Research article

An improved atomic search algorithm for optimization and application in ML DOA estimation of vector hydrophone array

  • Received: 30 September 2021 Revised: 22 December 2021 Accepted: 03 January 2022 Published: 10 January 2022
  • MSC : 65K05, 65K10

  • The atom search optimization (ASO) algorithm has the characteristics of fewer parameters and better performance than the traditional intelligent optimization algorithms, but it is found that ASO may easily fall into local optimum and its accuracy is not higher. Therefore, based on the idea of speed update in particle swarm optimization (PSO), an improved atomic search optimization (IASO) algorithm is proposed in this paper. Compared with traditional ASO, IASO has a faster convergence speed and higher precision for 23 benchmark functions. IASO algorithm has been successfully applied to maximum likelihood (ML) estimator for the direction of arrival (DOA), under the conditions of the different number of signal sources, different signal-to-noise ratio (SNR) and different population size, the simulation results show that ML estimator with IASO algorithum has faster convergence speed, fewer iterations and lower root mean square error (RMSE) than ML estimator with ASO, sine cosine algorithm (SCA), genetic algorithm (GA) and particle swarm optimization (PSO). Therefore, the proposed algorithm holds great potential for not only guaranteeing the estimation accuracy but also greatly reducing the computational complexity of multidimensional nonlinear optimization of ML estimator.

    Citation: Peng Wang, Weijia He, Fan Guo, Xuefang He, Jiajun Huang. An improved atomic search algorithm for optimization and application in ML DOA estimation of vector hydrophone array[J]. AIMS Mathematics, 2022, 7(4): 5563-5593. doi: 10.3934/math.2022308

    Related Papers:

    [1] Pinghua Yang, Caixia Yang . The new general solution for a class of fractional-order impulsive differential equations involving the Riemann-Liouville type Hadamard fractional derivative. AIMS Mathematics, 2023, 8(5): 11837-11850. doi: 10.3934/math.2023599
    [2] Hasanen A. Hammad, Hassen Aydi, Manuel De la Sen . The existence and stability results of multi-order boundary value problems involving Riemann-Liouville fractional operators. AIMS Mathematics, 2023, 8(5): 11325-11349. doi: 10.3934/math.2023574
    [3] Bashir Ahmad, Manal Alnahdi, Sotiris K. Ntouyas, Ahmed Alsaedi . On a mixed nonlinear boundary value problem with the right Caputo fractional derivative and multipoint closed boundary conditions. AIMS Mathematics, 2023, 8(5): 11709-11726. doi: 10.3934/math.2023593
    [4] Md. Asaduzzaman, Md. Zulfikar Ali . Existence of positive solution to the boundary value problems for coupled system of nonlinear fractional differential equations. AIMS Mathematics, 2019, 4(3): 880-895. doi: 10.3934/math.2019.3.880
    [5] Asghar Ahmadkhanlu, Hojjat Afshari, Jehad Alzabut . A new fixed point approach for solutions of a p-Laplacian fractional q-difference boundary value problem with an integral boundary condition. AIMS Mathematics, 2024, 9(9): 23770-23785. doi: 10.3934/math.20241155
    [6] Dumitru Baleanu, Muhammad Samraiz, Zahida Perveen, Sajid Iqbal, Kottakkaran Sooppy Nisar, Gauhar Rahman . Hermite-Hadamard-Fejer type inequalities via fractional integral of a function concerning another function. AIMS Mathematics, 2021, 6(5): 4280-4295. doi: 10.3934/math.2021253
    [7] Iman Ben Othmane, Lamine Nisse, Thabet Abdeljawad . On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems. AIMS Mathematics, 2024, 9(6): 14106-14129. doi: 10.3934/math.2024686
    [8] Snezhana Hristova, Antonia Dobreva . Existence, continuous dependence and finite time stability for Riemann-Liouville fractional differential equations with a constant delay. AIMS Mathematics, 2020, 5(4): 3809-3824. doi: 10.3934/math.2020247
    [9] Khalid K. Ali, K. R. Raslan, Amira Abd-Elall Ibrahim, Mohamed S. Mohamed . On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type. AIMS Mathematics, 2023, 8(8): 18206-18222. doi: 10.3934/math.2023925
    [10] Ahmed Alsaedi, Bashir Ahmad, Afrah Assolami, Sotiris K. Ntouyas . On a nonlinear coupled system of differential equations involving Hilfer fractional derivative and Riemann-Liouville mixed operators with nonlocal integro-multi-point boundary conditions. AIMS Mathematics, 2022, 7(7): 12718-12741. doi: 10.3934/math.2022704
  • The atom search optimization (ASO) algorithm has the characteristics of fewer parameters and better performance than the traditional intelligent optimization algorithms, but it is found that ASO may easily fall into local optimum and its accuracy is not higher. Therefore, based on the idea of speed update in particle swarm optimization (PSO), an improved atomic search optimization (IASO) algorithm is proposed in this paper. Compared with traditional ASO, IASO has a faster convergence speed and higher precision for 23 benchmark functions. IASO algorithm has been successfully applied to maximum likelihood (ML) estimator for the direction of arrival (DOA), under the conditions of the different number of signal sources, different signal-to-noise ratio (SNR) and different population size, the simulation results show that ML estimator with IASO algorithum has faster convergence speed, fewer iterations and lower root mean square error (RMSE) than ML estimator with ASO, sine cosine algorithm (SCA), genetic algorithm (GA) and particle swarm optimization (PSO). Therefore, the proposed algorithm holds great potential for not only guaranteeing the estimation accuracy but also greatly reducing the computational complexity of multidimensional nonlinear optimization of ML estimator.



    A particular feature of the fractional calculus that can be grasped by comprehending tautochrone problem is that scientists and engineers can create novel models containing fractional differential equations. Another outstanding feature that makes fractional operators important is that it can be applied eligibly in various disciplines such as physics, economics, biology, engineering, chemistry, mechanics and so on. In such models as epidemic, logistic, polymers and proteins, human tissue, biophysical, transmission of ultrasound waves, integer-order calculus seems to lagging behind the requirement of those applications when compared with the fractional versions of such models. Under the rigorous mathematical justification, it is possible to investigate many complex processes by means of the non-local fractional derivatives and integrals which enable us to observe past history owing to having memory effect represented by time-fractional derivative. One of the scopes of the fractional calculus is to provide flexibility in modelling under favour of real, complex or variable order. Interestingly enough, fractional operators can also be utilized in mathematical psychology in which the behavior of humankind is modeled by using the fact that they have past experience and memories. So, it is clear that to benefit from non-integer order derivatives and integrals is beneficial for modelling memory-dependent processes due to non-locality represented by space-fractional derivative. A great amount of phenomena in nature are created to provide more accurate and more flexible results thanks to non-integer derivatives. Some of the most common fractional operators capturing many advantageous instruments for modeling in numerous fields are that Riemann-Liouville (RL) developed firstly in literature and Caputo fractional derivatives which are the convolution of first-order derivative and power law. The former constitutes some troubles when applying to the real world problems whereas the latter has the privilege of being compatible with the initial conditions in applications. One can look for [1] for more information about RL and Caputo fractional derivatives.

    We shall remark that some fractional operators are composed by the idea of fractional derivative and integral of a function with respect to another function presented by Kilbas in [1]. The left and right fractional integrals of the function f with respect to the g on (a,b) are as below:

    gIαaf(t)=1Γ(α)ta(g(t)g(x))α1g(x)f(x)dx, (1.1)

    and

    bIαgf(t)=1Γ(α)bt(g(x)g(t))α1g(x)f(x)dx. (1.2)

    where Re(α)>0, g(t) is an increasing and positive monotone function on (a,b] and have a continuous derivative g(t) on (a,b). Also, the left and right fractional derivatives of f with respect to g are presented by

    gDαaf(t)=(1g(t)ddt)ngInαaf(t),bDαgf(t)=(1g(t)ddt)nbInαgf(t), (1.3)

    where Re(α)>0, n=[Re(α)]+1 and g(t)0. Note that by choosing the convenient g(t), one can get Riemann-Liouville, Hadamard, Katugampola fractional operators. So, an open problem is that it is possible to create novel fractional operators by choosing other productive and suitable function g(t), which allow us to utilize more variety of non-local fractional operators. Moreover, for these generalized fractional derivatives and integrals, Jarad and Abdeljawad in [2,3] have introduced the generalized LT which is the strong and useful method for many fractional differential equations. On the other hand, there also some non-local frational operators with non-singular kernel, for instance, Caputo-Fabrizio (CF) defined by the convolution of exponential function and first-order derivative and Atangana-Baleanu (AB) fractional derivative obtained by the convolution of Mittag-Leffler function and first-order derivative. By making use of aforementioned fractional operators, many authors have addressed fractional models in various areas. For example, Bonyah and Atangana in [4] have submitted the 3D IS-LM macroeconomic system model in economics in which past fluctuations or changes in market can be observed much better by non-local fractional operators with memory than classical counterparts. Also, the fractional Black-Scholes model has been presented by Yavuz and Ozdemir in [5]. In [6], Atangana and Araz have submitted modified Chuan models by means of three different kind of non-local fractional derivatives including Caputo, CF and AB. The fractional chickenpox disease model among school children by using real data for 25 weeks and the modeling of deforestation on wildlife species in terms of Caputo fractional operator have been investigated by Qureshi and Yusuf in [7,8]. Yavuz and Bonyah in [9] have examined the fractional schistosomiasis disease models which target to prevent the spread of infection by virtue of the CF and AB fractional derivatives. A fractional epidemic model having time-delay has discussed by Rihan et all in [10]. All of these fractional models mentioned above are only a few of the studies using an advantage of fractional operators. In these studies and in many other studies, the authors aim to find the most appropriate fractional derivative that they can utilize, to understand which fractional derivative works better for their objective under favour of real data and to determine which fractional derivative tends to approach the integer-order derivative more rapidly. Therefore, having several fractional operator definitions is of great importance in order to apply them to different type of models and to state much more accurate results. For more application on fractional operators, we refer the readers to [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29].

    Generally, in order to obtain fractional solutions of some models similar to the above-mentioned models, the authors replace the integer order derivative by a fractional derivative. However, when it comes to applying to physical models, this approach is not exactly correct due to the need to maintain the dimension fractional equation. For example, in [30], the authors have introduced the fractional falling body problem by preserving the dimension. They have done this as follows:

    ddt1σ1αdαdtα,0<α1, (1.4)

    where σ has the dimension of seconds. Also, in [31,32], the falling body problem by means fractional operators with exponential kernel has been investigated. In this study, we also examine the falling body problem relied on the Newton's second law which expresses the acceleration of a particle is depended on the mass of the particle and the net force action on the particle.

    Let us consider an object of mass m falling through the air from a height h with velocity v0 in a gravitational field. By utilizing the Newton's second law, we get

    mdvdt+mkv=mg, (1.5)

    where k is positive constant rate, g represents the gravitational constant. The solution of the equation (1.5) is

    v(t)=gk+ekt(v0+gk), (1.6)

    and by integrating for z(0)=h, we have

    z(t)=hgtk+1k(1ekt)(v0+gk). (1.7)

    Considering all the information presented above, we organize the article as follows: In section 2, some basic definitions and theorems about non-local fractional calculus are given. In section 3, the fractional falling body problem is investigated by means of ABC, generalized fractional derivative and generalized ABC including Mittag-Leffler function with three parameters. Also, we carry out simulation analysis by plotting some graphs in section 4. In section 5, some outstanding consequences are clarified.

    Before coming to the main results, we provide some significant definitions, theorems and properties of fractional calculus in order to establish a mathematically sound theory that will serve the purpose of the current article.

    Definition 2.1. [1] The Mittag-Leffler (ML) function including one parameter α is defined as follows

    Eα(t)=k=0tkΓ(αk+1)(tC,Re(α)>0), (2.1)

    whereas the ML function with two parameters α,β is

    Eα,β(t)=k=0tkΓ(αk+β)(t,βC,Re(α)>0). (2.2)

    As seen clearly, Eα,β(t) corresponds to the ML function (2.1) when β=1.

    Definition 2.2. [33] The generalized ML function is defined by

    Eρα,β(t)=k=0tk(ρ)kΓ(αk+β)k! (tC,α,β,ρC,Re(α)>0), (2.3)

    where (ρ)k=ρ(ρ+1)...(ρ+k1) is the Pochhammer symbol introduced by Prabhakar. Note that (1)k=k!, and so E1α,β(t)=Eα,β(t).

    Definition 2.3. [33] The ML function for a special function is given by

    Eα(λ,t)=k=0λktαkΓ(αk+1)(0λR,tC,Re(α)>0), (2.4)

    and

    Eα,β(λ,t)=k=0λktαk+β1Γ(αk+β)(0λR,t,βC,Re(α)>0). (2.5)

    It should be noticed that Eα,1(λ,t)=Eα(λ,t). Also, the modified ML function with three parameters can be written as

    Eρα,β(λ,t)=k=0λktαk+β1(ρ)kΓ(αk+β)k!(0λR,t,βC,Re(α)>0). (2.6)

    Definition 2.4. [1] The left and right Caputo fractional derivative are defined as below

    CaDαf(t)=1Γ(nα)ta(tx)nα1f(n)(x)dx, (2.7)

    and

    CDαbf(t)=(1)nΓ(nα)bt(xt)nα1f(n)(x)dx, (2.8)

    where αC, Re(α)>0, n=[Re(α)]+1.

    Definition 2.5. [34] The left and right Caputo-Fabrizio fractional derivative in the Caputo sense (CFC) are given by

    CFCaDαf(t)=M(α)1αtaf(x)exp(λ(tx))dx, (2.9)

    and

    CFCDαbf(t)=M(α)1αbtf(x)exp(λ(xt))dx, (2.10)

    where 0<α<1, M(α) is a normalization function and λ=α1α.

    Definition 2.6. [35] The left and right ABC fractional derivative are

    ABCaDαf(t)=B(α)1αtaf(x)Eα(λ(tx)α)dx, (2.11)

    and the right one

    ABCDαbf(t)=B(α)1αbtf(x)Eα(λ(xt)α)dx, (2.12)

    where 0<α<1, B(α) is a normalization function and λ=α1α.

    Definition 2.7. [33] The left and right ABC fractional derivative containing generalized ML function Eγα,μ(λtα) such that γR, Re(μ)>0, 0<α<1 and λ=α1α are defined by

    ABCaDα,μ,γf(t)=B(α)1αtaEγα,μ(λ(tx)α)f(x)dx, (2.13)

    and also

    ABCDα,μ,γbf(t)=B(α)1αbtEγα,μ(λ(xt)α)f(x)dx. (2.14)

    Definition 2.8. [36] The generalized left and right fractional integrals are defined by

    aIα,ρf(t)=1Γ(α)ρα1ta(tρxρ)α1f(x)xρ1dx, (2.15)

    and

    Iα,ρbf(t)=1Γ(α)ρα1bt(xρtρ)α1f(x)xρ1dx, (2.16)

    respectively.

    Definition 2.9. [37] The generalized left and right fractional derivatives in the Caputo sense are given respectively by

    CaDα,ρf(t)=aInα,ρ(t1ρddt)nf(t)=1Γ(nα)ρnα1ta(tρxρ)nα1(t1ρddt)nf(x)xρ1dx, (2.17)

    and

    CDα,ρbf(t)=Inα,ρb(t1ρddt)nf(t)=1Γ(nα)ρnα1bt(xρtρ)nα1(t1ρddt)nf(x)xρ1dx. (2.18)

    Definition 2.10. [33] Let υ,ω:[0,)R, then the convolution of υ and ω is

    (υω)(t)=t0υ(ts)ω(s)ds. (2.19)

    Proposition 2.11. [33] Assume that υ,ω:[0,)R, then the following property is valid

    L{(υω)(t)}=L{υ(t)}L{ω(t)}. (2.20)

    Theorem 2.1. [38] The LT of Caputo fractional derivative is presented by

    L{CDαf(t)}=sαF(s)n1k=0sαk1f(k)(0), (2.21)

    where F(s)=L{f(t)}.

    Theorem 2.2. [34] The LT of CFC fractional derivative is given as

    L{CFCDα}=M(α)1αsF(s)s+α1αM(α)1αf(0)s+α1α. (2.22)

    Theorem 2.3. [39] The LT of the ABC is as below

    L{ABCDαf(t)}=B(α)1αsαF(s)sα1f(0)sα+α1α. (2.23)

    Theorem 2.4. [3] Let fACnγ[0,a], a>0, α>0 and γk=(t1ρddt)kf(t), k=0,1,...,n has exponential order ectρρ, then we have

    L{C0Dα,ρf(t)}=sα[L{f(t)}n1k=0sk1(t1ρddt)kf(0)], (2.24)

    where s>0.

    Theorem 2.5. [33] The LT of the generalized ABC can be presented by

    L{ABCDα,μ,γf(t)}=B(α)1αs1μF(s)(1λsα)γB(α)1αf(0)sμ(1λsα)γ. (2.25)

    Lemma 2.12. The LT of some special functions are as below

    L{Eα(atα)}=sαs(sα+a).

    L{1Eα(atα)}=as(sα+a).

    L{tα1Eα,α(atα)}=1sα+a.

    Lemma 2.13. [40] Let α,μ,γ,λ,sC, Re(μ)>0, Re(s)>0, |λsα|<1, then the Laplace transform of Eγα,μ(λtα) is as follows

    L{Eγα,μ(λtα)}=sμ(1λsα)γ. (2.26)

    The purpose of this section is to introduce the solutions for fractional falling body problem by means of some non-local fractional derivative operators such as ABC, Katugampola and generalized ABC. We put a condition for ABC type falling body problem in order to achieve right result. Also, dimensionality of the physical parameter in the model is kept by using different auxiliary parameters for each fractional operator.

    The ABC type fractional falling body problem relied on Newton's second law is presented as follows

    mσ1αABC0Dαv(t)+mkv(t)=mg, (3.1)

    where the initial velocity v(0)=v0, g represents the gravitational constant, the mass of body is indicated by m and k is the positive constant rate.

    If we apply LT to the Eq (3.1), then we have

    L{ABC0Dαv(t)}+kσ1αL{v(t)}=L{gσ1α}, (3.2)
    B(α)1αsαL{v(t)}sα1v(0)sα+α1α+kσ1αL{v(t)}=gσ1αs, (3.3)
    L{v(t)}(B(α)1αsαsα+α1α+kσ1α)=B(α)1αsα1v(0)sα+αα1gσ1αs, (3.4)
    L{v(t)}=B(α)1αsαs(sα(B(α)1α+kσ1α)+kσ1αα1α)v(0)gσ1αs+α1αs(sα(B(α)1α+kσ1α)+kσ1αα1α), (3.5)
    L{v(t)}=B(α)B(α)+kσ1α(1α)sαs(sα+kασ1αB(α)+kσ1α(1α))v(0)gσ1α(1α)B(α)+kσ1α(1α)sαs(sα+kασ1αB(α)+kσ1α(1α))gkkασ1αB(α)+kσ1α(1α)s(sα+kασ1αB(α)+kσ1α(1α)), (3.6)

    and applying the inverse LT to the both side of the (3.6) and using the condition v(0)=v0, we obtain the velocity as follows

    v(t)=B(α)B(α)+kσ1α(1α)Eα(kασ1αB(α)+kσ1α(1α)tα)v0gσ1α(1α)B(α)+kσ1α(1α)Eα(kασ1αB(α)+kσ1α(1α)tα)gk[1Eα(kασ1αB(α)+kσ1α(1α)tα)]. (3.7)

    Because α=σk, 0<σ1k, the velocity v(t) can be written in the form below

    v(t)=B(α)B(α)+α1αkα(1α)Eα(α2αB(α)+α1αkα(1α)(kt)α)v0gα1αkα1(1α)B(α)+α1αkα(1α)Eα(α2αB(α)+α1αkα(1α)(kt)α)gk[1Eα(α2αB(α)+α1αkα(1α)(kt)α)], (3.8)

    where Eα(.) is the ML function. Note that we put the condition v0=gk in order to satisfy initial condition v(0)=v0. By benefiting from the velocity (3.7), vertical distance z(t) can be get in the following way

    ABC0Dαz(t)=B(α)σ1αB(α)+kσ1α(1α)Eα(kασ1αB(α)+kσ1α(1α)tα)v0gσ2(1α)(1α)B(α)+kσ1α(1α)Eα(kασ1αB(α)+kσ1α(1α)tα)gσ1αk[1Eα(kασ1αB(α)+kσ1α(1α))tα]. (3.9)

    By applying the LT to the Eq (3.9), we have

    L{ABC0Dαz(t)}=B(α)σ1αv0B(α)+kσ1α(1α)L{Eα(kασ1αB(α)+kσ1α(1α)tα)}gσ2(1α)(1α)B(α)+kσ1α(1α)L{Eα(kασ1αB(α)+kσ1α(1α)tα)}L{gσ1αk}+gσ1αkL{Eα(kασ1αB(α)+kσ1α(1α)tα)}, (3.10)
    B(α)1αsαL{z(t)}sα1z(0)sα+α1α=B(α)σ1αv0B(α)+kσ1α(1α)sαs(sα+kασ1αB(α)+kσ1α(1α))gσ2(1α)(1α)B(α)+kσ1α(1α)sαs(sα+kασ1αB(α)+kσ1α(1α))gσ1αks+gσ1αksαs(sα+kασ1αB(α)+kσ1α(1α)), (3.11)
    L{z(t)}=z(0)s+σ1α(1α)v0B(α)+kσ1α(1α)sαs(sα+kασ1αB(α)+kσ1α(1α))+v0kkασ1αB(α)+kσ1α(1α)s(sα+kασ1αB(α)+kσ1α(1α))gσ2(1α)(1α)2B(α)[B(α)+kσ1α(1α)]sαs(sα+kασ1αB(α)+kσ1α(1α))gσ1α(1α)kB(α)kασ1αB(α)+kσ1α(1α)s(sα+kασ1αB(α)+kσ1α(1α))gσ1α(1α)kB(α)1sgασ1αkB(α)1sα+1+gσ1α(1α)kB(α)sαs(sα+kασ1αB(α)+kσ1α(1α))+gB(α)+kgσ1α(1α)k2B(α)kασ1αB(α)+kσ1α(1α)s(sα+kασ1αB(α)+kσ1α(1α)), (3.12)

    by utilizing the inverse LT for the Eq (3.12) and taking the z(0)=h, we obtain the vertical distance z(t) as below

    z(t)=h+σ1α(1α)v0B(α)+kσ1α(1α)Eα(kασ1αB(α)+kσ1α(1α)tα)+v0k[1Eα(kασ1αB(α)+kσ1α(1α)tα)]gσ2(1α)(1α)2B(α)[B(α)+kσ1α(1α)]Eα(kασ1αB(α)+kσ1α(1α)tα)gσ1α(1α)kB(α)[1Eα(kασ1αB(α)+kσ1α(1α)tα)]gσ1αkB(α)[1α+αtαΓ(1+α)]+gσ1α(1α)kB(α)Eα(kασ1αB(α)+kσ1α(1α)tα)+gB(α)+kgσ1α(1α)k2B(α)[1Eα(kασ1αB(α)+kσ1α(1α)tα)], (3.13)

    where v0=gσ1αB(α). Due to the fact that α=σk, 0<σ1k, the vertical distance z(t) can be written as follows

    z(t)=h+α1αkα1(1α)v0B(α)+α1αkα(1α)Eα(α2αB(α)+α1αkα(1α)(kt)α)+v0k[1Eα(α2αB(α)+α1αkα(1α)(kt)α)]gα2(1α)k2(α1)(1α)2B(α)[B(α)+α1αkα(1α)]Eα(α2αB(α)+α1αkα(1α)(kt)α)gα1αkα1(1α)kB(α)[1Eα(α2αB(α)+α1αkα(1α)(kt)α)]gα1αkαk2B(α)[1α+αtαΓ(1+α)]+gα1αkα(1α)kB(α)Eα(α2αB(α)+α1αkα1(1α)(kt)α)+gB(α)+gα1αkα(1α)k2B(α)[1Eα(α2αB(α)+α1αkα(1α)(kt)α)]. (3.14)

    The fractional falling body problem relied on Newton's second law by means of generalized fractional derivative introduced by Katugampola is given by

    mσ1αρC0Dα,ρv(t)+mkv(t)=mg, (3.15)

    where the initial velocity v(0)=v0, g is the gravitational constant, the mass of body is represented by m and k is the positive constant rate.

    Applying the LT to the both side of the Eq (3.15), we have

    L{C0Dα,ρv(t)}+kσ1αρL{v(t)}=L{gσ1αρ}, (3.16)
    sαL{v(t)}sα1v(0)+kσ1αρL{v(t)}=gσ1αρs, (3.17)
    L{v(t)}=sαs(sα+kσ1αρ)v(0)gkkσ1αρs(sα+kσ1αρ). (3.18)

    If the inverse LT is utilized for (3.18), one can obtain the following velocity

    v(t)=v0Eα(kσ1αρ(tρρ)α)gk[1Eα(kσ1αρ(tρρ)α)], (3.19)

    by inserting the α=σk, 0<σ1k, we get

    v(t)=v0Eα(α1αρkαρ(tρρ)α)gk[1Eα(α1αρkαρ(tρρ)α)]. (3.20)

    From the velocity (3.19), we obtain the vertical distance z(t) in terms of generalized fractional derivative after some essential calculations below

    C0Dα,ρz(t)=σ1αρv0Eα(kσ1αρ(tρρ)α)σ1αρgk[1Eα(kσ1αρ(tρρ)α)], (3.21)

    applying the LT to the both side of (3.21), one can have

    L{C0Dα,ρz(t)}=σ1αρv0L{Eα(kσ1αρ(tρρ)α)}L{gσ1αρk}+gσ1αρkL{Eα(kσ1αρ(tρρ)α)}, (3.22)
    L{z(t)}=z(0)s+v0kkσ1αρs(sα+kσ1αρ)gσ1αρksα+1+gk2kσ1αρs(sα+kσ1αρ), (3.23)

    after applying the inverse LT to the (3.23) and for z(0)=h, we get

    z(t)=h+v0k[1Eα(kσ1αρ(tρρ)α)]gσ1αρkΓ(α+1)(tρρ)α+gk2[1Eα(kσ1αρ(tρρ)α)], (3.24)

    substituting the α=σk, 0<σ1k to the Eq (3.24), we obtain as follows

    z(t)=h+v0k[1Eα(α1αρkαρ(tρρ)α)]gα1αρk2αρΓ(α+1)(tρρ)α+gk2[1Eα(α1αρkαρ(tρρ)α)]. (3.25)

    The fractional falling body problem relied on Newton's second law in terms of generalized ABC including ML function with three parameters is as follows

    mσ1αμABC0Dα,μ,γv(t)+mkv(t)=mg, (3.26)

    where the initial velocity v(0)=v0, g represents the gravitational constant, the mass of body is indicated by m and k is the positive constant rate.

    If we apply the LT to the (3.26), we have

    L{ABC0Dα,μ,γv(t)}+kσ1αμL{v(t)}=L{gσ1αμ}, (3.27)
    B(α)1αs1μ(1λsα)γL{v(t)}B(α)1αsμv0(1λsα)γ+kσ1αμL{v(t)}=gσ1αμs, (3.28)
    L{v(t)}=v0s+(kσ1αμ(1α)B(α)sμ(1λsα)γ)+1sgσ1αμB(α)1αs1μ(1λsα)γ+kσ1αμ. (3.29)

    In order to obtain inverse LT of the (3.29), this equation should be expanded as below

    L{v(t)}=v0sj=0(kσ1αμ)j(1αB(α))js(μ1)j(1λsα)γj+gσ1αμ1sj=0(kσ1αμ)j(1αB(α))j+1s(μ1)(j+1)(1λsα)γ(j+1), (3.30)

    by applying inverse LT to the expression (3.30), one can get the following velocity

    v(t)=v0j=0(kσ1αμ)j(1αB(α))jEγjα,(1μ)j+1(λ,t)+gσ1αμj=0(kσ1αμ)j(1αB(α))j+1Eγ(j+1)α,(1μ)(j+1)+1(λ,t), (3.31)

    plugging the α=σk, 0<σ1k to the (3.31), we reach

    v(t)=v0j=0(kαμα1αμ)j(1αB(α))jEγjα,(1μ)j+1(λ,t)+gα1αk1αj=0(kαα1α)j(1αB(α))j+1Eγ(j+1)α,(1μ)(j+1)+1(λ,t). (3.32)

    We can obtain the vertical distance z(t) in terms of generalized ABC by benefiting from the velocity (3.31) after the following calculations

    ABC0Dα,μ,γz(t)=v0σ1αμj=0(kσ1αμ)j(1αB(α))jEγjα,(1μ)j+1(λ,t)+gσ2(1αμ)j=0(kσ1αμ)j(1αB(α))j+1Eγ(j+1)α,(1μ)(j+1)+1(λ,t), (3.33)
    L{z(t)}=z(0)s+v0j=0(kσ1αμ)j(1αB(α))j+1s(μ1)(j+1)1(1λsα)γ(j+1)+gσ2(1αμ)j=0(kσ1αμ)j(1αB(α))j+2s(μ1)(j+2)1(1λsα)γ(j+2), (3.34)

    utilizing the inverse LT for the Eq (3.34) and when z(0)=h, one can have

    z(t)=h+v0j=0(kσ1αμ)j(1αB(α))j+1Eγ(j+1)α,(1μ)(j+1)+1(λ,t)+gσ2(1αμ)j=0(kσ1αμ)j(1αB(α))j+2Eγ(j+2)α,(1μ)(j+2)+1(λ,t), (3.35)

    after inserting the α=σk, 0<σ1k to the (3.35), we get

    z(t)=h+v0j=0(α1αμkαμ)j(1αB(α))j+1Eγ(j+1)α,(1μ)(j+1)+1(λ,t)+gα2(1αμ)k2(1αμ)j=0(α1αμkαμ)j(1αB(α))j+2Eγ(j+2)α,(1μ)(j+2)+1(λ,t). (3.36)

    This section is dedicated to demonstrate a comparison between such non-local fractional operators and traditional derivative. We compare these fractional operators with traditional derivative to observe which fractional derivative approaches the classical derivative faster. By this way, the behavior of each non-integer order derivative is shown by plotting. Additionaly, the main objective is to elaborate and expatiate the main findings of our results via graphical illustrations. To this aim, we set some suitable values of α and ρ to see the actual characteristic of behavior of our model. The comparison we made is between ABC, generalized ABC, generalized fractional derivative, Caputo, CFC and their corresponding classical version. So it can be seen that the presented graphs availed the main difference between the mentioned non-local fractional operators and classical version with the help of different parameter values.

    In order to comprehend the exact advantage of non-local fractional derivative operators for some governing models, one should utilize the real data. So, without using real data we can only observe the behavior of the solution curves and see the accuracy of our results. As can be seen in [30,31,32], the Caputo and CF type fractional falling body problem are handled by some authors. By benefiting from them, we discuss the relation between these fractional operators and our results obtained by ABC, generalized ABC and generalized fractional derivative.

    In Figure 1, the vertical notion of a falling body is demonstrated by means of ABC fractional derivative when α=0.5,0.6,0.7,0.8,1. Caputo and ABC fractional operators are compared with classical derivative for α=0.9 in Figure 2 and for α=0.8 in Figure 3. It can be noticed clearly that ABC tends to approach the integer-order case faster. In Figure 4, we show the vertical motion of a falling body in terms of CF fractional operator when α=0.5,0.6,0.7,0.8,1. Also, CFC, Caputo and classical derivative are compared with each other when α=0.9,0.95,0.8 in Figures 57 while CFC, generalized fractional derivative, ABC and Caputo are compared with integer-order derivative for ρ=0.9 and α=0.7, ρ=0.9 and α=0.9, ρ=0.9 and α=0.95. In Figures 810 CFC, generalized fractional derivative, ABC and Caputo operators are compared when ρ=0.9, α=0.7,0.9,0.95. Similarly, ABC fractional derivative operator tends approach the classical derivative faster then other counterparts.

    Figure 1.  Comparative analysis with ABC fractional derivative.
    Figure 2.  Comparative analysis for α=0.9.
    Figure 3.  Comparative analysis for α=0.8.
    Figure 4.  Comparative analysis with CFC fractional derivative.
    Figure 5.  Comparative analysis for α=0.9.
    Figure 6.  Comparative analysis for α=0.95.
    Figure 7.  Comparative analysis for α=0.8.
    Figure 8.  Comparative analysis for ρ=0.9 and α=0.7.
    Figure 9.  Comparative analysis for ρ=0.9 and α=0.9.
    Figure 10.  Comparative analysis ρ=0.9 and α=0.95.

    In recent years, fractional derivative operators have been utilized frequently in the solution of many physical models. On the other hand, various physical problems investigated using real data show that problems solved by means of fractional operators exhibit closer behavior to real data. So, we have analyzed an outstanding physical model called falling body problem in terms of some beneficial non-local fractional operators such as ABC, generalized ABC and generalized fractional derivative. Also, we have noticed that in order to solve a constant coefficient linear differential equation with initial condition, we have to put a convenient condition to satisfy the initial condition. Thereby, when solving the ABC type fractional falling body problem, we put a condition for velocity and vertical distance of falling body.

    In order to keep the dimensionality of the physical parameter, an auxiliary parameter σ has been used in different forms like σ1α, σ1αρ and σ1αμ for each fractional operator. Moreover, for generalized ABC type fractional falling body problem containing the Mittag-Leffler function with three parameters, power series has been used to apply inverse Laplace transform for getting velocity and vertical distance. Ultimately, all results obtained in this study have been strengthened by graphs.

    It is worth pointing out that in all graphs, the case of α=1 and ρ=1 corresponds to the traditional solutions and by comparing the classical solutions with the fractional solutions, each with different parameters, we can see clearly that our solutions behaves similar to the traditional one and as α and ρ values approach 1, the solution curves tends to approach classical solutions. This shows that our fractional solutions are accurate. So, the characteristic behavior of solution curves has been observed by comparing the solutions obtained above-stated operators.

    The authors declare no conflict of interest in this paper.



    [1] B. Jiao, Z. Lian, X. Gu, A dynamic inertia weight particle swarm optimization algorithm, Chaos Soliton Fract, 37 (2008), 698–705. https://doi.org/10.1016/j.chaos.2006.09.063 doi: 10.1016/j.chaos.2006.09.063
    [2] D. E. Goldberg, Genetic algorithm in search optimization and machine learning, Addison Wesley, 8 (1989), 2104–2116. https://dl.acm.org/doi/book/10.5555/534133 doi: 10.5555/534133
    [3] S. Kirpatrick, C. D. Gelatt, M. P. Vecchi, Optimization by simulated annealing, Readings Computer Vision, 220 (1983), 671–680. https://doi.org/10.1126/science.220.4598.671 doi: 10.1126/science.220.4598.671
    [4] S. Mirjalili, SCA: A Sine Cosine Algorithm for solving optimization problems, Knowl-based Syst., 96 (2016), 120–133. https://doi.org/10.1016/j.knosys.2015.12.022 doi: 10.1016/j.knosys.2015.12.022
    [5] Z. Zhang, J. Lin, Y. Shi, Application of artificial bee colony algorithm to maximum likelihood DOA estimation, J. Bionic. Eng., 10 (2013), 100–109. https://doi.org/10.1016/S1672-6529(13)60204-8 doi: 10.1016/S1672-6529(13)60204-8
    [6] S. Feng, Z. Zhang, Y. Shi, Introduction of bat algorithm into maximum likelihood DOA estimation, Modern Electronics Technique, 39 (2016), 26–29. https://doi.org/10.16652/j.issn.1004-373x.2016.08.007 doi: 10.16652/j.issn.1004-373x.2016.08.007
    [7] X. Fan, L. Pang, P. Shi, G. Li, X. Zhang, Application of bee evolutionary genetic algorithm to maximum likelihood direction-of-arrival estimation, Math. Probl. Eng., 2019 (2019), 1–11. https://doi.org/10.1155/2019/6035870 doi: 10.1155/2019/6035870
    [8] M. Jain, V. Singh, A. Rani, A novel nature-inspired algorithm for optimization: Squirrel search algorithm, Swarm Evol. Comput., 44 (2018), 148–175. https://doi.org/10.1016/j.swevo.2018.02.013 doi: 10.1016/j.swevo.2018.02.013
    [9] W. Zhao, L. Wang, Z. Zhang, Atom search optimization and its application to solve a hydrogeologic parameter estimation problem, Knowl-based Syst., 163 (2018), 283–304. https://doi.org/10.1016/j.knosys.2018.08.030 doi: 10.1016/j.knosys.2018.08.030
    [10] H. C. Corben, P. Stehle, Classical Mechanics, Physics Today, 6 (1953). https://doi.org/10.1063/1.3061288 doi: 10.1063/1.3061288
    [11] J. P. Ryckaert, G. Ciccotti, H. J. C Berendsen, Numerical integration of the cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes, J. Comput. Phys., 23 (1977), 327–341. https://doi.org/10.1016/0021-9991(77)90098-5 doi: 10.1016/0021-9991(77)90098-5
    [12] A. Stone, The theory of intermolecular forces, Pure. Appl. Chem., 51 (1979), 1627–1636. https://doi.org/10.1351/pac197951081627 doi: 10.1351/pac197951081627
    [13] J. E. Jones, On the determination of molecular fields Ⅱ. From the equation of state of a gas, P. Roy. Soc. A-Math. Phy., 106 (1924), 463–477. https://doi.org/10.2307/94265 doi: 10.2307/94265
    [14] W. Zhao, L. Wang, Z. Zhang, A novel atom search optimization for dispersion coefficient estimation in groundwater, Future Gener Comp. Sy., 91 (2018), 601–610. https://doi.org/10.1016/j.future.2018.05.037 doi: 10.1016/j.future.2018.05.037
    [15] A. M. Agwa, A. A. El-Fergany, G. M. Sarhan, Steady-State modeling of fuel cells based on atom search optimizer, Energies, 12 (2019), 1884. https://doi.org/10.3390/en12101884 doi: 10.3390/en12101884
    [16] A. Almagboul Mohammed, F. Shu, Y. Qian, X. Zhou, J. Wang, J. Hu, Atom search optimization algorithm based hybrid antenna array receive beamforming to control sidelobe level and steering the null, Aeu-int J. Electron. C., 111 (2019), 152854. https://doi.org/10.1016/j.aeue.2019.152854 doi: 10.1016/j.aeue.2019.152854
    [17] S. Barshandeh, A new hybrid chaotic atom search optimization based on tree-seed algorithm and Levy flight for solving optimization problems, Eng. Comput-germany, 37 (2021), 3079–3122. https://doi.org/10.1007/s00366-020-00994-0 doi: 10.1007/s00366-020-00994-0
    [18] K. K. Ghosh, R. Guha, S. Ghosh, S. K. Bera, R. Sarkar, Atom Search Optimization with Simulated Annealing-a Hybrid Metaheuristic Approach for Feature Selection, arXiv preprint arXiv: 2005.08642, (2020). https://arXiv.org/pdf/2005.08642v1
    [19] M. A. Elaziz, N. Nabil, A. A. Ewees, S. Lu, Automatic data clustering based on hybrid atom search optimization and Sine-Cosine algorithm, 2019 IEEE Congress on Evolutionary Computation (CEC), (2019), 2315–2322. https://doi.org/10.1109/CEC.2019.8790361 doi: 10.1109/CEC.2019.8790361
    [20] P. Sun, H. Liu, Y. Zhang, L. Tu, Q. Meng, An intensify atom search optimization for engineering design problems, Appl. Math. Model., 89 (2021), 837–859. https://doi.org/10.1016/j.apm.2020.07.052 doi: 10.1016/j.apm.2020.07.052
    [21] L. Xu, J. Chen, Y. Gao, Off-Grid DOA estimation based on sparse representation and rife algorithm, Microelectron J., 59 (2017), 193–201. https://doi.org/10.2528/PIERM17070404 doi: 10.2528/PIERM17070404
    [22] A. Peyman, Z. Kordrostami, K. Hassanli, Design of a MEMS bionic vector hydrophone with piezo-gated MOSFET readout, Prog. Electromagn Res. M., 98 (2020), 104748. https://doi.org/10.1016/j.mejo.2020.104748 doi: 10.1016/j.mejo.2020.104748
    [23] H. Song, M. Diao, T. Tang, J. Qin, Vector-Sensor Array DOA Estimation Based on Spatial Time-Frequency Distribution, 2018 Eighth International Conference on Instrumentation & Measurement, Computer, Communication and Control (IMCCC), (2020), 1351–1356. https://doi.org/10.1109/IMCCC.2018.00280 doi: 10.1109/IMCCC.2018.00280
    [24] M. Cao, X. Mao, L. Huang, Elevation, azimuth, and polarization estimation with nested electromagnetic vector-sensor arrays via tensor modeling, Eurasip J. Wirel. Comm., 2020 (2020), 153. https://doi.org/10.1186/s13638-020-01764-8 doi: 10.1186/s13638-020-01764-8
    [25] V. Baron, A. Finez, S. Bouley, F. Fayet, J. I. Mars, B. Nicolas, Hydrophone array optimization, conception, and validation for localization of acoustic sources in deep-Sea mining, IEEE J. Oceanic. Eng., 46 (2021), 555–563. https://doi.org/10.1109/JOE.2020.3004018 doi: 10.1109/JOE.2020.3004018
    [26] W. Wand, Q. Zhang, W. Shi, J. Shi, X. Wang, Iterative sparse covariance matrix fitting direction of arrival estimation method based on vector hydrophone array, Xibei Gongye Daxue Xuebao, 38 (2020), 14–23. https://doi.org/10.1051/jnwpu/20203810014 doi: 10.1051/jnwpu/20203810014
    [27] K. Aghababaiyan, R. G.Zefreh, V. Shah-Mansouri, 3D-OMP and 3D-FOMP algorithms for DOA estimation, Phys. Commun-amst, 31 (2018), 87–95. https://doi.org/10.1016/j.phycom.2018.10.005 doi: 10.1016/j.phycom.2018.10.005
    [28] K. Aghababaiyan, V. Shah-Mansouri, B. Maham, High-precision OMP-based direction of arrival estimation scheme for hybrid non-uniform array, IEEE Commun. Lett., 24 (2019), 354–357. https://doi.org/10.1109/LCOMM.2019.2952595 doi: 10.1109/LCOMM.2019.2952595
    [29] A. Nehorai, E. Paldi, , Acoustic vector-sensor array processing, IEEE T. Signal. Proces., 42 (1994), 2481–2491. https://doi.org/10.1109/ACSSC.1992.269285 doi: 10.1109/ACSSC.1992.269285
    [30] K. T. Wong, M. D. Zoltowski, Root-MUSIC-based azimuth-elevation angle-of-arrival estimation with uniformly spaced but arbitrarily oriented velocity hydrophones, IEEE T. Signal. Proces., 47 (1999), 3250–3260. https://doi.org/10.1109/78.806070 doi: 10.1109/78.806070
    [31] K. T. Wong, M. D. Zoltowski, Uni-vector-sensor ESPRIT for multisource azimuth, elevation, and polarization estimatio, IEEE T. Antenn. Propag., 45 (1997), 1467–1474. https://doi.org/10.1109/8.633852 doi: 10.1109/8.633852
    [32] I. Ziskind, M. Wax, Maximum likelihood localization of multiple sources by alternating projection, IEEE Trans. Acoust. Speech Signal Process, 36 (1988), 1553–1560. https://doi.org/10.1109/29.7543 doi: 10.1109/29.7543
    [33] M. Feder, E. Weinstein, Parameter estimation of superimposed signals using the EM algorithm, IEEE Trans. Acoust. Speech Signal Process, 36 (1988), 477–489. https://doi.org/10.1109/29.1552 doi: 10.1109/29.1552
    [34] Y. Zheng, L. Liu, X. Yang, SPICE-ML Algorithm for Direction-of-Arrival Estimation, Sensors, 20 (2019), 119. https://doi.org/10.3390/s20010119 doi: 10.3390/s20010119
    [35] Y. Hu, J. Lu, X. Qiu, Direction of arrival estimation of multiple acoustic sources using a maximum likelihood method in the spherical harmonic domain, Appl. Acoust., 135 (2018), 85–90. https://doi.org/10.1016/j.apacoust.2018.02.005 doi: 10.1016/j.apacoust.2018.02.005
    [36] J. W. Paik, K. H. Lee, J. H. Lee, Asymptotic performance analysis of maximum likelihood algorithm for direction-of-arrival estimation: Explicit expression of estimation error and mean square error, Applied Sciences, 10 (2020), 2415. https://doi.org/10.3390/app10072415 doi: 10.3390/app10072415
    [37] S. Jesus, Efficient ML direction of arrival estimation assuming unknown sensor noise powers, arXiv preprint arXiv: 2001.01935, (2020), https: //arXiv: 2001.01935
    [38] Y. Yoon, Y. H. Kim, Optimizing taxon addition order and branch lengths in the construction of phylogenetic trees using maximum likelihood, J. Bioinf. Comput. Biol., 18 (2020), 837–859. https://doi.org/10.1142/S0219720020500031 doi: 10.1142/S0219720020500031
    [39] P. Vishnu, C. S. Ramalingam, An improved LSF-based algorithm for sinusoidal frequency estimation that achieves maximum likelihood performance, 2020 International Conference on Signal Processing and Communications (SPCOM), (2020), 1–5. https://doi.org/10.1109/SPCOM50965.2020.9179546
    [40] M. Li, Y. Lu, Genetic algorithm based maximum likelihood DOA estimation, RADAR 2002, 2002 (2002), 502–506. https://doi.org/10.1109/RADAR.2002.1174766 doi: 10.1109/RADAR.2002.1174766
    [41] A. Sharma, S. Mathur, Comparative analysis of ML-PSO DOA estimation with conventional techniques in varied multipath channel environment, Wireless Pers. Commun., 100 (2018), 803–817. https://doi.org/10.1007/s11277-018-5350-0 doi: 10.1007/s11277-018-5350-0
    [42] P. Wang, Y. Kong, X. He, M. Zhang, X. Tan, An improved squirrel search algorithm for maximum likelihood DOA estimation and application for MEMS vector hydrophone array, IEEE Access, 7 (2019), 118343–118358. https://doi.org/10.1109/ACCESS.2019.2936823 doi: 10.1109/ACCESS.2019.2936823
    [43] L. Cai, H. Tian, H. Chen, J. Hu, A random maximum likelihood algorithm based on limited PSO initial space, Computer Modernization, 282 (2019), 60–65. https://doi.org/10.3969/j.issn.1006-2475.2019.02.011 doi: 10.3969/j.issn.1006-2475.2019.02.011
  • This article has been cited by:

    1. Ateq Alsaadi, Mieczysław Cichoń, Mohamed M. A. Metwali, Integrable Solutions for Gripenberg-Type Equations with m-Product of Fractional Operators and Applications to Initial Value Problems, 2022, 10, 2227-7390, 1172, 10.3390/math10071172
    2. Bashir Ahmad, Manal Alnahdi, Sotiris K. Ntouyas, Existence Results for a Differential Equation Involving the Right Caputo Fractional Derivative and Mixed Nonlinearities with Nonlocal Closed Boundary Conditions, 2023, 7, 2504-3110, 129, 10.3390/fractalfract7020129
    3. Bashir Ahmad, Manal Alnahdi, Sotiris K. Ntouyas, Ahmed Alsaedi, On a mixed nonlinear boundary value problem with the right Caputo fractional derivative and multipoint closed boundary conditions, 2023, 8, 2473-6988, 11709, 10.3934/math.2023593
    4. Mehran Ghaderi, Shahram Rezapour, On an m-dimensional system of quantum inclusions by a new computational approach and heatmap, 2024, 2024, 1029-242X, 10.1186/s13660-024-03125-1
    5. Ahmed Alsaedi, Manal Alnahdi, Bashir Ahmad, Sotiris K. Ntouyas, On a nonlinear coupled Caputo-type fractional differential system with coupled closed boundary conditions, 2023, 8, 2473-6988, 17981, 10.3934/math.2023914
    6. AHMED ALSAEDI, HANA AL-HUTAMI, BASHIR AHMAD, INVESTIGATION OF A NONLINEAR MULTI-TERM IMPULSIVE ANTI-PERIODIC BOUNDARY VALUE PROBLEM OF FRACTIONAL q-INTEGRO-DIFFERENCE EQUATIONS, 2023, 31, 0218-348X, 10.1142/S0218348X23401916
    7. Ahmed Alsaedi, Bashir Ahmad, Hana Al-Hutami, Nonlinear Multi-term Impulsive Fractional q-Difference Equations with Closed Boundary Conditions, 2024, 23, 1575-5460, 10.1007/s12346-023-00934-5
    8. Ravi P. Agarwal, Bashir Ahmad, Hana Al-Hutami, Ahmed Alsaedi, Existence results for nonlinear multi-term impulsive fractional q-integro-difference equations with nonlocal boundary conditions, 2023, 8, 2473-6988, 19313, 10.3934/math.2023985
    9. Bashir Ahmad, Manal Alnahdi, Sotiris K. Ntouyas, Ahmed Alsaedi, On a Mixed Nonlinear Fractional Boundary Value Problem with a New Class of Closed Integral Boundary Conditions, 2023, 22, 1575-5460, 10.1007/s12346-023-00781-4
    10. Madeaha Alghanmi, Ravi P. Agarwal, Bashir Ahmad, Existence of Solutions for a Coupled System of Nonlinear Implicit Differential Equations Involving ϱ
    -Fractional Derivative with Anti Periodic Boundary Conditions, 2024, 23, 1575-5460, 10.1007/s12346-023-00861-5
    11. Bashir Ahmad, Muhammed Aldhuain, Ahmed Alsaedi, Existence Results for a Right-Caputo Type Fractional Differential Equation with Mixed Nonlinearities and Nonlocal Multipoint Sub-strips Type Closed Boundary Conditions, 2024, 45, 1995-0802, 6457, 10.1134/S1995080224606969
    12. Ahmed Alsaedi, Hafed A. Saeed, Hamed Alsulami, Existence and stability of solutions for a nonlocal multi-point and multi-strip coupled boundary value problem of nonlinear fractional Langevin equations, 2025, 15, 1664-3607, 10.1142/S1664360724500140
  • Reader Comments
  • © 2022 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(2857) PDF downloads(117) Cited by(6)

Figures and Tables

Figures(12)  /  Tables(7)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog