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Research article Special Issues

Unsteady-state turbulent flow field predictions with a convolutional autoencoder architecture

  • Received: 15 June 2023 Revised: 19 October 2023 Accepted: 19 October 2023 Published: 02 November 2023
  • MSC : 76F99, 65Z05

  • Traditional numerical methods, such as computational fluid dynamics (CFD), demand large computational resources and memory for modeling fluid dynamic systems. Hence, deep learning (DL) and, specifically Convolutional Neural Networks (CNN) autoencoders have resulted in accurate tools to obtain approximations of the streamwise and vertical velocities and pressure fields, when stationary flows are considered. The novelty of this paper consists of predicting the future instants from an initial one with a CNN autoencoder architecture when an unsteady flow is considered. Two neural models are proposed: The former predicts the future instants on the basis of an initial sample and the latter approximates the initial sample. The inputs of the CNNs take the signed distance function (SDF) and the flow region channel (FRC), and, for the representation of the temporal evolution, the previous CFD sample is added. To increment the amount of training data of the second neural model, a data augmentation technique based on the similarity principle for fluid dynamics is implemented. As a result, low absolute error rates are obtained in the prediction of the first samples near the shapes surfaces. Even in the most advanced time instants, the prediction of the vortices zone is quite reliable. 62.12 and 9000 speed-up ratios are achieved by the predictions of the first and second neural models, respectively, compared to the computational cost regarded by the CFD simulations.

    Citation: Álvaro Abucide, Koldo Portal, Unai Fernandez-Gamiz, Ekaitz Zulueta, Iker Azurmendi. Unsteady-state turbulent flow field predictions with a convolutional autoencoder architecture[J]. AIMS Mathematics, 2023, 8(12): 29734-29758. doi: 10.3934/math.20231522

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  • Traditional numerical methods, such as computational fluid dynamics (CFD), demand large computational resources and memory for modeling fluid dynamic systems. Hence, deep learning (DL) and, specifically Convolutional Neural Networks (CNN) autoencoders have resulted in accurate tools to obtain approximations of the streamwise and vertical velocities and pressure fields, when stationary flows are considered. The novelty of this paper consists of predicting the future instants from an initial one with a CNN autoencoder architecture when an unsteady flow is considered. Two neural models are proposed: The former predicts the future instants on the basis of an initial sample and the latter approximates the initial sample. The inputs of the CNNs take the signed distance function (SDF) and the flow region channel (FRC), and, for the representation of the temporal evolution, the previous CFD sample is added. To increment the amount of training data of the second neural model, a data augmentation technique based on the similarity principle for fluid dynamics is implemented. As a result, low absolute error rates are obtained in the prediction of the first samples near the shapes surfaces. Even in the most advanced time instants, the prediction of the vortices zone is quite reliable. 62.12 and 9000 speed-up ratios are achieved by the predictions of the first and second neural models, respectively, compared to the computational cost regarded by the CFD simulations.



    In his survey-cum-expository review article, Srivastava [1] presented and motivated about brief expository overview of the classical q -analysis versus the so-called (p,q)-analysis with an obviously redundant additional parameter p. We also briefly consider several other families of such extensively and widely-investigated linear convolution operators as (for example) the Dziok-Srivastava, Srivastava-Wright and Srivastava-Attiya linear convolution operators, together with their extended and generalized versions. The theory of (p,q)-analysis has important role in many areas of mathematics and physics. Our usages here of the q-calculus and the fractional q-calculus in geometric function theory of complex analysis are believed to encourage and motivate significant further developments on these and other related topics (see Srivastava and Karlsson [2,pp. 350-351], Srivastava [3,4]). Our main objective in this survey-cum-expository article is based chiefly upon the fact that the recent and future usages of the classical q-calculus and the fractional q-calculus in geometric function theory of complex analysis have the potential to encourage and motivate significant further researches on many of these and other related subjects. Jackson [5,6] was the first that gave some application of q -calculus and introduced the q-analogue of derivative and integral operator (see also [7,8]), we apply the concept of q -convolution in order to introduce and study the general Taylor-Maclaurin coefficient estimates for functions belonging to a new class of normalized analytic in the open unit disk, which we have defined here.

    Let A denote the class of analytic functions of the form

    f(z):=z+m=2amzm,zΔ:={zC:|z|<1} (1.1)

    and let SA consisting on functions that are univalent in Δ. If the function hA is given by

    h(z):=z+m=2bmzm,(zΔ).                                  (1.2)

    The Hadamard product (or convolution) of f and h, given by (1.1) and (1.2), respectively, is defined by

    (fh)(z):=z+m=2ambmzm,zΔ. (1.3)

    If f and F are analytic functions in Δ, we say that f is subordinate to F, written as f(z)F(z), if there exists a Schwarz function s, which is analytic in Δ, with s(0)=0, and |s(z)|<1 for all zΔ, such that f(z)=F(s(z)), zΔ. Furthermore, if the function F is univalent in Δ, then we have the following equivalence ([9,10])

    f(z)F(z)f(0)=F(0)andf(Δ)F(Δ).

    The Koebe one-quarter theorem (see [11]) prove that the image of Δ under every univalent function fS contains the disk of radius 14. Therefore, every function fS has an inverse f1 that satisfies

    f(f1(w))=w,(|w|<r0(f),r0(f)14),

    where

    g(w)=f1(w)=wa2w2+(2a22a3)w3(5a325a2a3+a4)w4+.=w+m=2Amwm

    A function fA is said to be bi-univalent in Δ if both f and f1 are univalent in Δ. Let Σ represent the class of bi-univalent functions in Δ given by (1.1). The class of analytic bi-univalent functions was first familiarised by Lewin [12], where it was shown that |a2|<1.51. Brannan and Clunie [13] enhanced Lewin's result to |a2|<2 and later Netanyahu [14] proved that |a2|<43. 

    Note that the functions

    f1(z)=z1z,f2(z)=12log1+z1z,f3(z)=log(1z)

    with their corresponding inverses

    f11(w)=w1+w,f12(w)=e2w1e2w+1,f13(w)=ew1ew

    are elements of Σ (see [15,16]). For a brief history and exciting examples in the class Σ (see [17]). Brannan and Taha [18] (see also [16]) presented certain subclasses of the bi-univalent functions class Σ similar to the familiar subclasses S(α) and K(α) of starlike and convex functions of order α (0α<1), respectively (see [17,19,20]). Ensuing Brannan and Taha [18], a function fA is said to be in the class SΣ(α) of bi-starlike functions of order α (0<α1), if each of the following conditions are satisfied:

    fΣ,with|argzf(z)f(z)|<απ2(zΔ),

    and

    |argwg(w)g(w)|<απ2(wΔ),

    where the function g is the analytic extension of f1 to Δ, given by

    g(w)=wa2w2+(2a22a3)w3(5a325a2a3+a4)w4+(wΔ). (1.4)

    A function fA is said to be in the class KΣ(α) of bi-convex functions of order α (0<α1), if each of the following conditions are satisfied:

    fΣ,with|arg(1+zf(z)f(z))|<απ2(zΔ),

    and

    |arg(1+wg(w)g(w))|<απ2(wΔ).

    The classes SΣ(α) and KΣ(α) of bi-starlike functions of order α and bi-convex functions of order α (0<α1), corresponding to the function classes S(α) and K(α), were also introduced analogously. For each of the function classes SΣ(α) and KΣ(α), they found non-sharp estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3| ([16,18]).

    The Faber polynomials introduced by Faber [21] play an important role in various areas of mathematical sciences, especially in Geometric Function Theory of Complex Analysis (see, for details, [22]). In 2013, Hamidi and Jahangiri [23,24,25] took a new approach to show that the initial coefficients of classes of bi- starlike functions e as well as provide an estimate for the general coefficients of such functions subject to a given gap series condition.Recently, their idea of application of Faber polynomials triggered a number of related publications by several authors (see, for example, [26,27,28] and also references cited threin) investigated some interesting and useful properties for analytic functions. Using the Faber polynomial expansion of functions fA has the form (1.1), the coefficients of its inverse map may be expressed as

    g(w)=f1(w)=w+m=21mKmm1(a2,a3,...)wm, (1.5)

    where

    Kmm1(a2,a3,...)=(m)!(2m+1)!(m1)!am12+(m)!(2(m+1))!(m3)!am32a3+(m)!(2m+3)!(m4)!am42a4+(m)!(2(m+2))!(m5)!am52[a5+(m+2)a23]+(m)!(2m+5)!(m6)!am62[a6+(2m+5)a3a4]+i7ami2Ui, (1.6)

    such that Ui with 7im is a homogeneous polynomial in the variables a2,a3,...,am, In particular, the first three terms of Kmm1 are

    K21=2a2,K32=3(2a22a3),K43=4(5a325a2a3+a4).

    In general, an expansion of Knm (nN) is (see [29,30,31,32,33])

    Knm=nam+n(n1)2D2m+n!3!(n3)!D3m+...+n!m!(nm)!Dmm,

    where Dnm=Dnm(a2,a3,...) and

    Dpm(a1,a2,...am)=m=1p!i1!...im!ai11...aimm,

    while a1=1 and the sum is taken over all non-negative integers i1...im satisfying

    i1+i2+...+im=pi1+2i2+...+mim=m.

    Evidently

    Dmm(a1,a2,...am)=am1.

    Srivastava [1] made use of several operators of q-calculus and fractional q-calculus and recollecting the definition and representations. The q-shifted factorial is defined for κ,qC and nN0=N{0} as follows

    (κ;q)m={1,m=0(1κ)(1κq)(1κqk1),mN.

    By using the q-Gamma function Γq(z), we get

    (qκ;q)m=(1q)m Γq(κ+m)Γq(κ)(mN0),

    where (see [34])

    Γq(z)=(1q)1z(q;q)(qz;q)(|q|<1).

    Also, we note that

    (κ;q)=m=0(1κqm)(|q|<1),

    and, the q-Gamma function Γq(z) is known

    Γq(z+1)=[z]q Γq(z),

    where [m]q symbolizes the basic q-number defined as follows

    [m]q:={1qm1q,mC1+m1j=1qj,mN. (1.7)

    Using the definition formula (1.7) we have the next two products:

    (i) For any non-negative integer m, the q-shifted factorial is given by

    [m]q!:={1,ifm=0,mn=1[n]q,  ifmN.

    (ii) For any positive number r, the q-generalized Pochhammer symbol is defined by

    [r]q,m:={1,ifm=0,r+m1n=r[n]q,ifmN.

    It is known in terms of the classical (Euler's) Gamma function Γ(z), that

    Γq(z)Γ(z)     asq1.

    Also, we observe that

    limq1{(qκ;q)m(1q)m}=(κ)m,

    where (κ)m is the familiar Pochhammer symbol defined by

    (κ)m={1,ifm=0,κ(κ+1)...(κ+m1),ifmN.

    For 0<q<1, the q-derivative operator (or, equivalently, the q- difference operator) El-Deeb et al. [35] defined Dq for fh given by (1.3) is defined by (see [5,6])

    Dq(fh)(z):=Dq(z+m=2ambmzm)=(fh)(z)(fh)(qz)z(1q)=1+m=2[m]qambmzm1(zΔ),

    where, as in the definition (1.7)

    [m]q:={1qm1q=1+m1j=1qj      (mN),0                               (m=0). (1.8)

    For κ>1 and 0<q<1, El-Deeb et al. [35] (see also) defined the linear operator Hκ,qh:AA by

    Hκ,qhf(z)Mq,κ+1(z)=zDq(fh)(z)(zΔ),

    where the function Mq,κ+1 is given by

    Mq,κ+1(z):=z+m=2[κ+1]q,m1[m1]q!zm(zΔ).

    A simple computation shows that

    Hκ,qhf(z):=z+m=2[m]q![κ+1]q,m1ambm zm(κ>1,0<q<1, zΔ). (1.9)

    From the definition relation (1.9), we can easily verify that the next relations hold for all fA:

    (i) [κ+1]qHκ,qhf(z)=[κ]qHκ+1,qhf(z)+qκz Dq(Hκ+1,qhf(z))(zΔ);(ii)Iκhf(z):=limq1Hκ,qhf(z)=z+m=2m!(κ+1)m1ambmzm(zΔ). (1.10)

    Remark 1. Taking precise cases for the coefficients bm we attain the next special cases for the operator Hκ,qh:

    (ⅰ) For bm=1, we obtain the operator Iκq defined by Srivastava [32] and Arif et al. [36] as follows

    Iκqf(z):=z+m=2[m]q![κ+1]q,m1amzm(κ>1,0<q<1, zΔ); (1.11)

    (ⅱ) For bm=(1)m1Γ(υ+1)4m1(m1)!Γ(m+υ), υ>0, we obtain the operator Nκυ,q defined by El-Deeb and Bulboacă [37] and El-Deeb [38] as follows

    Nκυ,qf(z):=z+m=2(1)m1Γ(υ+1)4m1(m1)!Γ(m+υ)[m]q![κ+1]q,m1amzm=z+m=2[m]q![κ+1]q,m1ψmamzm(υ>0,κ>1,0<q<1, zΔ), (1.12)

    where

    ψm:=(1)m1Γ(υ+1)4m1(m1)!Γ(m+υ); (1.13)

    (ⅲ) For bm=(n+1n+m)α, α>0, n0, we obtain the operator Mκ,αn,q defined by El-Deeb and Bulboacă [39] and Srivastava and El-Deeb [40] as follows

    Mκ,αn,qf(z):=z+m=2(n+1n+m)α[m]q![κ+1]q,m1amzm(zΔ); (1.14)

    (ⅳ) For bm=ρm1(m1)!eρ, ρ>0, we obtain the q-analogue of Poisson operator defined by El-Deeb et al. [35] (see [41]) as follows

    Iκ,ρqf(z):=z+m=2ρm1(m1)!eρ[m]q![κ+1]q,m1amzm(zΔ). (1.15)

    (ⅴ) For bm=[1++μ(m1)1+]n, nZ, 0, μ0, we obtain the q-analogue of Prajapat operator defined by El-Deeb et al. [35] (see also [42]) as follows

    Jκ,nq,,μf(z):=z+m=2[1++μ(m1)1+]n[m,q]![κ+1,q]m1amzm(zΔ); (1.16)

    (ⅵ) For bm=(n+m2m1)θm1(1θ)n nN, 0θ1, we obtain the q-analogue of the Pascal distribution operator defined by Srivastava and El-Deeb [28] (see also [35,43,44]) as follows

    κ,nq,θf(z):=z+m=2(n+m2m1)θm1(1θ)n[m,q]![κ+1,q]m1amzm(zΔ). (1.17)

    The purpose of the paper is to present a new subclass of functions Lq,κΣ(η;h;Φ) of the class Σ, that generalize the previous defined classes. This subclass is defined with the aid of a general Hκ,qh linear operator defined by convolution products composed with the aid of q-derivative operator. This new class extend and generalize many preceding operators as it was presented in Remark 1, and the main goal of the paper is find estimates on the coefficients |a2|, |a3|, and for the Fekete-Szegö functional for functions in these new subclasses. These classes will be introduced by using the subordination and the results are obtained by employing the techniques used earlier by Srivastava et al. [16]. This last work represents one of the most important study of the bi-univalent functions, and inspired many investigations in this area including the present paper, while many other recent papers deals with problems initiated in this work, like [33,44,45,46,47,48], and many others. Inspired by the work of Silverman and Silvia [49] (also see[50]) and recent study by Srivastava et al [51], in this article, we define the following new subclass of bi-univalent functions Mq,κΣ(ϖ,ϑ,h) as follows:

    Definition 1. Let ϖ(π,π] and let the function fΣ be of the form (1.1) and h is given by (1.2), the function f is said to be in the class Mq,κΣ(ϖ,ϑ,h) if the following conditions are satisfied:

    ((Hκ,qhf(z))+(1+eiϖ)2z(Hκ,qhf(z)))>ϑ, (1.18)

    and

    ((Hκ,qhg(w))+(1+eiϖ)2w(Hκ,qhg(w)))>ϑ (1.19)

    with κ>1, 0<q<1, 0ϑ<1 and z,wΔ, where the function g is the analytic extension of f1 to Δ, and is given by (1.4).

    Definition 2. Let ϖ=0 and let the function fΣ be of the form (1.1) and h is given by (1.2), the function f is said to be in the class Mq,κΣ(ϑ,h) if the following conditions are satisfied:

    ((Hκ,qhf(z))+z(Hκ,qhf(z)))>ϑ, (1.20)

    and

    ((Hκ,qhg(w))+w(Hκ,qhg(w)))>ϑ (1.21)

    with κ>1, 0<q<1, 0ϑ<1 and z,wΔ, where the function g is the analytic extension of f1 to Δ, and is given by (1.4).

    Definition 3. Let ϖ=π and let the function fΣ be of the form (1.1) and h is given by (1.2), the function f is said to be in the class HMq,κΣ(ϑ,h) if the following conditions are satisfied:

    ((Hκ,qhf(z)))>ϑand((Hκ,qhg(w)))>ϑ (1.22)

    with κ>1, 0<q<1, 0ϑ<1 and z,wΔ, where the function g is the analytic extension of f1 to Δ, and is given by (1.4).

    Remark 2. (ⅰ) Putting q1 we obtain that limq1Mq,κΣ(ϖ,ϑ;h)=:GκΣ(ϖ,ϑ;h), where GκΣ(ϖ,ϑ;h) represents the functions fΣ that satisfy (1.18) and (1.19) for Hκ,qh replaced with Iκh (1.10).

    (ⅱ) Fixing bm=(1)m1Γ(υ+1)4m1(m1)!Γ(m+υ), υ>0, we obtain the class Bq,κΣ(ϖ,ϑ,υ), that represents the functions fΣ that satisfy (1.18) and (1.19) for Hκ,qh replaced with Nκυ,q (1.12).

    (ⅲ) Taking bm=(n+1n+m)α, α>0, n0, we obtain the class Lq,κΣ(ϖ,ϑ,n,α), that represents the functions fΣ that satisfy (1.18) and (1.19) for Hκ,qh replaced with Mκ,αn,q (1.14).

    (ⅳ) Fixing bm=ρm1(m1)!eρ, ρ>0, we obtain the class Mq,κΣ(ϖ,ϑ,ρ), that represents the functions fΣ that satisfy (1.18) and (1.19) for Hκ,qh replaced with Iκ,ρq (1.15).

    (ⅴ) Choosing bm=[1++μ(m1)1+]n, nZ, 0, μ0, we obtain the class Mq,κΣ(ϖ,ϑ,n,,μ), that represents the functions fΣ that satisfy (1.18) and (1.19) for Hκ,qh replaced with Jκ,nq,,μ (1.16).

    Throughout this paper, we assume that

    ϖ(π;π],κ>1,0ϑ<1,0<q<1.

    Recall the following Lemma which will be needed to prove our results.

    Lemma 1. (Caratheodory Lemma [11]) If ϕP and ϕ(z)=1+n=1cnzn then |cn|2 for each n, this inequality is sharp for all n where P is the family of all functions ϕ analytic and having positive real part in Δ with ϕ(0)=1.

    We firstly introduce a bound for the general coefficients of functions belong to the class Mq,κΣ(ϖ,ϑ;h).

    Theorem 2. Let the function f given by (1.1) belongs to the class Mq,κΣ(ϖ,ϑ;h). If ak=0 for 2km1, then

    |am|4(1ϑ)[κ+1,q]m1m|2+(1+eiϖ)(m1)| [m,q]!bm.

    Proof. If fMq,κΣ(ϖ,ϑ;h), from (1.18), (1.19), we have

    ((Hκ,qhf(z))+(1+eiϖ)2z(Hκ,qhf(z)))=1+m=2m2[2+(1+eiϖ)(m1)][m,q]![κ+1,q]m1bmamzm1(zΔ), (2.1)

    and

    ((Hκ,qhg(w))+(1+eiϖ)2z(Hκ,qhg(w)))=1+m=2m2[2+(1+eiϖ)(m1)][m,q]![κ+1,q]m1bm Amwm1
    =1+m=2m2[2+(1+eiϖ)(m1)][m,q]![κ+1,q]m1bm 1mKmm1(a2,...,am)wm1(wΔ). (2.2)

    Since

    fMq,κΣ(ϖ,ϑ;h) and g=f1Mq,κΣ(γ,η,ϑ;h),

    we know that there are two positive real part functions:

    U(z)=1+m=1cmzm,

    and

    V(w)=1+m=1dmwm,

    where

    (U(z))>0and (V(w))>0(z,wΔ),

    so that

    (Hκ,qhf(z))+(1+eiθ)2z(Hκ,qhf(z))=ϑ+(1ϑ)U(z)
    =1+(1ϑ)m=1cmzm, (2.3)

    and

    (Hκ,qhg(w))+(1+eiθ)2z(Hκ,qhg(w))=ϑ+(1ϑ)V(w)
    =1+(1ϑ)m=1dmwm. (2.4)

    Using (2.1) and comparing the corresponding coefficients in (2.3), we obtain

    m2[2+(1+eiϖ)(m1)][m,q]![κ+1,q]m1bmam=(1ϑ)cm1, (2.5)

    and similarly, by using (2.2) in the equality (2.4), we have

    m2[2+(1+eiϖ)(m1)][m,q]![κ+1,q]m1bm1mKmm1(a2,a3,...am)=(1ϑ)dm1, (2.6)

    under the assumption ak=0 for 0km1, we obtain Am=am and so

    m2[2+(1+eiϖ)(m1)][m,q]![κ+1,q]m1bmam=(1ϑ)cm1, (2.7)

    and

    m2[2+(1+eiϖ)(m1)][m,q]![κ+1,q]m1bmam=(1ϑ)dm1, (2.8)

    Taking the absolute values of (2.7) and (2.8), we conclude that

    |am|=|2(1ϑ)[κ+1,q]m1cm1m[2+(1+eiϖ)(m1)] [m,q]!bm|=|2(1ϑ)[κ+1,q]m1dm1m[2+(1+eiϖ)(m1)] [m,q]!bm|.

    Applying the Caratheodory Lemma 1, we obtain

    |am|4(1ϑ)[κ+1,q]m1m|2+(1+eiϖ)(m1)| [m,q]!bm,

    which completes the proof of Theorem.

    Theorem 3. Let the function f given by (1.1) belongs to the class Mq,κΣ(ϖ,ϑ;h), then

    |a2|{2(1ϑ)[κ+1,q]|3+eiϖ|[2,q]!b2,0ϑ<1|3+eiϖ|2 ([2,q]!)2[κ+2,q]b223|2+eiϖ| [3,q]![κ+1,q]b32(1ϑ)[κ+1,q]23|2+eiϖ| [3,q]!b3,1|3+eiϖ|2 ([2,q]!)2[κ+2,q]b223|2+eiϖ| [3,q]![κ+1,q]b3ϑ<1, (2.9)
    |a3|2(1ϑ)[κ+1,q]23|2+eiϖ|[3,q]!b3, (2.10)

    and

    |a32a22|2(1ϑ)[κ+1,q]23|2+eiϖ| [3,q]!b3. (2.11)

    Proof. Fixing m=2 and m=3 in (2.5), (2.6), we have

    (3+eiϖ) [2,q]![κ+1,q]b2a2=(1ϑ)c1, (2.12)
    3(2+eiϖ) [3,q]![κ+1,q]2b3a3=(1ϑ)c2, (2.13)
    (3+eiϖ) [2,q]![κ+1,q]b2a2=(1ϑ)d1, (2.14)

    and

    3(2+eiϖ) [3,q]![κ+1,q]2b3(2a22a3)=(1ϑ)d2. (2.15)

    From (2.12) and (2.14), by using the Caratheodory Lemma1, we obtain

    |a2|=(1ϑ)[κ+1,q]|c1||3+eiϖ|[2,q]!b2=(1ϑ)[κ+1,q]|d1||3+eiϖ|[2,q]!b22(1ϑ)[κ+1,q]|3+eiϖ|[2,q]!b2. (2.16)

    Also, from (2.13) and (2.15), we have

    6(2+eiϖ) [3,q]![κ+1,q]2b3a22=(1ϑ)(c2+d2),
     a22=(1ϑ)[κ+1,q]26(2+eiϖ)[3,q]!b3(c2+d2), (2.17)

    and by using the Caratheodory Lemma 1, we obtain

    |a2|2(1ϑ)[κ+1,q]23|2+eiϖ| [3,q]!b3. (2.18)

    From (2.16) and (2.18), we obtain the desired estimate on the coefficient as asserted in (2.9).

    To find the bound on the coefficient |a3|, we subtract (2.15) from (2.13). we get

    6(2+eiϖ) [3,q]![κ+1,q]2b3(a3a22)=(1ϑ)(c2d2),

    or

    a3=a22+(1ϑ)(c2d2)[κ+1,q]26(2+eiϖ)[3,q]!b3, (2.19)

    substituting the value of a22 from (2.12) into (2.19), we obtain

    a3=(1ϑ)2[κ+1,q]2c21(3+eiϖ)2([2,q]!)2b22+(1ϑ)(c2d2)[κ+1,q]26(2+eiϖ)[3,q]!b3.

    Using the Caratheodory Lemma 1, we find that

    |a3|4(1ϑ)2[κ+1,q]2|3+eiϖ|2([2,q]!)2b22+2(1ϑ)[κ+1,q]23|2+eiϖ|[3,q]!b3, (2.20)

    and from (2.13), we have

    a3=(1ϑ)[κ+1,q]2 c23(2+eiϖ)[3,q]!b3.

    Appling the Caratheodory Lemma 1, we obtain

    |a3|2(1ϑ)[κ+1,q]23|2+eiϖ|[3,q]!b3. (2.21)

    Combining (2.20) and (2.21), we have the desired estimate on the coefficient |a3| as asserted in (2.10).

    Finally, from (2.15), we deduce that

    |a32a22|(1ϑ)[κ+1,q]2|d2|3|2+eiϖ| [3,q]!b3=2(1ϑ)[κ+1,q]23|2+eiϖ| [3,q]!b3.

    Thus the proof of Theorem 3 was completed.

    Fekete and Szegö [52] introduced the generalized functional |a3a22|, where is some real number. Due to Zaprawa [53], (also see [54]) in the following theorem we determine the Fekete-Szegö functional for fMq,κΣ(ϖ,ϑ;h).

    Theorem 4. Let the function f given by (1.1) belongs to the class Mq,κΣ(ϖ,ϑ;h) and R. Then we have

    |a3a22|((1ϑ)[κ+1,q]23|2+eiϖ|[3,q]!b3){|2|+||}.

    Proof. From (2.17) and (2.19)we obtain

    a3a22=(1)(1ϑ)[κ+1,q]26(2+eiϖ)[3,q]!b3(c2+d2)+(1ϑ)[κ+1,q]26(2+eiϖ)[3,q]!b3(c2d2),=((1ϑ)[κ+1,q]26(2+eiϖ)[3,q]!b3){[(1)+1]c2+[(1)1]d2}.

    So we have

    \begin{equation} a_{3}-\aleph a_{2}^{2} = \left(\frac{\left( 1-\vartheta \right)[\kappa +1, q]_{2}}{6\left( 2+e^{i\varpi}\right)[3, q]!\, b_{3}}\right)\{(2-\aleph)c_2+(-\aleph)d_2\}. \end{equation} (3.1)

    Then, by taking modulus of (3.1), we conclude that

    \begin{equation*} |a_{3}-\aleph a_{2}^{2}|\leq\left(\frac{\left( 1-\vartheta \right)[\kappa +1, q]_{2}}{3\left| 2+e^{i\varpi}\right|[3, q]!\, b_{3}}\right)\{|2-\aleph|+|\aleph|\} \end{equation*}

    Taking \aleph = 1 , we have the following result.

    \begin{equation*} |a_{3}- a_{2}^{2}|\leq\frac{2\left( 1-\vartheta \right)[\kappa +1, q]_{2}}{3\left| 2+e^{i\varpi}\right|[3, q]!\, b_{3}}. \end{equation*}

    In the current paper, we mainly get upper bounds of the initial Taylors coefficients of bi-univalent functions related with q- calculus operator. By fixing b_m as demonstrated in Remark 1, one can effortlessly deduce results correspondents to Theorems 2 and 3 associated with various operators listed in Remark 1. Further allowing q\rightarrow 1^{-} as itemized in Remark 2 we can outspread the results for new subclasses stated in Remark 2. Moreover by fixing \varpi = 0 and \varpi = \pi in Theorems 2 and 3, we can easily state the results for f\in\mathcal{M}_{\Sigma }^{q, \kappa }\left(\vartheta; h\right) and f\in\mathcal{HM}_{\Sigma }^{q, \kappa }\left(\vartheta; h\right) . Further by suitably fixing the parameters in Theorem 4, we can deduce Fekete-Szegö functional for these function classes. By using the subordination technique, we can extend the study by defining a new class

    \begin{equation*} \left[ \left( \mathcal{H}_{h}^{\kappa , q}f(z)\right)^{\prime }+\left( \frac{1+e^{i\varpi}}{2}\right) z\left( \mathcal{H} _{h}^{\kappa , q}f(z)\right) ^{\prime \prime }\right]\prec\Psi(z) \end{equation*}

    where \Psi(z) the function \Psi is an analytic univalent function such that \Re \; \left(\Psi\right) > 0 \; \; \mathrm{in }\; \; \Delta with \Psi(0) = 1, \; \Psi^{\prime }(0) > 0 and \Psi maps \Delta onto a region starlike with respect to 1 and symmetric with respect to the real axis and is given by \Psi(z) = z+B_1z+B_2z^2+B_3z^3+\cdots, (B_1 > 0). Also, motivating further researches on the subject-matter of this, we have chosen to draw the attention of the interested readers toward a considerably large number of related recent publications (see, for example, [1,2,4]). and developments in the area of mathematical analysis. In conclusion, we choose to reiterate an important observation, which was presented in the recently-published review-cum-expository review article by Srivastava ([1], p. 340), who pointed out the fact that the results for the above-mentioned or new q- analogues can easily (and possibly trivially) be translated into the corresponding results for the so-called (p; q)- analogues(with 0 < |q| < p \leq 1 )by applying some obvious parametric and argument variations with the additional parameter p being redundant.

    The researcher(s) would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.The authors are grateful to the reviewers for their valuable remarks, comments, and advices that help us to improve the quality of the paper.

    The authors declare that they have no competing interests.



    [1] S. L. Brunton, B. R. Noack, P. Koumoutsakos, Machine learning for fluid mechanics, Annu. Rev. Fluid Mech., 52 (2020), 477–508. https://doi.org/10.1146/annurev-fluid-010719-060214 doi: 10.1146/annurev-fluid-010719-060214
    [2] S. Qin, S. Wang, L. Wang, C. Wang, G. Sun, Y. Zhong, Multi-objective optimization of cascade blade profile based on reinforcement learning, Appl. Sci., 11 (2021), 106. https://doi.org/10.3390/app11010106 doi: 10.3390/app11010106
    [3] Y. Qiu, J. Bai, N. Liu, C. Wang, Global aerodynamic design optimization based on data dimensionality reduction, Chinese J. Aeronaut., 31 (2018), 643–659. https://doi.org/10.1016/j.cja.2018.02.005 doi: 10.1016/j.cja.2018.02.005
    [4] B. N. Hanna, N. T. Dinh, R. W. Youngblood, I. A. Bolotnov, Coarse-grid computational fluid dynamic (CG-CFD) error prediction using machine learning, preprint paper, 2017. https://doi.org/10.48550/arXiv.1710.09105 doi: 10.48550/arXiv.1710.09105
    [5] H. Bao, J. Feng, N. Dinh, H. Zhang, Computationally efficient CFD prediction of bubbly flow using physics-guided deep learning, Int. J. Multiphase Flow, 131 (2020), 103378. https://doi.org/10.1016/j.ijmultiphaseflow.2020.103378 doi: 10.1016/j.ijmultiphaseflow.2020.103378
    [6] K. Tlales, K. E. Otmani, G. Ntoukas, G. Rubio, E. Ferrer, Machine learning adaptation for laminar and turbulent flows: applications to high order discontinuous Galerkin solvers, preprint paper, 2022. https://doi.org/10.48550/arXiv.2209.02401 doi: 10.48550/arXiv.2209.02401
    [7] X. Guo, W. Li, F. Iorio, Convolutional neural networks for steady flow approximation, In: Proceedings of the Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2016,481–490. https://doi.org/10.1145/2939672.2939738
    [8] M. D. Ribeiro, A. Rehman, S. Ahmed, A. Dengel, DeepCFD: Efficient steady-state laminar flow approximation with deep convolutional neural networks, preprint paper, 2020. https://doi.org/10.48550/arXiv.2004.08826 doi: 10.48550/arXiv.2004.08826
    [9] A. Kashefi, D. Rempe, L. J. Guibas, A point-cloud deep learning framework for prediction of fluid flow fields on irregular geometries, Phys. Fluids, 33 (2021), 027104. https://doi.org/10.1063/5.0033376 doi: 10.1063/5.0033376
    [10] T. Murata, K. Fukami, K. Fukagata, Nonlinear mode decomposition with convolutional neural networks for fluid dynamics, J. Fluid Mech., 882 (2020), A13. https://doi.org/10.1017/jfm.2019.822 doi: 10.1017/jfm.2019.822
    [11] J. Ling, A. Kurzawski, J. Templeton, Reynolds averaged turbulence modelling using deep neural networks with embedded invariance, J. Fluid Mech., 807 (2016), 155–166. https://doi.org/10.1017/jfm.2016.615 doi: 10.1017/jfm.2016.615
    [12] S. Lee, D. You, Prediction of laminar vortex shedding over a cylinder using deep learning, preprint paper, 2017. https://doi.org/10.48550/arXiv.1712.07854 doi: 10.48550/arXiv.1712.07854
    [13] Y. Liu, Y. Lu, Y. Wang, D. Sun, L. Deng, F. Wang, et al., A CNN-based shock detection method in flow visualization, Comput. Fluids, 184 (2019), 1–9. https://doi.org/10.1016/j.compfluid.2019.03.022 doi: 10.1016/j.compfluid.2019.03.022
    [14] L. Deng, Y. Wang, Y. Liu, F. Wang, S. Li, J. Liu, A CNN-based vortex identification method, J. Vis., 22 (2019), 65–78, https://doi.org/10.1007/s12650-018-0523-1 doi: 10.1007/s12650-018-0523-1
    [15] H. Nowruzi, H. Ghassemi, M. Ghiasi, Performance predicting of 2D and 3D submerged hydrofoils using CFD and ANNs, J. Mar. Sci. Technol., 22 (2017), 710–733. https://doi.org/10.1007/s00773-017-0443-0 doi: 10.1007/s00773-017-0443-0
    [16] A. Mohan, D. Daniel, M. Chertkov, D. Livescu, Compressed convolutional LSTM: An efficient deep learning framework to model high fidelity 3D turbulence, preprint paper, 2019. https://doi.org/10.48550/arXiv.1903.00033 doi: 10.48550/arXiv.1903.00033
    [17] K. Portal-Porras, U. Fernandez-Gamiz, A. Ugarte-Anero, F. Zulueta, A. Zulueta, Alternative artificial neural network structures for turbulent flow velocity field prediction, Mathematics, 9 (2021), 1939. https://doi.org/10.3390/math9161939 doi: 10.3390/math9161939
    [18] A. Abucide-Armas, K. Portal-Porras, U. Fernandez-Gamiz, E. Zulueta, A. Teso-Fz-Betoño, A data augmentation-based technique for deep learning applied to CFD simulations, Mathematics, 9 (2021), 1843. https://doi.org/10.3390/math9161843 doi: 10.3390/math9161843
    [19] N. Thuerey, K. Weißenow, L. Prantl, X. Hu, Deep learning methods for Reynolds-averaged Navier–Stokes simulations of airfoil flows, AIAA J., 58 (2020), 25–36, https://doi.org/10.2514/1.J058291 doi: 10.2514/1.J058291
    [20] R. Fang, D. Sondak, P. Protopapas, S. Succi, Deep learning for turbulent channel flow, preprint paper, 2018. https://doi.org/10.48550/arXiv.1812.02241 doi: 10.48550/arXiv.1812.02241
    [21] K. Champion, B. Lusch, J. N. Kutz, S. L. Brunton, Data-driven discovery of coordinates and governing equations, Proc. Natl. Acad. Sci. USA, 116 (2019), 22445–22451. https://doi.org/10.1073/pnas.1906995116 doi: 10.1073/pnas.1906995116
    [22] K. Fukami, T. Murata, K. Zhang, K. Fukagata, Sparse identification of nonlinear dynamics with low-dimensionalized flow representations, J. Fluid Mech., 926 (2021), A10. https://doi.org/10.1017/jfm.2021.697 doi: 10.1017/jfm.2021.697
    [23] R. Maulik, T. Botsas, N. Ramachandra, L. R. Mason, I. Pan, Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation, Phys. D Nonlinear Phenom., 416 (2021), 132797. https://doi.org/10.1016/j.physd.2020.132797 doi: 10.1016/j.physd.2020.132797
    [24] L. Agostini, Exploration and prediction of fluid dynamical systems using auto-encoder technology, Phys. Fluids, 32 (2020), 067103. https://doi.org/10.1063/5.0012906 doi: 10.1063/5.0012906
    [25] R. King, O. Hennigh, A. Mohan, M. Chertkov, From deep to physics-informed learning of turbulence: Diagnostics, preprint paper, 2018. https://doi.org/10.48550/arXiv.1810.07785 doi: 10.48550/arXiv.1810.07785
    [26] R. Maulik, B. Lusch, P. Balaprakash, Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders, Phys. Fluids, 33 (2021), 037106. https://doi.org/10.1063/5.0039986 doi: 10.1063/5.0039986
    [27] F. J. Gonzalez, M. Balajewicz, Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems, preprint paper, 2018. https://doi.org/10.48550/arXiv.1808.01346 doi: 10.48550/arXiv.1808.01346
    [28] G. Iaccarino, A. Ooi, P. A. Durbin, M. Behnia, Reynolds averaged simulation of unsteady separated flow, Int. J. Heat Fluid Flow, 24 (2003), 147–156. https://doi.org/10.1016/S0142-727X(02)00210-2 doi: 10.1016/S0142-727X(02)00210-2
    [29] S. Osher, S. Chakravarthy, Upwind schemes and boundary conditions with applications to Euler equations in general geometries, J. Comput. Phys., 50 (1983), 447–481, https://doi.org/10.1016/0021-9991(83)90106-7 doi: 10.1016/0021-9991(83)90106-7
    [30] Siemens Software, 2023. Available from: https://www.plm.automation.siemens.com/global/en/.
    [31] F. R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA J., 32 (1994), 1598–1605. https://doi.org/10.2514/3.12149 doi: 10.2514/3.12149
    [32] B. N. Rajani, A. Kandasamy, S. Majumdar, Numerical simulation of laminar flow past a circular cylinder, Appl. Math. Model., 33 (2009), 1228–1247. https://doi.org/10.1016/j.apm.2008.01.017 doi: 10.1016/j.apm.2008.01.017
    [33] M. M. Rahman, M. M. Karim, M. A. Alim, Numerical investigation of unsteady flow past a circular cylinder using 2-D finite volume method, J. Nav. Arch. Mar. Engg., 4 (1970), 27–42. https://doi.org/10.3329/jname.v4i1.914 doi: 10.3329/jname.v4i1.914
    [34] I. Aramendia, U. Fernandez-Gamiz, E. Zulueta Guerrero, J. Lopez-Guede, J. Sancho, Power control optimization of an underwater piezoelectric energy harvester, Appl. Sci., 8 (2018), 389. https://doi.org/10.3390/app8030389 doi: 10.3390/app8030389
    [35] S. Bhatnagar, Y. Afshar, S. Pan, K. Duraisamy, S. Kaushik, Prediction of aerodynamic flow fields using convolutional neural networks, Comput. Mech., 64 (2019), 525–545. https://doi.org/10.1007/s00466-019-01740-0 doi: 10.1007/s00466-019-01740-0
    [36] L. F. Richardson, J. A. Gaunt, Ⅷ. The deferred approach to the limit, Philos. Trans. Royal Soc. London. Series A Containing Papers Math. Phys. Char., 226 (1927), 299–361. https://doi.org/10.1098/rsta.1927.0008 doi: 10.1098/rsta.1927.0008
    [37] A. Roshko, Vortex shedding from circular cylinders at low Reynolds numbers, J. Fluid Mech., 46 (1971), 749–756. https://doi.org/10.1017/S002211207100082X doi: 10.1017/S002211207100082X
    [38] Y. Lecun, L. Bottou, Y. Bengio, P. Haffner, Gradient-based learning applied to document recognition, Proc. IEEE, 86 (1998), 2278–2324. https://doi.org/10.1109/5.726791 doi: 10.1109/5.726791
    [39] O. Ronneberger, P. Fischer, T. Brox, U-net: Convolutional networks for biomedical image segmentation, In: Navab, N., Hornegger, J., Wells, W., Frangi, A. (eds) Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015. MICCAI 2015. Lecture Notes in Computer Science, 9351 (2015), 234–241. https://doi.org/10.1007/978-3-319-24574-4_28
    [40] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686–707. https://doi.org/10.1016/j.jcp.2018.10.045 doi: 10.1016/j.jcp.2018.10.045
    [41] A. Kashefi, T. Mukerji, Physics-informed PointNet: A deep learning solver for steady-state incompressible flows and thermal fields on multiple sets of irregular geometries, J. Comput. Phys., 468 (2022), 111510. https://doi.org/10.1016/j.jcp.2022.111510 doi: 10.1016/j.jcp.2022.111510
    [42] X. Jin, S. Cai, H. Li, G. E. Karniadakis, NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations, J. Comput. Phys., 426 (2021), 109951. https://doi.org/10.1016/j.jcp.2020.109951 doi: 10.1016/j.jcp.2020.109951
    [43] A. Kashefi, T. Mukerji, Prediction of fluid flow in porous media by sparse observations and physics-informed PointNet, Neural Networks, 167 (2022), 80–91. https://doi.org/10.1016/j.neunet.2023.08.006 doi: 10.1016/j.neunet.2023.08.006
    [44] A. Kashefi, T. Mukerji, Chatgpt for programming numerical methods, J. Mach. Learn. Model. Comput., 4 (2023), 1–74. https://doi.org/10.1615/JMachLearnModelComput.2023048492 doi: 10.1615/JMachLearnModelComput.2023048492
    [45] V. Kumar, L. Gleyzer, A. Kahana, K. Shukla, G. E. Karniadakis, MyCrunchGPT: A chatGPT assisted framework for scientific machine learning, J. Mach. Learn. Model. Comput., 2023. https://doi.org/10.1615/JMachLearnModelComput.20230495182023 doi: 10.1615/JMachLearnModelComput.20230495182023
    [46] D. P. Kingma, J. Ba, Adam: A method for stochastic optimization, preprint paper, 2014. https://doi.org/10.48550/arXiv.1412.6980 doi: 10.48550/arXiv.1412.6980
    [47] I. Loshchilov, F. Hutter, Decoupled weight decay regularization, preprint paper, 2017. https://doi.org/10.48550/arXiv.1711.05101 doi: 10.48550/arXiv.1711.05101
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