. H. Kao, Computer methods for estimating Weibull parameters in reliability studies, IRE T. Reliab. Qual. Control, PGRQC-13 (1958), 15–22. https://doi.org/10.1109/IRE-PGRQC.1958.5007164 doi: 10.1109/IRE-PGRQC.1958.5007164
    [26] J. M. Amigó, S. G. Balogh, S. Hernández, A brief review of generalized entropies, Entropy, 20 (2018), 813. https://doi.org/10.3390/e20110813 doi: 10.3390/e20110813
    [27] S. H. Abid, U. J. Quaez, J. E. Contreras-Reyes, An information-theoretic approach for multivariate skew-t distributions and applications, Mathematics, 9 (2021), 146. https://doi.org/10.3390/math9020146 doi: 10.3390/math9020146
    [28] C. G. Small, Expansions and asymptotics for statistics, 1 Ed., Chapman and Hall/CRC, 2010. https://doi.org/10.1201/9781420011029
    [29] J. J. Swain, S. Venkatraman, J. R. Wilson, Least-squares estimation of distribution functions in Johnson's translation system, J. Stat. Comput. Simul., 29 (1988), 271–297. https://doi.org/10.1080/00949658808811068 doi: 10.1080/00949658808811068
    [30] P. Macdonald, Comments and queries comment on "An estimation procedure for mixtures of distributions" by Choi and Bulgren, J. Royal Stat. Soc.: Ser. B (Methodol.), 33 (1971), 326–329. https://doi.org/10.1111/j.2517-6161.1971.tb00884.x doi: 10.1111/j.2517-6161.1971.tb00884.x
    [31] T. W. Anderson, D. A. Darling, Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes, Ann. Math. Stat., 23 (1952), 193–212. https://doi.org/10.1214/aoms/1177729437 doi: 10.1214/aoms/1177729437
    [32] M. Mansour, H. M. Yousof, W. Shehata, M. Ibrahim, A new two parameter Burr XII distribution: properties, copula, different estimation methods and modeling acute bone cancer data, J. Nonlinear Sci. Appl., 13 (2020), 223-238. http://dx.doi.org/10.22436/jnsa.013.05.01 doi: 10.22436/jnsa.013.05.01
    [33] H. M. Okasha, M. Shrahili, A new extended Burr XII distribution with applications, J. Comput. Theor. Nanosci., 14 (2017), 5261–5269. https://doi.org/10.1166/jctn.2017.6930 doi: 10.1166/jctn.2017.6930
    [34] A. M. Almarashi, K. Khan, C. Chesneau, F. Jamal, Group acceptance sampling plan using Marshall-Olkin Kumaraswamy exponential (MOKw-E) distribution, Processes, 9 (2021), 1066. https://doi.org/10.3390/pr9061066 doi: 10.3390/pr9061066
    [35] C. H. Jun, S. Balamurali, S. H. Lee, Variables sampling plans for Weibull distributed lifetimes under sudden death testing, IEEE T. Reliab., 55 (2006), 53–58. https://doi.org/10.1109/TR.2005.863802 doi: 10.1109/TR.2005.863802
    [36] C. W. Wu, W. L. Pearn, A variables sampling plan based on Cpmk for product acceptance determination, Eur. J. Oper. Res., 184 (2008), 549–560. https://doi.org/10.1016/j.ejor.2006.11.032 doi: 10.1016/j.ejor.2006.11.032
    [37] J. Chen, S. T. B. Choy, K. H. Li, Optimal Bayesian sampling acceptance plan with random censoring, Eur. J. Oper. Res., 155 (2004), 683–694. https://doi.org/10.1016/S0377-2217(02)00889-5 doi: 10.1016/S0377-2217(02)00889-5
    [38] A. J. Fernández, Progressively censored variables sampling plans for two-parameter exponential distributions, J. Appl. Stat., 32 (2005), 823–829. https://doi.org/10.1080/02664760500080074 doi: 10.1080/02664760500080074
    [39] W. L. Pearn, C. W. Wu, Variables sampling plans with PPM fraction of defectives and process loss consideration, J. Oper. Res. Soc., 57 (2006), 450–459. https://doi.org/10.1057/palgrave.jors.2602013 doi: 10.1057/palgrave.jors.2602013
    [40] A. J. Fernández, C. J. Pérez-González, M. Aslam, C. H. Jun, Design of progressively censored group sampling plans for Weibull distributions: an optimization problem, Eur. J. Oper. Res., 211 (2011), 525–532. https://doi.org/10.1016/j.ejor.2010.12.002 doi: 10.1016/j.ejor.2010.12.002
  • This article has been cited by:

    1. Xin XIA, Ying ZHANG, Zhigang HUANG, On Limiting Directions of Julia Sets of Entire Solutions of Complex Differential Equations, 2024, 29, 1007-1202, 357, 10.1051/wujns/2024294357
  • Reader Comments
  • © 2023 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(2007) PDF downloads(59) Cited by(4)

Figures and Tables

Figures(12)  /  Tables(10)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog