Research article Special Issues

Real-time tracking of moving objects from scattering matrix in real-world microwave imaging

  • Received: 26 February 2024 Revised: 27 March 2024 Accepted: 07 April 2024 Published: 12 April 2024
  • MSC : 78A46

  • The problem of the real-time microwave imaging of small, moving objects from a scattering matrix without diagonal elements, whose elements are measured scattering parameters, is considered herein. An imaging algorithm based on a Kirchhoff migration operated at single frequency is designed, and its mathematical structure is investigated by establishing a relationship with an infinite series of Bessel functions of integer order and antenna configuration. This is based on the application of the Born approximation to the scattering parameters of small objects. The structure explains the reason for the detection of moving objects via a designed imaging function and supplies some of its properties. To demonstrate the strengths and weaknesses of the proposed algorithm, various simulations with real-data are conducted.

    Citation: Seong-Ho Son, Kwang-Jae Lee, Won-Kwang Park. Real-time tracking of moving objects from scattering matrix in real-world microwave imaging[J]. AIMS Mathematics, 2024, 9(6): 13570-13588. doi: 10.3934/math.2024662

    Related Papers:

    [1] M. Mossa Al-Sawalha, Rasool Shah, Adnan Khan, Osama Y. Ababneh, Thongchai Botmart . Fractional view analysis of Kersten-Krasil'shchik coupled KdV-mKdV systems with non-singular kernel derivatives. AIMS Mathematics, 2022, 7(10): 18334-18359. doi: 10.3934/math.20221010
    [2] Ihsan Ullah, Aman Ullah, Shabir Ahmad, Hijaz Ahmad, Taher A. Nofal . A survey of KdV-CDG equations via nonsingular fractional operators. AIMS Mathematics, 2023, 8(8): 18964-18981. doi: 10.3934/math.2023966
    [3] Saima Rashid, Rehana Ashraf, Fahd Jarad . Strong interaction of Jafari decomposition method with nonlinear fractional-order partial differential equations arising in plasma via the singular and nonsingular kernels. AIMS Mathematics, 2022, 7(5): 7936-7963. doi: 10.3934/math.2022444
    [4] Aslı Alkan, Halil Anaç . The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method. AIMS Mathematics, 2024, 9(9): 25333-25359. doi: 10.3934/math.20241237
    [5] Rasool Shah, Abd-Allah Hyder, Naveed Iqbal, Thongchai Botmart . Fractional view evaluation system of Schrödinger-KdV equation by a comparative analysis. AIMS Mathematics, 2022, 7(11): 19846-19864. doi: 10.3934/math.20221087
    [6] Khalid Khan, Amir Ali, Manuel De la Sen, Muhammad Irfan . Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels. AIMS Mathematics, 2022, 7(2): 1580-1602. doi: 10.3934/math.2022092
    [7] Saima Rashid, Sobia Sultana, Bushra Kanwal, Fahd Jarad, Aasma Khalid . Fuzzy fractional estimates of Swift-Hohenberg model obtained using the Atangana-Baleanu fractional derivative operator. AIMS Mathematics, 2022, 7(9): 16067-16101. doi: 10.3934/math.2022880
    [8] Muammer Ayata, Ozan Özkan . A new application of conformable Laplace decomposition method for fractional Newell-Whitehead-Segel equation. AIMS Mathematics, 2020, 5(6): 7402-7412. doi: 10.3934/math.2020474
    [9] Khalid Khan, Amir Ali, Muhammad Irfan, Zareen A. Khan . Solitary wave solutions in time-fractional Korteweg-de Vries equations with power law kernel. AIMS Mathematics, 2023, 8(1): 792-814. doi: 10.3934/math.2023039
    [10] Musawa Yahya Almusawa, Hassan Almusawa . Numerical analysis of the fractional nonlinear waves of fifth-order KdV and Kawahara equations under Caputo operator. AIMS Mathematics, 2024, 9(11): 31898-31925. doi: 10.3934/math.20241533
  • The problem of the real-time microwave imaging of small, moving objects from a scattering matrix without diagonal elements, whose elements are measured scattering parameters, is considered herein. An imaging algorithm based on a Kirchhoff migration operated at single frequency is designed, and its mathematical structure is investigated by establishing a relationship with an infinite series of Bessel functions of integer order and antenna configuration. This is based on the application of the Born approximation to the scattering parameters of small objects. The structure explains the reason for the detection of moving objects via a designed imaging function and supplies some of its properties. To demonstrate the strengths and weaknesses of the proposed algorithm, various simulations with real-data are conducted.



    Fractional order Kersten-Krasil'shchik (KK) coupled Korteweg-de Vries (KdV)-modified KdV (mKdV) systems have served as the focus of extensive research in recent years due to their potential use in a variety of disciplines, including fluid dynamics, nonlinear optics and plasma physics [1,2,3,4]. These systems are characterized by the presence of fractional derivatives, which introduce non-local and memory effects into the dynamics of the system [5,6,7,8]. The KK coupled KdV-mKdV system is a system of two coupled nonlinear partial differential equations, which describe the evolution of two waves in a dispersive medium. The first equation is the well-known KdV equation, which describes the propagation of small amplitude, long wavelength waves. The second equation is the mKdV equation, which describes the propagation of larger amplitude, shorter wavelength waves. The KK coupling term, which is a nonlinear and non-local term, describes the interaction between the two waves [3,4,7]. Li et al. delved into an epidemic model's analysis and comparisons with other mechanisms in 2018 [12], while Jin and Wang explored chemotaxis phenomena in 2016 [13]. He et al. focused on fixed-point and variational inequality problems for Hadamard manifolds in 2022 [14]; this was followed by He et al. discussing nonexpansive mapping algorithms in 2023 [15]. Chen et al. contributed to particle physics, discovering hidden-charm pentaquarks in 2021 [16]. Lyu et al. analyzed cavity dynamics in water entries [17], whereas Yang and Kai delved into nonlinear Schr¨odinger equations in 2023.

    Time fractional Kersten-Krasil'shchik coupled KdV-mKdV nonlinear system and homogeneous two component time fractional coupled third order KdV systems are very important fractional nonlinear systems for describing the behaviour of waves in multi-component plasma and elaborate various nonlinear phenomena in plasma physics. Other studies have focused on the stability, existence and uniqueness of solutions for KK coupled KdV-mKdV systems of fractional order. In addition to these studies, there have been many other works that have explored the properties of fractional order KK coupled KdV-mKdV systems, such as their integrability, conservation laws and soliton interactions. Overall, the literature on fractional order KK coupled KdV-mKdV systems is rich and diverse, and it continues to grow as researchers explore new properties and applications of these systems [18,19,20].

    There are several methods that have been proposed to solve the fractional KdV (fKdV) equation and the mKdV equation. One of these methods is the homotopy perturbation method; this method uses a perturbation series and a homotopy approach to solve nonlinear differential equations. It has been used to find approximate solutions to the fKdV equation [21]. The variational iteration method uses a variation of a trial solution to find approximate solutions to nonlinear differential equations. The homotopy analysis method uses a homotopy approach and a perturbation series to solve nonlinear differential equations. It has been used to find approximate solutions to the fKdV equation [22] and the mKdV equation [23]. The Adomian decomposition method uses a decomposition of the nonlinear term of a differential equation into a series of simpler functions. It has been used to find approximate solutions to the fKdV equation and the mKdV equation [24,25]. Yang and Kai, dynamical properties, modulation instability analysis and chaotic behaviors to the nonlinear coupled Schrodinger equation in fiber Bragg gratings [26]. Chen et al. presented a linear free energy relationship in chemistry in 2020 [27]. Luo et al. proposed a new gradient method for force identification in vehicle-bridge systems in 2022 [28]. Additionally, Chen et al. focused on adaptive control of underwater vehicles in 2022 [29]. Lastly, Gu, Li and Liao developed an evolutionary multitasking approach for solving nonlinear equation systems in 2024 [30]. These studies collectively offer significant insights and advancements across a broad spectrum of scientific research areas, enriching our understanding and methodologies in their respective domains.

    The ADM is a powerful technique for solving nonlinear differential equations. Developed by George Adomian in the late 1980s, the ADM is based on the idea of decomposing the solution of a nonlinear equation into a series of simpler functions, known as Adomian polynomials. These polynomials are obtained by iteratively applying the nonlinear operator to a constant function [31,32]. The ADM has been applied to a wide range of nonlinear problems, including partial differential equations, integral equations and stochastic differential equations. One of the key advantages of the ADM is its ability to handle equations with singularities, such as those that arise in physics and engineering. In recent years, researchers have been exploring the use of the ADM in combination with other techniques, such as the ZZ transform. The ZZ transform is a mathematical tool that can be used to transform a nonlinear equation into a linear equation, making it easier to solve. By combining the ADM with the ZZ transform, researchers have been able to solve a wide range of nonlinear problems with greater efficiency and accuracy. Many researchers have used the ADM together with the ZZ transform to solve, for example, the nonlinear fractional partial differential equations in fluid dynamics, nonlinear integral equations in quantum mechanics and nonlinear fractional stochastic differential equations in finance. Overall, the ADM with the ZZ transform has been shown to be an efficient and flexible strategy for addressing nonlinear problems, with numerous potential applications in various fields [33,34].

    The current work is organized as follows. In Section 2, some fundamental definitions of fractional calculus are provided. The basic ideas of the Aboodh transform and the ADM are described in Section 3. In Section 4, we build approximate solutions to fractional Kersten-Krasil'shchik coupled KdV-mKdV systems of partial differential equations. Section 5 contains the conclusions.

    Definition 2.1. For functions, the Aboodh transformation is achieved as follow:

    B={U(ϱ):M,n1,n2>0,|U(ϱ)|<Meεϱ},

    which is described as follows [33,34]:

    A{U(ϱ)}=1ε0U(ϱ)eεϱdϱ,  ϱ>0 and n1εn2.

    Theorem 2.2. Consider G and F as the Aboodh and Laplace transformations, respectively, of U(ϱ) over the set B [35,36]. Then

    G(ε)=F(ε)ε. (2.1)

    Generalizing the Laplace and Aboodh integral transformations, Zain Ul Abadin Zafar created the ZZ transformation [37]. The ZZ transform is described as follows.

    Definition 2.3. For all values of ϱ0, the Z-transform for the function U(ϱ) is Z(κ,ε), which can be expressed as follows [37]:

    ZZ(U(ϱ))=Z(κ,ε)=ε0U(κϱ)eεϱdϱ.

    The Z-transform is linear in nature, just as the Laplace and Aboodh transforms. On the other hand, the Mittag-Leffler function (MLF) is an expansion of the exponential function:

    Eδ(z)=m=0zmΓ(1+mδ),Re(δ)>0.

    Definition 2.4. The Atangana-Baleanu-Caputo (ABC) derivative of a function U(φ,ϱ) in the space H1(a,b) for β(0,1) has the following definition [38]:

    ABCaDβϱU(φ,ϱ)=B(β)β+1ϱaU(φ,ϱ)Eβ(β(ϱη)ββ+1)dη.

    Definition 2.5. The Atangana-Baleanu Riemann-Liouville (ABR) derivative is a part of the space H1(a,b). The derivative can be represented for any value of β(0,1) as follows [38]:

    ABRaDβϱU(φ,η)=B(β)β+1ddϱϱaU(φ,η)Eβ(β(ϱη)ββ+1)dη.

    The property of the function B(β) is that it tends to 1 for both 0 and 1. Additionally, β>0, B(β)>a.

    Theorem 2.6. The Laplace transformation for the ABR derivative and ABC derivative are given by [38]:

    L{ABCaDβϱU(φ,ϱ)}(ε)=B(β)β+1εβL{U(φ,ϱ)}εβ1U(φ,0)εβ+ββ+1 (2.2)

    and

    L{ABRaDβϱU(φ,ϱ)}(ε)=B(β)β+1εβL{U(φ,ϱ)}εβ+ββ+1. (2.3)

    In the theorems below we assume that U(ϱ)H1(a,b), where b>a and β(0,1).

    Theorem 2.7. The Aboodh transform gives rise to a new ABR derivative, which is known as the Aboodh transformed ABR derivative [36]

    G(ε)=A{ABRaDβϱU(φ,ϱ)}(ε)=1ε[B(β)β+1εβL{U(φ,ϱ)}εβ+ββ+1]. (2.4)

    Theorem 2.8. The Aboodh transformation of ABC derivative is defined as follows [36]:

    G(ε)=A{ABCaDβϱU(φ,ϱ)}(ε)=1ε[B(β)β+1εβL{U(φ,ϱ)}εβ1U(φ,0)εβ+ββ+1]. (2.5)

    Theorem 2.9. The ZZ transformation of U(ϱ)=ϱβ1 is defined as

    Z(κ,ε)=Γ(β)(κε)β1. (2.6)

    Proof. The Aboodh transformation of U(ϱ)=ϱβ,  β0 is given by

    G(ε)=Γ(β)εβ+1.
     Now, G(εκ)=Γ(β)κβ+1εβ+1

    Applying Eq (2.6), we obtain

    Z(κ,ε)=ε2κ2G(εκ)=ε2κ2Γ(β)κβ+1εβ+1=Γ(β)(κε)β1.

    Theorem 2.10. Let β and ω be complex numbers and assume that the real part of β is greater than 0. The ZZ transformation of Eβ(ωϱβ) can be defined as follows [36]:

    ZZ{(Eβ(ωϱβ))}=Z(κ,ε)=(1ω(κε)β)1. (2.7)

    Proof. The Aboodh transformation of Eβ(ωϱβ) is defined as follows:

    G(ε)=F(ε)ε=εβ1ε(εβω). (2.8)

    So,

    G(εκ)=(εκ)β1(εκ)((εκ)βω) (2.9)

    we obtain

    Z(κ,ε)=(εκ)2G(εκ)=(εκ)2(εκ)β1(εκ)((εκ)βω)=(εκ)β(εκ)βω=(1ω(κε)β)1.

    Theorem 2.11. The ZZ transform of the ABC derivative can be defined as follows: If G(ε) and Z(κ,ε) are the ZZ and Aboodh transformations of U(ϱ), respectively [36], they we have

    ZZ{ABC0DβϱU(ϱ)}=[B(β)β+1εa+2κβ+2G(εκ)εβκβf(0)εβκβ+ββ+1]. (2.10)

    Proof. Applying this Eqs (2.1) and (2.5), we get

    G(εκ)=κε[B(β)β+1(εκ)β+1G(εκ)(εκ)β1f(0)(εκ)β+ββ+1]. (2.11)

    The ABC Z transformation is represented as follows:

    Z(κ,ε)=(εκ)2G(εκ)=(εκ)2κε[B(β)β+1(εκ)β+1G(εκ)(εκ)β1f(0)(εκ)β+ββ+1]=[B(β)β+1(εκ)β+2G(εκ)(εκ)βf(0)(εκ)β+ββ+1].

    Theorem 2.12. Let us assume that the ZZ transformation of U(ϱ) is represented by G(ε) and the Aboodh transformation of U(ϱ) is represented by Z(κ,ε). Then, the ZZ transformation of the ABR derivative is defined as [36]

    ZZ{ABR0Dβϱf(ϱ)}=[B(β)β+1εβ+2κβ+2G(εκ)εμκμ+ββ+1]. (2.12)

    Proof. Applying Eqs (2.1) and (2.4), we get

    G(εκ)=κε[B(β)β+1(εκ)β+1G(εκ)(εκ)β+ββ+1]. (2.13)
    Z(κ,ε)=(εκ)2G(εκ)=(εκ)2(κε)[B(β)β+1(εκ)β+1G(εκ)(εκ)β+ββ+1]=[B(β)β+1(εκ)β+2G(εκ)(εκ)β+ββ+1].

    In this section, we will examine the equation known as the fractional partial differential equation:

    DβU(φ,)=L(U(φ,))+N(U(φ,))+h(φ,)=M(φ,), (3.1)

    with the initial condition

    U(φ,0)=ϕ(φ), (3.2)

    where L(φ,) represents linear terms, N(φ,) represents nonlinear terms and h(φ,) represents the source term.

    Using the ZZ transform and ABC fractional derivatives, Eq (3.1) can be re-expressed as follows:

    1q(β,κ,ε)(Z[U(φ,)]ϕ(φ)ε)=Z[M(φ,)], (3.3)

    with

    q(β,κ,ε)=1β+β(κε)βB(β). (3.4)

    By taking the inverse ZZ transform, we get

    U(φ,)=Z1(ϕ(φ)ε+q(β,κ,ε)Z[M(φ,)]). (3.5)

    In terms of Adomain decomposition, we have

    i=0Ui(φ,)=Z1(ϕ(φ)ε+q(β,κ,ε)Z[h(φ,)])+Z1(q(β,κ,ε)Z[i=0L(Ui(φ,))+A]), (3.6)
    UABC0(φ,)=Z1(ϕ(φ)ε+q(β,κ,ε)Z[h(φ,)]),UABC1(φ,)=Z1(q(β,κ,ε)Z[L(U0(φ,))+A0]),UABCl+1(φ,)=Z1(q(β,κ,ε)Z[L(Ul(φ,))+Al]),  l=1,2,3,. (3.7)

    The solution to Eq (3.1) can be expressed by using ADMABC.

    UABC(φ,)=UABC0(φ,)+UABC1(φ,)+UABC2(φ,)+. (3.8)

    Example 4.1. Let us examine the following fractional KK joined KdV-mKdV nonlinear system:

    DβU+U3φ6UUφ+3VV3φ+3VφV2φ3UφV2+6UVVφ=0,  >0,  φR,  0<β1,DβV+V3φ3V2Vφ3UVφ+3UφV=0, (4.1)

    with the initial conditions given by

    U(φ,0)=c2c sech2(cφ),    c>0,V(φ,0)=2c sech(cφ). (4.2)

    By taking the ZZ transform, we get

    Z[DβU(φ,)]=Z[U3φ6UUφ+3VV3φ+3VφV2φ3UφV2+6UVVφ],Z[DβV(φ,)]=Z[V3φ3V2Vφ3UVφ+3UφV]. (4.3)

    Thus we have

    1εβZ[U(φ,)]ε2βU(φ,0)=Z[U3φ6UUφ+3VV3φ+3VφV2φ3UφV2+6UVVφ],1εβZ[V(φ,)]ε2βU(φ,0)=Z[V3φ3V2Vφ3UVφ+3UφV]. (4.4)

    By simplification we get

    Z[U(φ,)]=ε2[c2c sech2(cφ)](1β+β(κε)β)B(β)Z[U3φ6UUφ+3VV3φ+3VφV2φ3UφV2+6UVVφ],Z[V(φ,)]=ε2[2c sech(cφ)](1β+β(κε)β)B(β)Z[V3φ3V2Vφ3UVφ+3UφV]. (4.5)

    By taking the inverse ZZ transformation, we have

    U(φ,)=[c2c sech2(cφ)]Z1[(1β+β(κε)β)B(β)Z{U3φ6UUφ+3VV3φ+3VφV2φ3UφV2+6UVVφ}],V(φ,)=[2c sech(cφ)]Z1[(1β+β(κε)β)B(β)Z{V3φ3V2Vφ3UVφ+3UφV}]. (4.6)

    Assume that for the unknown functions U(φ,) and V(φ,), the series form solution is given as

    U(φ,)=l=0Ul(φ,),V(φ,)=l=0Ul(φ,), (4.7)

    The nonlinear components of the Adomian polynomials can be represented as follows: 6UUφ+3VV3φ=m=0Am, 3VφV2φ3UφV2=m=0Bm, 6UVVφ=m=0Cm and 3V2Vφ3UVφ+3UφV=m=0Dm. With the help of these terms, Eq (4.6) can be expressed as follows:

    l=0Ul+1(φ,)=c2c sech2(cφ)Z1[(1β+β(κε)β)B(β)Z{U3φ+l=0Al+l=0Bl+l=0Cl}],l=0Vl+1(φ,)=2c sech(cφ)Z1[(1β+β(κε)β)B(β)Z{U3φ+l=0Dl}]. (4.8)

    On comparing both sides of Eq (4.8), we have

    U0(φ,)=c2c sech2(cφ),V0(φ,)=2c sech(cφ),
    U1(φ,)=8c52sinh(cφ) sech3(cφ)(1β+ββΓ(β+1)),V1(φ,)=4c2sinh(cφ) sech2(cφ)(1β+ββΓ(β+1)), (4.9)
    U2(φ,)=16c4[2cosh2(cφ)3] sech4(cφ)[β22βΓ(2β+1)+2β(1β)βΓ(β+1)+(1β)2],V2(φ,)=8c72[cosh2(cφ)2] sech3(cφ)[β22βΓ(2β+1)+2β(1β)βΓ(β+1)+(1β)2]. (4.10)

    In this manner, the terms Ul and Vl for (l3) can be easily obtained. As a result, the series solution can be expressed as follows:

    U(φ,)=l=0Ul(φ,)=U0(φ,)+U1(φ,)+U2(φ,)+,U(φ,)=c2c sech2(cφ)+8c52sinh(cφ) sech3(cφ)(1β+ββΓ(β+1))16c4[2cosh2(cφ)3] sech4(cφ)[β22βΓ(2β+1)+2β(1β)βΓ(β+1)+(1β)2]+V(φ,)=l=0Vl(φ,)=V0(φ,)+V1(φ,)+V2(φ,)+,V(φ,)=2c sech(cφ)4c2sinh(cφ) sech2(cφ)(1β+ββΓ(β+1))+8c72[cosh2(cφ)2] sech3(cφ)[β22βΓ(2β+1)+2β(1β)βΓ(β+1)+(1β)2]+ (4.11)

    When β=1, we get the exact solution as

    U(φ,)=c2c sech2(c(φ+2c)),V(φ,)=2c sech(c(φ+2c)). (4.12)

    The graphical discussion involves several key figures illustrate the solutions for U(φ,) and V(φ,) in Example 4.1 at different parameter values. Figure 1 showcases the analytical and exact solutions at β=1 for U(φ,). In Figure 2, the approximate solutions are depicted at β=0.8,0.6.

    Figure 1.  The analytical and exact solutions at β=1 in U(φ,) for Example 4.1.
    Figure 2.  The approximate solutions at β=0.8,0.6 in U(φ,) for Example 4.1.

    Figure 3 extends the analysis by presenting analytical solutions at various values of β for U(φ,). Moving on to V(φ,), Figure 4 exhibits the analytical and exact solutions at β=1, while Figure 5 displays the analytical results at β=0.8,0.6.

    Figure 3.  The analytical solutions at various values of β in U(φ,) for Example 4.1.
    Figure 4.  The analytical and exact solutions at β=1 in V(φ,) for Example 4.1.
    Figure 5.  The analytical result at β=0.8,0.6 in V(φ,) for Example 4.1.

    Lastly, Figure 6 provides a comprehensive overview, presenting analytical results at various values of β, including β=1,0.8,0.6,0.4, for V(φ,). These figures collectively offer a detailed visual representation of the solutions under different conditions, facilitating a thorough understanding of the system's behavior.

    Figure 6.  The analytical result at various values in β for V(φ,) for Example 4.1.

    Example 4.2. Let us examine a homogeneous two-component KdV system of third order with a time-fractional component, as follows:

    DβUU3φUUφVVφ=0,  >0,  φR,  0<β1,DβV+2V3φUVφ=0, (4.13)

    with the initial conditions given by

    U(φ,0)=36tanh2(φ2),V(φ,0)=3c2tanh(φ2). (4.14)

    By taking the ZZ transform, we get

    Z[DβU(φ,)]=Z[U3φUUφVVφ],Z[DβV(φ,)]=Z[2V3φUVφ]. (4.15)

    Thus we have

    1εβZ[U(φ,)]ε2βU(φ,0)=Z[U3φUUφVVφ],1εβZ[V(φ,)]ε2βU(φ,0)=Z[2V3φUVφ]. (4.16)

    By simplification we get

    Z[U(φ,)]=ε2[36tanh2(φ2)](1β+β(κε)β)B(β)Z[U3φUUφVVφ],Z[V(φ,)]=ε2[3c2tanh(φ2)](1β+β(κε)β)B(β)Z[2V3φUVφ]. (4.17)

    By taking the inverse ZZ transform, we have

    U(φ,)=36tanh2(φ2)Z1[(1β+β(κε)β)B(β)Z{U3φUUφVVφ}],V(φ,)=[3c2tanh(φ2)]Z1[(1β+β(κε)β)B(β)Z{2V3φUVφ}]. (4.18)

    Assume that for the unknown functions U(φ,) and V(φ,), the series form solution is given as

    U(φ,)=l=0Ul(φ,),V(φ,)=l=0Ul(φ,). (4.19)

    The representation of nonlinear components using Adomian polynomials is shown as follows: UUφVVφ=m=0Am and UVφ=m=0Bm. With these terms, Eq (4.18) can be expressed as follows:

    l=0Ul+1(φ,)=36tanh2(φ2)+Z1[(1β+β(κε)β)B(β)Z{U3φ+l=0Al}],l=0Vl+1(φ,)=3c2tanh(φ2)+Z1[(1β+β(κε)β)B(β)Z{2V3φl=0Bl}]. (4.20)

    On comparing both sides of Eq (4.20), we have

    U0(φ,)=36tanh2(φ2),V0(φ,)=3c2tanh(φ2),
    U1(φ,)=6 sech2(φ2)tanh(φ2)(1β+ββΓ(β+1)),V1(φ,)=3c2 sech2(φ2)tanh(φ2)(1β+ββΓ(β+1)), (4.21)
    U2(φ,)=3[2+7 sech2(φ2)15 sech4(φ2)] sech2(φ2)[β22βΓ(2β+1)+2β(1β)βΓ(β+1)+(1β)2],V2(φ,)=3c22[2+21 sech2(φ2)24 sech4(φ2)] sech2(φ2)[β22βΓ(2β+1)+2β(1β)βΓ(β+1)+(1β)2]. (4.22)

    By using this method, the terms Ul and Vl can be easily obtained for l3. Therefore, the solution in the form of a series is as follows:

    U(φ,)=l=0Ul(φ,)=U0(φ,)+U1(φ,)+U2(φ,)+,U(φ,)=36tanh2(φ2)+6 sech2(φ2)tanh(φ2)(1β+ββΓ(β+1))+3[2+7 sech2(φ2)15 sech4(φ2)] sech2(φ2)[β22βΓ(2β+1)+2β(1β)βΓ(β+1)+(1β)2]+.V(φ,)=l=0Vl(φ,)=V0(φ,)+V1(φ,)+V2(φ,)+,V(φ,)=3c2tanh(φ2)+3c2 sech2(φ2)tanh(φ2)(1β+ββΓ(β+1))+3c22[2+21 sech2(φ2)24 sech4(φ2)] sech2(φ2)[β22βΓ(2β+1)+2β(1β)βΓ(β+1)+(1β)2]+. (4.23)

    When β=1, we get the exact solution as

    U(φ,)=36tanh2(φ+2),V(φ,)=3c2tanh(φ+2). (4.24)

    The graphical discussion involves several key figures that illustrate the solutions for U(φ,) and V(φ,) in Example 4.2 at different parameter values. Figure 7 showcases the analytical and exact solutions at β=1 for U(φ,). In Figure 8, the approximate solutions are depicted at β=0.8,0.6.

    Figure 7.  The analytical and exact solutions at β=1 in U(φ,) for Example 4.2.
    Figure 8.  The analytical result at β=0.8,0.6 in U(φ,) for Example 4.2.

    Figure 9 extends the analysis by presenting analytical solutions at various values of β for U(φ,). We will moving on to V(φ,).

    Figure 9.  The analytical result at various values of β in U(φ,) for Example 4.2.

    Figure 10 exhibits analytical and exact solutions at β=1, while Figure 11 displays the analytical results at β=0.8,0.6. Lastly, Figure 12 provides a comprehensive overview, presenting the analytical results at various values of β, including β=1,0.8,0.6,0.4, for V(φ,). These figures collectively offer a detailed visual representation of the solutions under different conditions, facilitating a thorough understanding of the system's behavior.

    Figure 10.  The analytical and exact result at β=1 in V(φ,) for Example 4.2.
    Figure 11.  The analytical solution of β=0.8,0.6 in V(φ,) for Example 4.2.
    Figure 12.  The analytical result at various values in β for V(φ,) for Example 4.2.

    In summary, the combination of the ADM and the ZZ transform has demonstrated its effectiveness in the analysis of the fractional KK coupled KdV-mKdV system encountered in multi-component plasmas. The utilization of this method has yielded accurate and efficient solutions, offering valuable insights into the intricate behavior of these complex systems. Additionally, the incorporation of the ZZ transform has enabled this frequency domain analysis, contributing supplementary information regarding the system's dynamics. This integrated approach stands as a valuable tool for comprehending multi-component plasma behaviors, and it holds the potential for application to analogous systems in future investigations. Future work may explore extending this methodology to different plasma models or investigating the impact of additional physical parameters, thereby broadening the scope of its applicability.

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

    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. 5081).

    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. 5081).

    The authors declare that they have no competing interests.



    [1] K. Agarwal, X. Chen, Y. Zhong, A multipole-expansion based linear sampling method for solving inverse scattering problems, Opt. Express, 18 (2010), 6366–6381. https://doi.org/10.1364/OE.18.006366 doi: 10.1364/OE.18.006366
    [2] H. Ammari, J. Garnier, H. Kang, M. Lim, K. Sølna, Multistatic imaging of extended targets, SIAM J. Imaging Sci., 5 (2012), 564–600. https://doi.org/10.1137/10080631X doi: 10.1137/10080631X
    [3] H. Ammari, J. Garnier, H. Kang, W.-K. Park, K. Sølna, Imaging schemes for perfectly conducting cracks, SIAM J. Appl. Math., 71 (2011), 68–91. https://doi.org/10.1137/100800130 doi: 10.1137/100800130
    [4] H. Ammari, E. Iakovleva, D. Lesselier, G. Perrusson, MUSIC type electromagnetic imaging of a collection of small three-dimensional inclusions, SIAM J. Sci. Comput., 29 (2007), 674–709. https://doi.org/10.1137/050640655 doi: 10.1137/050640655
    [5] T. Atay, M. Kaplan, Y. Kilic, N. Karapinar, A-Track: A new approach for detection of moving objects in FITS images, Comput. Phys. Commun., 207 (2016), 524–530. https://doi.org/10.1016/j.cpc.2016.07.023 doi: 10.1016/j.cpc.2016.07.023
    [6] E. J. Baranoski, Through-wall imaging: historical perspective and future directions, J. Franklin Inst., 345 (2008), 556–569. https://doi.org/10.1016/j.jfranklin.2008.01.005 doi: 10.1016/j.jfranklin.2008.01.005
    [7] M. Bonnet, Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems, Eng. Anal. Bound. Elem., 35 (2011), 223–235. https://doi.org/10.1016/j.enganabound.2010.08.007 doi: 10.1016/j.enganabound.2010.08.007
    [8] C. Cai, W. Liu, J. S. Fu, Y. Lu, A new approach for ground moving target indication in foliage environment, Signal Process., 86 (2006), 84–97. https://doi.org/10.1016/j.sigpro.2005.04.011 doi: 10.1016/j.sigpro.2005.04.011
    [9] E. Castro-Camus, M. Koch, D. M. Mittleman, Recent advances in terahertz imaging: 1999 to 2021, Appl. Phys. B, 128 (2022), 12. https://doi.org/10.1007/s00340-021-07732-4 doi: 10.1007/s00340-021-07732-4
    [10] L. Collins, P. Gao, D. Schofield, J. P. Moulton, L. C. Majakowsky, D. M. Reidy, et al., A statistical approach to landmine detection using broadband electromagnetic data, IEEE Trans. Geosci. Remote, 40 (2002), 950–962. https://doi.org/10.1109/TGRS.2002.1006387 doi: 10.1109/TGRS.2002.1006387
    [11] D. Colton, H. Haddar, P. Monk, The linear sampling method for solving the electromagnetic inverse scattering problem, SIAM J. Sci. Comput., 24 (2002), 719–731. https://doi.org/10.1137/S1064827501390467 doi: 10.1137/S1064827501390467
    [12] D. Colton, R. Kress, Inverse acoustic and electromagnetic scattering problems, New York: Springer, 1998. https://doi.org/10.1007/978-1-4614-4942-3
    [13] S. Coşğun, E. Bilgin, M. Çayören, Microwave imaging of breast cancer with factorization method: SPIONs as contrast agent, Med. Phys., 47 (2020), 3113–3122. https://doi.org/10.1002/mp.14156 doi: 10.1002/mp.14156
    [14] H. Diao, H. Liu, L. Wang, On generalized Holmgren's principle to the Lamé operator with applications to inverse elastic problems, Calc. Var., 59 (2020), 179. https://doi.org/10.1007/s00526-020-01830-5 doi: 10.1007/s00526-020-01830-5
    [15] J. R. Fienup, Detecting moving targets in SAR imagery by focusing, IEEE Trans. Aero. Elec. Syst., 37 (2001), 794–809. https://doi.org/10.1109/7.953237 doi: 10.1109/7.953237
    [16] A. Foudazix, A. Mirala, M. T. Ghasr, K. M. Donnell, Active microwave thermography for nondestructive evaluation of surface cracks in metal structures, IEEE Trans. Instrum. Meas., 68 (2019), 576–585. https://doi.org/10.1109/TIM.2018.2843601 doi: 10.1109/TIM.2018.2843601
    [17] A. Franchois, C. Pichot, Microwave imaging-complex permittivity reconstruction with a Levenberg-Marquardt method, IEEE Trans. Antenn. Propag., 45 (1997), 203–215. https://doi.org/10.1109/8.560338 doi: 10.1109/8.560338
    [18] B. B. Guzina, F. Pourahmadian, Why the high-frequency inverse scattering by topological sensitivity may work, Proc. R. Soc. A, 471 (2015), 20150187. https://doi.org/10.1098/rspa.2015.0187 doi: 10.1098/rspa.2015.0187
    [19] H. Haddar, P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem, Inverse Probl., 18 (2002), 891–906. https://doi.org/10.1088/0266-5611/18/3/323 doi: 10.1088/0266-5611/18/3/323
    [20] I. Harris, D.-L. Nguyen, Orthogonality sampling method for the electromagnetic inverse scattering problem, SIAM J. Sci. Comput., 42 (2020), B722–B737. https://doi.org/10.1137/19M129783X doi: 10.1137/19M129783X
    [21] M. Haynes, J. Stang, M. Moghaddam, Real-time microwave imaging of differential temperature for thermal therapy monitoring, IEEE Trans. Biomed. Eng., 61 (2014), 1787–1797. https://doi.org/10.1109/TBME.2014.2307072 doi: 10.1109/TBME.2014.2307072
    [22] K. Ito, B. Jin, J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Probl., 28 (2012), 025003. https://doi.org/10.1088/0266-5611/28/2/025003 doi: 10.1088/0266-5611/28/2/025003
    [23] K. Ito, B. Jin, J. Zou, A direct sampling method for inverse electromagnetic medium scattering, Inverse Probl., 29 (2013), 095018. https://doi.org/10.1088/0266-5611/29/9/095018 doi: 10.1088/0266-5611/29/9/095018
    [24] L. Jofre, A. Broquetas, J. Romeu, S. Blanch, A. P. Toda, X. Fabregas, et al., UWB tomographic radar imaging of penetrable and impenetrable objects, Proc. IEEE, 97 (2009), 451–464. https://doi.org/10.1109/JPROC.2008.2008854 doi: 10.1109/JPROC.2008.2008854
    [25] S. Kang, S. Chae, W.-K. Park, A study on the orthogonality sampling method corresponding to the observation directions configuration, Res. Phys., 33 (2022), 105108. https://doi.org/10.1016/j.rinp.2021.105108 doi: 10.1016/j.rinp.2021.105108
    [26] S. Kang, W.-K. Park, A novel study on the bifocusing method in two-dimensional inverse scattering problem, AIMS Mathematics, 8 (2023), 27080–27112. https://doi.org/10.3934/math.20231386 doi: 10.3934/math.20231386
    [27] S. Kang, W.-K. Park, S.-H. Son, A qualitative analysis of the bifocusing method for a real-time anomaly detection in microwave imaging, Comput. Math. Appl., 137 (2023), 93–101. https://doi.org/10.1016/j.camwa.2023.02.017 doi: 10.1016/j.camwa.2023.02.017
    [28] J.-Y. Kim, K.-J. Lee, B.-R. Kim, S.-I. Jeon, S.-H. Son, Numerical and experimental assessments of focused microwave thermotherapy system at 925MHz, ETRI J., 41 (2019), 850–862. https://doi.org/10.4218/etrij.2018-0088 doi: 10.4218/etrij.2018-0088
    [29] A. Kirsch, S. Ritter, A linear sampling method for inverse scattering from an open arc, Inverse Probl., 16 (2000), 89–105. https://doi.org/10.1088/0266-5611/16/1/308 doi: 10.1088/0266-5611/16/1/308
    [30] F. L. Louër, M.-L. Rapún, Topological sensitivity for solving inverse multiple scattering problems in 3D electromagnetism. Part Ⅰ: one step method, SIAM J. Imaging Sci., 10 (2017), 1291–1321. https://doi.org/10.1137/17M1113850 doi: 10.1137/17M1113850
    [31] J. J. Mallorqui, N. Joachimowicz, A. Broquetas, J. C. Bolomey, Quantitative images of large biological bodies in microwave tomography by using numerical and real data, Electron. Lett., 32 (1996), 2138–2140. https://doi.org/10.1049/el:19961409 doi: 10.1049/el:19961409
    [32] A. T. Mobashsher, A. M. Abbosh, On-site rapid diagnosis of intracranial hematoma using portable multi-slice microwave imaging system, Sci. Rep., 6 (2016), 37620. https://doi.org/10.1038/srep37620 doi: 10.1038/srep37620
    [33] J. R. Moreira, W. Keydel, A new MTI-SAR approach using the reflectivity displacement method, IEEE. Trans. Geosci. Remote, 33 (1995), 1238–1244. https://doi.org/10.1109/36.469488 doi: 10.1109/36.469488
    [34] G. Oliveri, N. Anselmi, A. Massa, Compressive sensing imaging of non-sparse 2D scatterers by a total-variation approach within the Born approximation, IEEE Trans. Antenn. Propag., 62 (2014), 5157–5170. https://doi.org/10.1109/TAP.2014.2344673 doi: 10.1109/TAP.2014.2344673
    [35] N. O. Önhon, M. Çetin, SAR moving object imaging using sparsity imposing priors, EURASIP J. Adv. Signal Process., 2017 (2017), 10. https://doi.org/10.1186/s13634-016-0442-z doi: 10.1186/s13634-016-0442-z
    [36] W.-K. Park, Asymptotic properties of MUSIC-type imaging in two-dimensional inverse scattering from thin electromagnetic inclusions, SIAM J. Appl. Math., 75 (2015), 209–228. https://doi.org/10.1137/140975176 doi: 10.1137/140975176
    [37] W.-K. Park, Multi-frequency subspace migration for imaging of perfectly conducting, arc-like cracks in full- and limited-view inverse scattering problems, J. Comput. Phys., 283 (2015), 52–80. https://doi.org/10.1016/j.jcp.2014.11.036 doi: 10.1016/j.jcp.2014.11.036
    [38] W.-K. Park, Performance analysis of multi-frequency topological derivative for reconstructing perfectly conducting cracks, J. Comput. Phys., 335 (2017), 865–884. https://doi.org/10.1016/j.jcp.2017.02.007 doi: 10.1016/j.jcp.2017.02.007
    [39] W.-K. Park, Direct sampling method for retrieving small perfectly conducting cracks, J. Comput. Phys., 373 (2018), 648–661. https://doi.org/10.1016/j.jcp.2018.07.014 doi: 10.1016/j.jcp.2018.07.014
    [40] W.-K. Park, Real-time microwave imaging of unknown anomalies via scattering matrix, Mech. Syst. Signal Proc., 118 (2019), 658–674. https://doi.org/10.1016/j.ymssp.2018.09.012 doi: 10.1016/j.ymssp.2018.09.012
    [41] W.-K. Park, Application of MUSIC algorithm in real-world microwave imaging of unknown anomalies from scattering matrix, Mech. Syst. Signal Proc., 153 (2021), 107501. https://doi.org/10.1016/j.ymssp.2020.107501 doi: 10.1016/j.ymssp.2020.107501
    [42] W.-K. Park, Real-time detection of small anomaly from limited-aperture measurements in real-world microwave imaging, Mech. Syst. Signal Proc., 171 (2022), 108937. https://doi.org/10.1016/j.ymssp.2022.108937 doi: 10.1016/j.ymssp.2022.108937
    [43] W.-K. Park, A novel study on the orthogonality sampling method in microwave imaging without background information, Appl. Math. Lett., 145 (2023), 108766. https://doi.org/10.1016/j.aml.2023.108766 doi: 10.1016/j.aml.2023.108766
    [44] W.-K. Park, On the application of orthogonality sampling method for object detection in microwave imaging, IEEE Trans. Antenn. Propag., 71 (2023), 934–946. https://doi.org/10.1109/TAP.2022.3220033 doi: 10.1109/TAP.2022.3220033
    [45] W.-K. Park, On the identification of small anomaly in microwave imaging without homogeneous background information, AIMS Mathematics, 8 (2023), 27210–27226. https://doi.org/10.3934/math.20231392 doi: 10.3934/math.20231392
    [46] W.-K. Park, H. P. Kim, K.-J. Lee, S.-H. Son, MUSIC algorithm for location searching of dielectric anomalies from Sparameters using microwave imaging, J. Comput. Phys., 348 (2017), 259–270. http://doi.org/10.1016/j.jcp.2017.07.035 doi: 10.1016/j.jcp.2017.07.035
    [47] W.-K. Park, D. Lesselier, Reconstruction of thin electromagnetic inclusions by a level set method, Inverse Probl., 25 (2009), 085010. https://doi.org/10.1088/0266-5611/25/8/085010 doi: 10.1088/0266-5611/25/8/085010
    [48] R. Potthast, A study on orthogonality sampling, Inverse Probl., 26 (2010), 074015. https://doi.org/10.1088/0266-5611/26/7/074015 doi: 10.1088/0266-5611/26/7/074015
    [49] Q. Rao, G. Xu, P. Wang, Z. Zheng, Study of the propagation characteristics of terahertz waves in a collisional and inhomogeneous dusty plasma with a ceramic substrate and oblique angle of incidence, Int. J. Antenn. Propag., 2021 (2021), 6625530. https://doi.org/10.1155/2021/6625530 doi: 10.1155/2021/6625530
    [50] T. Rubæk, P. M. Meaney, P. Meincke, K. D. Paulsen, Nonlinear microwave imaging for breast-cancer screening using Gauss–Newton's method and the CGLS inversion algorithm, IEEE Trans. Antenn. Propag., 55 (2007), 2320–2331. https://doi.org/10.1109/TAP.2007.901993 doi: 10.1109/TAP.2007.901993
    [51] M. Slaney, A. C. Kak, L. E. Larsen, Limitations of imaging with first-order diffraction tomography, IEEE Trans. Microwave Theory Tech., 32 (1984), 860–874. https://doi.org/10.1109/TMTT.1984.1132783 doi: 10.1109/TMTT.1984.1132783
    [52] S.-H. Son, K.-J. Lee, W.-K. Park, Application and analysis of direct sampling method in real-world microwave imaging, Appl. Math. Lett., 96 (2019), 47–53. https://doi.org/10.1016/j.aml.2019.04.016 doi: 10.1016/j.aml.2019.04.016
    [53] S.-H. Son, W.-K. Park, Application of the bifocusing method in microwave imaging without background information, J. Korean Soc. Ind. Appl. Math., 27 (2023), 109–122. https://doi.org/10.12941/jksiam.2023.27.109 doi: 10.12941/jksiam.2023.27.109
    [54] S.-H. Son, N. Simonov, H.-J. Kim, J.-M. Lee, S.-I. Jeon, Preclinical prototype development of a microwave tomography system for breast cancer detection, ETRI J., 32 (2010), 901–910. https://doi.org/10.4218/etrij.10.0109.0626 doi: 10.4218/etrij.10.0109.0626
    [55] W. Son, W.-K. Park, S.-H. Son, Microwave imaging method using neural networks for object localization, J. Electromagn. Eng. Sci., 22 (2022), 576–579. https://doi.org/10.26866/jees.2022.5.r.125 doi: 10.26866/jees.2022.5.r.125
    [56] A. E. Souvorov, A. E. Bulyshev, S. Y. Semenov, R. H. Svenson, A. G. Nazarov, Y. E. Sizov, et al., Microwave tomography: a two-dimensional Newton iterative scheme, IEEE Trans. Microwave Theory Tech., 46 (1998), 1654–1659. https://doi.org/10.1109/22.734548 doi: 10.1109/22.734548
    [57] I. Stojanovic, W. C. Karl, Imaging of moving targets with multi-static SAR using an overcomplete dictionary, IEEE J. Sel. Topics Signal Process., 4 (2010), 164–176. https://doi.org/10.1109/JSTSP.2009.2038982 doi: 10.1109/JSTSP.2009.2038982
    [58] G. Xu, Z. Song, Interaction of terahertz waves propagation in a homogeneous, magnetized, and collisional plasma slab, Wave. Random Media, 29 (2019), 665–677. https://doi.org/10.1080/17455030.2018.1462542 doi: 10.1080/17455030.2018.1462542
    [59] Y. Yin, W. Yin, P. Meng, H. Liu, On a hybrid approach for recovering multiple obstacles, Commun. Comput. Phys., 31 (2022), 869–892. https://doi.org/10.4208/cicp.OA-2021-0124 doi: 10.4208/cicp.OA-2021-0124
    [60] H. Yu, G. Xu, Z. Zheng, Transmission characteristics of terahertz waves propagation in magnetized plasma using the WKB method, Optik, 188 (2019), 244–250. https://doi.org/10.1016/j.ijleo.2019.05.061 doi: 10.1016/j.ijleo.2019.05.061
    [61] P. Zhang, P. Meng, W. Yin, H. Liu, A neural network method for time-dependent inverse source problem with limited-aperture data, J. Comput. Appl. Math., 421 (2023), 114842. https://doi.org/10.1016/j.cam.2022.114842 doi: 10.1016/j.cam.2022.114842
  • This article has been cited by:

    1. M. Sivakumar, M. Mallikarjuna, R. Senthamarai, A kinetic non-steady state analysis of immobilized enzyme systems with external mass transfer resistance, 2024, 9, 2473-6988, 18083, 10.3934/math.2024882
    2. Sajad Iqbal, Francisco Martínez, An innovative approach to approximating solutions of fractional partial differential equations, 2024, 99, 0031-8949, 065259, 10.1088/1402-4896/ad4928
    3. Musawa Yahya Almusawa, Hassan Almusawa, Dark and bright soliton phenomena of the generalized time-space fractional equation with gas bubbles, 2024, 9, 2473-6988, 30043, 10.3934/math.20241451
    4. Nazik J. Ahmed, Abdulghafor M. Al-Rozbayani, 2024, Chapter 43, 978-3-031-70923-4, 571, 10.1007/978-3-031-70924-1_43
    5. Panpan Wang, Xiufang Feng, Shangqin He, Lie Symmetry Analysis of Fractional Kersten–Krasil’shchik Coupled KdV–mKdV System, 2025, 24, 1575-5460, 10.1007/s12346-024-01152-3
    6. Khudhayr A. Rashedi, Musawa Yahya Almusawa, Hassan Almusawa, Tariq S. Alshammari, Adel Almarashi, Lump-type kink wave phenomena of the space-time fractional phi-four equation, 2024, 9, 2473-6988, 34372, 10.3934/math.20241637
    7. Manohar R. Gombi, B. J. Gireesha, P. Venkatesh, M. L. Keerthi, G. K. Ramesh, Fractional-order energy equation of a fully wet longitudinal fin with convective–radiative heat exchange through Sumudu transform analysis, 2025, 29, 1385-2000, 10.1007/s11043-025-09773-0
    8. Galal M. Moatimid, Mona A. A. Mohamed, Khaled Elagamy, Ahmed A. Gaber, Analysis of Eyring–Powell couple stress nanofluid flow with motile microorganisms over a rough sphere: Modified Adomian decomposition, 2025, 105, 0044-2267, 10.1002/zamm.202400981
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

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

Metrics

Article views(1263) PDF downloads(75) Cited by(4)

Figures and Tables

Figures(5)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog