Review Special Issues

Recent advances in mechanism/data-driven fault diagnosis of complex engineering systems with uncertainties

  • Received: 26 August 2024 Revised: 23 September 2024 Accepted: 09 October 2024 Published: 21 October 2024
  • MSC : 65C20, 68T10

  • The relentless advancement of modern technology has given rise to increasingly intricate and sophisticated engineering systems, which in turn demand more reliable and intelligent fault diagnosis methods. This paper presents a comprehensive review of fault diagnosis in uncertain environments, focusing on innovative strategies for intelligent fault diagnosis. To this end, conventional fault diagnosis methods are first reviewed, including advances in mechanism-driven, data-driven, and hybrid-driven diagnostic models and their strengths, limitations, and applicability across various scenarios. Subsequently, we provide a thorough exploration of multi-source uncertainty in fault diagnosis, addressing its generation, quantification, and implications for diagnostic processes. Then, intelligent strategies for all stages of fault diagnosis starting from signal acquisition are highlighted, especially in the context of complex engineering systems. Finally, we conclude with insights and perspectives on future directions in the field, emphasizing the need for the continued evolution of intelligent diagnostic systems to meet the challenges posed by modern engineering complexities.

    Citation: Chong Wang, Xinxing Chen, Xin Qiang, Haoran Fan, Shaohua Li. Recent advances in mechanism/data-driven fault diagnosis of complex engineering systems with uncertainties[J]. AIMS Mathematics, 2024, 9(11): 29736-29772. doi: 10.3934/math.20241441

    Related Papers:

    [1] Nadiyah Hussain Alharthi, Abdon Atangana, Badr S. Alkahtani . Numerical analysis of some partial differential equations with fractal-fractional derivative. AIMS Mathematics, 2023, 8(1): 2240-2256. doi: 10.3934/math.2023116
    [2] Abdon Atangana, Ali Akgül . Analysis of a derivative with two variable orders. AIMS Mathematics, 2022, 7(5): 7274-7293. doi: 10.3934/math.2022406
    [3] Hasib Khan, Jehad Alzabut, Anwar Shah, Sina Etemad, Shahram Rezapour, Choonkil Park . A study on the fractal-fractional tobacco smoking model. AIMS Mathematics, 2022, 7(8): 13887-13909. doi: 10.3934/math.2022767
    [4] Khaled M. Saad, Manal Alqhtani . Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear. AIMS Mathematics, 2021, 6(4): 3788-3804. doi: 10.3934/math.2021225
    [5] Amir Ali, Abid Ullah Khan, Obaid Algahtani, Sayed Saifullah . Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels. AIMS Mathematics, 2022, 7(8): 14975-14990. doi: 10.3934/math.2022820
    [6] Abdon Atangana, Seda İğret Araz . Extension of Chaplygin's existence and uniqueness method for fractal-fractional nonlinear differential equations. AIMS Mathematics, 2024, 9(3): 5763-5793. doi: 10.3934/math.2024280
    [7] Rahat Zarin, Amir Khan, Pushpendra Kumar, Usa Wannasingha Humphries . Fractional-order dynamics of Chagas-HIV epidemic model with different fractional operators. AIMS Mathematics, 2022, 7(10): 18897-18924. doi: 10.3934/math.20221041
    [8] Muhammad Farman, Ali Akgül, Sameh Askar, Thongchai Botmart, Aqeel Ahmad, Hijaz Ahmad . Modeling and analysis of fractional order Zika model. AIMS Mathematics, 2022, 7(3): 3912-3938. doi: 10.3934/math.2022216
    [9] Muhammad Farman, Ali Akgül, Kottakkaran Sooppy Nisar, Dilshad Ahmad, Aqeel Ahmad, Sarfaraz Kamangar, C Ahamed Saleel . Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel. AIMS Mathematics, 2022, 7(1): 756-783. doi: 10.3934/math.2022046
    [10] Muhammad Farman, Aqeel Ahmad, Ali Akgül, Muhammad Umer Saleem, Kottakkaran Sooppy Nisar, Velusamy Vijayakumar . Dynamical behavior of tumor-immune system with fractal-fractional operator. AIMS Mathematics, 2022, 7(5): 8751-8773. doi: 10.3934/math.2022489
  • The relentless advancement of modern technology has given rise to increasingly intricate and sophisticated engineering systems, which in turn demand more reliable and intelligent fault diagnosis methods. This paper presents a comprehensive review of fault diagnosis in uncertain environments, focusing on innovative strategies for intelligent fault diagnosis. To this end, conventional fault diagnosis methods are first reviewed, including advances in mechanism-driven, data-driven, and hybrid-driven diagnostic models and their strengths, limitations, and applicability across various scenarios. Subsequently, we provide a thorough exploration of multi-source uncertainty in fault diagnosis, addressing its generation, quantification, and implications for diagnostic processes. Then, intelligent strategies for all stages of fault diagnosis starting from signal acquisition are highlighted, especially in the context of complex engineering systems. Finally, we conclude with insights and perspectives on future directions in the field, emphasizing the need for the continued evolution of intelligent diagnostic systems to meet the challenges posed by modern engineering complexities.



    We all know Fermat's last theorem: For an integer n>2, the equation xn+yn=zn for x,y,z has no positive integer solution. It took 356 years from when it was proposed in 1637, to 1993 when Wiles conquered it. This equation can also be extended to functional equations. In complex analysis, the researchers began to focus on meromorphic solutions of the equation fn(z)+gn(z)=hn(z). As far as we know, Montel [22] was the first scholar to study this problem, and later Gross and Baker carried out the follow-up research work [2,9,10]. Then, applying Nevanlinna theory to the study of this kind of functional equation became a hot topic. For example, Yang et al. [31] studied the transcendental meromorphic solution of the functional equation f(z)2+f(z)2=1 and found that the solution to this equation must have the form of f(z)=12(peλz+1peλz), where p,λ are nonzero constants. With the establishment of the Nevanlinna theory with the difference of meromorphic function [8,11], more attention was paid to Fermat-type functional equation with the difference of meromorphic function. Liu et al. [15,16,17] studied the finite order transcendental entire solutions of Fermat-type difference equations f(z+c)2+f(z)2=1 and f(z+c)2+f(z)2=1, and they obtained that the solutions of these two equations are sine functions. Subsequently, more attention has been paid to this area of research [14,18,19,20,21,23,24,25,28,29,30,33,34].

    In 2016, Liu and Yang [18] studied the existence of solutions to quadratic trinomial functional equations, as well as the entire function and its derivatives and differences, and they converted the equations in the following two theorems into Fermat- type equations by transformation.

    Theorem 1.1. ([18, Theorem 1.4]) If α0,±1, then the finite order transcendental entire solution of equation

    f(z)2+2αf(z)f(z+c)+f(z+c)2=1

    is of order one.

    Theorem 1.2. ([18, Theorem 1.6]) If α0,±1, then the equation

    f(z)2+2αf(z)f(z)+f(z)2=1

    has no transcendental meromorphic solutions.

    On the other hand, Han and Lü [12] studied the existence of solutions to the Fermat-type equation when the right side was an exponential function. Here we only list the n=2 case in their results.

    Theorem 1.3. ([12, Theorem 1.1]) The meromorphic solutions of f of the following differential equation

    f(z)2+f(z)2=eαz+β

    are

    f(z)=eβ2sin(z+b)

    if α=0, and

    f(z)=deαz+β2

    if α0 with d2(1+(α2)2)=1.

    In the same article, they also studied the case of replacing f(z) with f(z+c) in the above equation, and found that the solution of equation

    f(z)2+f(z+c)2=eαz+β

    is f(z)=deαz+β2 with d2(1+eac)=1.

    Combining these conclusions above, Luo et al. [20] studied the following three equations with finite order transcendental entire solutions. These three equations are

    f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), (1.1)
    f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z) (1.2)

    and

    f(z)2+2αf(z)f(z)+f(z)2=eg(z), (1.3)

    where α0,±1, c are constants and g(z) is a polynomial. If all these equations admit finite order transcendental entire solutions, g(z) must be a polynomial with the degree of one, and the solutions f of these equations are all exponential functions or the sum of two exponential functions whose exponents are polynomials of the degree of one, as seen in Theorems 2.1–2.3 [20]. Each of these three equations contains only two terms of f(z),f(z) or f(z+c), so can we consider a quadratic equation that contains all three of these terms?

    Inspired by this, we shall study the problem of finite order transcendental entire solutions of functional equations involving the quadratic of f, its derivative and its difference. In fact, we studied the finite order transcendental entire solution for (1.4) below.

    Theorem 1.4. Suppose that α0,±1, β0, γ0 and c0 are four constants such that α2+β2+γ21+2αβγ, and g(z) is a nonconstant polynomial. If the complex equation

    [f(z)]2+[f(z+c)]2+f2(z)+2αf(z)f(z+c)+2βf(z)f(z+c)+2γf(z)f(z)=eg(z) (1.4)

    admits a transcendental entire solution f(z) of finite order, then for

    δ=1α2β2γ2+2αβγ1α2,

    the solution has two forms:

    (1)

    f(z)=deaz+b22iδ,

    where a0 and b is an arbitrary constant. Moreover, g(z)=az+b, and d is a constant with

    1+a24+eac+(aα+2β)eac/2+aγ=4δd2.

    (2)

    f(z)=ea1z+b1ea2z+b22iδ,

    where a1,a2(a1a2) are nonzero constants satisfying (1.5), and b1,b2 are arbitrary constants, g(z)=(a1+a2)z+b1+b2.

    {a21+2γa1+e2a1c+(2αa1+2β)ea1c+1=0;a22+2γa2+e2a2c+(2αa2+2β)ea2c+1=0;[ea1cea2c+α(a1a2)]2+(1α2)(a1a2)2+4δ=0. (1.5)

    Let's give two examples to show that Theorem 1.4 is true.

    Example 1.5. Suppose α=1/3, β=γ=1 and c=2 in (1.4), then δ=1/2. Set a=2,b=2, then by the relationship of a and d, we have

    d=24+83e2+e4.

    We can verify that the entire function

    f(z)=±14+83e2+e4ez+1

    is a solution of

    [f(z)]2+[f(z+2)]2+f2(z)+23f(z)f(z+2)+2f(z)f(z+2)+2f(z)f(z)=e2z+2.

    Example 1.6. Suppose

    α=3πi4,β=1+3π22,γ=5πi4

    and c=1 in (1.4), then δ=π2. a1=πi,a2=3πi, b1,b2 are arbitrary constants. We can verify that the entire function

    f(z)=eπiz+b1e3πiz+b2±2πi

    is a solution of

    [f(z)]2+[f(z+1)]2+f2(z)+3πi2f(z)f(z+1)+(2+3π2)f(z)f(z+1)5πi2f(z)f(z)=e4πiz+b1+b2.

    From Theorem 1.4 we have the following corollary.

    Corollary 1.7. Under the assumption of Theorem 1.4, if the degree of polynomial g(z) is greater than one, then (1.4) does not have a transcendental solution with finite order.

    If q(0,1) is a constant, then f(qz) is called the q-difference of meromorphic function f(z). The q-difference is also an important research content in the value distribution theory, and the research on it can be traced back to the early 20th century [5,13].

    In recent decades, with the establishment of Nevanlinna theory related to it [3], the research on q-difference has been vigorously developed, and this theory has been applied to many q-difference equations to get a lot of results [4,6,7,16,26,27]. Therefore, we considered to replace f(z+c) in (1.4) by f(qz) as to get a q-difference functional equation, and then studied the finite order transcendental entire solution of this equation. Through the complicated discussion and calculation of different cases, we came to the following conclusion.

    Theorem 1.8. Suppose that α0,±1, β0, γ0,±1 and q0,1 are four constants such that α2+β2+γ21+2αβγ, and g(z) is a nonconstant polynomial. If the complex equation

    [f(z)]2+[f(qz)]2+f2(z)+2αf(z)f(qz)+2βf(z)f(qz)+2γf(z)f(z)=eg(z) (1.6)

    admits a transcendental entire solution f(z) of finite order, then

    f(z)=±eaz+b2

    and g(z)=aqz+b, b is an arbitrary constant, a0 and γ21 such that

    {a24+γa+1=0,αa+2β=0. (1.7)

    Here is an example to test the truth of the Theorem 1.8.

    Example 1.9. Suppose α=1/2,β=1,γ=5/4 and q is any constant except 0,1 in (1.6), then we can verify that the entire function f(z)=±e2z+1 is a solution of

    [f(z)]2+[f(qz)]2+f2(z)+f(z)f(qz)2f(z)f(qz)52f(z)f(z)=e4qz+2.

    A corollary also can be obtained from Theorem 1.8.

    Corollary 1.10. Under the assumption of Theorem 1.8, if the degree of polynomial g(z) is greater than one, then (1.6) does not admit transcendental entire solution with finite order.

    Remark 1.11. Equations (1.4) and (1.6) can be transformed into three term quadratic equations by linear transformation. The purpose of restrictions α21 and α2+β2+γ21+2αβγ in Theorems 1.4 and 1.8 is to not allow these three term quadratic equations to degenerate into quadratic equations with two or one terms, which have been studied in previous literatures. This can be seen easily from (3.2) in the proof below.

    Remark 1.12. From the proof of Theorems 1.4 and 1.8 and the above three examples, we can find that if the two equations have finite order transcendental entire solutions, then the solutions of both equations are exponential functions and their exponents are polynomials with the degree of one. For (1.4), after the solution was substituted into the equation, the terms of the equation contained the common factor eg(z). After dividing both sides of the equation by eg(z), the relationship between the coefficients of the equation and the coefficients of the exponent was obtained. For (1.6), when one substitutes the solution into it, the term eg(z) in the right side of the equation is equal to [f(qz)]2 in the left side, which can be subtracted from both sides of the equation. The signs of the other two mixed terms containing f(qz) are opposite to each other, so these two mixed terms were canceled out.

    The following lemma played a key role in the proofs of this paper. It is about the factorization of an entire function. In particular, if f(z) was a finite order entire function without zero, then f(z)=eh(z) where h(z) was a nonconstant polynomial, as seen in Theorems 1.42 and 1.44 [32].

    Lemma 2.1. (Hadamard's factorization theorem) [32, Theorem 2.5] Let f be an entire function of finite order λ(f) with zeros {a1,a2,}C{0} and a k-fold zero at the origin. Then,

    f(z)=zkP(z)eQ(z)

    where P(z) is the canonical product of f formed with the non-null zeros of f,

    P(z)=n=1(1zan)ezan+12(zan)2++1h(zan)h,

    and h is the smallest integer for which this series converges, called the genus of the canonical product. Q(z) is a polynomial of degree λ(f) and hλ.

    The second lemma belongs to Borel. It's about the combination of entire functions, and we'll use it repeatedly in the proofs in Sections 3 and 4. When using it, the key is to verify the second condition below.

    Lemma 2.2. [32, Theorem 1.52] If fj(z)(j=1,2,,n) and gj(z)(j=1,2,,n)(n2) are entire functions satisfying

    (1) nj=1fj(z)egj(z)0,

    (2) the orders of fj are less than that of egh(z)gk(z) for 1jn,1h<kn.

    Then, fj0,(j=1,2,,n).

    According to the linear algebra, any quadratic form can be reduced to the standard form by a non-degenerate linear transformation. So, setting

    {f(z)=w,f(z)=uαv+αβγ1α2w,f(z+c)=vβαγ1α2w (3.1)

    and substituting it into (1.4), we obtain that

    u2+(1α2)v2+1α2β2γ2+2αβγ1α2w2=eg(z). (3.2)

    For simplicity and convenience, let's denote

    δ:=1α2β2γ2+2αβγ1α2,

    then (3.2) can convert into

    (u2+(1α2)v2eg(z)2)2+(δweg(z)2)2=1. (3.3)

    Consequently, we have

    (u2+(1α2)v2eg(z)2+iδweg(z)2)(u2+(1α2)v2eg(z)2iδweg(z)2)=1. (3.4)

    By Hadamard's factorization theorem, if the multiplicities of any zeros of the entire function u2+(1α2)v2 is even number, then u2+(1α2)v2 is also an entire function. The following (3.5) holds in the complex plane, where p(z) is a polynomial. If u2+(1α2)v2 have some zeros with odd number multiplicities, then u2+(1α2)v2 has branch points in the complex plane. Branches are obtained by connecting finite branch points and infinity points appropriately by line segments. These segments are called branch cuts, and u2+(1α2)v2 is analytic and univalent in every branch [1,35]. Since the two analytical factors on the left side of (3.4) have no zeros in each branch, there exists an analytical function p(z) such that the equations

    {u2+(1α2)v2eg(z)2+iδweg(z)2=ep(z),u2+(1α2)v2eg(z)2iδweg(z)2=ep(z) (3.5)

    hold in every branch. Denote

    λ1(z):=p(z)+g(z)2,   λ2(z):=p(z)+g(z)2,

    then,

    w=eλ1(z)eλ2(z)2iδ (3.6)

    hold in every branch. Moreover, noting that w=f(z) is an entire function with finite order, the righthand side of (3.6) can be extended to the whole complex plane. Therefore, one can supplement the definition of function p(z) at the points on branch cuts by the limiting values, and it is still called p(z) after supplementary definition. Thus, p(z) is analytic on the whole complex plane, so it's an entire function. Because ep(z) is of finite order, p(z) is actually a polynomial, so we get

    u2+(1α2)v2=(eλ1(z)+eλ2(z)2)2. (3.7)

    Noting that f(z)=w, we have

    {f(z)=eλ1(z)eλ2(z)2iδ,f(z)=λ1(z)eλ1(z)λ2(z)eλ2(z)2iδ,f(z+c)=eλ1(z+c)eλ2(z+c)2iδ. (3.8)

    From (3.1), we know that

    u=f(z)+αf(z+c)+γf(z)

    and

    v=f(z+c)+βαγ1α2f(z).

    Substituting the above u,v into (3.7) we get

    [f(z)]2+[f(z+c)]2+β2+γ22αβγ1α2f2(z)+2αf(z)f(z+c)+2βf(z+c)f(z)+2γf(z)f(z)=e2λ1(z)+e2λ2(z)+2eλ1(z)+λ2(z)4. (3.9)

    For simplicity and convenience, we give the following notation:

    λ1:=λ1(z),λ2:=λ2(z),¯λ1:=λ1(z+c),¯λ2:=λ2(z+c).

    Substituting (3.8) into (3.9) we obtain

    λ21e2λ1+λ22e2λ22λ1λ2eλ1+λ24δ+e2¯λ1+e2¯λ22e¯λ1+¯λ24δ+β2+γ22αβγ1α2e2λ1+e2λ22eλ1+λ24δ+2αλ1eλ1+¯λ1λ1eλ1+¯λ2λ2e¯λ1+λ2+λ2eλ2+¯λ24δ+2βeλ1+¯λ1eλ1+¯λ2e¯λ1+λ2+eλ2+¯λ24δ+2γλ1e2λ1λ1eλ1+λ2λ2eλ1+λ2+λ2e2λ24δ=e2λ1+e2λ2+2eλ1+λ24. (3.10)

    The transcendental terms appearing in the above equation have exponents

    2λ1,2λ2,λ1+λ2,2¯λ1,2¯λ2,¯λ1+¯λ2,λ1+¯λ1,λ1+¯λ2,¯λ1+λ2andλ2+¯λ2.

    In order to apply Lemma 2.2 to (3.10), we checked whether the pairwise difference between these exponents was constant. If λ1λ2, then f(z)0 this was impossible, so λ1λ2. The following two cases are discussed.

    Case 1. If λ1λ2 is a nonzero constant, then p(z) is a constant, denoted by p in the following. Consequently, for pkπi(kZ),

    f(z)=w=(epep)eg(z)/22iδ. (3.11)

    Then,

    f(z)=(epep)eg(z)/22iδg(z)2 (3.12)

    and

    f(z+c)=(epep)eg(z+c)/22iδ. (3.13)

    Substituting (3.11)–(3.13) into (1.4), the terms in the left side of (1.4) can be expressed respectively as

    {[f(z)]2=d2eg(z)4δ(g(z))24,[f(z+c)]2=d2eg(z+c)4δ,f2(z)=d2eg(z)4δ,2αf(z)f(z+c)=2αd2eg(z)+g(z+c)24δg(z)2,2βf(z)f(z+c)=2βd2eg(z)+g(z+c)24δ,2γf(z)f(z)=2γd2eg(z)4δg(z)2, (3.14)

    where d:=epep. If the degree of polynomial g(z) is greater than one, the three exponents g(z),g(z+c) and g(z)+g(z+c)2 are pairwise distinct. By Lemma 2.2, we obtained that after combining like terms, the coefficients of these three exponential terms eg(z),eg(z+c) and eg(z)+g(z+c)2 are zero. In particular, d24δ=0 since it's the coefficient of the sole term eg(z+c). This is impossible, because that means f0, so the degree of g(z) is one. Therefore, suppose g(z)=az+b, a(0),b are constants. Substitute it into (3.14), then into (1.4), and eliminate eg(z) from both sides of this equation. Then, we get

    1+a24+eac+(aα+2β)eac/2+aγ=4δd2. (3.15)

    This means if the constants α,β,γ,c in the original (1.4) are known, then the solution is

    f(z)=deaz+b22iδ,

    where constants a,d should satisfy the relationship of (3.15), and b is an arbitrary constant.

    Case 2. If λ1λ2 is not a constant, then p(z) is not a constant; instead, it is a nonconstant polynomial. For (3.10) we multiply 4δ on both sides, combine like terms and move all the terms to the left side of this equation, then the right side is just zero. Thus, the coefficients of the distinct transcendental terms can be listed in Table 1.

    Table 1.  Transcendental terms and corresponding coefficients.
    Transcendental terms Corresponding coefficients
    e2λ1 λ21+2γλ1+1
    e2λ2 λ22+2γλ2+1
    eλ1+λ2 2λ1λ22γ(λ1+λ2)+4δ2
    e2¯λ1 1
    e2¯λ2 1
    e¯λ1+¯λ2 2
    eλ1+¯λ1 2αλ1+2β
    eλ1+¯λ2 2αλ12β
    e¯λ1+λ2 2αλ22β
    eλ2+¯λ2 2αλ2+2β

     | Show Table
    DownLoad: CSV

    Because the difference between any two of 2λ1,2λ2,λ1+λ2 is not constant, the term containing e2λ1 cannot combine with terms containing e2λ2 or eλ1+λ2.

    Suppose that deg(λ1)=m>1 and deg(λ2)=n>1. If the term containing e2λ1 cannot combine with any other transcendental terms, then its coefficient has to be zero for any zC by Lemma 2.2. This is impossible, since its coefficients are nonconstant polynomials. The only term in the coefficient that may cancel out with λ21 is the term that contains λ2. They must have the same degree, so we have 2(m1)=n1. By the same arguments, we have m1=2(n1) by considering λ22 with λ1. Then, we get a contradiction, and it yields that deg(λ1),deg(λ2) are both at most one.

    Therefore, we can assume that λ1=a1z+b1 and λ2=a2z+b2 where a1a2 are constants, and b1,b2 are arbitrary constants. The transcendental terms and the corresponding coefficients in Table 1 can convert into those in Table 2. Since the three transcendental terms e2λ1, e2λ2 and eλ1+λ2 cannot be combined with each other, we get the following system of (3.16) with respect to the coefficients by Lemma 2.2.

    {a21+2γa1+e2a1c+(2αa1+2β)ea1c+1=0,a22+2γa2+e2a2c+(2αa2+2β)ea2c+1=0,2a1a22γ(a1+a2)2e(a1+a2)c(2αa1+2β)ea2c(2αa2+2β)ea1c+4δ2=0. (3.16)
    Table 2.  The transcendental term after the change and the corresponding coefficients.
    Before After Corresponding coefficients
    e2λ1 e2λ1 a21+2γa1+1
    e2¯λ1 e2λ1 e2a1c
    eλ1+¯λ1 e2λ1 (2αa1+2β)ea1c
    e2λ2 e2λ2 a22+2γa2+1
    e2¯λ2 e2λ2 e2a2c
    eλ2+¯λ2 e2λ2 (2αa2+2β)ea2c
    eλ1+λ2 eλ1+λ2 2a1a22γ(a1+a2)+4δ2
    e¯λ1+¯λ2 eλ1+λ2 2e(a1+a2)c
    eλ1+¯λ2 eλ1+λ2 (2αa12β)ea2c
    e¯λ1+λ2 eλ1+λ2 (2αa22β)ea1c

     | Show Table
    DownLoad: CSV

    Adding the three equations in (3.16) together, they convert into

    {a21+2γa1+e2a1c+(2αa1+2β)ea1c+1=0,a22+2γa2+e2a2c+(2αa2+2β)ea2c+1=0,[ea1cea2c+α(a1a2)]2+(1α2)(a1a2)2+4δ=0. (3.17)

    Therefore, the original (1.4) has solutions of the form

    f(z)=ea1z+b1ea2z+b22iδ,

    where a1,a2(a1a2) are nonzero constants satisfying (3.17), and b1,b2 are arbitrary constants.

    For an exponential polynomial f(z) with finite order, the exponents for each exponential terms of f(z) are the same as those of f(z), but the exponents of f(qz) are not the same as the exponents of f(z) for q0,1. The term eg(z) in the right side of (1.6) with coefficient one must combine with one of these two kinds of exponential terms, transcendental terms in f(z) or f(qz), by Lemma 2.2. The following are divided into two cases for discussion.

    Case 1. If eg(z) can combine with the exponential terms in f(z), then replacing f(z+c) by f(qz) in Section 3 and using the same methods in it we get

    {f(z)=eλ1(z)eλ2(z)2iδ,f(z)=λ1(z)eλ1(z)λ2(z)eλ2(z)2iδ,f(qz)=eλ1(qz)eλ2(qz)2iδ, (4.1)

    where λ1(z),λ2(z),δ are the same as in Section 3, and we also have

    u=f(z)+αf(qz)+γf(z),v=f(qz)+βαγ1α2f(z).

    Substituting the above u,v into

    u2+(1α2)v2=(eλ1(z)+eλ2(z)2)2, (4.2)

    it yields

    [f(z)]2+[f(qz)]2+β2+γ22αβγ1α2f2(z)+2αf(z)f(qz)+2βf(qz)f(z)+2γf(z)f(z)=e2λ1(z)+e2λ2(z)+2eλ1(z)+λ2(z)4. (4.3)

    For simplicity and convenience, we denote

    λ1:=λ1(z),λ2:=λ2(z),~λ1:=λ1(qz),~λ2:=λ2(qz).

    Substituting (4.1) into (4.3) we get

    λ21e2λ1+λ22e2λ22λ1λ2eλ1+λ24δ+e2~λ1+e2~λ22e~λ1+~λ24δ+β2+γ22αβγ1α2e2λ1+e2λ22eλ1+λ24δ+2αλ1eλ1+~λ1λ1eλ1+~λ2λ2e~λ1+λ2+λ2eλ2+~λ24δ+2βeλ1+~λ1eλ1+~λ2e~λ1+λ2+eλ2+~λ24δ+2γλ1e2λ1λ1eλ1+λ2λ2eλ1+λ2+λ2e2λ24δ=e2λ1+e2λ2+2eλ1+λ24. (4.4)

    The transcendental terms appearing in (4.4) have exponents

    2λ1,2λ2,λ1+λ2,2~λ1,2~λ2,~λ1+~λ2,λ1+~λ1,λ1+~λ2,~λ1+λ2andλ2+~λ2.

    In order to apply Lemma 2.2 to (4.4), we checked whether the pairwise difference between these exponents was constant. If λ1λ2, then f(z)0. This is impossible, so λ1λ2.

    Subcase 1.1. If λ1(z)λ2(z) is a nonzero constant, then p(z) is a nonzero constant, denoted by p in the following for simplicity. Consequently,

    f(z)=w=(epep)eg(z)/22iδ. (4.5)

    Then,

    f(z)=(epep)eg(z)/22iδg(z)2 (4.6)

    and

    f(qz)=(epep)eg(qz)/22iδ. (4.7)

    Substituting (4.5)–(4.7) into (1.6), the terms in the left side of this equation can be expressed as

    {[f(z)]2=d2eg(z)4δ(g(z))24,[f(qz)]2=d2eg(qz)4δ,f2(z)=d2eg(z)4δ,2αf(z)f(qz)=2αd2eg(z)+g(qz)24δg(z)2,2βf(z)f(qz)=2βd2eg(z)+g(qz)24δ,2γf(z)f(z)=2γd2eg(z)4δg(z)2, (4.8)

    where d:=epep.

    If polynomial g(z) contains at least two nonconstant terms, without loss of generality, we set

    g(z)=anzn++amzm++a0,n>m,

    then,

    g(qz)=an(qz)n++am(qz)m++a0

    and

    g(qz)g(z)=an(qn1)zn++am(qm1)zm+.

    If g(qz)g(z) is a constant and one has q=1, then this contradicts the assumption and g(qz)g(z) is not a constant. By the same argument, g(qz)g(z)+g(qz)2 is also not a constant. Therefore, the three exponential terms, eg(z),eg(qz) and eg(z)+g(qz)2, are pairwise distinct, even if we don't consider their constant coefficients. Substituting (4.8) into (1.6) and applying Lemma 2.2 to the obtained equation, we get that the coefficients of the three exponential terms are zero. In particular, d24δ=0 since it's the coefficient of the sole term eg(qz). This is impossible, because that means that f0.

    So, g(z) is the form of g(z)=anzn+b, where an(0),b are constants. Substitute this into (4.8) and we obtain that

    {[f(z)]2=d2eanzn+b4δ(nanzn1)24,[f(qz)]2=d2eanqnzn+b4δ,f2(z)=d2eanzn+b4δ,2αf(z)f(qz)=2αnanzn12d2ean(1+qn)zn2+b4δ,2βf(z)f(qz)=2βd2ean(1+qn)zn2+b4δ,2γf(z)f(z)=2γd2eanzn+b4δnanzn12, (4.9)

    then take the above expressions into (1.6). If qn1, the expression of [f(qz)]2 has a zero coefficient by Lemma 2.2, that is d24δ=0, which is impossible, so qn=1. Then, for all zC we have

    (nanzn1)24+2+(α+β)nanzn1+2β4δd2

    by eliminating eg(z) from both sides of (1.6), so n has to be one, and q=qn=1, which contradicts the assumption.

    Subcase 1.2. If λ1λ2 is not a constant, then p(z) is not a constant; instead, it is a nonconstant polynomial. We multiply 4δ, combine like terms in (4.4), and move all the terms to the left side of the equation, then the right side is just zero. Thus, the coefficients of the distinct transcendental terms are listed in Table 3.

    Table 3.  Transcendental terms and corresponding coefficients.
    Transcendental terms Corresponding coefficients
    e2λ1 λ21+2γλ1+1
    e2λ2 λ22+2γλ2+1
    eλ1+λ2 2λ1λ22γ(λ1+λ2)+4δ2
    e2~λ1 1
    e2~λ2 1
    e~λ1+~λ2 2
    eλ1+~λ1 2αλ1+2β
    eλ1+~λ2 2αλ12β
    e~λ1+λ2 2αλ22β
    eλ2+~λ2 2αλ2+2β

     | Show Table
    DownLoad: CSV

    By the same method in Case 2 of Section 3 (proof of Theorem 1.4), the degree of λ1(z) and λ2(z) both are at most one. Set λ1(z)=a1z+b1 and λ2(z)=a2z+b2, where a1a2, b1,b2 are arbitrary constants, then ~λ1(z)=a1qz+b1 and ~λ2(z)=a2qz+b2. Substituting these into Table 3, we get the results in Table 4.

    Table 4.  Transcendental terms and corresponding coefficients.
    No. Before After Corresponding coefficients
    e2λ1 e2a1z+2b1 a21+2γa1+1
    e2λ2 e2a2z+2b2 a22+2γa2+1
    eλ1+λ2 e(a1+a2)z+b1+b2 2a1a22γ(a1+a2)+4δ2
    e2~λ1 e2a1qz+2b1 1
    e2~λ2 e2a2qz+2b2 1
    e~λ1+~λ2 e(a1+a2)qz+b1+b2 2
    eλ1+~λ1 ea1(1+q)z+2b1 2αa1+2β
    eλ1+~λ2 e(a1+a2q)z+b1+b2 2αa12β
    e~λ1+λ2 e(a1q+a2)z+b1+b2 2αa22β
    eλ2+~λ2 ea2(1+q)z+2b2 2αa2+2β

     | Show Table
    DownLoad: CSV

    The coefficient of term ④ in Table 4 is one and it must combine with some like terms by Lemma 2.2. 2a1q in term ④ may be equal to 2a2 in term ②, a1+a2 in term ③, a1+a2q in term ⑧ and a2+a2q in term ⑩ since q1,a1a2. Then, we also considered that terms ⑤ and ⑥ must combine with some other like terms, respectively, since their coefficients are both nonzero, so there are many cases that have to be discussed. Through discussion for all possible cases, it is impossible to have a finite order entire solution (see the Appendix).

    Case 2. If eg(z) can combine with the exponential terms in f(qz), then

    {u=f(z)+αf(qz)+γf(z),v=f(z)+βαγ1γ2f(qz),w=f(qz), (4.10)

    and (1.6) can convert into

    u2+(1γ2)v2+δw2=eg(z), (4.11)

    where

    δ:=1α2β2γ2+2αβγ1γ2.

    By the same method in Section 3, we have

    {f(qz)=eλ1(z)eλ2(z)2iδ,f(z)=eλ1(z/q)eλ2(z/q)2iδ,f(z)=λ1eλ1(z/q)λ2eλ2(z/q)2qiδ, (4.12)

    where

    λ1(z)=p(z)+g(z)/2,  λ2(z)=p(z)+g(z)/2,

    p(z) is a polynomial. Here, λ1(z/q), λ2(z/q) are composite functions with respect to z, and according to the chain rule for derivatives, λ1 and λ2 represent the derivative of the outer function. By the same method in Case 1, we can divide into two subcases.

    Subcase 2.1. If λ1(z)λ2(z) is a constant, then (4.12) can be rewritten as

    {f(z)=deg(z/q)22iδ,f(z)=deg(z/q)22iδg2q,f(qz)=deg(z)22iδ, (4.13)

    where d=epep. Here, g(z/q) is a composite function with respect to z, and according to the chain rule for derivatives, g here represents the derivative of the outer function. Substituting (4.13) into each term in the right side of (1.6), we have

    {[f(z)]2=d2eg(z/q)4δ(g)24q2,[f(qz)]2=d2eg(z)4δ,f2(z)=d2eg(z/q)4δ,2αf(z)f(qz)=2αd2eg(z)+g(z/q)24δg2q,2βf(z)f(qz)=2βd2eg(z)+g(z/q)24δ,2γf(z)f(z)=2γd2eg(z/q)4δg2q. (4.14)

    If polynomial g(z) contains at least two nonconstant terms, without loss of generality we set

    g(z)=anzn++amzm++a0,   n>m,

    then

    g(z/q)=an(z/q)n++am(z/q)m++a0

    and

    g(z/q)g(z)=an(1qn1)zn++am(1qm1)zm+.

    If g(z/q)g(z) is a constant and one has q=1, this contradicts the assumption and g(z/q)g(z) is not a constant. By the same argument, g(z/q)g(z)+g(z/q)2 is also not a constant. Therefore, the three exponential terms, eg(z),eg(qz) and eg(z)+g(qz)2, are distinct and can't combine like terms. Substituting (4.14) into (1.6) and applying Lemma 2.2 to the obtained equation, we get that the coefficients of the three exponential terms are zeroes after combining like terms:

    {d24δ10,d24δ(g)24q2+d24δ+2γd24δg2q0,2αd24δg2q+2βd24δ0. (4.15)

    Since the three equations hold for all zC, g(z/q) has a degree of one, and so does g(z).

    Therefore, we can set g(z/q)=az+b, then g(z)=aqz+b, where a0 and b is an arbitrary constant. Noting that g represents the derivative of the outer function, so g=aq, then the above equations convert into

    {d24δ=1,a24+1+2γa0,αa+2β0. (4.16)

    Thus, it yields α2+β22αβγ=0 and γ21.

    In other words, (1.6) admits a solution in this case with the form f(z)=eaz+b2 and g(z)=aqz+b such that

    {a24+1+2γa0,αa+2β0, (4.17)

    then α2+β22αβγ=0 and γ21.

    Subcase 2.2. If λ1(z)λ2(z) is nonconstant, then from (4.11) we have

    u2+(1γ2)v2=(eλ1(z)+eλ2(z)2)2. (4.18)

    Substituting (4.10) into (4.18) we deduce that

    [f(z)]2+[f(z)]2+α2+β22αβγ1γ2[f(qz)]2+2αf(z)f(qz)+2βf(qz)f(z)+2γf(z)f(z)=e2λ1(z)+e2λ2(z)+2eλ1(z)+λ2(z)4. (4.19)

    For simplicity and convenience, we denote

    λ1:=λ1(z),λ2:=λ2(z),^λ1:=λ1(z/q),^λ2:=λ2(z/q).

    Substituting (4.12) into (4.19) we obtain

    1q2λ21e2^λ1+λ22e2^λ22λ1λ2e^λ1+^λ24δ+e2^λ1+e2^λ22e^λ1+^λ24δ+α2+β22αβγ1γ2e2λ1+e2λ22eλ1+λ24δ+2α1qλ1eλ1+^λ1λ1e^λ1+λ2λ2eλ1+^λ2+λ2eλ2+^λ24δ+2βeλ1+^λ1eλ1+^λ2e^λ1+λ2+eλ2+^λ24δ+2γ1qλ1e2^λ1λ1e^λ1+^λ2λ2e^λ1+^λ2+λ2e2^λ24δ=e2λ1+e2λ2+2eλ1+λ24. (4.20)

    Let's multiply both sides of (4.20) by 4δ and move all terms to the left side of the equation. After combining the terms of the same kind, we get the results in Table 5. Using the method in Subcase 1.2, we can also obtain that there are no suitable finite order transcendental entire solutions for (1.6) in this case. The details are omitted here.

    Table 5.  Transcendental terms and corresponding coefficients.
    Transcendental terms Corresponding coefficients
    e2λ1 1
    e2λ2 1
    eλ1+λ2 4δ2
    e2^λ1 1q2λ21+2γqλ1+1
    e2^λ2 1q2λ22+2γqλ2+1
    e^λ1+^λ2 2q2λ1λ22γq(λ1+λ2)2
    eλ1+^λ1 2αqλ1+2β
    eλ1+^λ2 2αqλ22β
    e^λ1+λ2 2αqλ12β
    eλ2+^λ2 2αqλ2+2β

     | Show Table
    DownLoad: CSV

    In this paper we proved two theorems (Theorems 1.4 and 1.8), studied the finite order entire solutions of (1.4) and (1.6), respectively and found the concrete forms of solutions of these two equations, both of which were exponential functions. Examples 1.5 and 1.6 verified the two cases of solutions of the equation in Theorem 1.4, and Example 1.9 verified the truth of Theorem 1.8. The equations studied in this paper can be transformed into the Fermat-type equation with three quadratic terms by linear transformation, which improves the previous Fermat-type equations with only two quadratic terms, so it is very novel.

    The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by Natural Science Foundation of Henan (No. 232300420129) and the Key Scientific Research Project of Colleges and Universities in Henan Province (No. 22A110004), China. The author would like to thank the referee for a careful reading of the manuscript and valuable comments.

    The author declares no conflict of interest.

    We divided Table 4 into the following four cases with 10 subcases, but in each case there is no finite order transcendental entire solution. The classification is based on the fact that terms ④, ⑤ and ⑥ in Table 4 must be combined with other terms, since their coefficients are nonzero. Otherwise, they contradict with Lemma 2.2.

    Case A1. Term ④ can combine with term ② and ⑨ in Table 4, that is, a1q=a2. Since term ⑤ should also combine with other terms, we split into some subcases.

    Subcase A1.1. Term ⑤ may combine with term ① in Table 4. Thus, we get the results in Table A1.

    Table A1.  q=a2a1=1,a2=a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e2a1z e2b2(a212γa1+1)
    e0 eb1+b2(2a21+4δ2)
    e2a1z e2b1
    e2a1z e2b2
    e0 2eb1+b2
    e0 e2b1(2αa1+2β)
    e2a1z eb1+b2(2αa12β)
    e2a1z eb1+b2(2αa12β)
    e0 e2b2(2αa1+2β)

     | Show Table
    DownLoad: CSV

    After combining terms of the same kind in Table A1, according to Lemma 2.2, we know that the coefficients must always be zero, and the following equations were obtained

    {e2b1(a21+2γa1+1)+e2b2+eb1+b2(2αa12β)=0,e2b2(a212γa1+1)+e2b1+eb1+b2(2αa12β)=0,eb1+b2(a21+2δ2)+e2b1(αa1+β)+e2b2(αa1+β)=0.

    Thus, we have

    {eb1=±eb2,α=±γ,β=±1,

    it yields δ=0, which is impossible.

    Subcase A1.2. Term ⑤ may combine with term ③ in Table 4. Then, we obtained the results in Table A2 as follows.

    Table A2.  q=a2a1=1/2,a2=a1/2.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    ea1z e2b2(a214γa1+1)
    ea12z eb1+b2(a21γa1+4δ2)
    ea1z e2b1
    ea12z e2b2
    ea14z 2eb1+b2
    ea12z e2b1(2αa1+2β)=0
    e5a14z eb1+b2(2αa12β)=0
    ea1z eb1+b2(αa12β)=eb1+b2(3β)
    ea14z e2b2(αa1+2β)

     | Show Table
    DownLoad: CSV

    Combining terms of the same kind and according to Lemma 2.2, we know that the coefficients must always be zero, and the following equations are obtained

    {e2b1(a21+2γa1+1)=0,e2b2(a214γa1+1)+e2b1+eb1+b2(αa12β)=0,eb1+b2(a21γa1+4δ2)+e2b2+e2b1(2αa1+2β)=0,2eb1+b2+e2b2(αa1+2β)=0,eb1+b2(2αa12β)=0.

    For the above equation system, there is no suitable solution a1.

    Case A2. Term ④ can combine with term ③ in Table 4, that is, 2a1q=a1+a2. Since term ⑤ should also combine with other terms, we split it into three subcases.

    Subcase A2.1. Term ⑤ may combine with term ① in Table 4. We get the results in Table A3.

    Table A3.  q=a1+a22a1=1/2,a2=2a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e4a1z e2b2(4a214γa1+1)
    ea1z eb1+b2(4a21+2γa1+4δ2)
    ea1z e2b1
    e2a1z e2b2
    ea12z 2eb1+b2
    ea12z e2b1(2αa1+2β)
    e2a1z eb1+b2(2αa12β)
    e5a12z eb1+b2(4αa12β)
    ea1z e2b2(4αa1+2β)

     | Show Table
    DownLoad: CSV

    From Table A3, after combining terms of the same kind, according to Lemma 2.2, we know that the coefficients must always be zero, and the following equations are obtained

    {e2b1(a21+2γa1+1)+e2b2+eb1+b2(2αa12β)=0,e2b2(4a214γa1+1)=0,eb1+b2(4a21+2γa1+4δ2)+e2b1+e2b2(4αa1+2β)=0,2eb1+b2+e2b1(2αa1+2β)=0,eb1+b2(4αa12β)=0.

    For the above equation system, there is no suitable solution a1.

    Subcase A2.2. Term ⑤ may combine with term ⑦ in Table 4. Then, we get the results in Table A4.

    Table A4.  q=a1+a22a1=14,a2=32a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e3a1z e2b2(94a213γa1+1)
    e12a1z eb1+b2(3a21+γa1+4δ2)
    e12a1z e2b1
    e34a1z e2b2
    e18a1z 2eb1+b2
    e34a1z e2b1(2αa1+2β)
    e118a1z eb1+b2(2αa12β)
    e7a14z eb1+b2(3αa12β)
    e98a1z e2b2(3αa1+2β)

     | Show Table
    DownLoad: CSV

    In Table A4, the term ⑥ cannot combine with other transcendental terms; it's impossible.

    Subcase A2.3. Term ⑤ may combine with term ⑨ in Table 4. Then, we deduce the results in Table A5.

    Table A5.  q=a1+a22a1=14,a2=12a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    ea1z e2b2(a22+2γa2+1)
    e12a1z eb1+b2(2a1a22γ(a1+a2)+4δ2)
    e12a1z e2b1
    e14a1z e2b2
    e18a1z 2eb1+b2
    e54a1z e2b1(2αa1+2β)
    e78a1z eb1+b2(2αa12β)
    e14a1z eb1+b2(2αa22β)
    e58a1z e2b2(2αa2+2β)

     | Show Table
    DownLoad: CSV

    In Table A5, the term ⑥ cannot combine with other transcendental terms; it's impossible.

    Case A3. Term ④ can combine with term ⑧ in Table 4, that is, 2a1q=a1+a2q. Since term ⑤ should also combine with other terms, we split it into two subcases.

    Subcase A3.1. Term ⑤ may combine with term ③ in Table 4. We get the results in Table A6.

    Table A6.  q=a12a1a2=14,a2=2a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e4a1z e2b2(a22+2γa2+1)
    ea1z eb1+b2(2a1a22γ(a1+a2)+4δ2)
    e12a1z e2b1
    ea1z e2b2
    e14a1z 2eb1+b2
    e54a1z e2b1(2αa1+2β)
    e12a1z eb1+b2(2αa12β)
    e74a1z eb1+b2(2αa22β)
    e52a1z e2b2(2αa2+2β)

     | Show Table
    DownLoad: CSV

    The term ⑥ cannot combine with other transcendental terms; this is impossible.

    Subcase A3.2. Term ⑤ may combine with term ⑨ in Table 4. We get the results in Table A7.

    Table A7.  q=a12a1a2=13,a2=a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e2a1z e2b2(a22+2γa2+1)
    e0 eb1+b2(2a1a22γ(a1+a2)+4δ2)
    e23a1z e2b1
    e23a1z e2b2
    e0 2eb1+b2
    e43a1z e2b1(2αa1+2β)
    e23a1z eb1+b2(2αa12β)
    e23a1z eb1+b2(2αa22β)
    e43a1z e2b2(2αa2+2β)

     | Show Table
    DownLoad: CSV

    By ⑦ and ⑩ in Table A7, we have a1=a2, which is a contradiction.

    Case A4. Term ④ can combine with term ⑩ in Table 4, that is, 2a1q=a2+a2q. Since term ⑤ should also combine with other terms, we split it into three subcases.

    Subcase A4.1. Term ⑤ may combine with term ① in Table 4. We get the results in Table A8.

    Table A8.  q=a22a1a2=12,a2=2a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e4a1z e2b2(4a214γa1+1)
    ea1z eb1+b2(4a21+2γa1+4δ2)
    ea1z e2b1
    e2a1z e2b2
    e12a1 2eb1+b2
    e12a1z e2b1(2αa1+2β)
    e2a1z eb1+b2(2αa12β)
    e52a1z eb1+b2(4αa12β)
    ea1z e2b2(4αa1+2β)

     | Show Table
    DownLoad: CSV

    Subcase A4.2. Term ⑤ may combine with term ③ in Table 4. Then, we have the results in Table A9.

    Table A9.  q=a22a1a2=14,a2=23a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e43a1z e2b2(a22+2γa2+1)
    e13a1z eb1+b2(2a1a22γ(a1+a2)+4δ2)
    e12a1z e2b1
    e13a1z e2b2
    e112a1 2eb1+b2
    e34a1z e2b1(2αa1+2β)
    e76a1z eb1+b2(2αa12β)
    e1112a1z eb1+b2(2αa22β)
    e12a1z e2b2(2αa2+2β)

     | Show Table
    DownLoad: CSV

    The term ⑥ cannot combine with other transcendental terms; it's impossible.

    Subcase A4.3. Term ⑤ may combine with term ⑦ in Table 4. Then, we get the results in Table A10.

    Table A10.  q=a22a1a2=13,a2=a1.
    No. Transcendental terms Corresponding coefficients
    e2a1z e2b1(a21+2γa1+1)
    e2a1z e2b2(a22+2γa2+1)
    e0 eb1+b2(2a1a22γ(a1+a2)+4δ2)
    e23a1z e2b1
    e23a1z e2b2
    e0 2eb1+b2
    e23a1z e2b1(2αa1+2β)
    e43a1z eb1+b2(2αa12β)
    e43a1z eb1+b2(2αa22β)
    e23a1z e2b2(2αa2+2β)

     | Show Table
    DownLoad: CSV

    From ⑧ and ⑨ in Table A10, we have a1=a2; it's impossible.



    [1] L. Jia, Q. Gao, Z. P. Liu, H. B. Tan, L. W. Zhou, Multidisciplinary fault diagnosis of complex engineering systems: A case study of nuclear power plants, Int. J. Ind. Ergon., 80 (2020), 103060. https://doi.org/10.1016/j.ergon.2020.103060 doi: 10.1016/j.ergon.2020.103060
    [2] Y. B. Li, B. Li, J. C. Ji, H. Kalhori, Advanced fault diagnosis and health monitoring techniques for complex engineering systems, Sensors, 22 (2022), 10002. https://doi.org/10.3390/s222410002 doi: 10.3390/s222410002
    [3] C. Wang, H. G. Matthies, Random model with fuzzy distribution parameters for hybrid uncertainty propagation in engineering systems, Comput. Meth. Appl. Mech. Eng., 359 (2020), 112673. https://doi.org/10.1016/j.cma.2019.112673 doi: 10.1016/j.cma.2019.112673
    [4] F. Villecco, A. Pellegrino, Evaluation of uncertainties in the design process of complex mechanical systems, Entropy, 19 (2017), e19090475. https://doi.org/10.3390/e19090475 doi: 10.3390/e19090475
    [5] E. Hüllermeier, W. Waegeman, Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods, Mach. Learn., 110 (2021), 457–506. https://doi.org/10.1007/s10994-021-05946-3 doi: 10.1007/s10994-021-05946-3
    [6] H. R. Fan, C. Wang, S. H. Li, Novel method for reliability optimization design based on rough set theory and hybrid surrogate model, Comput. Meth. Appl. Mech. Eng., 429 (2024), 117170. https://doi.org/10.1016/j.cma.2024.117170 doi: 10.1016/j.cma.2024.117170
    [7] M. Mansouri, R. Fezai, M. Trabelsi, M. Hajji, M.-F. Harkat, H. Nounou, et al., A novel fault diagnosis of uncertain systems based on interval Gaussian process regression: Application to wind energy conversion systems, IEEE Access, 8 (2020), 219672–219679. https://doi.org/10.1109/access.2020.3042101 doi: 10.1109/access.2020.3042101
    [8] X. X. Liu, Y. T. Ju, X. H. Liu, S. Miao, W. G. Zhang, An imu fault diagnosis and information reconstruction method based on analytical redundancy for autonomous underwater vehicle, IEEE Sens. J., 22 (2022), 12127–12138. https://doi.org/10.1109/jsen.2022.3174340 doi: 10.1109/jsen.2022.3174340
    [9] D. Yu, Fault diagnosis for a hydraulic drive system using a parameter-estimation method, Control Eng. Practice, 5 (1997), 1283–1291. https://doi.org/10.1016/s0967-0661(97)84367-5 doi: 10.1016/s0967-0661(97)84367-5
    [10] G. C. Zhang, L. Chen, K. K. Liang, Fault monitoring and diagnosis of aerostat actuator based on pca and state observer, Int. J. Model. Identif. Control, 32 (2019), 145. https://doi.org/10.1504/ijmic.2019.102367 doi: 10.1504/ijmic.2019.102367
    [11] Y. Song, M. Y. Zhong, J. Chen, Y. Liu, An alternative parity space-based fault diagnosability analysis approach for linear discrete time systems, IEEE Access, 6 (2018), 16110–16118. https://doi.org/10.1109/access.2018.2816970 doi: 10.1109/access.2018.2816970
    [12] V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, A review of process fault detection and diagnosis, Comput. Chem. Eng., 27 (2003), 313–326. https://doi.org/10.1016/s0098-1354(02)00161-8 doi: 10.1016/s0098-1354(02)00161-8
    [13] S. W. Pan, D. Xiao, S. T. Xing, S. S. Law, P. Y. Du, Y. J. Li, A general extended kalman filter for simultaneous estimation of system and unknown inputs, Eng. Struct., 109 (2016), 85–98. https://doi.org/10.1016/j.engstruct.2015.11.014 doi: 10.1016/j.engstruct.2015.11.014
    [14] E. Walker, S. Rayman, R. E. White, Comparison of a particle filter and other state estimation methods for prognostics of lithium-ion batteries, J. Power Sources, 287 (2015), 1–12. https://doi.org/10.1016/j.jpowsour.2015.04.020 doi: 10.1016/j.jpowsour.2015.04.020
    [15] S. Nolan, A. Smerzi, L. Pezzè, A machine learning approach to Bayesian parameter estimation, npj Quantum Inform., 7 (2021), 169. https://doi.org/10.1038/s41534-021-00497-w doi: 10.1038/s41534-021-00497-w
    [16] R. Tarantino, F. Szigeti, E. Colina-Morles, Generalized luenberger observer-based fault-detection filter design: An industrial application, Control Eng. Practice, 8 (2000), 665–671. https://doi.org/10.1016/s0967-0661(99)00181-1 doi: 10.1016/s0967-0661(99)00181-1
    [17] L. A. Rusinov, N. V. Vorobiev, V. V. Kurkina, Fault diagnosis in chemical processes and equipment with feedbacks, Chemometrics Intell. Lab. Syst., 126 (2013), 123–128. https://doi.org/10.1016/j.chemolab.2013.03.015 doi: 10.1016/j.chemolab.2013.03.015
    [18] F. Pierri, G. Paviglianiti, F. Caccavale, M. Mattei, Observer-based sensor fault detection and isolation for chemical batch reactors, Eng. Appl. Artif. Intell., 21 (2008), 1204–1216. https://doi.org/10.1016/j.engappai.2008.02.002 doi: 10.1016/j.engappai.2008.02.002
    [19] H. M. Odendaal, T. Jones, Actuator fault detection and isolation: An optimised parity space approach, Control Eng. Practice, 26 (2014), 222–232. https://doi.org/10.1016/j.conengprac.2014.01.013 doi: 10.1016/j.conengprac.2014.01.013
    [20] C. J. Duan, Z. Y. Fei, J. C. Li, A variable selection aided residual generator design approach for process control and monitoring, Neurocomputing, 171 (2016), 1013–1020. https://doi.org/10.1016/j.neucom.2015.07.042 doi: 10.1016/j.neucom.2015.07.042
    [21] P. Zhang, S. X. Ding, Disturbance decoupling in fault detection of linear periodic systems, Automatica, 43 (2007), 1410–1417. https://doi.org/10.1016/j.automatica.2007.01.005 doi: 10.1016/j.automatica.2007.01.005
    [22] Q. Wang, C. Taal, O. Fink, Integrating expert knowledge with domain adaptation for unsupervised fault diagnosis, IEEE Trans. Instrum. Meas., 71 (2022), 1–12. https://doi.org/10.1109/tim.2021.3127654 doi: 10.1109/tim.2021.3127654
    [23] P. Zhao, X. D. Mu, Z. R. Yin, Z. X. Yi, An approach of fault diagnosis for system based on fuzzy fault tree, 2008 International Conference on MultiMedia and Information Technology, Three Gorges, China, 2008,697–700. https://doi.org/10.1109/mmit.2008.142
    [24] Z. N. Lin, Y. X. Wang, H. Q. Xu, F. R. Wei, A novel reduced-order analytical fault diagnosis model for power grid, IEEE Access, 12 (2024), 59521–59532. https://doi.org/10.1109/access.2024.3392905 doi: 10.1109/access.2024.3392905
    [25] C. Cheng, X. Y. Qiao, H. Luo, W. X. Teng, M. L. Gao, B. C. Zhang, et al., A semi-quantitative information based fault diagnosis method for the running gears system of high-speed trains, IEEE Access, 7 (2019), 38168–38178. https://doi.org/10.1109/access.2019.2906976 doi: 10.1109/access.2019.2906976
    [26] J. P. Shi, W. G. Tong, D. L. Wang, Design of the transformer fault diagnosis expert system based on fuzzy reasoning, 2009 International Forum on Computer Science-Technology and Applications, Chongqing, China, 2009,110–114. https://doi.org/10.1109/ifcsta.2009.34
    [27] A. R. Sahu, S. K. Palei, A. Mishra, Data-driven fault diagnosis approaches for industrial equipment: A review, Expert Syst., 41 (2024), 13360. https://doi.org/10.1111/exsy.13360 doi: 10.1111/exsy.13360
    [28] G. Wang, J. Y. Zhao, J. H. Yang, J. F. Jiao, J. L. Xie, F. Feng, Multivariate statistical analysis based cross voltage correlation method for internal short-circuit and sensor faults diagnosis of lithium-ion battery system, J. Energy Storage, 62 (2023), 106978. https://doi.org/10.1016/j.est.2023.106978 doi: 10.1016/j.est.2023.106978
    [29] Z. Zhang, X. He, Active fault diagnosis for linear systems: Within a signal processing framework, IEEE Trans. Instrum. Meas., 71 (2022), 1–9. https://doi.org/10.1109/tim.2022.3150889 doi: 10.1109/tim.2022.3150889
    [30] R. N. Liu, B. Y. Yang, E. Zio, X. F. Chen, Artificial intelligence for fault diagnosis of rotating machinery: A review, Mech. Syst. Signal Proc., 108 (2018), 33–47. https://doi.org/10.1016/j.ymssp.2018.02.016 doi: 10.1016/j.ymssp.2018.02.016
    [31] Y. Q. Liu, B. Liu, X. J. Zhao, M. Xie, A mixture of variational canonical correlation analysis for nonlinear and quality-relevant process monitoring, IEEE Trans. Ind. Electron., 65 (2018), 6478–6486. https://doi.org/10.1109/tie.2017.2786253 doi: 10.1109/tie.2017.2786253
    [32] G. Lee, C. H. Han, E. S. Yoon, Multiple-fault diagnosis of the tennessee eastman process based on system decomposition and dynamic pls, Ind. Eng. Chem. Res., 43 (2004), 8037–8048. https://doi.org/10.1021/ie049624u doi: 10.1021/ie049624u
    [33] G. Yu, C. N. Li, J. Sun, Machine fault diagnosis based on Gaussian mixture model and its application, Int. J. Adv. Manuf. Technol., 48 (2010), 205–212. https://doi.org/10.1007/s00170-009-2283-5 doi: 10.1007/s00170-009-2283-5
    [34] W. Deng, S. J. Zhang, H. M. Zhao, X. H. Yang, A novel fault diagnosis method based on integrating empirical wavelet transform and fuzzy entropy for motor bearing, IEEE Access, 6 (2018), 35042–35056. https://doi.org/10.1109/access.2018.2834540 doi: 10.1109/access.2018.2834540
    [35] J. B. Guo, Fault diagnosis method of flexible converter valve equipment based on ensemble empirical mode decomposition and temporal convolutional networks, J. Electr. Syst., 20 (2024), 344–352. https://doi.org/10.52783/jes.2386 doi: 10.52783/jes.2386
    [36] D. J. Yu, M. Wang, X. M. Cheng, A method for the compound fault diagnosis of gearboxes based on morphological component analysis, Measurement, 91 (2016), 519–531. https://doi.org/10.1016/j.measurement.2016.05.087 doi: 10.1016/j.measurement.2016.05.087
    [37] L. Ciabattoni, F. Ferracuti, A. Freddi, A. Monteriu, Statistical spectral analysis for fault diagnosis of rotating machines, IEEE Trans. Ind. Electron., 65 (2018), 4301–4310. https://doi.org/10.1109/tie.2017.2762623 doi: 10.1109/tie.2017.2762623
    [38] W. E. Sanders, T. Burton, A. Khosousi, S. Ramchandani, Machine learning: At the heart of failure diagnosis, Curr. Opin. Cardiol., 36 (2021), 227–233. https://doi.org/10.1097/hco.0000000000000833 doi: 10.1097/hco.0000000000000833
    [39] Y. G. Lei, B. Yang, X. W. Jiang, F. Jia, N. P. Li, A. K. Nandi, Applications of machine learning to machine fault diagnosis: A review and roadmap, Mech. Syst. Signal Proc., 138 (2020), 106587. https://doi.org/10.1016/j.ymssp.2019.106587 doi: 10.1016/j.ymssp.2019.106587
    [40] Z. N. An, F. Wu, C. Zhang, J. H. Ma, B. Sun, B. H. Tang, et al., Deep learning-based composite fault diagnosis, IEEE Jour. Emer. Select. Top. Circu. Syste., 13 (2023), 572–581. https://doi.org/10.1109/jetcas.2023.3262241 doi: 10.1109/jetcas.2023.3262241
    [41] D. T. Hoang, H. J. Kang, A survey on deep learning based bearing fault diagnosis, Neurocomputing, 335 (2019), 327–335. https://doi.org/10.1016/j.neucom.2018.06.078 doi: 10.1016/j.neucom.2018.06.078
    [42] X. Y. Fan, J. Li, H. Hao, Review of piezoelectric impedance based structural health monitoring: Physics-based and data-driven methods, Adv. Struct. Eng., 24 (2021), 3609–3626. https://doi.org/10.1177/13694332211038444 doi: 10.1177/13694332211038444
    [43] Q. Ni, X. M. Li, Z. W. Chen, Z. L. Zhao, L. L. Lai, A mechanism and data hybrid-driven method for main circuit ground fault diagnosis in electrical traction system, IEEE Trans. Ind. Electron., 70 (2023), 12806–12815. https://doi.org/10.1109/tie.2023.3260356 doi: 10.1109/tie.2023.3260356
    [44] D. An, N. H. Kim, J. H. Choi, Practical options for selecting data-driven or physics-based prognostics algorithms with reviews, Reliab. Eng. Syst. Saf., 133 (2015), 223–236. https://doi.org/10.1016/j.ress.2014.09.014 doi: 10.1016/j.ress.2014.09.014
    [45] J. Guo, Z. Y. Li, M. Y. Li, A review on prognostics methods for engineering systems, IEEE Trans. Reliab., 69 (2020), 1110–1129. https://doi.org/10.1109/tr.2019.2957965 doi: 10.1109/tr.2019.2957965
    [46] L. Kou, C. Liu, G. W. Cai, J. N. Zhou, Q. D. Yuan, S. M. Pang, Fault diagnosis for open-circuit faults in npc inverter based on knowledge-driven and data-driven approaches, IET Power Electron., 13 (2020), 1236–1245. https://doi.org/10.1049/iet-pel.2019.0835 doi: 10.1049/iet-pel.2019.0835
    [47] X. X. Xiao, C. H. Li, J. Huang, T. Yu, Fault diagnosis of rolling bearing based on knowledge graph with data accumulation strategy, IEEE Sens. J., 22 (2022), 18831–18840. https://doi.org/10.1109/JSEN.2022.3201839 doi: 10.1109/JSEN.2022.3201839
    [48] K. Sachin, M. Torres, Y. C. Chan, M. Pecht, A hybrid prognostics methodology for electronic products, 2008 IEEE International Joint Conference on Neural Networks, Hong Kong, China, 2008, 3479–3485. https://doi.org/10.1109/IJCNN.2008.4634294
    [49] S. F. Cheng, M. Pecht, A fusion prognostics method for remaining useful life prediction of electronic products, 2009 IEEE International Conference on Automation Science and Engineering, Bangalore, India, 2009,102–107. https://doi.org/10.1109/COASE.2009.5234098
    [50] H. G. Zhang, R. Kang, M. Pecht, A hybrid prognostics and health management approach for condition-based maintenance, 2009 IEEE International Conference on Industrial Engineering and Engineering Management, Hong Kong, China, 2009, 1165–1169. https://doi.org/10.1109/ieem.2009.5372976
    [51] M. A. Chao, C. Kulkarni, K. Goebel, O. Fink, Fusing physics-based and deep learning models for prognostics, Reliab. Eng. Syst. Saf., 217 (2022), 107961. https://doi.org/10.1016/j.ress.2021.107961 doi: 10.1016/j.ress.2021.107961
    [52] T. T. Li, Y. Zhao, C. B. Zhang, J. Luo, X. J. Zhang, A knowledge-guided and data-driven method for building hvac systems fault diagnosis, Build. Environ., 198 (2021), 107850. https://doi.org/10.1016/j.buildenv.2021.107850 doi: 10.1016/j.buildenv.2021.107850
    [53] L. H. Ye, X. Ma, C. L. Wen, Rotating machinery fault diagnosis method by combining time-frequency domain features and cnn knowledge transfer, Sensors, 21 (2021), 8168. https://doi.org/10.3390/s21248168 doi: 10.3390/s21248168
    [54] W. Xu, Y. Wan, T. Y. Zuo, X. M. Sha, Transfer learning based data feature transfer for fault diagnosis, IEEE Access, 8 (2020), 76120–76129. https://doi.org/10.1109/ACCESS.2020.2989510 doi: 10.1109/ACCESS.2020.2989510
    [55] X. P. Niu, R. Z. Wang, D. Liao, S. P. Zhu, X. C. Zhang, B. Keshtegar, Probabilistic modeling of uncertainties in fatigue reliability analysis of turbine bladed disks, Int. J. Fatigue, 142 (2021), 105912. https://doi.org/10.1016/j.ijfatigue.2020.105912 doi: 10.1016/j.ijfatigue.2020.105912
    [56] M. Valdenegro-Toro, D. S. Mori, A deeper look into aleatoric and epistemic uncertainty disentanglement, 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), New Orleans, LA, USA, 2022, 1508–1516. https://doi.org/10.1109/cvprw56347.2022.00157
    [57] C. Wang, H. G. Matthies, M. H. Xu, Y. L. Li, Dual interval-and-fuzzy analysis method for temperature prediction with hybrid epistemic uncertainties via polynomial chaos expansion, Comput. Meth. Appl. Mech. Eng., 336 (2018), 171–186. https://doi.org/10.1016/j.cma.2018.03.013 doi: 10.1016/j.cma.2018.03.013
    [58] A. D. Kiureghian, O. Ditlevsen, Aleatory or epistemic? Does it matter?, Struct. Saf., 31 (2009), 105–112. https://doi.org/10.1016/j.strusafe.2008.06.020 doi: 10.1016/j.strusafe.2008.06.020
    [59] M. E. Paté-Cornell, Uncertainties in risk analysis: Six levels of treatment, Reliab. Eng. Syst. Saf., 54 (1996), 95–111. https://doi.org/10.1016/s0951-8320(96)00067-1 doi: 10.1016/s0951-8320(96)00067-1
    [60] C. Wang, H. R. Fan, X. Qiang, A review of uncertainty-based multidisciplinary design optimization methods based on intelligent strategies, Symmetry-Basel, 15 (2023), 1875. https://doi.org/10.3390/sym15101875 doi: 10.3390/sym15101875
    [61] C. Wang, X. Qiang, M. H. Xu, T. Wu, Recent advances in surrogate modeling methods for uncertainty quantification and propagation, Symmetry-Basel, 14 (2022), 1219. https://doi.org/10.3390/sym14061219 doi: 10.3390/sym14061219
    [62] D. Di Francesco, M. Girolami, A. B. Duncan, M. Chryssanthopoulos, A probabilistic model for quantifying uncertainty in the failure assessment diagram, Struct. Saf., 99 (2022), 102262. https://doi.org/10.1016/j.strusafe.2022.102262 doi: 10.1016/j.strusafe.2022.102262
    [63] P. Manfredi, Probabilistic uncertainty quantification of microwave circuits using Gaussian processes, IEEE Trans. Microw. Theory Tech., 71 (2023), 2360–2372. https://doi.org/10.1109/TMTT.2022.3228953 doi: 10.1109/TMTT.2022.3228953
    [64] J. S. Wu, G. E. Apostolakis, D. Okrent, Uncertainties in system analysis: Probabilistic versus nonprobabilistic theories, Reliab. Eng. Syst. Saf., 30 (1990), 163–181. https://doi.org/10.1016/0951-8320(90)90093-3 doi: 10.1016/0951-8320(90)90093-3
    [65] B. Hu, Q. M. Gong, Y. Q. Zhang, Y. H. Yin, W. J. Chen, Characterizing uncertainty in geotechnical design of energy piles based on Bayesian theorem, Acta Geotech., 17 (2022), 4191–4206. https://doi.org/10.1007/s11440-022-01535-3 doi: 10.1007/s11440-022-01535-3
    [66] K. Yao, J. Gao, Law of large numbers for uncertain random variables, IEEE Trans. Fuzzy Syst., 24 (2016), 615–621. https://doi.org/10.1109/TFUZZ.2015.2466080 doi: 10.1109/TFUZZ.2015.2466080
    [67] C. Zhang, Q. Liu, B. Zhou, C. Y. Chung, J. Li, L. Zhu, et al., A central limit theorem-based method for dc and ac power flow analysis under interval uncertainty of renewable power generation, IEEE Trans. Sustain. Energy, 14 (2023), 563–575. https://doi.org/10.1109/TSTE.2022.3220567 doi: 10.1109/TSTE.2022.3220567
    [68] C. Wang, Z. K. Song, H. R. Fan, Novel evidence theory-based reliability analysis of functionally graded plate considering thermal stress behavior, Aerosp. Sci. Technol., 146 (2024), 108936. https://doi.org/10.1016/j.ast.2024.108936 doi: 10.1016/j.ast.2024.108936
    [69] C. Wang, Evidence-theory-based uncertain parameter identification method for mechanical systems with imprecise information, Comput. Meth. Appl. Mech. Eng., 351 (2019), 281–296. https://doi.org/10.1016/j.cma.2019.03.048 doi: 10.1016/j.cma.2019.03.048
    [70] F. Arévalo, M. P. C. Alison, M. T. Ibrahim, A. Schwung, Adaptive information fusion using evidence theory and uncertainty quantification, IEEE Access, 12 (2024), 2236–2259. https://doi.org/10.1109/ACCESS.2023.3348270 doi: 10.1109/ACCESS.2023.3348270
    [71] H. R. Bae, R. V. Grandhi, R. A. Canfield, Uncertainty quantification of structural response using evidence theory, AIAA J., 41 (2003), 2062–2068. https://doi.org/10.2514/2.1898 doi: 10.2514/2.1898
    [72] Y. He, M. Mirzargar, R. M. Kirby, Mixed aleatory and epistemic uncertainty quantification using fuzzy set theory, Int. J. Approx. Reasoning, 66 (2015), 1–15. https://doi.org/10.1016/j.ijar.2015.07.002 doi: 10.1016/j.ijar.2015.07.002
    [73] C. Wang, H. G. Matthies, Hybrid evidence-and-fuzzy uncertainty propagation under a dual-level analysis framework, Fuzzy Sets Syst., 367 (2019), 51–67. https://doi.org/10.1016/j.fss.2018.10.002 doi: 10.1016/j.fss.2018.10.002
    [74] R. M. Rodríguez, L. Martínez, V. Torra, Z. S. Xu, F. Herrera, Hesitant fuzzy sets: State of the art and future directions, Int. J. Intell. Syst., 29 (2014), 495–524. https://doi.org/10.1002/int.21654 doi: 10.1002/int.21654
    [75] S. H. Khairuddin, M. H. Hasan, M. A. Hashmani, M. H. Azam, Generating clustering-based interval fuzzy type-2 triangular and trapezoidal membership functions: A structured literature review, Symmetry-Basel, 13 (2021), 239. https://doi.org/10.3390/sym13020239 doi: 10.3390/sym13020239
    [76] C. Wang, H. R. Fan, T. Wu, Novel rough set theory-based method for epistemic uncertainty modeling, analysis and applications, Appl. Math. Model., 113 (2023), 456–474. https://doi.org/10.1016/j.apm.2022.09.002 doi: 10.1016/j.apm.2022.09.002
    [77] X. Y. Zhang, Y. Y. Yao, Tri-level attribute reduction in rough set theory, Expert Syst. Appl., 190 (2022), 116187. https://doi.org/10.1016/j.eswa.2021.116187 doi: 10.1016/j.eswa.2021.116187
    [78] F. Y. Li, Z. Luo, G. Y. Sun, N. Zhang, An uncertain multidisciplinary design optimization method using interval convex models, Eng. Optimiz., 45 (2013), 697–718. https://doi.org/10.1080/0305215x.2012.690871 doi: 10.1080/0305215x.2012.690871
    [79] H. Lü, K. Yang, X. T. Huang, W.-B. Shangguan, K. G. Zhao, Uncertainty and correlation propagation analysis of powertrain mounting systems based on multi-ellipsoid convex model, Mech. Syst. Signal Proc., 173 (2022), 109058. https://doi.org/10.1016/j.ymssp.2022.109058 doi: 10.1016/j.ymssp.2022.109058
    [80] X. Qiang, C. Wang, H. R. Fan, Hybrid interval model for uncertainty analysis of imprecise or conflicting information, Appl. Math. Model., 129 (2024), 837–856. https://doi.org/10.1016/j.apm.2024.02.014 doi: 10.1016/j.apm.2024.02.014
    [81] C. Wang, X. Qiang, H. R. Fan, T. Wu, Y. L. Chen, Novel data-driven method for non-probabilistic uncertainty analysis of engineering structures based on ellipsoid model, Comput. Meth. Appl. Mech. Eng., 394 (2022), 114889. https://doi.org/10.1016/j.cma.2022.114889 doi: 10.1016/j.cma.2022.114889
    [82] C. Wang, H. G. Matthies, A modified parallelepiped model for non-probabilistic uncertainty quantification and propagation analysis, Comput. Meth. Appl. Mech. Eng., 369 (2020), 113209. https://doi.org/10.1016/j.cma.2020.113209 doi: 10.1016/j.cma.2020.113209
    [83] C. Wang, L. Hong, X. Qiang, M. H. Xu, Novel numerical method for uncertainty analysis of coupled vibro-acoustic problem considering thermal stress, Comput. Meth. Appl. Mech. Eng., 420 (2024), 116727. https://doi.org/10.1016/j.cma.2023.116727 doi: 10.1016/j.cma.2023.116727
    [84] L. X. Cao, J. Liu, L. Xie, C. Jiang, R. G. Bi, Non-probabilistic polygonal convex set model for structural uncertainty quantification, Analog Integr. Circuits Process., 89 (2021), 504–518. https://doi.org/10.1016/j.apm.2020.07.025 doi: 10.1016/j.apm.2020.07.025
    [85] L. P. Zhu, I. Elishakoff, J. H. Starnes, Derivation of multi-dimensional ellipsoidal convex model for experimental data, Math. Comput. Model., 24 (1996), 103–114. https://doi.org/10.1016/0895-7177(96)00094-5 doi: 10.1016/0895-7177(96)00094-5
    [86] C. Jiang, X. Han, G. Y. Lu, J. Liu, Z. Zhang, Y. C. Bai, Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique, Comput. Meth. Appl. Mech. Eng., 200 (2011), 2528–2546. https://doi.org/10.1016/j.cma.2011.04.007 doi: 10.1016/j.cma.2011.04.007
    [87] J. Liu, Z. B. Yu, D. Q. Zhang, H. Liu, X. Han, Multimodal ellipsoid model for non-probabilistic structural uncertainty quantification and propagation, Int. J. Mech. Mater. Des., 17 (2021), 633–657. https://doi.org/10.1007/s10999-021-09551-z doi: 10.1007/s10999-021-09551-z
    [88] Z. Kang, W. B. Zhang, Construction and application of an ellipsoidal convex model using a semi-definite programming formulation from measured data, Comput. Meth. Appl. Mech. Eng., 300 (2016), 461–489. https://doi.org/10.1016/j.cma.2015.11.025 doi: 10.1016/j.cma.2015.11.025
    [89] L. Wang, J. X. Liu, Dynamic uncertainty quantification and risk prediction based on the grey mathematics and outcrossing theory, Appl. Sci.-Basel, 12 (2022), 5389. https://doi.org/10.3390/app12115389 doi: 10.3390/app12115389
    [90] Y. H. Yan, X. J. Wang, Y. L. Li, Non-probabilistic credible set model for structural uncertainty quantification, Structures, 53 (2023), 1408–1424. https://doi.org/10.1016/j.istruc.2023.05.011 doi: 10.1016/j.istruc.2023.05.011
    [91] T. Zhang, J. Y. Jiao, J. Lin, H. Li, J. D. Hua, D. He, Uncertainty-based contrastive prototype-matching network towards cross-domain fault diagnosis with small data, Knowledge-Based Syst., 254 (2022), 109651. https://doi.org/10.1016/j.knosys.2022.109651 doi: 10.1016/j.knosys.2022.109651
    [92] J. Chen, D. Zhou, Z. Guo, J. Lin, C. Lyu, C. Lu, An active learning method based on uncertainty and complexity for gearbox fault diagnosis, IEEE Access, 7 (2019), 9022–9031. https://doi.org/10.1109/ACCESS.2019.2890979 doi: 10.1109/ACCESS.2019.2890979
    [93] H. Ma, C. Ekanayake, T. K. Saha, Power transformer fault diagnosis under measurement originated uncertainties, IEEE Trns. Dielectr. Electr. Insul., 19 (2012), 1982–1990. https://doi.org/10.1109/tdei.2012.6396956 doi: 10.1109/tdei.2012.6396956
    [94] X. J. Shi, H. B. Gu, B. Yao, Fuzzy Bayesian network fault diagnosis method based on fault tree for coal mine drainage system, IEEE Sens. J., 24 (2024), 7537–7547. https://doi.org/10.1109/jsen.2024.3354415 doi: 10.1109/jsen.2024.3354415
    [95] R. X. Duan, Y. N. Lin, Y. N. Zeng, Fault diagnosis for complex systems based on reliability analysis and sensors data considering epistemic uncertainty, Eksploat. Niezawodn., 20 (2018), 558–566. https://doi.org/10.17531/ein.2018.4.7 doi: 10.17531/ein.2018.4.7
    [96] J. Wang, H. Peng, W. P. Yu, J. Ming, M. J. Pérez-Jiménez, C. Y. Tao, et al., Interval-valued fuzzy spiking neural p systems for fault diagnosis of power transmission networks, Eng. Appl. Artif. Intell., 82 (2019), 102–109. https://doi.org/10.1016/j.engappai.2019.03.014 doi: 10.1016/j.engappai.2019.03.014
    [97] A. Hoballah, D. E. A. Mansour, I. B. M. Taha, Hybrid grey wolf optimizer for transformer fault diagnosis using dissolved gases considering uncertainty in measurements, IEEE Access, 8 (2020), 139176–139187. https://doi.org/10.1109/access.2020.3012633 doi: 10.1109/access.2020.3012633
    [98] K. Zhou, J. Tang, Probabilistic gear fault diagnosis using Bayesian convolutional neural network, IFAC-PapersOnLine, 55 (2022), 795–799. https://doi.org/10.1016/j.ifacol.2022.11.279 doi: 10.1016/j.ifacol.2022.11.279
    [99] H. T. Zhou, W. H. Chen, L. S. Cheng, J. Liu, M. Xia, Trustworthy fault diagnosis with uncertainty estimation through evidential convolutional neural networks, IEEE Trans. Ind. Inform., 19 (2023), 10842–10852. https://doi.org/10.1109/TⅡ.2023.3241587 doi: 10.1109/TⅡ.2023.3241587
    [100] S. Huang, R. Duan, J. He, T. Feng, Y. Zeng, Fault diagnosis strategy for complex systems based on multi-source heterogeneous information under epistemic uncertainty, IEEE Access, 8 (2020), 50921–50933. https://doi.org/10.1109/ACCESS.2020.2980397 doi: 10.1109/ACCESS.2020.2980397
    [101] S. X. Liu, S. Y. Zhou, B. Y. Li, Z. H. Niu, M. Abdullah, R. R. Wang, Servo torque fault diagnosis implementation for heavy-legged robots using insufficient information, ISA Transactions, 147 (2024), 439–452. https://doi.org/10.1016/j.isatra.2024.02.004 doi: 10.1016/j.isatra.2024.02.004
    [102] T. Zhang, S. He, J. Chen, T. Pan, Z. Zhou, Toward small sample challenge in intelligent fault diagnosis: Attention-weighted multidepth feature fusion net with signals augmentation, IEEE Trans. Instrum. Meas., 71 (2022), 1–13. https://doi.org/10.1109/TIM.2021.3134999 doi: 10.1109/TIM.2021.3134999
    [103] A. Kulkarni, J. Terpenny, V. Prabhu, Sensor selection framework for designing fault diagnostics system, Sensors, 21 (2021), 6470. https://doi.org/10.3390/s21196470 doi: 10.3390/s21196470
    [104] C. Herrojo, F. Paredes, J. Mata-Contreras, F. Martín, Chipless-rfid: A review and recent developments, Sensors, 19 (2019), 3385. https://doi.org/10.3390/s19153385 doi: 10.3390/s19153385
    [105] T. Kalsoom, N. Ramzan, S. Ahmed, M. Ur-Rehman, Advances in sensor technologies in the era of smart factory and industry 4.0, Sensors, 20 (2020), 6783. https://doi.org/10.3390/s20236783 doi: 10.3390/s20236783
    [106] A. Leal, J. Casas, C. Marques, M. J. Pontes, A. Frizera, Application of additive layer manufacturing technique on the development of high sensitive fiber bragg grating temperature sensors, Sensors, 18 (2018), 4120. https://doi.org/10.3390/s18124120 doi: 10.3390/s18124120
    [107] G. D. Lewis, P. Merken, M. Vandewal, Enhanced accuracy of cmos smart temperature sensors by nonlinear curvature correction, Sensors, 18 (2018), 4087. https://doi.org/10.3390/s18124087 doi: 10.3390/s18124087
    [108] H. Landaluce, L. Arjona, A. Perallos, F. Falcone, I. Angulo, F. Muralter, A review of iot sensing applications and challenges using rfid and wireless sensor networks, Sensors, 20 (2020), 2495. https://doi.org/10.3390/s20092495 doi: 10.3390/s20092495
    [109] S. L. Wei, W. B. Qin, L. W. Han, F. Y. Cheng, The research on compensation algorithm of infrared temperature measurement based on intelligent sensors, Cluster Comput., 22 (2019), 6091–6100. https://doi.org/10.1007/s10586-018-1828-5 doi: 10.1007/s10586-018-1828-5
    [110] M. Tessarolo, L. Possanzini, E. G. Campari, R. Bonfiglioli, F. S. Violante, A. Bonfiglio, et al., Adaptable pressure textile sensors based on a conductive polymer, Flex. Print. Electron., 3 (2018), 034001. https://doi.org/10.1088/2058-8585/aacbee doi: 10.1088/2058-8585/aacbee
    [111] K. A. Mathias, S. M. Kulkarni, Investigation on influence of geometry on performance of a cavity-less pressure sensor, IOP Conf. Ser.: Mater. Sci. Eng., 417 (2018), 012035. https://doi.org/10.1088/1757-899x/417/1/012035 doi: 10.1088/1757-899x/417/1/012035
    [112] W. P. Eaton, J. H. Smith, Micromachined pressure sensors: Review and recent developments, Smart Mater. Struct., 6 (1997), 30–41. https://doi.org/10.1117/12.276606 doi: 10.1117/12.276606
    [113] M. Mousavi, M. Alzgool, S. Towfighian, A mems pressure sensor using electrostatic levitation, IEEE Sens. J., 21 (2021), 18601–18608. https://doi.org/10.1109/JSEN.2021.3091665 doi: 10.1109/JSEN.2021.3091665
    [114] A. P. Cherkun, G. V. Mishakov, A. V. Sharkov, E. I. Demikhov, The use of a piezoelectric force sensor in the magnetic force microscopy of thin permalloy films, Ultramicroscopy, 217 (2020), 113072. https://doi.org/10.1016/j.ultramic.2020.113072 doi: 10.1016/j.ultramic.2020.113072
    [115] A. Nastro, M. Ferrari, V. Ferrari, Double-actuator position-feedback mechanism for adjustable sensitivity in electrostatic-capacitive mems force sensors, Sens. Actuator A-Phys., 312 (2020), 112127. https://doi.org/10.1016/j.sna.2020.112127 doi: 10.1016/j.sna.2020.112127
    [116] M. L. Gödecke, C. M. Bett, D. Buchta, K. Frenner, W. Osten, Optical sensor design for fast and process-robust position measurements on small diffraction gratings, Opt. Lasers Eng., 134 (2020), 106267. https://doi.org/10.1016/j.optlaseng.2020.106267 doi: 10.1016/j.optlaseng.2020.106267
    [117] Y. J. Chan, A. R. Carr, S. Charkhabi, M. Furnish, A. M. Beierle, N. F. Reuel, Wireless position sensing and normalization of embedded resonant sensors using a resonator array, Sens. Actuator A-Phys., 303 (2020), 111853. https://doi.org/10.1016/j.sna.2020.111853 doi: 10.1016/j.sna.2020.111853
    [118] J. A. Kim, J. W. Kim, C. S. Kang, J. Y. Lee, J. Jin, On-machine calibration of angular position and runout of a precision rotation stage using two absolute position sensors, Measurement, 153 (2020), 107399. https://doi.org/10.1016/j.measurement.2019.107399 doi: 10.1016/j.measurement.2019.107399
    [119] L. E. Helseth, On the accuracy of an interdigital electrostatic position sensor, J. Electrost., 107 (2020), 103480. https://doi.org/10.1016/j.elstat.2020.103480 doi: 10.1016/j.elstat.2020.103480
    [120] K. Palmer, H. Kratz, H. Nguyen, G. Thornell, A highly integratable silicon thermal gas flow sensor, J. Micromech. Microeng., 22 (2012), 065015. https://doi.org/10.1088/0960-1317/22/6/065015 doi: 10.1088/0960-1317/22/6/065015
    [121] A. Moreno-Gomez, C. A. Perez-Ramirez, A. Dominguez-Gonzalez, M. Valtierra-Rodriguez, O. Chavez-Alegria, J. P. Amezquita-Sanchez, Sensors used in structural health monitoring, Arch. Comput. Method Eng., 25 (2018), 901–918. https://doi.org/10.1007/s11831-017-9217-4 doi: 10.1007/s11831-017-9217-4
    [122] A. M. Shkel, Smart mems: Micro-structures with error-suppression and self-calibration control capabilities, Proceedings of the 2001 American Control Conference, Arlington, VA, USA, 2001, 1208–1213. https://doi.org/10.1109/ACC.2001.945886
    [123] X. Insausti, M. Zárraga-Rodríguez, C. Nolasco-Ferencikova, J. Gutiérrez-Gutiérrez, In-network algorithm for passive sensors in structural health monitoring, IEEE Signal Process. Lett., 30 (2023), 952–956. https://doi.org/10.1109/lsp.2023.3298279 doi: 10.1109/lsp.2023.3298279
    [124] B. Jeon, J. S. Yoon, J. Um, S. H. Suh, The architecture development of industry 4.0 compliant smart machine tool system (smts), J. Intell. Manuf., 31 (2020), 1837–1859. https://doi.org/10.1007/s10845-020-01539-4 doi: 10.1007/s10845-020-01539-4
    [125] M. H. Zhu, J. Li, W. B. Wang, D. P. Chen, Self-detection and self-diagnosis methods for sensors in intelligent integrated sensing system, IEEE Sens. J., 21 (2021), 19247–19254. https://doi.org/10.1109/JSEN.2021.3090990 doi: 10.1109/JSEN.2021.3090990
    [126] J. Chen, P. Li, G. B. Song, Z. Ren, Y. Tan, Y. J. Zheng, Feedback control for structural health monitoring in a smart aggregate based sensor network, Int. J. Struct. Stab. Dyn., 18 (2017), 1850064. https://doi.org/10.1142/S0219455418500645 doi: 10.1142/S0219455418500645
    [127] C. Wang, Z. M. Peng, R. Liu, C. Chen, Research on multi-fault diagnosis method based on time domain features of vibration signals, Sensors, 22 (2022), 8164. https://doi.org/10.3390/s22218164 doi: 10.3390/s22218164
    [128] Z. F. Du, R. H. Zhang, H. Chen, Characteristic signal extracted from a continuous time signal on the aspect of frequency domain, Chin. Phys. B, 28 (2019), 090502. https://doi.org/10.1088/1674-1056/ab344a doi: 10.1088/1674-1056/ab344a
    [129] Y. Lu, J. Tang, On time-frequency domain feature extraction of wave signals for structural health monitoring, Measurement, 114 (2018), 51–59. https://doi.org/10.1016/j.measurement.2017.09.016 doi: 10.1016/j.measurement.2017.09.016
    [130] M. Imani, Modified pca, lda and lpp feature extraction methods for polsar image classification, Multimed. Tools Appl., 83 (2024), 41171–41192. https://doi.org/10.1007/s11042-023-17269-7 doi: 10.1007/s11042-023-17269-7
    [131] Z. Xia, Y. Chen, C. Xu, Multiview pca: A methodology of feature extraction and dimension reduction for high-order data, IEEE T. Cybern., 52 (2022), 11068–11080. https://doi.org/10.1109/TCYB.2021.3106485 doi: 10.1109/TCYB.2021.3106485
    [132] Y. Aliyari Ghassabeh, F. Rudzicz, H. A. Moghaddam, Fast incremental lda feature extraction, Pattern Recognit., 48 (2015), 1999–2012. https://doi.org/10.1016/j.patcog.2014.12.012 doi: 10.1016/j.patcog.2014.12.012
    [133] E. Parsaeimehr, M. Fartash, J. A. Torkestani, Improving feature extraction using a hybrid of cnn and lstm for entity identification, Neural Process. Lett., 55 (2023), 5979–5994. https://doi.org/10.1007/s11063-022-11122-y doi: 10.1007/s11063-022-11122-y
    [134] P. Wang, X. M. Zhang, Y. Hao, A method combining cnn and elm for feature extraction and classification of sar image, J. Sens., 2019 (2019), 6134610. https://doi.org/10.1155/2019/6134610 doi: 10.1155/2019/6134610
    [135] O. İrsoy, E. Alpaydın, Unsupervised feature extraction with autoencoder trees, Neurocomputing, 258 (2017), 63–73. https://doi.org/10.1016/j.neucom.2017.02.075 doi: 10.1016/j.neucom.2017.02.075
    [136] Y. Y. Wang, D. J. Song, W. T. Wang, S. X. Rao, X. Y. Wang, M. N. Wang, Self-supervised learning and semi-supervised learning for multi-sequence medical image classification, Neurocomputing, 513 (2022), 383–394. https://doi.org/10.1016/j.neucom.2022.09.097 doi: 10.1016/j.neucom.2022.09.097
    [137] W. X. Sun, J. Chen, J. Q. Li, Decision tree and pca-based fault diagnosis of rotating machinery, Mech. Syst. Signal Proc., 21 (2007), 1300–1317. https://doi.org/10.1016/j.ymssp.2006.06.010 doi: 10.1016/j.ymssp.2006.06.010
    [138] N. R. Sakthivel, V. Sugumaran, S. Babudevasenapati, Vibration based fault diagnosis of monoblock centrifugal pump using decision tree, Expert Syst. Appl., 37 (2010), 4040–4049. https://doi.org/10.1016/j.eswa.2009.10.002 doi: 10.1016/j.eswa.2009.10.002
    [139] Y. Y. Li, L. Y. Song, Q. C. Sun, H. Xu, X. G. Li, Z. J. Fang, et al., Rolling bearing fault diagnosis based on quantum ls-svm, EPJ Quantum Technol., 9 (2022), 18. https://doi.org/10.1140/epjqt/s40507-022-00137-y doi: 10.1140/epjqt/s40507-022-00137-y
    [140] A. H. Zhang, D. L. Yu, Z. Q. Zhang, Tlsca-svm fault diagnosis optimization method based on transfer learning, Processes, 10 (2022), 362. https://doi.org/10.3390/pr10020362 doi: 10.3390/pr10020362
    [141] T. Huang, Q. Zhang, X. A. Tang, S. Y. Zhao, X. N. Lu, A novel fault diagnosis method based on cnn and lstm and its application in fault diagnosis for complex systems, Artif. Intell. Rev., 55 (2022), 1289–1315. https://doi.org/10.1007/s10462-021-09993-z doi: 10.1007/s10462-021-09993-z
    [142] H. Fang, H. Liu, X. Wang, J. Deng, J. An, The method based on clustering for unknown failure diagnosis of rolling bearings, IEEE Trans. Instrum. Meas., 72 (2023), 1–8. https://doi.org/10.1109/TIM.2023.3251406 doi: 10.1109/TIM.2023.3251406
    [143] A. Rodríguez-Ramos, A. J. da Silva Neto, O. Llanes-Santiago, An approach to fault diagnosis with online detection of novel faults using fuzzy clustering tools, Expert Syst. Appl., 113 (2018), 200–212. https://doi.org/10.1016/j.eswa.2018.06.055 doi: 10.1016/j.eswa.2018.06.055
    [144] L. K. Chang, S. H. Wang, M. C. Tsai, Demagnetization fault diagnosis of a pmsm using auto-encoder and k-means clustering, Energies, 13 (2020), 4467. https://doi.org/10.3390/en13174467 doi: 10.3390/en13174467
    [145] J. Du, S. P. Wang, H. Y. Zhang, Layered clustering multi-fault diagnosis for hydraulic piston pump, Mech. Syst. Signal Proc., 36 (2013), 487–504. https://doi.org/10.1016/j.ymssp.2012.10.020 doi: 10.1016/j.ymssp.2012.10.020
    [146] Y. Y. Li, J. D. Wang, H. Y. Zhao, C. Wang, Q. Shao, Adaptive dbscan clustering and gasa optimization for underdetermined mixing matrix estimation in fault diagnosis of reciprocating compressors, Sensors, 24 (2024), 167. https://doi.org/10.3390/s24010167 doi: 10.3390/s24010167
    [147] C. X. Jian, K. J. Yang, Y. H. Ao, Industrial fault diagnosis based on active learning and semi-supervised learning using small training set, Eng. Appl. Artif. Intell., 104 (2021), 104365. https://doi.org/10.1016/j.engappai.2021.104365 doi: 10.1016/j.engappai.2021.104365
    [148] S. Zheng, J. Zhao, A self-adaptive temporal-spatial self-training algorithm for semisupervised fault diagnosis of industrial processes, IEEE Trans. Ind. Inform., 18 (2022), 6700–6711. https://doi.org/10.1109/TⅡ.2021.3120686 doi: 10.1109/TⅡ.2021.3120686
    [149] J. Y. Long, Y. B. Chen, Z. Yang, Y. W. Huang, C. Li, A novel self-training semi-supervised deep learning approach for machinery fault diagnosis, Int. J. Prod. Res., 61 (2023), 8238–8251. https://doi.org/10.1080/00207543.2022.2032860 doi: 10.1080/00207543.2022.2032860
    [150] K. Yu, H. Z. Han, Q. Fu, H. Ma, J. Zeng, Symmetric co-training based unsupervised domain adaptation approach for intelligent fault diagnosis of rolling bearing, Meas. Sci. Technol., 31 (2020), 115008. https://doi.org/10.1088/1361-6501/ab9841 doi: 10.1088/1361-6501/ab9841
    [151] L. Wang, D. F. Zhou, H. Tian, H. Zhang, W. Zhang, Parametric fault diagnosis of analog circuits based on a semi-supervised algorithm, Symmetry-Basel, 11 (2019), 228. https://doi.org/10.3390/sym11020228 doi: 10.3390/sym11020228
    [152] C. X. Jian, Y. H. Ao, Imbalanced fault diagnosis based on semi-supervised ensemble learning, J. Intell. Manuf., 34 (2023), 3143–3158. https://doi.org/10.1007/s10845-022-01985-2 doi: 10.1007/s10845-022-01985-2
    [153] X. Li, F. L. Zhang, Classification of multi-type bearing fault features based on semi-supervised generative adversarial network (gan), Meas. Sci. Technol., 35 (2024), 025107. https://doi.org/10.1088/1361-6501/ad068e doi: 10.1088/1361-6501/ad068e
    [154] L. Wang, H. Tian, H. Zhang, Soft fault diagnosis of analog circuits based on semi-supervised support vector machine, Analog Integr. Circuits Process., 108 (2021), 305–315. https://doi.org/10.1007/s10470-021-01851-w doi: 10.1007/s10470-021-01851-w
    [155] P. Xu, L. X. Fu, K. Xu, W. B. Sun, Q. Tan, Y. P. Zhang, et al., Investigation into maize seed disease identification based on deep learning and multi-source spectral information fusion techniques, J. Food Compos. Anal., 119 (2023), 105254. https://doi.org/10.1016/j.jfca.2023.105254 doi: 10.1016/j.jfca.2023.105254
    [156] P. F. Zhang, T. R. Li, Z. Yuan, C. Luo, G. Q. Wang, J. Liu, et al., A data-level fusion model for unsupervised attribute selection in multi-source homogeneous data, Inf. Fusion, 80 (2022), 87–103. https://doi.org/10.1016/j.inffus.2021.10.017 doi: 10.1016/j.inffus.2021.10.017
    [157] M. B. Song, Y. F. Zhi, M. D. An, W. Xu, G. H. Li, X. L. Wang, Centrifugal pump cavitation fault diagnosis based on feature-level multi-source information fusion, Processes, 12 (2024), 196. https://doi.org/10.3390/pr12010196 doi: 10.3390/pr12010196
    [158] L. L. Liu, X. Wan, J. Y. Li, W. X. Wang, Z. G. Gao, An improved entropy-weighted topsis method for decision-level fusion evaluation system of multi-source data, Sensors, 22 (2022), 6391. https://doi.org/10.3390/s22176391 doi: 10.3390/s22176391
    [159] Y. W. Liu, Y. Q. Cheng, Z. Z. Zhang, J. J. Wu, Multi-information fusion fault diagnosis based on knn and improved evidence theory, J. Vib. Eng. Technol., 10 (2022), 841–852. https://doi.org/10.1007/s42417-021-00413-8 doi: 10.1007/s42417-021-00413-8
    [160] J. Xu, Y. Sui, T. Dai, A Bayesian network inference approach for dynamic risk assessment using multisource-based information fusion in an interval type-2 fuzzy set environment, IEEE Trans. Fuzzy Syst., 32 (2024), 5702–5713. https://doi.org/10.1109/TFUZZ.2024.3425495 doi: 10.1109/TFUZZ.2024.3425495
    [161] Y. C. Jie, Y. Chen, X. S. Li, P. Yi, H. S. Tan, X. Q. Cheng, Fufusion: Fuzzy sets theory for infrared and visible image fusion, In: Pattern recognition and computer vision, Singapore: Springer, 2024,466–478. https://doi.org/10.1007/978-981-99-8432-9_37
    [162] F. Y. Xiao, Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy, Inf. Fusion, 46 (2019), 23–32. https://doi.org/10.1016/j.inffus.2018.04.003 doi: 10.1016/j.inffus.2018.04.003
    [163] G. Koliander, Y. El-Laham, P. M. Djuric, F. Hlawatsch, Fusion of probability density functions, Proceedings of the IEEE, 110 (2022), 404–453. https://doi.org/10.1109/jproc.2022.3154399 doi: 10.1109/jproc.2022.3154399
    [164] Y. J. Pan, R. Q. An, D. Z. Fu, Z. Y. Zheng, Z. H. Yang, Unsupervised fault detection with a decision fusion method based on Bayesian in the pumping unit, IEEE Sens. J., 21 (2021), 21829–21838. https://doi.org/10.1109/jsen.2021.3103520 doi: 10.1109/jsen.2021.3103520
    [165] K. V. Kumar, A. Sathish, Medical image fusion based on type-2 fuzzy sets with teaching learning based optimization, Multimed. Tools Appl., 83 (2024), 33235–33262. https://doi.org/10.1007/s11042-023-16859-9 doi: 10.1007/s11042-023-16859-9
    [166] P. F. Zhang, T. R. Li, G. Q. Wang, C. Luo, H. M. Chen, J. B. Zhang, et al., Multi-source information fusion based on rough set theory: A review, Inf. Fusion, 68 (2021), 85–117. https://doi.org/10.1016/j.inffus.2020.11.004 doi: 10.1016/j.inffus.2020.11.004
    [167] Y. S. Wang, M. Y. He, L. Sun, D. Wu, Y. Wang, X. L. Qing, Weighted adaptive kalman filtering-based diverse information fusion for hole edge crack monitoring, Mech. Syst. Signal Proc., 167 (2022), 108534. https://doi.org/10.1016/j.ymssp.2021.108534 doi: 10.1016/j.ymssp.2021.108534
    [168] N. Guenther, M. Schonlau, Support vector machines, Stata J., 16 (2016), 917–937. https://doi.org/10.1177/1536867x1601600407 doi: 10.1177/1536867x1601600407
    [169] P. Cunningham, S. J. Delany, K-nearest neighbour classifiers-a tutorial, ACM Comput. Surv., 54 (2021), 128. https://doi.org/10.1145/3459665 doi: 10.1145/3459665
    [170] Z. Liu, S. B. Zhong, Q. Liu, C. X. Xie, Y. Z. Dai, C. Peng, et al., Thyroid nodule recognition using a joint convolutional neural network with information fusion of ultrasound images and radiofrequency data, Eur. Radiol., 31 (2021), 5001–5011. https://doi.org/10.1007/s00330-020-07585-z doi: 10.1007/s00330-020-07585-z
    [171] A. Y. Chen, F. Wang, W. H. Liu, S. Chang, H. Wang, J. He, et al., Multi-information fusion neural networks for arrhythmia automatic detection, Comput. Meth. Programs Biomed., 193 (2020), 105479. https://doi.org/10.1016/j.cmpb.2020.105479 doi: 10.1016/j.cmpb.2020.105479
  • This article has been cited by:

    1. Hira Soomro, Nooraini Zainuddin, Hanita Daud, Joshua Sunday, Noraini Jamaludin, Abdullah Abdullah, Mulono Apriyanto, Evizal Abdul Kadir, Variable Step Block Hybrid Method for Stiff Chemical Kinetics Problems, 2022, 12, 2076-3417, 4484, 10.3390/app12094484
    2. Zeeshan Ali, Faranak Rabiei, Kamyar Hosseini, A fractal–fractional-order modified Predator–Prey mathematical model with immigrations, 2023, 207, 03784754, 466, 10.1016/j.matcom.2023.01.006
    3. Sümeyra Uçar, Analysis of hepatitis B disease with fractal–fractional Caputo derivative using real data from Turkey, 2023, 419, 03770427, 114692, 10.1016/j.cam.2022.114692
    4. Mohammad Partohaghighi, Ali Akgül, Rubayyi T. Alqahtani, New Type Modelling of the Circumscribed Self-Excited Spherical Attractor, 2022, 10, 2227-7390, 732, 10.3390/math10050732
    5. G.M. Vijayalakshmi, Roselyn Besi. P, A fractal fractional order vaccination model of COVID-19 pandemic using Adam’s moulton analysis, 2022, 8, 26667207, 100144, 10.1016/j.rico.2022.100144
    6. SHAIMAA A. M. ABDELMOHSEN, SHABIR AHMAD, MANSOUR F. YASSEN, SAEED AHMED ASIRI, ABDELBACKI M. M. ASHRAF, SAYED SAIFULLAH, FAHD JARAD, NUMERICAL ANALYSIS FOR HIDDEN CHAOTIC BEHAVIOR OF A COUPLED MEMRISTIVE DYNAMICAL SYSTEM VIA FRACTAL–FRACTIONAL OPERATOR BASED ON NEWTON POLYNOMIAL INTERPOLATION, 2023, 31, 0218-348X, 10.1142/S0218348X2340087X
    7. D.A. Tverdyi, R.I. Parovik, A.R. Hayotov, A.K. Boltaev, Распараллеливание численного алгоритма решения задачи Коши для нелинейного дифференциального уравнения дробного переменного порядка с помощью технологии OpenMP, 2023, 20796641, 87, 10.26117/2079-6641-2023-43-2-87-110
    8. Khalid Hattaf, A New Class of Generalized Fractal and Fractal-Fractional Derivatives with Non-Singular Kernels, 2023, 7, 2504-3110, 395, 10.3390/fractalfract7050395
    9. Ali Raza, Ovidiu V. Stadoleanu, Ahmed M. Abed, Ali Hasan Ali, Mohammed Sallah, Heat transfer model analysis of fractional Jeffery-type hybrid nanofluid dripping through a poured microchannel, 2024, 22, 26662027, 100656, 10.1016/j.ijft.2024.100656
    10. Mubashir Qayyum, Efaza Ahmad, Hijaz Ahmad, Bandar Almohsen, New solutions of time-space fractional coupled Schrödinger systems, 2023, 8, 2473-6988, 27033, 10.3934/math.20231383
    11. Muhammad Shahzad, Soma Mustafa, Sarbaz H A Khoshnaw, Inference of complex reaction mechanisms applying model reduction techniques, 2024, 99, 0031-8949, 045242, 10.1088/1402-4896/ad3291
    12. B. El Ansari, E. H. El Kinani, A. Ouhadan, Symmetry analysis of the time fractional potential-KdV equation, 2025, 44, 2238-3603, 10.1007/s40314-024-02991-1
    13. Muhammad Farman, Changjin Xu, Perwasha Abbas, Aceng Sambas, Faisal Sultan, Kottakkaran Sooppy Nisar, Stability and chemical modeling of quantifying disparities in atmospheric analysis with sustainable fractal fractional approach, 2025, 142, 10075704, 108525, 10.1016/j.cnsns.2024.108525
    14. Hira Khan, Gauhar Rahman, Muhammad Samraiz, Kamal Shah, Thabet Abdeljawad, On Generalized Fractal-Fractional Derivative and Integral Operators Associated with Generalized Mittag-Leffler Function, 2025, 24058440, e42144, 10.1016/j.heliyon.2025.e42144
    15. Emmanuel Kengne, Ahmed Lakhssassi, Dynamics of stochastic nonlinear waves in fractional complex media, 2025, 542, 03759601, 130423, 10.1016/j.physleta.2025.130423
  • 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(1726) PDF downloads(132) Cited by(1)

Figures and Tables

Figures(10)  /  Tables(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog