Research article

New weighted generalizations for differentiable exponentially convex mapping with application

  • Received: 19 December 2019 Accepted: 25 March 2020 Published: 10 April 2020
  • MSC : 26D10, 26D15, 26E60

  • The main aim of the present paper is to present a novel approach base on the exponentially convex function to broaden the utilization of celebrated Hermite-Hadamard type inequality. The proposed technique presents an auxiliary result of constructing the set of base functions and gives deformation equations in a simple form. The auxiliary result in the convexity has provided a convenient way of establishing the convergence region of several novel results. The strategy is not limited to the small parameter, such as in the classical method. The numerical examples obtained by the proposed approach indicate that the approach is easy to implement and computationally very attractive. The implementation of this numerical scheme clearly exhibits its effectiveness, reliability, and easiness regarding the applications in error estimates for weighted mean, the integral formula, rth moments of a continuous random variable, application to weighted special means and in developing the variants by extraordinary choices of n and θ as well as its better approximation.

    Citation: Saima Rashid, Rehana Ashraf, Muhammad Aslam Noor, Khalida Inayat Noor, Yu-Ming Chu. New weighted generalizations for differentiable exponentially convex mapping with application[J]. AIMS Mathematics, 2020, 5(4): 3525-3546. doi: 10.3934/math.2020229

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  • The main aim of the present paper is to present a novel approach base on the exponentially convex function to broaden the utilization of celebrated Hermite-Hadamard type inequality. The proposed technique presents an auxiliary result of constructing the set of base functions and gives deformation equations in a simple form. The auxiliary result in the convexity has provided a convenient way of establishing the convergence region of several novel results. The strategy is not limited to the small parameter, such as in the classical method. The numerical examples obtained by the proposed approach indicate that the approach is easy to implement and computationally very attractive. The implementation of this numerical scheme clearly exhibits its effectiveness, reliability, and easiness regarding the applications in error estimates for weighted mean, the integral formula, rth moments of a continuous random variable, application to weighted special means and in developing the variants by extraordinary choices of n and θ as well as its better approximation.


    Nonlinear evolution equations (NEEs) are used extensively in engineering and scientific fields, including wave propagation phenomena, quantum mechanics, shallow water wave propagation, chemical kinematics, solid-state physics, optical fibers, fluid mechanics, plasma physics, heat flow and so on. Recently, much research on NEEs has focused on existence, uniqueness, convergence and finding solutions: for example, [1,2,3,4,5,6,7,8,9,10,11], and the references therein. One of the essential physical issues for NEEs is obtaining traveling wave solutions. Therefore, the looking for mathematical techniques to generate exact solutions to NEEs has become a significant and necessary task in nonlinear sciences. Recently, various techniques for dealing with NEEs have been established, such as the exp-function method [12], perturbation method [13,14], sine-cosine method [15,16], spectral methods [17], tanh-sech method [18,19], Jacobi elliptic function [20], Hirota's method [21], exp(ϕ(ς))-expansion method [22], extended trial equation method [23,24], (G/G)-expansion method [25,26], etc.

    In recent years, the use of fractional differential equations has grown due to their wide range of applications in fields such as control theory, fluid flow, finance, electrical networks, solid state physics, chemical kinematics, optical fiber, plasma physics, signal processing, and biological populations. A number of mathematicians have proposed various types of fractional derivatives. These types include, the new truncated M-fractional derivative, Caputo fractional derivative, Riemann-Liouville fractional derivative, Grunwald-Letnikov fractional derivative, He's fractional derivative, Riesz fractional derivatives, the Weyl derivative and conformable fractional definitions [27,28,29,30,31,32,33,34].

    Khalil et al. [32] have developed a new derivative operator known as the conformable derivative (CD). From this point, let us define the CD for the function u:[0,)R of order β(0,1] as follows:

    Dβzu(z)=limh0u(z+hz1β)u(z)h.

    The CD satisfies the following properties for any constants a and b:

    1) Dβz[au(z)+bv(z)]=aDβzu(z)+bDβzv(z) , 2) Dβz[a]=0,

    3) Dβz[za]=azaβ, 4) Dβzu(z)=z1βdudz.

    In contrast, stochastic partial differential equations (SPDEs) are models for spatiotemporal physical, biological and chemical systems that are sensitive to random influences. In the past few decades, these models have been the subject of extensive research. It has been emphasized how crucial it is to take stochastic effects into account when modeling complex systems. For instance, there is rising interest in employing SPDEs to represent complex phenomena mathematically in the fields of finance, mechanical and electrical engineering, biophysics, information systems, materials sciences, condensed matter physics, and climate systems [35,36].

    Therefore, it is crucial to consider NEEs with fractional derivatives and for some stochastic force. Here, we consider the fractional-stochastic Fokas-Lenells equation (FSFLE):

    DαxΦtγ1DαxxΦ2iγ2DαxΦ+ϑ|Φ|2(Φ+iρDαxΦ)+σDαxΦWt=0, (1.1)

    where Φ(x,t) gives the complex field, γ1, γ2 and ρ are positive constants, ϑ=±1, W is the standard Wiener process,  σ is the strength of the noise, and ΦdW is multiplicative noise in the Stratonovich sense.

    If we put σ=0 and α=1, then we get the Fokas-Lenells equation [37,38,39]:

    Φxtγ1Φxx2iγ2Φx+ϑ|Φ|2(Φ+iρΦx)=0. (1.2)

    Equation (1.2) is one of the most significant equations, and it has many applications in telecommunication models, complex system theory, quantum field theory and quantum mechanics. Also, it appears as a pattern that identifies nonlinear pulse propagation in optical fibers. Demiray and Bulut [37] obtained the exact solutions of Eq (1.2) by utilizing the extended trial equation and generalized Kudryashov methods. Meanwhile, Xu and Fan [38] used the Riemann-Hilbert problem to obtain the long-time asymptotic behavior of the solution of Eq (1.2).

    It is important to note that Stratonovich and Itô [40] are the two versions of stochastic integrals that are most frequently used. Modeling problems essentially establish which form is acceptable; nevertheless, once that form is chosen, an equivalent equation of the alternate form can be created utilizing the same solutions. The following correlation can therefore be used to switch between Stratonovich (denoted as t0ΦdW) and Itô (denoted as t0ΦdW):

    t0σΦ(τ)dW(τ)=t0σΦ(τ)dW(τ)+σ22t0Φ(τ)dτ. (1.3)

    Our aim in this study is to derive the analytical fractional-stochastic solutions of the FSFLE (1.1). The modified mapping method is what we employ to obtain these solutions. Physics researchers would find the solutions very helpful in defining several major physical processes because of the stochastic term and fractional derivatives present in Eq (1.1). Additionally, by using MATLAB software, we introduce numerous graphs to investigate the effects of noise and the fractional derivative on the exact solution of the FSFLE (1.1).

    The outline of this paper is as follows: In Section 2, we get the wave equation for the FSFLE (1.1). In Section 3, the modified mapping method is used to get the exact solutions of the FSFLE (1.1). In Section 4, we can see how white noise and the fractional derivative affect the acquired FSFLE solutions. At last, the conclusions of the paper are given.

    The wave equation for FSFLE (1.1) is obtained by using the wave transformation

    Φ(x,t)=Ψ(η)eiΘ(μ)σW(t)σ2t, Θ(μ)=μ1αxα+μ2t  and η=η1αxα+η2t, (2.1)

    where the function Ψ is deterministic, and μ1, μ2, η1 and η2 are non-zero constants. We note that

    Φt=[η2Ψ+iμ2ΨσΨWt+12σ2Ψσ2Ψ]eiΘ(μ)σW(t)σ2t,=[η2Ψ+iμ2ΨσΨWt12σ2Ψ]eiΘ(μ)σW(t)σ2t,=[η2Ψ+iμ2ΨσΨWt]eiΘ(μ)σW(t)σ2t, (2.2)

    where we used Eq (1.3), and the term 12σ2Ψ is the Itô correction.

    DαxΦt=[η1η2Ψ+i(η1μ2+μ1η2)Ψσ(η1Ψ+iμ1Ψ)Wtμ1μ2Ψ]eiΘ(μ)σW(t)σ2t, (2.3)

    and

    DαxΦ=(η1Ψ+iμ1Ψ)eiΘ(μ)+σW(t)σ2tDαxxΦ=(η21Ψ+2iμ1η1Ψμ21Ψ)eiΘ(μ)σW(t)σ2t. (2.4)

    Inserting Eqs (2.3) and (2.4) into Eq (1.1), we have the following system:

    (η1η2γ1η21)Ψ+(νμ1μ2+γ21μ1+2γ2μ1)Ψνρμ1Ψ3e[2σW(t)2σ2t]=0, (2.5)

    and

    i[(η1μ2+μ1η22γ1μ1η12γ2η1)Ψ+νρη1Ψ2Ψe[2σW(t)2σ2t]]=0. (2.6)

    We have, by taking the expectation on both sides,

    (η1η2γ1η21)Ψ+(νμ1μ2+γ21μ1+2γ2μ1)Ψνρμ1Ψ3e2σ2tEe2σW(t)=0, (2.7)

    and

    i[(η1μ2+μ1η22γ1μ1η12γ2η1)Ψ+νρη1Ψ2Ψe2σ2tEe2σW(t)]=0. (2.8)

    Since W(t) is normal distribution, then E(e2kW(t))=e2k2t for any real number k. Therefore, Eqs (2.7) and (2.8) become

    (η1η2γ1η21)Ψ+(νμ1μ2+γ21μ1+2γ2μ1)Ψνρμ1Ψ3=0, (2.9)
    i[(η1μ2+μ1η22γ1μ1η12γ2η1)Ψ+νρη1Ψ2Ψ]=0. (2.10)

    From the imaginary part of Eq (2.10), we obtained

    η2=1μ1(η1μ2+2γ1μ1η1+2γ2η1νρη1Ψ2).

    while the real part is given by

    Ψ+AΨBΨ3=0, (2.11)

    where

    A=(νμ1μ2+γ21μ1+2γ2μ1)(η1η2γ1η21), and B=νρμ1(η1η2γ1η21).

    In this section, we apply the modified mapping method stated in [41]. Assuming that the solutions of Eq (2.11) have the form

    Ψ(η)=Ni=0iφi(η)+Ni=1iφi(η), (3.1)

    where i and i are unknown constants to be evaluated for i=1,2,..N, and φ satisfies the first type of the elliptic equation

    φ=r+qφ2+pφ4, (3.2)

    where r,q and p are real parameters.

    First, let us balance Ψ with Ψ3 in Eq (2.11) to find the parameter N as

    N+2=3NN=1.

    With N=1, Eq (3.2) takes the form

    Ψ(η)=0+1φ(η)+1φ(η). (3.3)

    Differentiating Eq (3.3) twice and using (3.2), we get

    Ψ=1(qφ+2pφ3)+1(qφ1+2rφ3). (3.4)

    Putting Eqs (3.3) and (3.4) into Eq (2.11) we have

    (21pB31)φ33B021φ2+(1q3B2013B121+1A)φ+(A0B306B011)+(A1+1q3B2013B121)φ13B21φ2+(2r1B31)φ3=0.

    Comparing each coefficient of φk and φk with zero for k=3,2,1,0, we attain

    21pB31=0,
    3B021=0,
    1q3B2013B121+1A=0,
    A0B306B011=0,
    A1+1q3B2013B121=0,
    3B021=0

    and

    2r1B31=0.

    When we solve these equations, we get three different families:

    First family:

    0=0,  1=±2pB, 1=0,  q=A. (3.5)

    Second family:

    0=0,  1=0, 1=±2rB,  q=A. (3.6)

    Third family:

    0=0,  1=±2pB, 1=±2rB,  q=6prA. (3.7)

    First family: The solution of Eq (2.11), by utilizing Eqs (3.3) and (3.5), takes the form

    Φ(x,t)=±2pBφ(η)eiΘ(μ)σW(t)σ2t. (3.8)

    There are many cases depending on p>0:

    Case 1-1: If p=κ2, q=(1+κ2) and r=1, then φ(η)=sn(η). In this case the solution of FSFLE (1.1), by utilizing Eq (3.8), is

    Φ(x,t)=±κ2Bsn(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.9)

    If κ1, then Eq (3.9) transfers to

    Φ(x,t)=±2Btanh(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.10)

    Case 1-2: If p=1, q=2κ21 and r=κ2(1κ2), then φ(η)=ds(η). In this case the solution of FSFLE (1.1), by using Eq (3.8), is

    Φ(x,t)=±2Bds(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.11)

    If κ1, then Eq (3.11) transfers to

    Φ(x,t)=±2Bcsch(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.12)

    If κ0, then Eq (3.11) transfers to

    Φ(x,t)=±2Bcsc(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.13)

    Case 1-3: If p=1, q=2κ2 and r=(1κ2), then φ(η)=cs(η). In this case the solution of FSFLE (1.1), by utilizing Eq (3.8), is

    Φ(x,t)=±2Bcs(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.14)

    If κ1, then Eq (3.14) transfers to

    Φ(x,t)=±2Bcsch(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.15)

    When κ0, then Eq (3.14) transfers to

    Φ(x,t)=±2Bcot(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.16)

    Case 1-4: If p=κ24, q=(κ22)2 and r=14, then φ(η)=sn(η)1+dn(η). In this case the solution of FSFLE (1.1), by using Eq (3.8), is

    Φ(x,t)=±κ12Bsn(η1αxα+η2t)1+dn(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.17)

    If κ1, then Eq (3.17) transfers to

    Φ(x,t)=±12Btanh(η1αxα+η2t)1+sech(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.18)

    Case 1-5: If p=(1κ2)24, q=(1κ2)22 and r=14, then φ(η)=sn(η)dn(η)+cn(η). In this case the solution of FSFLE (1.1), by using Eq (3.8), is

    Φ(x,t)=±(1κ2)12B[sn(η1αxα+η2t)dn(η1αxα+η2t)+cn(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.19)

    If κ0, then Eq (3.19) transfers to

    Φ(x,t)=±12B[sin(η1αxα+η2t)1+cos(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.20)

    Case 1-6: If p=1κ24, q=(1κ2)2 and r=1κ24, then φ(η)=cn(η)1+sn(η). In this case the solution of FSFLE (1.1), by using Eq (3.8), is

    Φ(x,t)=±2Bcn(η1αxα+η2t)1+sn(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.21)

    When κ0, then Eq (3.21) transfers to

    Φ(x,t)=±122Bcos(η1αxα+η2t)1+sin(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.22)

    Case 1-7: If p=1, q=0 and r=0, then φ(η)=cη. In this case the solution of FSFLE (1.1), by utilizing Eq (3.8), is

    Φ(x,t)=±2Bc(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.23)

    Second family: The solution of Eq (2.11), by using Eqs (3.3) and (3.6), takes the form

    Φ(x,t)=±2rB1φ(η)eiΘ(μ)σW(t)σ2t. (3.24)

    There are many cases depending on r>0:

    Case 2-1: If p=κ2, q=(1+κ2) and r=1, then φ(η)=sn(η). In this case the solution of FSFLE (1.1), by utilizing Eq (3.24), is

    Φ(x,t)=±2B1sn(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.25)

    If κ1, then Eq (3.25) transfers to

    Φ(x,t)=±2Bcoth(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.26)

    Case 2-2: If p=1, q=2κ2 and r=(1κ2), then φ(η)=cs(η). In this case the solution of FSFLE (1.1), by utilizing Eq (3.24), is

    Φ(x,t)=±2(1κ2)B1cs(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.27)

    When κ0, then Eq (3.27) transfers to

    Φ(x,t)=±2Btan(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.28)

    Case 2-3: If p=κ2, q=2κ21 and r=(1κ2), then φ(η)=cn(μ). In this case the solution of FSFLE (1.1), by utilizing Eq (3.24), is

    Φ(x,t)=±2(1κ2)B1cn(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.29)

    If κ0, then Eq (3.31) transfers to

    Φ(x,t)=±2Bsec(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.30)

    Case 2-4: If p=κ24, q=(κ22)2 and r=14, then φ(η)=sn(η)1+dn(η). In this case the solution of FSFLE (1.1), by using Eq (3.24), is

    Φ(x,t)=±12B1+dn(η1αxα+η2t)sn(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.31)

    If κ1, then Eq (3.31) transfers to

    Φ(x,t)=±12B[coth(η1αxα+η2t)+csch(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.32)

    Case 2-5: If p=1κ24, q=(1κ2)2 and r=1κ24, then φ(η)=cn(η)1+sn(η). In this case the solution of FSFLE (1.1), by utilizing Eq (3.8), is

    Φ(x,t)=±1κ22B1+sn(η1αxα+η2t)cn(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.33)

    When κ0, then Eq (3.33) transfers to

    Φ(x,t)=±2B[sec(η1αxα+η2t)±tancn(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.34)

    Case 2-6: If p=(1κ2)24, q=(1κ2)22 and r=14, then φ(η)=sn(η)dn(η)+cn(η). In this case the solution of FSFLE (1.1), by using Eq (3.24), is

    Φ(x,t)=±12B[dn(η1αxα+η2t)+cn(η1αxα+η2t)sn(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.35)

    If κ0, then Eq (3.35) transfers to

    Φ(x,t)=±12B[csc(η1αxα+η2t)+cot(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.36)

    If κ1, then Eq (3.35) transfers to

    Φ(x,t)=±2Bcsch(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.37)

    Third family: The solution of Eq (2.11), using Eqs (3.3) and (3.7), takes the form

    Φ(x,t)=[±2pBφ(η)±2rB1φ(η)]eiΘ(μ)σW(t)σ2t. (3.38)

    There are many cases depending on r>0:

    Case 3-1: If p=κ2, q=(1+κ2) and r=1, then φ(η)=sn(η). In this case the solution of FSFLE (1.1), by utilizing Eq (3.38), is

    Φ(x,t)=±2B[κsn(η1αxα+η2t)+1sn(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.39)

    If κ1, then Eq (3.39) transfers to

    Φ(x,t)=±[2Btanh(η1αxα+η2t)+2Bcoth(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.40)

    Case 3-2: If p=1, q=2κ2 and r=(1κ2), then φ(η)=cs(η). In this case the solution of FSFLE (1.1), by using Eq (3.38), is

    Φ(x,t)=±[2Bcs(η)+2(1κ2)B1cs(η)eiΘ(μ)σW(t)σ2t, (3.41)

    where η=η1αxα+η2t. When κ0, then Eq (3.41) transfers to

    Φ(x,t)=±2B[cot(η1αxα+η2t)+tan(η1αxα+η2t)]eiΘ(μ)σW(t)σ2t. (3.42)

    Case 3-3: If p=κ24, q=(κ22)2 and r=14, then φ(η)=sn(η)1±dn(η). In this case the solution of FSFLE (1.1), by utilizing Eq (3.38), is

    Φ(x,t)=±12B[κsn(η1+dn(η)+1+dn(η)sn(η)]eiΘ(μ)σW(t)σ2t, (3.43)

    where η=η1αxα+η2t. If κ1, then Eq (3.43) transfers to

    Φ(x,t)=±2Bcoth(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.44)

    Case 3-4: If p=1κ24, q=(1κ2)2 and r=1κ24, then φ(η)=cn(η)1+sn(η). In this case the solution of FSFLE (1.1), by utilizing Eq (3.38), is

    Φ(x,t)=±1κ22B[cn(η)1+sn(η)+1+sn(η)cn(η)]eiΘ(μ)σW(t)σ2t, (3.45)

    where η=η1αxα+η2t. When κ0, then Eq (3.45) transfers to

    Φ(x,t)=±2Bsec(η)eiΘ(μ)σW(t)σ2t. (3.46)

    Case 3-5: If p=(1κ2)24, q=(1κ2)22 and r=14, then φ(η)=sn(η)dn(η)+cn(η). In this case the solution of the FSFLE (1.1), by using Eq (3.38), is

    Φ(x,t)=±12B[(1κ2)sn(η)dn(η)+cn(η)+dn(η)+cn(η)sn(η)]eiΘ(μ)σW(t)σ2t, (3.47)

    where η=η1αxα+η2t. If κ0, then Eq (3.35) transfers to

    Φ(x,t)=±2Bcsc(η1αxα+η2t)eiΘ(μ)σW(t)σ2t. (3.48)

    In deterministic systems, the stabilizing and destabilizing consequences of noisy terms are well known at this time, based on the research done on the issue [42,43]. There is no longer any doubt that these effects are critical to comprehending the long-term behavior of real systems. Recently, there have been studies on the stabilization problem of stochastic nonlinear delay systems; see, for instance [44,45]. Now, we examine the effect of white noise and fractional derivatives on the exact solution of the FSFLE (1.1). To describe the behavior of these solutions, we present a number of diagrams. For certain obtained solutions such as Eqs (3.9) and (3.10), let us fix the parameters ρ= γ1=μ1=η1=1, η2=2, μ2=2, x[0,4] and t[0,2] to simulate these diagrams.

    First, the fractional derivative effects: In Figures 1 and 2, if σ=0, we can see that the graph's shape is compressed as the value of β decreases:

    Figure 1.  (a–c) 3D graph of solution |Φ(x,t)| in Eq (3.9) with σ=0 and different values of α=1, 0.7, 0.5 (d) 2D graph of Eq (3.9) with different values of α=1, 0.7, 0.5.
    Figure 2.  (a–c) indicate 3D-graph of solution |Φ(x,t)| in Eq (3.10) with σ=0 and different values of α=1, 0.7, 0.5 (d) denotes 2D-graph of Eq (3.10) for different values of α=1, 0.7, and 0.5.

    We deduced from Figures 1 and 2 that there is no overlap between the curves of the solutions. Furthermore, as the order of the fractional derivative decreases, the surface moves to the right.

    Second, the noise effects: In Figure 3, the surface is not flat and contains various fluctuations when σ=0 (i.e., there is no noise).

    Figure 3.  3D diagram of solution |Φ(x,t)| in Eqs (3.9) and (3.10).

    While we can see in Figures 4 and 5, after small transit behaviors, the surface has become more planar:

    Figure 4.  3D graph of solution |Φ(x,t)| in Eq (3.9) for σ=1, 2.
    Figure 5.  3D graph of solution |Φ(x,t)| in Eq (3.10) for σ=1, 2.

    In the end, we can deduce from Figures 4 and 5 that, when the noise is ignored (i.e., at σ=0), there are some different types of solutions, such as a periodic solution, kink solution, etc. After minor transit patterns, the surface becomes significantly flatter when noise is included and its strength is increased by σ=1,2. This demonstrates that the multiplicative white noise has an effect on the FSFLE solutions and stabilizes them around zero.

    We looked at FSFLE derived in the Itô sense by multiplicative white noise in this paper. By using a modified mapping technique, we were able to acquire the exact fractional-stochastic solutions. These solutions play a crucial role in the explanation of a wide range of exciting and complex physical phenomena. In addition, the fractional derivative and multiplicative white noise effects on the analytical solution of FSFLE (1.1) were demonstrated using MATLAB software. We came to the conclusion that the multiplicative white noise stabilized the solutions around zero and the fractional-derivative pushed the surface to the right when the fractional-order derivative declined.

    This research has been funded by Deputy for Research & Innovation, Ministry of Education through Initiative of Institutional Funding at University of Ha'il-Saudi Arabia through project number IFP-22029.

    The authors declare that there are no conflicts of interest.



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