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Dynamical behavior of fractionalized simply supported beam: An application of fractional operators to Bernoulli-Euler theory

  • Kashif Ali Abro EMAIL logo , Abdon Atangana and Ali Raza Khoso
Published/Copyright: October 8, 2021
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Nonlinear Engineering
From the journal Nonlinear Engineering Volume 10 Issue 1

Abstract

The complex structures usually depend upon unconstrained and constrained simply supported beams because the passive damping is applied to control vibrations or dissipate acoustic energies involved in aerospace and automotive industries. This manuscript aims to present an analytic study of a simply supported beam based on the modern fractional approaches namely Caputo-Fabrizio and Atanagna-Baleanu fractional differential operators. The governing equation of motion is fractionalized for knowing the vivid effects of principal parametric resonances. The powerful techniques of Laplace and Fourier sine transforms are invoked for investigating the exact solutions with fractional and non-fractional approaches. The analytic solutions are presented in terms of elementary as well as special functions and depicted for graphical illustration based on embedded parameters. Finally, effects of the amplitude of vibrations and the natural frequency are discussed based on the sensitivities of dynamic characteristics of simply supported beam.

Keywords: simply supported beam; integral transform; analytic solutions; fractional approaches; rheological analysis

1 Introduction

A beam is a load-bearing, standard and rigid element which is frequently utilized in structural engineering; this is because they have several applications, ranging from complex structure to simple structure for resisting vertical loads, bending moments and shear forces. Engineers use the different concepts of beams like simply supported beam, cantilever beam, uniformly distributed beam, continuously supported beam, fixed beam and many others in mechanical analyses for capturing the realistic simplicity of distribution of forces [1, 2, 3, 4, 5, 6, 7]. Zhu et al. [8] presented an analytical study of viscoelastic Timoshenko beam based on the fractional derivative in which three-dimensional fractional derivative constitutive relation has been traced on the basis of quasi-static behavior of the viscoelastic Timoshenko beam with step loading. Here they focused the dynamical response of deflection subjected to a periodic excitation. Hedrih [9] analyzed the vibrations of the beam based on nonhomogeneous continuously creeping material subject to the modulus of elasticity. Here the mathematical modeling has been performed on the basis of partial differential equation through constitutive relation of each layer. The fractional techniques were invoked for knowing the rotation of inertia for different boundary conditions. Freundlich [10] observed the simply supported beam for vibration analysis at steady-state position of beam. They modeled the fractional differential equations for knowing the amplitude-frequency characteristics based on the memory effects from Bernoulli-Euler beam model. They compared simply supported beam for vibration analysis for integer order derivative with fractional order derivative. Di-paola et al. [11] investigated the dynamic and quasi-static loads for the fractional viscoelastic Euler-Bernoulli beam in which axial strain and axial stress was focused. They traced out the fractional analytical solutions subject to mechanical boundary conditions for investigating gradient of curvature, shearing force, curvature and bending moment. Martin [12] observed an interested study on simply supported beam on the basis of uniformly distributed load with fractional and non-fractional approaches. The Laplace transform was invoked on classical and fractional Zener model for knowing the rheological influences and dynamic responses of the structure. Zhu and Chung [13] worked on simply supported beam with spinning and axially moving motion for checking the stability and vibration in which their main focus was to investigate the natural frequencies. They concluded on the basis of the rotary inertia that computed dynamic responses and natural frequencies are more reliable than the previous equations. Stepa et al. [14] examined the study of fractional viscoelastic beam by invoking the Galerkin method for finding approximate solutions of system of coupled fractional order differential equations. They emphasized the concentrated masses and base excitation by the comparison of classical and fractional order model. The numerical study of Euler-Bernoulli beams based on the fractional constitutive equations of viscoelastic beam has been analyzed by Yu et al. [15]. Here, Quasi-Legendre polynomial in the time domain has been invoked on the constitutive equation of the beam for transferring into matrix equation and then discretized and solved via numerical solutions for finding the displacements under different external loads. Qin et al. [16] examined the comparative analysis of numerical and analytical approaches for the simply supported horizontally composite curved I-beam. They found numerical solutions by employing finite element method and analytical solution by using trigonometric series for knowing the accuracy of deflection. In this continuity, debuting modern fractional differential operators have become a burning topic due the influence of memory effects during the deformation of certain types of beams. The modern fractional derivatives vary from singular to non-singular kernel and local to nonlocal kernel. The kernel depends upon the domain of definition involved in the modern fractional derivatives. The well-known modern fractional derivatives are (Atangana–Baleanu fractional derivative, Caputo–Fabrizio fractional derivative, Riemann–Liouville fractional derivative, Caputo fractional derivative, Hadamard fractional derivative, Riesz fractional derivative, Weyl fractional derivative, Hilfer fractional derivative and Erdérlyi–Kober fractional derivative) and few others [17, 18, 19, 20, 36, 37, 38, 39, 40, 41, 42, 43]. Additionally, the fractionalized mathematical models [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58] with different mathematical techniques have been studied in different perspectives [59, 60, 61, 62, 63, 64, 65, 66, 67, 68] and varying conditions. In brevity, our aim is to present an analytic study of a simply supported beam based on the modern fractional approaches namely Caputo-Fabrizio and Atanagna-Baleanu fractional differential operators. The governing equation of motion is fractionalized for knowing the vivid effects of principal parametric resonances. The powerful techniques of Laplace and Fourier sine transforms are invoked for investigating the exact solutions with fractional and non-fractional approaches. The analytic solutions are presented in terms of elementary as well as special functions and depicted for graphical illustration based on embedded parameters. Finally, effects of the amplitude of vibrations and the natural frequency are discussed based on the sensitivities of dynamic characteristics of simply supported beam.

2 Fractional modeling of simply supported beam with theoretical background

Different levels of accuracy have lead several beam theories due to various assumptions in which Euler-Bernoulli beam theory one of the most useful and simplest. This is because, Euler-Bernoulli beam theory arises from a combination of four distinct terminologies namely the force resultant, kinematic, equilibrium and constitutive, equations. An idealized problem of a long beam subjected to two bending moments which are usually constant properties in structures. The cross-section of the beam is assumed to be symmetric and bending takes place in that plane of symmetry. This type of loading is often referred to as pure bending as shown in Fig. 1 which reflects before and after deformation:

Figure 1 Structure of beam with before and after deformation
Figure 1

Structure of beam with before and after deformation

By neglecting shear deformation and rotary inertia, the equation of motion of the beam is derived under the assumptions of the Bernoulli-Euler theory, for which the governing equation of beam is described along with the initial and boundary conditions as [10]:

(1) ∂2u(y,t)∂y2(1+bddt)=−a2∂2u ∂t2(y,t)

(2) u(y,0)=u′(y,0)=0,u(0,t)=tp,

Here, the letting parameters for equation (1) are b = μE−1J−1 and = AρE−1J−1. The passive parameters for Eq. (1) are defined as μ is damping parameter, Young modulus of the beam material is represented by E, J refers moment of inertia of the beam cross-section, the cross-section area of the beam is measured by A, ρ denotes material mass density of the beam, t is a time, y presents longitudinal coordinate and u(y, t) is transversal displacement of a beam. There is no denying fact that several deficiencies have been admitted by some fractional derivatives consisting kernels of singular nature. The main significance of kernels is to explain the entire memory effects. Different fractional derivatives have different limitations based on the nature of kernels. All of them, the fractional derivative of Caputo-Fabrizio is based on non-singular exponential kernel and Atangana-Baleanu fractional derivative is based on non-singular, non-local Mittag-Leffler kernel. In order to highlight the hidden phenomenon of both fractional operators, the governing equation of beam is developed in terms of fractional differentiations through which explanation of the full effect of the memory can be traced out. The fractionalized governing equation of beam is described as:

(3) ∂2u(y,t)∂y2(1+bdαdtα)=−a2∂2αu(y,t)∂t2α

(4) ∂2u(y,t)∂y2(1+bdβdtβ)=−a2∂2βu(y,t)∂t2β

Where, dαdtα and dβdtβ are the fractional operators namely Caputo-Fabrizio fractional operator [19, 21, 22, 23, 24, 25] and Atangana-Baleanu fractional operator [20, 26, 27, 28, 29, 30] respectively. The fractionalized governing equations of beam (3–4) are capable to model the intermediate material properties, for instance deflection and stress at any point, modulus of elasticity, moment of inertia of the cross-section, moment of resistance and few others. The modeling of fractionalized governing equations of beam (3–4) is based on Mittage-Leffler function as a kernel and exponential function as a kernel respectively and also proposed by [19, 20]:

(5) Dtαu(y,t)=∫0τ(1−α)−1exp(−α(z−t)1−α)u′(y,t)dt.0≤α≤1,

(6) Dtβu(y,t)=∫0τ(1−β)−1Eβ(−β(z−t)β1−β)u'(y,t)dt,0≤β≤1.

While, the normalization functions for equations (5–6) are M (α) = M (β) = M (0) = M (1) = 1.

3 Methodology for analytic solutions

In this section, fractional order linear differential equations of beam (3–4) under imposed conditions (2) are investigated by employing Fourier sine and Laplace transforms. The basic principle of these methods is to analyze the problem for spatial and time domain variable. In brief, definitions of Laplace and Fourier sine transforms are as Fs{f(x)}=2π∫0∞sin(kx)f(x)dx=Fs(k) , L{f(t)}=∫0∞e−qt f(t)dt=F(q) respectively.

3.1 Caputo-Fabrizio fractional solution for simply supported beam

Applying the Fourier sine transform [31, 32] and appendix (A1–A2) on the governing fractional differential equation of simply supported beam (3), we obtain

(7) −ξ2us(ξ,t)+ξ2π  tp−ξ2bdαdtαus(ξ,t)+ξb2πtpdαdtα=−a2∂2αus(ξ,t)∂t2α,

By imposing conditions (21,2,3) and Laplace transform [33, 34] on Eqs. (7). Here, ūs (ξ, τ) is an image of Laplace transform of us (ξ, t), we have,

(8) u¯s(ξ,τ)=2π ξp!τp+1(M1τ2+M2τ+M3M4τ2+M5τ+M6),

Where, the letting notations from (M0 − M6) are defined in the equation (9) as

(9) M0=(1−α)−1,M1=1+bM0,M2=2αM0+bαM02,M3=M02α2,  M4=ξ2+M0ξ2b−M02a2,M5=2M0ξ2α+bξ2M02α,M6=M02ξ2α2

In order to justify the imposed conditions, we write Eq.(8) into suitable format, we arrive at

(10) u¯s(ξ,τ)=2π ξp!τp+1+M7M42π ξp!τp(τ2+M10τ+M11τ(τ−M14)(τ−M15)),

The simplified form of Eq. (10) is obtained with following expressions defined in Eq. (11),

(11) M7=M1−M4,M8=M2−M5,M9=M3−M6,M10=M8M7,M11=M9M7,M12=M5M4,M13=M6M4,(τ2+M12τ+M13)=(τ−M14)(τ−M15) .

Now, inverting Eq.(10) by means of Fourier sine transform and using appendix (A3), we get Eq.(10) in spatial variable format as

(12) u¯(y,τ)=p!τp+1+2πM7M4p!∫0∞ξ2sin(yξ)1τp(τ2+M10τ+M11τ(τ−M14)(τ−M15))dξ,

For converting Eq.(12) from frequency domain to time domain, we invert Eq. (12) via Laplace transform and invoking appendix (A4–A6), we have final expression in terms of product of convolution as

(13) u(y,t)=tp+2πM7M4p!∫0∞ξ2sin(yξ)∫0t(t−z)p×{M11M14M15+(M142+M10M14+M11M14(M14+M15))×exp(M14t)+(M152+M10M15+M11M15(M15+M14))exp(M15t)}dξ dz

3.2 Atangana-Baleanu fractional solution for simply supported beam

Applying the Fourier sine transform and appendix (A1–A2) on the governing fractional differential equation of simply supported beam (4), we obtain

(14) −ξ2us(ξ,t)+ξ2π  tp−ξ2bdβdtβus(ξ,t)+ξb2πtpdβdtβ=−a2∂2βus(ξ,t)∂t2β,

By imposing conditions (21,2,3) and Laplace transform on Eqs. (14). Here, ūs (ξ, τ) is an image of Laplace transform of us (ξ, t), we have,

(15) u¯s(ξ,τ)=2π ξp!τp+1(N1τ2β+N2τβ+N3N4τ2β+N5τβ+N6),

Where, the letting notations from (N0 − N6) are defined in the equation (9) as

(16) N0=(1−β)−1,N1=1+bN0,N2=2βN0+bβN02,N3=N02β2,  N4=ξ2+N0ξ2b−N02a2,N5=2N0ξ2β+bξ2N02β,N6=N02ξ2β2 ,

In order to justify the imposed conditions, we write Eq.(15) into suitable format, we arrive at

(17) u¯s(ξ,τ)=2π ξp!τp+1+N7N42π ξp!τp+1(τ2β+N10τβ+N11τ2β+N12τβ+N13),

The simplified form of Eq. (10) is obtained with following expressions defined in Eq. (17),

(18) N7=N1−N4,N8=N2−N5,N9=N3−N6,N10=N8N7,N11=N9N7,N12=N5N4,N13=N6N4,               .

The concepts of infinite series [35] are utilized on Eq.(17) for separating the transformed variables as

(19) u¯s(ξ,τ)=2πξ p!τp+1+2π ξp!τp+1N7N4∑q0=0∞(−N13)−q0∑q1=0∞(−N12)q1Γ(q0+1)q1!Γ(q0−q1+1)×(τ2β+N10τβ+N11)τ2q0β−βq1,

Inverting Eq.(17) by means of Fourier sine transform for converting the transformed variable into spatial variable, we have

(20) u¯(y,τ)=p!τp+1+2N7πN4∫0∞ξsin(yξ)∑q0=0∞(−N13)−q0∑q1=0∞(−N12)q1Γ(q0+1)q1!Γ(q0−q1+1)τ2q0β−βq1−2β−p−1 dξ+N7N102−1πN4∫0∞ξsin(yξ)∑q0=0∞(−N13)−q0∑q1=0∞(−N12)q1Γ(q0+1)q1!Γ(q0−q1+1)τ2q0β−βq1−β−p−1dξ+2πN7N11N4∫0∞ξsin(yξ)∑q0=0∞(−N13)−q0∑q1=0∞(−N12)q1Γ(q0+1)q1!Γ(q0−q1+1)τ2q0β−βq1−p−1dξ,

Applying inverse Laplace transform on Eq. (18), we find final solution as

(21) u(y,t)=tp+2N7πN4∫0∞ξsin(yξ)∑q0=0∞(−N13)−q0∑q1=0∞(−N12)q1Γ(q0+1)t2q0β−βq1−2β−pq1!Γ(q0−q1+1)Γ(2q0β−βq1−2β−p−1)dξ+N7N102−1πN4∫0∞ξsin(yξ)∑q0=0∞(−N13)−q0∑q1=0∞(−N12)q1Γ(q0+1)t2q0β−βq1−β−pq1!Γ(q0−q1+1)Γ(2q0β−βq1−β−p−1)dξ+2πN7N11N4∫0∞ξsin?yξ)∑q0=0∞(−N13)−q0∑q1=0∞(−N12)q1Γ(q0+1)t2q0β−βq1−pq1!Γ(q0−q1+1)Γ(2q0β−βq1−p−1)dξ.

is the analytical solution of transversal displacement of a beam via Atangana-Baleanu fractional operator. Furthermore, the analytical solutions (13) and (18) obtained through both fractional operators can be retrieved for classical solution by letting α = β = 1 in Eqs. (13) and (18).

4 Results with parametric conclusion

In this manuscript, an analytic study of a simply supported beam based on the modern fractional approaches namely Caputo-Fabrizio and Atanagna-Baleanu fractional differential operators is investigated for knowing the effective role of non-singular and non-local kernels involved in fractional differentiations. Due to this fact, governing equation of simply supported beam is fractionalized and then their solutions have been investigated by means of Laplace and Fourier sine transforms. Such solutions are presented in terms of elementary as well as special functions and depicted for graphical illustration based on embedded parameters. The main findings regarding amplitude of vibrations and the natural frequency are discussed as per following results:

4.1 Role of damping parameter in fractional solutions

In order to characterize the energy dissipation through damping, the dynamic response of simply supported beam is analyzed in Figure 2. It is also well known fact that the damping ratio of the beam changes with moving load location, hence we depicted Figure 2 for effective role of damping on displacement by choosing physical values as μ = 0.05, 0.07, 0.09. It is clear from Figure 2 that displacement obtained through Caputo-Fabrizio fractional operator has rapid cycling in comparison with displacement obtained through Caputo-Fabrizio fractional operator. Physically, both fractional operators decay in amplitude of motion of simply supported beam. One can observe that energy dissipation through fractional mechanisms cannot be ceased for the vibratory analysis of simply supported beam. On a view of criticism, classical models of simply supported beam are not capable of critical damping which provides the quickest approach to zero amplitude.

Figure 2 Graph of displacement via CF and AB fractional operator with three different variants of damping parameter μ
Figure 2

Graph of displacement via CF and AB fractional operator with three different variants of damping parameter μ

4.2 Role of moment of inertia in fractional solutions

The ability to resist bending of simply supported beam is usually measured by the geometric property so called moment of inertia which depends up on a reference axis. We depicted the measurement of resistance of simply supported beam in Figure 3 by choosing larger specific values of moment of inertia because more is the moment of inertia then more is the resistance offered by the simply supported beam to rotation. Here, the displacement obtained through both CF and AB fractional differentiations have opposite trends when the specific values of moment of inertia increase. Physically, displacement obtained through both CF and AB fractional differentiations have measured the tendency to move back to a low energy state or rest reciprocally. Due to this obvious fact, CF and AB fractional differentiations give an idea of how much force is needed to maintain or alter the current state.

Figure 3 Graph of displacement via CF and AB fractional operator with three different variants of moment of inertia J.
Figure 3

Graph of displacement via CF and AB fractional operator with three different variants of moment of inertia J.

4.3 Role of young modulus of elasticity in fractional solutions

For measuring the fractional stiffness whether tensile or compressive deformation in simply supported beam, we elucidate the structural implant of simply supported beam for deformation by the fractionally obtained displacement. Figure 4 is prepared for measuring the resistance of a simply supported beam to its elasticity in which the specific values are taken from 0–40 as a concrete. It is noted that displacement obtained via CF fractional operator has larger number of oscillations in comparison AB fractional operator. As the temperature increases the elasticity of a simply supported beam decreases. This may be due to the fact memory effects of exponential and Mittage-Leffler kernels involved in both types of fractional operators.

Figure 4 Graph of displacement via CF and AB fractional operator with three different variants of young modulus of elasticity E.
Figure 4

Graph of displacement via CF and AB fractional operator with three different variants of young modulus of elasticity E.

4.4 Role of comparative analysis of fractional solutions at smaller and larger time

Figures 5–6 present the comparison of displacement investigated via CF fractional derivative and AB fractional derivative at two different times i–e (larger time) t = 10 s and (smaller time) t = 0.4 s with three variants of fractional parameters. Figure 5 is prepared for smaller time t = 0.4 s to compare the two different displacements in which solution obtained via CF fractional derivative could not collected full memory effects in comparison with AB fractional derivative. On the contrary, Figure 6 is depicted for larger time t = 10 s for the comparison of displacements. It is obvious from the oscillations depicted in Figure 6 that AB fractional derivative is more dominant in collections of entire memory effects. Physically, such results for displacements reflect the non-local spatial description and prevention of bending rigidities within the qualitative and quantitative behavior of simply supported beam for fractional calculus.

Figure 5 Graph of comparison of displacement via CF and AB fractional operator at smaller time t = 0.04 s.
Figure 5

Graph of comparison of displacement via CF and AB fractional operator at smaller time t = 0.04 s.

Figure 6 Graph of comparison of displacement via CF and AB fractional operator at larger time t = 10 s.
Figure 6

Graph of comparison of displacement via CF and AB fractional operator at larger time t = 10 s.

Acknowledgements

The author Kashif Ali Abro is highly thankful and grateful to Mehran university of Engineering and Technology, Jamshoro, Pakistan for generous support and facilities of this research work.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data Availability Statement: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

[1] Fung RF, Huang JS, Chen YC, Yao CM. Nonlinear dynamic analysis of the viscoelastic string with a harmonically varying transport speed. Comput Struc. 1998;66(6):777–84.10.1016/S0045-7949(98)00001-7Search in Google Scholar

[2] Rossikhin YA, Shitikova MV. Application of fractional calculus for analysis of nonlinear damped vibrations of suspension bridges. J Eng Mech. 1998;124(9):1029–36.10.1061/(ASCE)0733-9399(1998)124:9(1029)Search in Google Scholar

[3] Agrawal OP. Formulation of Euler-Lagrange equations for fractional variational problems. J Math Anal Appl. 2002;272(1):368–79.10.1016/S0022-247X(02)00180-4Search in Google Scholar

[4] Chen LQ, Zhao WJ, Zu JW. Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law. J Sound Vibrat. 2004;278:861–71.10.1016/j.jsv.2003.10.012Search in Google Scholar

[5] Galucio AC, Deu JF, Ohayon R. A fractional derivative viscoelastic model for hybrid active-passive damping treatments in time domain-application to sandwich beams. J Intell Mater Syst Struct. 2005;16(1):33–45.10.1177/1045389X05046685Search in Google Scholar

[6] Rossikhin YA, Shitikova MV. Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl Mech Rev. 2010;63(1):010801.10.1115/1.4000563Search in Google Scholar

[7] Kashif AA, Ahmet Y. Fractional treatment of vibration equation through modern analogy of fractional differentiations using integral transforms. Iran J Sci Technol Trans A Sci. 2019;43(5):1–8.10.1007/s40995-019-00687-4Search in Google Scholar

[8] Zhu Z, Li G, Cheng C. Quasi-static and dynamical analysis for viscoelastic Timoshenko beam with fractional derivative constitutive relation. Appl Math Mech. 2002;23(1):1–12.10.1007/BF02437724Search in Google Scholar

[9] Hedrih K. The transversal creeping vibrations of a fractional derivative order constitutive relation of nonhomogeneous beam. Math Probl Eng. 2006;2006:1–18.10.1155/MPE/2006/46236Search in Google Scholar

[10] Freundlich J. Vibrations of a simply supported beam with a fractional viscoelastic material model-supports movement excitation. Shock Vib. 2013;20(6):1103–12.10.1063/1.4765517Search in Google Scholar

[11] Dipaola M, Heuer R, Pirrotta A. Fractional viscoelastic Euler-Bernoulli beam. Int J Solids Struct. 2013;50(22–23):3505–10.10.1016/j.ijsolstr.2013.06.010Search in Google Scholar

[12] Martin O. Nonlinear dynamic analysis of viscoelastic beams using a fractional rheological model. Appl Math Model. 2017;43:351–9.10.1016/j.apm.2016.11.033Search in Google Scholar

[13] Zhu K, Chung J. Vibration and stability analysis of a simply-supported Rayleigh beam with spinning and axial motions. Appl Math Model. 2019;66:362–82.10.1016/j.apm.2018.09.021Search in Google Scholar

[14] Stepa P, Milan C, Danilo K, Marina M. A novel approach for vibration analysis of fractional viscoelastic beams with attached masses and base excitation. J Sound Vibrat. 2019;463:114955.10.1016/j.jsv.2019.114955Search in Google Scholar

[15] Yu C, Zhang J, Chen Y, Feng Y, Yang A. A numerical method for solving fractional-order viscoelastic Euler–Bernoulli beams. Chaos Solitons Fractals. 2019;128:275–9.10.1016/j.chaos.2019.07.035Search in Google Scholar

[16] Qin X, Liu H, Wu C, Gu Z. A trigonometric analytical solution of simply supported horizontally curved composite I-beam considering tangential slips. Math Probl Eng. 2016;2016:2465025.10.1155/2016/2465025Search in Google Scholar

[17] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier; 2006.Search in Google Scholar

[18] Hilfer R. Applications of Fractional Calculus in Physics. New York: World Scientific Publishing; 2000. https://doi.org/10.1142/3779.10.1142/3779Search in Google Scholar

[19] Caputo M, Fabrizio MA. New definition of fractional derivative without singular kernel. Prog. Fract. Diff. Appl. 2015;1:73–85.Search in Google Scholar

[20] Atangana A, Baleanu D. New fractional derivative with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm Sci. 2016;20(2):763–9.10.2298/TSCI160111018ASearch in Google Scholar

[21] Kashif AA, Muhammad NM, Gomez-Aguilar JF. Functional application of Fourier sine transform in radiating gas flow with nonsingular and nonlocal kernel. J Braz Soc Mech Sci Eng. 2019;41(10):400.10.1007/s40430-019-1899-0Search in Google Scholar

[22] Kashif AA, Ilyas K, José FG. Thermal effects of magnetohydrodynamic micropolar fluid embedded in porous medium with Fourier sine transform technique. J Braz Soc Mech Sci Eng. 2019;41(4):174–81.10.1007/s40430-019-1671-5Search in Google Scholar

[23] Gomez-Aguilar JF, Kashif AA, Olusola K, Ahmet Y. Chaos in a calcium oscillation model via Atangana-Baleanu operator with strong memory. The European Physical Journal Plus. Eur Phys J Plus. 2019;134(4):140.10.1140/epjp/i2019-12550-1Search in Google Scholar

[24] Abro KA, Ilyas K, Nisar KS. Novel technique of Atangana and Baleanu for heat dissipation in transmission line of electrical circuit. Chaos Solitons Fractals. 2019;129:40–5.10.1016/j.chaos.2019.08.001Search in Google Scholar

[25] Abro KA, Gomez-Aguilar JF. A comparison of heat and mass transfer on a Walter's-B fluid via Caputo-Fabrizio versus Atangana-Baleanu fractional derivatives using the Fox-H function. Eur Phys J Plus. 2019;134(3):101.10.1140/epjp/i2019-12507-4Search in Google Scholar

[26] Abro KA, Mohammad MR, Ilyas K, Irfan AA, Asifa T. Analysis of Stokes’ Second Problem for Nanofluids Using Modern Fractional Derivatives. J Nanofluids. 2018;7:738–47.10.1166/jon.2018.1486Search in Google Scholar

[27] Kashif AA, Ali AM, Anwer AM. Functionality of Circuit via Modern Fractional Differentiations. Analog Integr Circuits Signal Process. 2019;99(1):11–21.10.1007/s10470-018-1371-6Search in Google Scholar

[28] Ambreen S, Kashif AA, Muhammad AS. Thermodynamics of magnetohydrodynamic Brinkman fluid in porous medium: applications to thermal science. J Therm Anal Calorim. 2018. https://doi.org/10.1007/s10973-018-7897-0.10.1007/s10973-018-7897-0Search in Google Scholar

[29] Kashif AA, Ilyas K. Effects of CNTs on magnetohydrodynamic flow of methanol based nanofluids via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives. Therm Sci. 2019;23 2 Part B:883–98.10.2298/TSCI180116165ASearch in Google Scholar

[30] Ali AK, Abdon A. Role of Non-integer and Integer Order Differentiations on the Relaxation Phenomena of Viscoelastic Fluid. Phys Scr. 2020;95(3):035228.10.1088/1402-4896/ab560cSearch in Google Scholar

[31] Kashif AA, Ilyas K, José Francisco GA. Thermal effects of magnetohydrodynamic micropolar fluid embedded in porous medium with Fourier sine transform technique. J Braz Soc Mech Sci Eng. 2019;41(4):174–81.10.1007/s40430-019-1671-5Search in Google Scholar

[32] Abro KA. A Fractional and Analytic Investigation of Thermo-Diffusion Process on Free Convection Flow: An Application to Surface Modification Technology. Eur Phys J Plus. 2019;135(1):31–45.10.1140/epjp/s13360-019-00046-7Search in Google Scholar

[33] Kashif AA, Abdon A. Porous effects on the fractional modeling of magnetohydrodynamic pulsatile flow: an analytic study via strong kernels. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-10027-z.10.1007/s10973-020-10027-zSearch in Google Scholar

[34] Kashif AA, Anwar AM, Muhammad AU. A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives. Eur Phys J Plus. 2018;133(3):113.10.1140/epjp/i2018-11953-8Search in Google Scholar

[35] Bhojraj L, Kashif AA, Abdul WS. Thermodynamical analysis of heat transfer of gravitydriven fluid flow via fractional treatment: an analytical study. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09429-w.10.1007/s10973-020-09429-wSearch in Google Scholar

[36] Asıf Y, Hülya D, Kashif AA, Dogan K. Role of Gilson-Pickering equation for the different types of soliton solutions: A nonlinear analysis. Eur Phys J Plus. 2020;135(8):657.10.1140/epjp/s13360-020-00646-8Search in Google Scholar

[37] Aziz UA, Muhammad T, Kashif AA. Multiple soliton solutions with chiral nonlinear Schrödinger's equation in (2+1)-dimensions. Eur J Mech BFluids. 2020. https://doi.org/10.1016/j.euromechflu.2020.07.014.10.1016/j.euromechflu.2020.07.014Search in Google Scholar

[38] Kashif AA, Abdon A. Numerical study and chaotic analysis of meminductor and memcapacitor through fractal-fractional differential operator. Arab J Sci Eng. 2020. https://doi.org/10.1007/s13369-020-04780-4.10.1007/s13369-020-04780-4Search in Google Scholar

[39] Kashif AA, Atangana A. A comparative analysis of electromechanical model of piezoelectric actuator through Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Math Methods Appl Sci. 2020;43(17):1–11.10.1002/mma.6638Search in Google Scholar

[40] Abro KA, Jose FG. Role of Fourier sine transform on the dynamical model of tensioned carbon nanotubes with fractional operator. Math Methods Appl Sci. 2020:1–11.10.1002/mma.6655Search in Google Scholar

[41] Abro KA, Abdon A. Mathematical analysis of memristor through fractal-fractional differential operators: A numerical study. Math Methods Appl Sci. 2020;43(10):1–18.10.1002/mma.6378Search in Google Scholar

[42] Ali AK, Atangana A. A comparative study of convective fluid motion in rotating cavity via Atangana–Baleanu and Caputo–Fabrizio fractal–fractional differentiations. Eur Phys J Plus. 2020;135(2):226–42.10.1140/epjp/s13360-020-00136-xSearch in Google Scholar

[43] Kashif AA, Ambreen S, Atangana A. Thermal stratification of rotational second-grade fluid through fractional differential operators. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09312-8.10.1007/s10973-020-09312-8Search in Google Scholar

[44] Hasib K, Gomez-Aguilar JF, Aziz K, Tahir SK. Stability analysis for fractional order advection–reaction diffusion system. Physica A. 2019;521:737–51.10.1016/j.physa.2019.01.102Search in Google Scholar

[45] Aziz UA, Attia S, Kashif AA, Muhammad O, Takasar H. Dynamical Aspects of Smoking Model with Cravings to Smoke. Nonlinear Eng. 2021;10(1):91–108.10.1515/nleng-2021-0008Search in Google Scholar

[46] Gomez-Aguilar JF, Razo-Hernández R, Granados LD. A physical interpretation of fractional calculus in observables terms: analysis of the fractional time constant and the transitory response. Rev Mex Fis. 2014;60(1):32–8.Search in Google Scholar

[47] Aziz UA, Mashal A, Naeem U, Sohail N, Kashif AA. Thermal analysis of oblique stagnation point low with slip-page on second-order fluid. J Therm Anal Calorim. 2021. https://doi.org/10.1007/s10973-021-10760-z.10.1007/s10973-021-10760-zSearch in Google Scholar

[48] Gomez-Aguilar JF, Yepez MH, Escobar-Jiménez RF, Olivares-Peregrino VH, Reyes JM, Sosa IO. Series solution for the time-fractional coupled mKdV equation using the homotopy analysis method. Math Probl Eng. 2016;2016:1–8.10.1155/2016/7047126Search in Google Scholar

[49] Asıf Y, Hulya D, Kashif AA. Symbolic computation of Caudrey–Dodd–Gibbon equation subject to periodic trigonometric and hyperbolic symmetries. Eur Phys J Plus. 2021;136(4):358.10.1140/epjp/s13360-021-01350-xSearch in Google Scholar

[50] Gomez-Aguilar JF, Miranda-Hernández M, López-López MG, Alvarado-Martínez VM, Baleanu D. Modeling and simulation of the fractional space-time diffusion equation. Commun Nonlinear Sci Numer Simul. 2016;30(1–3):115–27.10.1016/j.cnsns.2015.06.014Search in Google Scholar

[51] Kashif AA, Abdon A, José FG. Role of biorder Atangana–Aguilar fractional differentiation on Drude model: an analytic study for distinct sources. Opt Quantum Electron. 2021;53(4):177.10.1007/s11082-021-02804-3Search in Google Scholar

[52] Morales-Delgado VF, Gomez-Aguilar JF, Khaled MS, Muhammad AK, Agarwal P. Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: a fractional calculus approach. Physica A. 2019;523:48–65.10.1016/j.physa.2019.02.018Search in Google Scholar

[53] Imran S, Nehad AS, Kashif AA. Thermography of ferromagnetic Walter's-B fluid through varying thermal stratification. S Afr J Chem Eng. 2021;36:118–26.10.1016/j.sajce.2020.12.004Search in Google Scholar

[54] Coronel-Escamilla A, Torres F, Gómez-Aguilar JF, Escobar-Jiménez RF, Guerrero-Ramírez GV. On the trajectory tracking control for an SCARA robot manipulator in a fractional model driven by induction motors with PSO tuning. Multibody Syst Dyn. 2018;43(3):257–77.10.1007/s11044-017-9586-3Search in Google Scholar

[55] Qasim A, Samia R, Aziz UA, Kashif AA. A mathematical model for thermography on viscous fluid based on damped thermal flux. Zeitschrift für Naturforschung A. 2021. https://doi.org/10.1515/zna-2020-0322.10.1515/zna-2020-0322Search in Google Scholar

[56] Rahimi-Gorji M, Van de Sande L, Debbaut C, Ghorbaniasl G, Braet H, Cosyns S, et al. Intraperitoneal aerosolized drug delivery: Technology, recent developments, and future outlook. Adv Drug Deliv Rev. 2020;160:105–14.10.1016/j.addr.2020.10.015Search in Google Scholar PubMed

[57] Kashif AA, Jose FG. Fractional modeling of fin on non-Fourier heat conduction via modern fractional differential operators. Arab J Sci Eng. 2021;46(3):2901–10.10.1007/s13369-020-05243-6Search in Google Scholar

[58] Van de Sande L, Rahimi-Gorji M, Giordano S, Davoli E, Matteo C, Detlefsen S, et al. Electrostatic Intraperitoneal Aerosol Delivery of Nanoparticles: Proof of Concept and Preclinical Validation. Adv Healthc Mater. 2020 Aug;9(16):e2000655.10.1002/adhm.202000655Search in Google Scholar PubMed

[59] Syed TS, Kashif AA, Sikandar A. Role of single slip assumption on the viscoelastic liquid subject to non-integer differentiable operators. Math Methods Appl Sci. 2021;44(7):1–16.10.1002/mma.7164Search in Google Scholar

[60] Mohammad RG, Charlotte D, Ghader G, Wouter W, Sarah C, Wim C. Electrostatic precipitation pressurized intraperitoneal aerosol chemotherapy (ePIPAC): finding the optimal electrical potential. Eur J Surg Oncol. 2021;47(2):e30.10.1016/j.ejso.2020.11.222Search in Google Scholar

[61] Arshad R, Aziz UA, Sajad H, Sami UK, Kashif AA. Effects of solid particles on fluid-particulate phase flow of non-Newtonian fluid through eccentric annuli having thin peristaltic walls. J Therm Anal Calorim. 2020: https://doi.org/10.1007/s10973-020-10447-x.10.1007/s10973-020-10447-xSearch in Google Scholar

[62] Mohammad RG, Helena B, Charlotte D, Ghader G, Sarah C, Wouter W, et al. Aerosolization of Thermosensitive Hydrogels using Pressurized Intraperitoneal Aerosol Chemotherapy (PIPAC). Ann Surgic Oncol SSO 2021: Int Conf Surgic Cancer Care. 2021;28:P178.Search in Google Scholar

[63] Kashif AA. Numerical study and chaotic oscillations for aerodynamic model of wind turbine via fractal and fractional differential operators. Numer Methods Partial Differential Eq; 2020. pp. 1–15.Search in Google Scholar

[64] Kashif AA, Sania Q, Abdon A. Mathematical and numerical optimality of non-singular fractional approaches on free and forced linear oscillator. Nonlinear Eng. 2020;9(1):449–56.10.1515/nleng-2020-0028Search in Google Scholar

[65] Mohammad RG, Charlotte D, Wouter W, Patrick S, Ghader G, Wim C. A numerical simulation of Pressurized intraperitoneal aerosol chemotherapy (PIPAC): validation part. 18th National Day Biomed Eng. Brussels, Belgium, 2019.Search in Google Scholar

[66] Mohammad RG, Leen VS, Charlotte D, Patrick S, Wouter W, Wim C. Computational fluid dynamics model of pressurized intraperitoneal aerosol chemotherapy: Gravity matters. Summer Biomech, Bioeng Biotransport Conf. Seven Springs, Pennsylvania, USA; 2019.Search in Google Scholar

[67] Takasar H, Aziz UA, Kashif AA, Muhammad O, Mehwish M. A mathematical and parametric study of epidemiological smoking model: a deterministic stability and optimality for solutions. Eur Phys J Plus. 2021;136(1):11.10.1140/epjp/s13360-020-00979-4Search in Google Scholar

[68] Imran QM, Kashif AA, Muhammad AS, Asif AS. Functional shape effects of nanoparticles on nanofluid suspended in ethylene glycol through Mittage-Leffler approach. Phys Scr. 2020;96(2):025005.10.1088/1402-4896/abd1b3Search in Google Scholar

Appendix

(A1) Fs{∂2u¯(y,q)∂t2}=−ξ2u¯s(ξ,q)+ξ2πu¯s(0,q),
(A2) Fs{∂u¯(y,q)∂t}=u¯s(ξ,q),
(A3) ∫0∞sin(yξ)ξdξ=π2,y>0,
(A4) L−1{(s2+M12s+M11)s(s−M14)(s−M15)}=M11M14M15+(M142+M12M14+M11M14(M14+M15))exp(M14t)+(M152+M12M15+M11M15(M15+M14))exp(M15t),
(A5) L−1{1sp}=tp−1Γ(p),
(A6) (f*g)(t)=∫0tf(t)g(t−u)du,
Received: 2021-05-01
Accepted: 2021-06-17
Published Online: 2021-10-08

© 2021 Kashif Ali Abro et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nleng-2021-0017
Keywords for this article
simply supported beam; integral transform; analytic solutions; fractional approaches; rheological analysis
Creative Commons
BY 4.0

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