A NEW FRACTIONAL MODEL FOR CONVECTIVE STRAIGHT FINS WITH TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY

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1 THERMAL SCIENCE: Year 18, Vol., No. 6B, pp A NEW FRACTIONAL MODEL FOR CONVECTIVE STRAIGHT FINS WITH TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY y Devendra KUMAR a, Jagdev SINGH, and Dumitru BALEANU *,c a Department of Mathematics, JECRC University, Jaipur, Rajasthan, India Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Etimesgut, Turkey c Institute of Space Sciences, Magurele, Bucharest, Romania Introduction Original scientific paper The key aim of this work is to present a new non-integer model for convective straight fins with temperature-dependent thermal conductivity associated with Caputo-Farizio fractional derivative. The fractional energy alance equation is solved y using homotopy perturation method coupled with Laplace transform method. The efficiency of straight fin has een derived in terms of thermo-geometric fin parameter. The numerical results derived y the application of suggested scheme are demonstrated graphically. The susequent correlation equations are very helpful for thermal design scientists and engineers to design straight fins having temperature-dependent thermal conductivity. Key words: fractional energy alance equation, thermal conductivity, straight fins, Caputo-Farizio fractional derivative, homotopy perturation method Most of scientific prolems such as heat transfer are modeled y using non-linear differential equations [1]. The concept of heat transfer is very important and useful in mechanical engineering as it is required in numerous ojects. The convective heat transfer rate can e enhanced with the aid of distinct schemes such as enlarging the surface area of heat transfer or coefficient of heat transfer. It is very popular that the heat transfer surface area can enlarged y affixing the fins faricated of materials having highly conductive on ase surface. Fins are designed in such a manner that it increases the heat transfer from the ase surface to its environment. Besides the well known utilizations such as heat exchanges, internal comustion engines and compressors, fins also insures efficacious in heat rejection systems in cooling of various types of electric instruments and space vehicles [, 3]. In this connection a detailed report was presented in a monograph y Kern and Kraus [4]. In an attempt Domairry and Fazeli [5] examined the efficiency of convective straight fins with the help of homotopy perturation method (HPM). Chiu and Chen [6] operated the Adomian s decomposition method (ADM) to investigate convective longitudinal fins having variale thermal conductivity. In another study Chiu and Chen [7] examined the convective-radiative with the help of decomposition technique. Bartas and Sellers [8] analyzed the heat-rejecting system. Furthermore, Coskun and Atay [9] employed the variational iteration technique to examine the convective straight and radial fins. Arslanturk [1] used and show the efficiency of ADM to examine the optimum design of * Corresponding author, dumitru@cankaya.edu.tr

2 79 THERMAL SCIENCE: Year 18, Vol., No. 6B, pp space radiators. In another investigation Aziz and Hug [11] employed the classical perturation scheme to examine the efficiency of convective straight fins. In this sequel Patra and Ray [1] studied the efficiency of convective straight fins possessing temperature-dependent thermal conductivity with the help of HPM y using sumudu transform scheme. In the last decade derivatives and integrals of fractional orders had notale development as revealed y several monographs dedicated to it (e. g. Hilfer [13], Podluny [14], Miller and Ross [15], Baleanu et al. [16], Kilas et al. [17], etc.), the plethora of research papers pulished in scientific journals (e. g. Kumar et al. [18] studied differential-difference equation of fractional order, Singh et al. [19] investigated the local fractional Tricomi equation, Bhrawy et al. [] examined the fractional Burgers equations, Area et al. [1] analyzed the Eola epidemic model of fractional order, Carvalho and Pinto [] presented a delay mathematical model of fractional order to determine the co-infection of malaria and the human immunodeficiency virus, Srivastava et al. [3] examined a fractional model of viration equation, Yang et al. [4] studied the fractional KdV equation involving local fractional derivative, Jafari et al. [5] investigated the differential equations pertaining to local fractional operators, Yang et al. [6] examined the local fractional diffusion and relaxation equations, He at al. [7] studied a new fractional derivative and its application to explanation of polar ear hairs, Wang and Liu [8] showed the applications of He s fractional derivative for non-linear fractional heat transfer equation, Liu et al. [9] used the He s fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy, Sayevand and Pichaghchi [3] studied a non-linear fractional KdV equation ased on He s fractional derivative, Hu et al. [31] studied fractal space-time and fractional calculus, Liu et al. [3] presented a fractional model for insulation clothings with cocoon-like porous structure, etc.) and definitions of various derivatives and integrals (e. g. Caputo [33], Yang [34], He [35, 36], etc.). In a recent work Caputo and Farizio [37] introduced a new fractional derivative. The importance of the newly derivative is due to the requisite of employing a mathematical model explaining the nature of various processes. In fact, the classical Caputo fractional derivative comes into view to e especially suitale for those mechanical processes, associated with plasticity, fatigue, damage and with electromagnetic hysteresis. The physical processes in which these effects asent it seem more suitale to employ the novel definition of fractional derivative. Atangana [38] used the newly fractional derivative to understand the nature of Fisher s reaction- diffusion equation. In another investigation Atangana and Koca [39] employed this approach to nonlinear Baggs and Freedman model and show the efficiency of the newly fractional derivative. In a series of papers Hristov [4, 41] has shown that the newly Caputo-Farizio fractional derivative can easily e otained from the Cattaneo concept of the flux relaxation if the damping function is the Jeffrey memory kernel. Sun et al. [4] examined the fractional relaxation and diffusion prolems with non-singular kernels. Yang et al. [43] proposed a new fractional derivative having non-singular kernel and verify its applications in steady heat flow. In another work Atangana and Baleanu [44] suggested a new non-integer order derivative having non-local and non-singular kernel. Mirza and Vieru [45] otained the solutions of advection-diffusion equation with time-fractional Caputo-Farizio derivative y using a comination of Laplace transform and Fourier transform. Atangana and Baleanu [46] show the application of Caputo-Farizio derivative in groundwater flow within confined aquifer. Ali et al. [47] used the Caputo-Farizio derivative to examine the MHD free convection flow of generalized Walters -B fluid model. Baleanu et al. [48] presented a comparative study of Caputo and Caputo-Farizio derivatives for advection differential equation. Algahtani [49] used the Atangana-Baleanu and Caputo-Farizio derivative with fractional order in Allen Cahn model.

3 THERMAL SCIENCE: Year 18, Vol., No. 6B, pp In view of the great importance of heat transfer and related prolems, we present a new fractional model for energy alance equation associated with newly Caputo-Farizio fractional derivative to calculate the efficiency of the convective straight fins possessing temperature-dependent thermal conductivity. The efficiency of HPM coupled with Laplace transform algorithm is employed to examine the energy alance equation of fractional order. We estimate the non-dimensional temperature distriution and fin tip temperature for the convective straight fins possessing thermal conductivity for the distinct values of various physical parameters. The used method is an extension of HPM y using Laplace transform [5-5] to handle non-linear PDE associated with newly Caputo-Farizio fractional derivative and is a comined form of HPM [53-55], Lapalce transform technique and He s polynomials [56]. The supremacy of applied scheme over the classical analytical techniques is that requires the less computer memory and reduces the computation time. Preliminaries Definition 1. Let us consider that θ H 1 (a, ), > a, β [, 1], then the Caputo-Farizio derivative of fractional order discovered y Caputo and Farizio [37] is written in the following manner: β M ( β) ' τ D [ θ ( )] = θ( τ)exp β dτ 1 β 1 β a (1) In eq. (1) M(β) is denoting normalization function, which holds the property M() = M(1) = 1 [37]. If θ H 1 (a, ) then the Caputo-Farizio fractional derivative can e re-expressed: if β βm ( β) τ D [ θ ( )] = [ θ ( ) θτ ( )]exp β dτ 1 β 1 β a () The eq. () also reduces in the following similar equation: t γ N( γ) ' τ D [ θ ( )] = θ( τ)exp d τ, N() N( ) 1 γ = = γ (3) a 1 β 1 γ = [, ), β = [,1] as presented y the Atangana [38]. β 1+ γ Moreover, 1 τ Lim exp = δτ ( ) γ γ γ The associate integral of the Caputo-Farizio fractional derivative was introduced y Losada and Nieto [57] expressed in the following manner. Definition. The integral operator of fractional order for the function θ() of order β, < β < 1 is written as [57]: β (1 β) β I [ θ ( )] = θ ( ) + θ( s)d s, ( β) M( β) ( β) M( β) (5) From eq. (5), the following result can e found: (1 β) β + = 1 ( β) M( β) ( β) M( β) (4) (6)

4 794 THERMAL SCIENCE: Year 18, Vol., No. 6B, pp It yields the following result: M ( β) =, β < 1 ( β ) Further Losada and Nieto [57] showed that the Caputo-Farizio derivative of order < β < 1 can e redefined y using previous discussed expression as: β 1 ' τ D [ θ ( )] = θ( τ)exp β dτ 1 β 1 β (8) a Now, we present the following important theorem for the newly Caputo-Farizio fractional derivative. Definition 3. If CF D β+1 θ() is the Caputo-Farizio fractional derivative of a function θ() then its Laplace transform formula is expressed as [37]: ' CF β 1 s θ( s) sθ() θ () L D θ ( ) + = s+ β (1 s) where θ (s) indicates the Laplace transform of the function θ(). x h, T a dx Figure 1. Schematic diagram of the prolem T Mathematical model of the prolem (7) (9) The schematic diagram of straight fin prolem possessing the aritrary cross-sectional area, A c, perimeter, P, and length,, is presented in fig. 1. The fin is joined with the ase surface having the temperature, T, and extends into fluid having temperature, T a, and its tip is insulated. Then the energy alance equation is written as [9, 1]: d dt Ac k( T ) Ph( T Ta) dx = dx In the eq. (1) k(t) indicates the temperature-dependent thermal conductivity and h represents the coefficient of heat transfer. It is considered that the thermal conductivity for the fin material is expressed: [ λ ] (1) kt ( ) = k 1 + ( T T) (11) In the expression (11) k represents the thermal conductivity at the amient fluid temperature of the fin and λ is standing for the variation of the thermal conductivity. Using the non-dimensional variales: 1/ Ph ψ T Ta x θ =, =, α = λ( T Ta), and = T Ta ka a c Consequently the eq. (1) reduces in the following form: d θ d θ dθ αθ α ψ θ + d + d d, 1 with the oundary conditions (1) (13)

5 THERMAL SCIENCE: Year 18, Vol., No. 6B, pp dθ = and θ 1 1 d = = (14) = Since we know that the integer order derivatives are in local in nature, so these derivatives can not descrie the prolem accurately. The Caputo-Farizio fractional derivative is more suitale to descrie to natural phenomena ecause its kernel is non-local and non-singular. Therefore, we replace the second order derivative d θ / d in eq. (13) y the newly Caputo-Farizio fractional derivative and eq. (13) converts to a fractional model of energy alance equation expressed: CF β 1 d θ dθ D + θ + αθ + α ψ θ, β 1 < (15) d d along with the oundary conditions (14). Basic idea of HPM coupled with Laplace transform method In order to demonstrate the fundamental plan of HPM coupled with Laplace transform algorithm, we take a non-linear differential equation involving Caputo-Farizio fractional derivative written: CF β + 1 D θ ( ) + Rθ ( ) + Nθ ( ) = g( ), < β 1 (16) with the initial conditions: θ ' () a, θ () = = (17) where CF D β+1 θ() represents the Caputo-Farizio derivative of non-integer order for the function θ(), R indicates the linear differential operator, N stands for the non-linear differential operator of general nature, and g() denotes the term due to the source. Firstly we operate the Laplace transform on fractional eq. (16), it yields: a s β(1 s) s β(1 s) L[ θ ( )] + Lg [ ( )] + = + + LR [ θ ( ) Nθ ( )] s s + s s On operating with the inverse of Laplace transform on eq. (18), it gives: 1 (1 ) ( ) ( ) s + β s θ = G L LR [ θ ( ) + Nθ ( )] s (19) In eq. (19) G() indicates the term occurring due to the source term and the initial conditions. Next we use the classical HPM: n= (18) n θ ( ) = p θ( ) () and the non-linear term Nθ() can e deformed in the following manner: n= where H n (θ) are the He s polynomials in [56] that are presented: n Nθ ( ) = ph n ( θ) (1) n ( θ, θ 1,..., θ i n n) = n i n! p θ i= p= n 1 H N p, n =, 1,, 3 ()

6 796 THERMAL SCIENCE: Year 18, Vol., No. 6B, pp On using the eqs. () and (1) in eq. (19), it yields: n 1 s+ β (1 s) n n pθn( ) = G( ) p L L R pθ ( ) ( ) n phn θ n= s + n= n= (3) The previous result is the comination of the standard Laplace transform scheme and the classical HPM employing the He s polynomials. On equating the coefficients of same powers of p, we have: p : θ ( ) = G( ), 1 1 s+ β (1 s) p : θ1( ) = L L [ Rθ( ) + H( θ) ] s 1 s+ β (1 s) p : θ( ) = L L [ Rθ1( xt, ) + H1( θ) ] s 3 1 s+ β (1 s) p : θ3( ) = L L [ Rθ( ) + H( θ) ] s Using the same way, the remaining the iterates θ n () can otained. Finally, the FHPTM solution θ() is expressed: N θ ( ) = Lim θ( ) (5) N n = 1 The HPM coupled with Laplace transform method for non-linear energy alance equation of fractional order First of all, we operate the Laplace transform on eq. (15), it yields: L K s+ (1 ) d d [ ( )] β s L θ θ αθ α θ = + ψ θ s s d d where K = θ(). On operating the inverse of Laplace transform on eq. (6), it gives: n 1 s+ β(1 s) d θ dθ θ ( ) = K L L αθ + α ψ θ s d d On applying the HPM, it yields: s+ β (1 s) n 1 n n ' n pθn = K p L L α phn + α phn ψ pθn n= s n= n= n= (8) In eq. (8) H n (θ) and H (θ) are He s polynomials given in the following manner: n d θ phn( θ ) = θ (9) d n= The initial few, components H (θ), H 1 (θ), H (θ),... are expressed: H d θ ( ) ( θ) = θ( ) d (4) (6) (7)

7 THERMAL SCIENCE: Year 18, Vol., No. 6B, pp for H (θ), we find that H H d θ1( ) d θ ( ) 1( θ) θ( ) θ 1( ) = d + d (3) d θ( ) d θ1( ) d θ ( ) ( θ) = θ( ) + θ 1( ) + θ ( ) d d d n ' phn n= dθ ( θ ) = d ' d θ ( ) H ( θ ) = d ' d θ ( ) d θ1( ) H1( θ ) = d d (31) ' d θ ( ) d θ( ) d θ1( ) H() θ = + d d d On equating the coefficients of the same powers of p, it yields: p p : θ ( ) = K Kψ 1 : θ1( ) = (1 β) + β Using the same way, the remaining iterates θ n () can e completely otained. Therefore, the solution is expressed: (3) θ( ) = θ ( ) + θ ( ) + θ ( ) + θ ( ) + θ ( ) +. (33) In eq. (9) K indicates the temperature at the fin tip and K [, 1]. The value of K can e easily otained y using the oundary condition θ =1 =1. Fin efficiency The rate of heat transfer from the straight fins is derived y employing the Newton s law of cooling: Q= PT ( Ta )dx (34) The most significant property of the fins is the fin efficiency. It is investigated in the prolems of heat and mass transfer. The fin efficiency is the ratio of the actual heat transferred y the fin and the heat transfer if the fin is entirely present in ase temperature: PT ( T )dx a 1 Q ρ= = = θη ( )dη Q P( T T ) ideal a (35)

8 798 THERMAL SCIENCE: Year 18, Vol., No. 6B, pp Now sustituting the value of θ from eqs. (33) and (35), we calculate the dimensionless fin efficiency for the straight fins. Numerical results and discussions Tale 1. Comparative study etween integer order derivative, classical Caputo fractional derivative and newly fractional derivative due to Caputo and Farizio for the non-dimensional temperature distriution within the fin at α =.3 and ψ = 1 Integer order derivative β = 1 Caputo fractional derivative β =.75 Caputo-Farizio fractional derivative β = Tale. The numerical results otained y using HPM [3], VIM [9] and present method for the non-dimensional temperature distriution within the fin at α =, ψ =., and β = 1 HPM [3] VIM [9] HPM coupled with Laplace transform method In this section, we compute the non-dimensional temperature distriution, θ(), and non-dimensional fin tip temperature, K. The comparison etween the results derived for integer order derivative, classical Caputo fractional derivative and newly Caputo-Farizio fractional derivative are presented in ta. 1. It can e perceived from ta. 1 that the numerical results associated with Caputo-Farizio fractional derivative show new characteristics compared to standard derivative and classical Caputo fractional derivative. The comparison etween the numerical simulations for non-dimensional temperature distriution otained with aid of distinct schemes is discussed in ta.. From ta., it can e easily noticed that the outcomes of the suggested scheme are perfectly agree with the results availale in the literature. The numerical results for non-dimensional temperature distriution, θ(), for various values of values of β, ψ, and α are displayed through figs.

9 THERMAL SCIENCE: Year 18, Vol., No. 6B, pp θ.9 θ.9 Ψ = β = 1 β = Ψ = 1.6 β = Figure. Response of dimensionless temperature distriution θ() for convective straight fins vs. for distinct values of β at α =.3, and ψ =1 θ Ψ = Figure 3. Behavior of non-dimensional temperature distriution θ() for convective straight fins corresponding to for distinct values of ψ at and α =.3, and β = 1 K α =.5 α =.3 α =.3 α = Figure 4. Nature of non-dimensional temperature distriution θ() for convective straight fins vs. for distinct values of α at ψ = 1 and β = 1-4, respectively. It can e detected from fig. that as the order of time-fractional derivative increases, it leads the increases in the value of the temperature, θ(). Figure 3 reveals that the temperature, θ(), is decreased y increasing the value of α. In can e oserved from fig. 4 that the temperature, θ(), is increased y increasing the value of α. The numerical results otained y using FHPTM for non-dimensional fin tip temperature, K, for various values of order of non-integer derivative and α are displayed in figs. 4 and 5, respectively. Figure 5 shows that as β increases, it leads to the corresponding increase in the value of fin tip temperature K. Figure 6 reveals that the value of K increases y increasing the value of α..4. β =.5 β = 1 β = ψ Figure 5. The nature of fin tip temperature K with respect to ψ for various values of β at α =. 1. K α =.5 α =.3 α =.3 α = ψ Figure 6. The nature of fin tip temperature K with respect to ψ for various values of α at β = 1

10 8 THERMAL SCIENCE: Year 18, Vol., No. 6B, pp Conclusions In this paper, a new non-integer model for convective straight fins possessing temperature-dependent thermal conductivity involving Caputo-Farizio fractional derivative is considered. The HPM coupled with Laplace transform method is successfully used to solve the energy alance equation of aritrary order. The HPM coupled with Laplace transform scheme is specially design to examine the non-linear differential equations pertaining to Caputo-Farizio fractional derivative. The most important part of this study is to use the newly Caputo-Farizio fractional derivative to investigate the convective straight fins possessing the temperature-dependent thermal conductivity. The numerical simulation is performed for non-dimensional temperature distriution and fin tip temperature that reveal that the selection of the order of fractional derivative remarkaly influence the outcomes. The results of this investigation are very helpful for engineers dealing with the heat conduction prolems of strongly non-linear nature. Thus, we can conclude that the use of newly Caputo-Farizio fractional derivative in modeling of the real world prolems is very interesting, gives very fruitful consequences and HPM coupled with Laplace transform method is very powerful and efficient scheme to study such type of nonlinear prolems of fractional order. References [1] Ganji, D. D., Rajai, A., Assessment of Homotopy Perturation Methods in Heat Radiation Equations, Int. Commun. Heat Mass Transfer, 33 (6), 3, pp [] Aziz, A., Nguyen, H., Two-Dimensional Performance of Convecting-Radiating Fins of Different Profile Shapes, Waerme Stoffueertrag, 8 (1993), 8, pp [3] Cuce, E., Cuce, P. M., Homotopy Perturation Method for Temperature Distriution, Fin Efficiency and Fin Effectiveness of Convective Straight Fines with Temperature-Dependent Thermal Conductivity, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci., 7 (1), 8, pp. 1-7 [4] Kern, D., Kraus, A., Extended Surface Heat Transfer, McGraw Hill Book Comp., New York, USA, 197 [5] Domairry, G., Fazeli, M., Homotopy Analysis Method to Determine the Fin Efficiency of Convective Straight Fins with Temperature-Dependent Thermal Conductivity, Commun. Nonlinear Sci. Numer. Simul., 14 (9),, pp [6] Chiu C. H., Chen, C. K., A Decomposition Method for Solving the Convective Longitudinal Fins with Variale Thermal Conductivity, Int. Journal Heat and Mass Transfer, 45 (), 1, pp [7] Chiu, C. H., Chen, C. K., Applications of Adomian s Decomposition Procedure to the Analysis of Convective-Radiative Fins, Journal of Heat Transfer, 15 (3),, pp [8] Bartas, J. G., Sellers, W. H., Radiation Fin Effectiveness, Journal of Heat Transfer, Series C, 8 (196), 1, pp [9] Coskun, S. B., Atay, M. D., Analysis of Convective Straight and Radial Fins with Temperature-Dependent Thermal Conductivity Using Variational Iteration Method with Comparison with Respect to Finite Element Analysis, Math. Pro. Eng., 7 (7), ID 47 [1] Arslanturk, C., Optimum Design of Space Radiators with Temperature-Dependent Thermal Conductivity, Applied Thermal Engineering, 6 (6), 11-1, pp [11] Aziz, A., Hug, S. M. E., Perturation Solution for Convecting Fin with Variale Thermal Conductivity, J. Heat Transfer, 97 (1975),, pp [1] Patra, A., Ray, S. S., Analysis for Fin Efficiency with Temperature Dependent Thermal Conductivity of Fractional Order Energy Balance Equation Using HPST Method, Alexandria Eng. J., 55 (16), 1, pp [13] Hilfer, R. (Ed.), Applications of Fractional Calculus in Physics, World Scientific Pulishing Company, Singapore-New Jersey-Hong Kong,, pp [14] Podluny, I., Fractional Differential Equations, Academic Press, New York, USA, 1999 [15] Miller, K. S., Ross, B., An Introduction to the fractional Calculus and Fractional Differential Equations, Wiley, New York, USA, 1993 [16] Baleanu, D., et al., (Eds.), New Trends in Nanotechnology and Fractional Calculus Applications, Springer, Dordrecht Heidelerg London New York, 1

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12 8 THERMAL SCIENCE: Year 18, Vol., No. 6B, pp [46] Atangana, A., Baleanu, D., Caputo-Farizio Derivative Applied to Groundwater Flow within Confined Aquifer, J. Eng. Mech., 143 (16), May, D4165 [47] Ali, F., et al., Application of Caputo-Farizio Derivatives to MHD Free Convection Flow of Generalized Walters -B Fluid Model, Eur. Phys. J. Plus, 131 (16), Oct., 377 [48] Baleanu, D., et al., Fractional Advection Differential Equation within Caputo and Caputo-Farizio Derivatives, Adv. Mech. Eng., 8 (17), 1, pp. 1-8 [49] Algahtani, O. J. J., Comparing the Atangana-Baleanu and Caputo-Farizio Derivative with Fractional Order: Allen Cahn Model, Chaos, Solitons & Fractals, 89 (16), Aug., pp [5] Khan, Y., Wu, Q., Homotopy Perturation Transform Method for Non-Linear Equations Using He s Polynomials, Computer and Mathematics with Applications, 61 (11), 8, pp [51] Goswami, A., et al., A Reliale Algorithm for KdV Equations Arising in Warm Plasma, Nonlinear Eng., 5 (16), 1, pp [5] Kumar, D., et al., Analytic and Approximate Solutions of Space and Time-Fractional Telegraph Equations via Laplace Transform, Walailak Journal of Science and Technology, 11 (14), 8, pp [53] He, J. H., Homotopy Perturation Technique, Computer Methods in Applied Mechanics and Engineering, 178 (1999), 3-4, pp [54] He, J. H., Homotopy Peruration Method: A New Non-Linear Analytical Technique, Appl. Math. Comput., 135 (3), 1, pp [55] He, J. H., New Interpretation of Homotopy Perturation Method, Int. J. Mod. Phys. B, (6), 18, pp [56] Ghorani, A., Beyond Adomian s Polynomials: He Polynomials, Chaos, Solitons & Fractals, 39 (9), 3, pp [57] Losada, J., Nieto, J. J., Properties of the New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1 (15), Apr., pp Paper sumitted: January 9, 17 Paper revised: March 11, 17 Paper accepted: March 4, Society of Thermal Engineers of Seria Pulished y the Vinča Institute of Nuclear Sciences, Belgrade, Seria. This is an open access article distriuted under the CC BY-NC-ND 4. terms and conditions