Integral transform of some fractional approaches

5.1 Laplace Transform of theRiemann-Liouville Fractional Integral:

The Riemann-Liouville Fractional Integral is given by D^−αf(x) = ¹

Γ(α) x − t _a^{x α−1}f t dt . Its Laplace transform [3,4,12-14] is given by

ℒ 𝐷^−𝛼f(x) =_Γ(α)¹ ℒ x^∝−1 ℒ f x = s^−∝F(s), α > 0.

5.2 Laplace Transform of the Riemann-Liouville Fractional Derivative The Riemann-Liouville Fractional differential operator is given by

D^αf(x) = 1 Γ(n −α)

d dx

n

x − t ^n−α−1f t dt ,

x

a

The Laplace transform of Riemann-Liouville Fractional differential operator [3, 4,12-14] is given by

ℒ D^αf(x): s = 𝑠^𝛼𝐹 𝑠 − 𝑠^𝑘 𝐷 ^{𝛼−𝑘−1} 𝑓 0

𝑛−1

𝑘=0

= 𝑠^𝛼𝐹 𝑠 − 𝑠^{𝑛−𝑘−1} 𝐷^𝑘𝐼^𝑛−𝛼𝑓 0

𝑛−1

𝑘=0

5.3 Laplace Transform of the Caputo Fractional Derivative The Caputo Fractional Differential operator is given by

𝐷_𝑥^𝛼

𝑎𝑐 𝑓(𝑥) = 1

𝛤(𝑛 − 𝛼) 𝑥 − 𝑡 ^{𝑛−𝛼−1}𝑓 ^𝑛 (𝑡) 𝑑𝑡 ,

𝑥

𝑎

The Laplace transform of Caputo Fractional Differential operator [3, 4, 12-14] is given by ℒ 𝐷_𝑎^𝑐 _𝑥^𝛼𝑓(𝑥): s = 𝑠^𝛼𝐹 𝑠 − 𝑠^{𝛼−𝑘−1}

𝑛−1

𝑘=0

𝑓 ^𝑘 0 , 𝑛 − 1 < 𝛼 < 𝑛 . 5.4 Mellin Transform of theRiemann-Liouville Fractional Integral:

Mellin transform of Riemann-Liouville fractional integral operator [3, 4, 12-14]

is given by

ℳ{𝐷^−𝛼f(x)} = F(s) =^{Γ(1−s−α)}

Γ(1−s) F(s + α)

5.5 Mellin Transform of theRiemann-Liouville and Caputo Fractional Derivative:

Mellin transform of Riemann-Liouville and Caputo fractional derivative operator[3, 4, 12-14] is given by

ℳ{𝐷^𝛼f(x)} = F(s) =^Γ(1−s+α)

Γ(1−s) F(s - α) 6. Applications of Fractional Calculus:

6.1 Diffusion Equation:

Diffusion equation is an interesting application of fractional calculus. The study of thermal flux on a given surface is important due to its influence on material wear and performance. In addition once the thermal flux is known, the temperature can be obtained. The brake disks are treated as semi-infinite bodies and assumed to have a constant temperature distribution.

Agrawal (2004) [15] published a paper which analyzes the effectiveness of using fractional order derivatives to obtain the heat flux at a given point. Traditionally this was achieved by performing a transient analysis of two nearby points. His motive was the thermal study of disk brakes. The following diffusion equations govern the thermal distribution of the body.

∂T(x,t)

∂t = ^K

ρc

∂²T(x,t)

∂x²

Where T(x, t) is the temperature at point x and time t, K is the thermal conductivity, ρ the mass density and c the specific heat of the disk material. After non-dimensional zing and applying Laplace Transform it is converted in fractional partial differential equation given by

1 ∝

∂^1/2θ(x,t)

∂t^1/2 = ^∂θ(x,t)_∂x

Using this fractional equation heat flux Q(t) and temperature at that point obtained.

Lot of Mathematicians work on diffusion equation some of them, Kulish gives more information on thermal flux analysis with fractional order derivatives in his paper [16], LokenathDebnathalso gives more detailed applications of fractional calculus relating to the diffusion equation in an paper [17].

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6.2 𝑷𝑰^𝝀𝑫^𝝁 Controller

The concept of a fractional order P I^λD^μ is proposed In a paper written by Igor Podlubny in 1999 [2], where the integrator and differentiator are of a fractional order. A fractional order transfer function is provided as

𝐺_𝑐 𝑠 =^𝑈(𝑠)_𝐸(𝑠) = 𝐾_𝑃+ 𝐾_𝐼𝑠^−𝜆+ 𝐾_𝐷𝑠^𝜇, 𝜆, 𝜇> 0

Here 𝜆 is the order of the integrator, 𝜇 is the order of differentiator, 𝐺_𝑐 𝑠 is the transfer of controller, U(s) is the controller‟s output and E(s) is an error. If 𝜆 =1 and 𝜇 = 1 equation becomes traditional P I D controller equation If 𝜆 =1 and 𝜇 = 0, equation converts to a P I controller equation. If 𝜆 =0 and 𝜇 = 1 equation converts to a P D controller equation.

In the time domain, it becomes an open-loop systemisdescribed by 𝑎_𝑘𝒟^𝛽𝑘𝑦 𝑡

𝑛

𝑘=0

= 𝐾_𝑃𝑤 𝑡 + 𝐾_𝐼𝒟^−𝜆𝑤 𝑡 + 𝐾_𝐷𝒟^𝜇𝑤 𝑡

Here w(t) is the input , y(t) is output of the system, 𝛽_𝑘 (k= 0,1,2….n) arbitrary real number and 𝑎_𝑘 (k=

0,1,2….n) arbitrary constants.

Effectiveness of this controller can be analyzes by an example of P𝐷^𝜇controller. The transfer function and timedomain fractional order differential equation are

𝐺 𝑠 = 1

𝑎₂𝑠^𝛽2+ 𝑎₁𝑠^𝛽1+ 𝑎₀ 𝑎₂𝑦^𝛽2 𝑡 + 𝑎₁𝑦^𝛽1 𝑡 + 𝑎₀𝑦 𝑡 = 𝑢(𝑡) With initial condition y(0) = 0, y‟(0) = 0, y‟‟(0) = 0.

Following figure shows the effectiveness of the controllers.

figure 1 is the comparison of conventional P D controller (thick line)and fractional P𝐷^𝜇 controller ( thin line).

7. Conclusion :

Fractional calculus acts as an powerful mathematical tool used to obtain solution to real world problems. Applicability of fractional calculus attracted mathematicians to work in their field using this branch of mathematics. The purpose of this paper is to attract new researcher to work in this field since though beginning of fractional calculus may be same as traditional calculus but actual work found in last three to four decades.

Refrences

[1] S. V. Nakade, R.N. Ingle, Study of some Special functions in Fractional Calculus,International Innovative Journal, ISSN 2319-8648. Special Issue for International conference on Applied Science 2017.

[2] I. Podlubny, Fractional Differential Equations, Academic Press, New York, (1999).

[3] S. V. Nakade, R.N. Ingle, Note on Integral transform of fractional calculus, International Journal of Research and Review, 2019, 6(2): 106-110.

[4] S. V. Nakade, R.N. Ingle,Study Of Some Mathematical Modeling Using Fractional Calculus, Ajanta An International Multi. Qtly Research Journal ISSN 2277-5730,pp84-92.

[5] S. Samko, A. Kilbas, O. Marichev; Fractional integrals and derivative: Theory and Applications, Gordon and Breach science publisher, 1993.

[6] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, (1974).

[7] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional differential Equations Wiley, 1993.

[8] Gorenflo R. and Mainardi F., Essentials of fractional calculus ,MaPhySto Center, 2000.

[9] Greenberg M., Foundations of applied mathematics, Prentice-Hall Inc., Englewood Cliffs, N.J. 07632, 1978.

[10] M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover, New York,1964.

[11] A. Erdélyi, et al. (Eds.), Higher Transcendental Functions, vols. 1_3, McGraw-Hill, New York, 1953.

[12] Sneddon Ian N. The use of Integral Transform, McGraw-Hill Book Co., NY1972.

[13] Erdelyi, A., W. Magnus, F. Oberhettinger and F.G.Tricomi, Tables of Integral Transfer, McGraw-Hill Book Co., NY1954.

[14] Davies G., Integral Transforms and Their Applications, 2^nd ed. Springer-Verlag, New York,1984.

[15] O. P. Agrawal. Application of fractional derivatives in therma analysis. Nonlinear Dynamics, 38:191–

206, 2004.

[16] V. V. Kulish and J. L. Lage. Fractional-diffusion solutions for transient local Temperature and heat flux.

Journal of Heat Transfer, 122, 2000.

[17] LokenathDebnath. Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences, 54:3413–3442, 2003.

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Two Fluid Cosmological Model Coupled with Mass Less Scalar Field in 𝒇(𝑻) Gravity

S.H.Shekh,V.R.Chirde*, S.V.Raut*

Department of Mathematics, S.P.M. Science and Gilani Arts and Commerce College, Ghatanji- 445301.India

*Department of Mathematics, G.S.G.Mahavidyalaya, Umarkhed-445206.India

Abstract:

We present a class of solutions of field equations inf(T)gravity describing the interaction and non-interaction between barotropic and dark fluid for spatially homogeneous and isotropic flat FRWuniverse with mass less scalar field.

In this universe, the field equation correspond to the particular choice of f(T)TTⁿ.An exact solution of field equations are obtained by applying the law of variation of Hubble’s parameter which yields a constant value of the deceleration parameter. The physical and geometrical parameters along with some kinematical test of the model discussed in detail.

Keywords: - Barotropic fluid, dark energy, f(T) theory of gravity, cosmology.

1. Introduction:

Experimental evidence [1, 2] has established that our universe undergoing a late-time accelerating expansion and the recent observations of SN Ia [3,4] have confirmed.The measurements of the cosmic microwave background (CMB) and large scale structure (LSS) strongly indicate that an expansion of the universe isacceleratingwhich is dominated by a component with negative pressure, dubbed as dark energy (DE)and has a flat geometry. The DE model has been characterizing in a conventional manner by the equation of state (EoS) parameter w_D p_D/_Dwhere _D and p_Drepresent the energy density and pressure of dark fluid. The EoS parameter of DE,w_D lies close to (1)and if it would be equal, a little bit upper or less than

)

(1 corresponds to standard CDM cosmology, the quintessence region or phantom region respectively while the possibilityw_D 1is ruled out by current cosmological data from SN Ia. Several authors [5-9] have examined and discussed the DE models in different context of use while some authors have investigated an interaction and non-interaction between DE and barotropic fluid using cosmological models like Chenand Wang [10] investigate the Evolution of the interacting viscous DE model in Einstein cosmology using FRW universe.Avellino [11] investigates an interaction between DE and bulk viscosity using spatially flat FRW universe. Saha [12] explored two-fluid scenario for DE models in FRW universe. Interacting two-fluid viscous DE models in non-flat universe and two-fluid DE models in an anisotropic universe inspected by Amirhashchi et al. [13, 14].

On the other hand, awesome abundance of observational evidence in favor the late-time accelerating expansion does not fit within the framework of General Relativity (GR). In order to explain accelerated expansion of the universe there is an alternative modification of general relativity, these alternatives goes back to 1928 with Einstein‟s attempt to unify gravity and electromagnetism through the introduction of a tetrad (vierbein) field along with the concept of absolute parallelism or teleparallelism [15] known as TeleparallelGravity (TG) or f(T)gravity. The gravitational field equation of TG is described in terms of torsion instead of curvature [16]. An advantage of f(T)theory is that its field equation is only second order. Various aspects of f(T)theory have been investigated by [17-20].Jamil et al. [21, 22] tried to resolve the Dark Matter (DM) problem in the light of f(T)gravity and successfully obtained the flat rotation curves of galaxies containing DM as component with the density profile of DM in galaxies. Also gives the interacting DE model in the framework of same theory for a particular choice off(T). Setare and Darabi [23]have studied the power- law solution when the universe enters in phantom phase and shown that such solutions may exist for some

) (T

f solutions.Particle creation in flat FriedmanRobertsonWalker universe in the framework of f(T)gravity

investigated by Setare and Houndjo [24] Chirde and Shekh [25] investigate barotropicbulk viscous cosmological model in f(T)gravity.

The study of interacting fields, one of them being zero-mass scalar field which is fundamental challenge to look into the yet unsolved problem of the unification of the gravitational and quantum theories. In the last few decades there has been renewed interest focused on the theory of gravitation representing zero-mass scalar fields coupled with gravitational field. The zero-mass scalar field has acquired particular importance.Maniharsingh [26], Singh &Bhamra [27] and Singh [28,29] studied different one-fluid models coupled with a scalar field. Singh and Deo [30] have investigated the problem of zero-mass scalar field interactions in the presence of a gravitational field for FRW space-time in GR and shown that the „Big-Bang‟ of universe at the initial stage can be avoided by introducing a zero-mass scalar field along with this some authors [31, 32] investigated some cosmological model with zero-mass scalar field.

Incited by above discussions, inthis paperwe investigate two-fluid model coupled with a scalar field in 𝑓(𝑇) gravity in order to be able to understand the hidden properties of such universe, in doing so we consider both non-interacting and interacting cases. The outline of the paper is as follows. In section 2, we describe the brief review of 𝑓(𝑇) theory. In section 3, the metric and the basic equations are described. Sections 4, deals with solution of field equations. In section 5, we consider both interacting and non-interacting cases. The physical stability of solutions analyzed in section 6.Section 7, deals with some kinematical properties of the universe and finally conclusions are summarized in the last section 8.

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