A HEURISTIC OPTIMIZATION METHOD OF FRACTIONAL CONVECTION REACTION An Application to Diffusion Process

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1 Khan, N. A., e al.: A Heurisic Opiizaion Mehod of Fracional Convecion Reacion... S43 A HEURISTIC OPTIMIZATION METHOD OF FRACTIONAL CONVECTION REACTION An Applicaion o Diffusion Process by Najeeb Ala KHAN a*, Tooba HAMEED a, Nadee Ala KHAN b, and Muhaad Asif Zahoor RAJA c a Deparen of Maheaics, Universiy of Karachi, Karachi, Pakisan b Deparen of Copuer Science, Iqra Universiy, Karachi, Pakisan c COMSATS Insiue of Inforaion Technology, Aock, Pakisan Original scienific paper hps://doi.org/10.98/tsci k Inroducion The convecion differenial odels play an essenial role in sudying differen cheical process and effecs of he diffusion process. This paper inends o provide opiized nuerical resuls of such equaions based on he conforable fracional derivaive. Subsequenly, a well-known heurisic opiizaion echnique, differenial evoluion algorih, is worked ou ogeher wih he Taylor s series epansion, o aain he opiized resuls. In he schee of he Taylor opiizaion ehod (TOM), afer epanding he funcions wih he Taylor s series, he unknown ers of he series are hen globally opiized using differenial evoluion. Moreover, o illusrae he applicabiliy of TOM, soe eaples of linear and non-linear fracional convecion diffusion equaions are eeplified graphically. The obained assessens and coparaive deonsraions divulged he rapid convergence of he esiaed soluions owards he eac soluions. Coprising wih an effecive epander and efficien opiizer, TOM reveals o be an appropriae approach o solve differen fracional differenial equaions odeling various probles of engineering. Key words: conforable fracional derivaive, Taylor s series, opiizaion Modeling dissiilar real-world phenoena using fracional definiions have becoe he os highly desiring areas of realisic sciences. For he reason ha he non-local properies of fracional operaor enable hese differenial odels o pu he inforaion, abou he recen and he hisorical siuaion, in a nushell [1]. For he las decades, any enlargeens have been ade in his regard o eplore enhanced definiions and properies in order o overcoe he inadequacies of previous definiions of fracional calculus, such as, He s fracional derivaive [3], Aangana-Baleanu fracional derivaive [4], Capuo and Fabrizio [5], conforable derivaive [6], ec. Consequenly, hese novel aspecs enrich he capabiliies of fracional differenial odels in bringing diverse physical significances o ligh [7-9]. Hence, by eans of differen heories of fracional derivaives, he behaviors of any fracional PDE have been sudied and various echniques have been developed [10-14] bu sill here are any hinks o be done in his direcion. * Corresponding auhor, e-ail: njbala@yahoo.co

2 S44 Khan, N. A., e al.: A Heurisic Opiizaion Mehod of Fracional Convecion Reacion... The fracional diffusion equaions are osly used in relaing abnoral slowly-diffusion phenoenon and describing he abnoral convecion phenoenon of liquid in he ediu. I is broadly applied in engineering and science, as aheaical odels ha are used o replicae copuing. Nuerous nuerical ehods have been designed in his regard. In [15], differenial ransfor ehod has been developed for fracional convecion-diffusion equaion. A new variaional ieraion ehod was applied o decipher fracional convecion-diffusion equaions in [16]. Waveles based ehod has been considered in [17] o solve he space-ie fracional convecion-diffusion equaions. An operaional ari of fracional-order Legendre funcions has been eployed o solve he ie fracional convecion diffusion equaion in [18]. Furherore, a lo of work has been discussed for he soluion of fracional diffusion equaions [19-]. In his anuscrip, eploying useful properies of Taylor s series wih he heurisic opiizaion echniques, we consider he following ypes of ie fracional convecion-diffusion equaion: λ z z z = A + N ( z) + G(, τ), a b, τ 0, 0 < λ 1 λ τ subjec o he iniial-boundary condiions: z (,0) = q( ), za (, τ) = s( τ), zb (, τ) = s( τ) () where N( z ) is a non-linear operaor represening he poenial energy, A is a fied paraeer, and λ denoes he fracional order derivaive. The physical undersanding of eq. (1) in heral engineering represens eperaure or species concenraion of he hea or ass ransferred inside he syse due o diffusion and convecion er. In lieraure, he heurisic opiizaion echniques have leaded he way o obain soluions of fracional differenial equaions. These global opiizing progras play a crucial par in assessing quick and accurae approiae soluions of he differenial odels, in graphical as well as in abular represenaion. Here, we have uilized differenial evoluion (DE) algorih [-4], which is relaed o he naural processes and geneics of populaions. For he os iporan par of DE algorih is ha i requires objecive funcions and is derivaives o be coninuous and akes a high nuber of evaluaions of he objecive funcion. Thus, for he governing proble, an error funcion is consruced by eans of Taylor s series epansion of he relaed funcions as he objecive funcion. The Taylor s series epansion, having he effecive abiliy o approiae he funcions ore accuraely han oher polynoial approiaions. I has been widely used o acquire he soluions of an ineger as well as fracional differenial equaions [5-6]. The aalgaaion of Taylor s series epansion and DE algorih, nae as TOM, considered in his aep, ehibis a rearkable ool o acquire he effecive soluions of funcions ogeher wih he globally opiized values of he error funcion. In addiion o his, he illusraive eaples, considered wih conforable fracional derivaive [5, 6, 10], elevaed he efficiency, sabiliy and appropriaeness of TOM. Fracional background Conforable fracional derivaive For any funcion z :[0, ) R, he conforable fracional derivaive of order α is given [5]: (1)

3 Khan, N. A., e al.: A Heurisic Opiizaion Mehod of Fracional Convecion Reacion... S45 1 α ( τ + τ ) z ( τ) T z ( ) li z α τ = (3) 0 for all τ > 0, α (0,1]. In addiion, if, is ( α + 1) -differeniable and coninuous a, hen four α ( α, α + 1] : ( ) ( ) ( ) ( α 1 ) ( α α) α z τ + τ z 1 τ T α z ( τ ) = li (4) 0 For furher deails and proofs, one would see [5, 6]. Taylor opiizaion ehod Consider a coninuous and differeniable funcion zτ (, ): R R, define over an inerval [0,1) [0,1), hen for any ineger ϑ > 0 he subsequen equaliy holds: (, τ) τ (, τ) z = w + v + w + AF (5) ( k where ) ( k vk = z (, τ ) and ) h wk = zτ (, τ ) are he k order derivaive (for k = 1,,..., N) of zτ (, ) a = 0 and τ = 0, respecively, and AF (, τ ) N = k = ( v+ wτ ) which represens he runcaed Taylor s series. Here, we concern wih he non-linear fracional iniial-boundary value proble, eqs. (1) and (), wih conforable fracional definiion wih fracional order 0< λ 1: k k! k ( ) (, τ ) k T z = z z + N z + G (6) λ where T λ represens he conforable fracional operaor. This odel has significan iporance in any physical siuaions [15-17]. Hence, he graphical inerpreaions and opiized soluions of he non-linear fracional odel of eq. (6) is he focal poin of his aep. Hence, in order o assess he soluions, we eploy he Taylor s opiizaion ehod. This ehod iniiaes wih he consrucion of rial soluions of he unknown funcions, on using runcaed Taylor s series epansion as defined in eq. (5) wih he given iniial condiions. Thus, he rial soluions zˆ rial (, τ ) of eq. (6) can be epressed: ( τ ) τ ( τ) zˆ,, Φ = w + v + w + AF, (7) rial where w 0, v 1, and w 1 are he iniial and boundary values and Φ is a vecor of vk and w k which h h are defined as he k derivaive of N funcion a = 0 and τ = 0, ha are o be deerined. Now, o copue he reaining ers of Taylor s series epansion for each of hese rial soluions, we se up an error funcion, which is hen opiized by using an opiizing echnique. In his endeavor, for he opiizaion purpose he DE algorih is uilized. This effecive heurisic opiizing echnique was proposed in [3]. Aong any oher global opiizers, DE is considered o be ore significan for is sipliciy and srong populaion-based sochasic search echnique over a coninuous doain. The key feaures of DE are he hree conrol paraeers, i. e., he populaion size NP, cross-over consan CR and he scaling facor Sf. These paraeers ay eensively affec he opiizaion perforance of he DE, herefore, in

4 S46 Khan, N. A., e al.: A Heurisic Opiizaion Mehod of Fracional Convecion Reacion... [-4] soe siple rules are defined for he selecion of hese paraeers. The DE algorih ainains he populaion of NP-diensional paraeer vecors, known as individuals, which akes new applican soluions by aking he paren individual and several oher individuals of he sae populaion. Using he uaion operaion, i randoly picks he generaed vecors fro he populaion o produce a uan vecor, which is said o be he arge vecor. The iniial populaion is esablished: ( )( ) L U L v + rand 0,1 v v (8) j j j Afer he copleion of uaion operaion, cross-over phase is considered in which each pair of arge vecor is aken wih is corresponding uan vecor o generae a rail vecor. Afer he cross-over operaion, selecion of he funcion value is perfored by coparing he funcion value of each rial vecor o ha of is corresponding arge vecor in he curren populaion. If he funcion value of each rial vecor becoes less han or equal o he arge vecor, hen he rial vecor will change he arge vecor and coes in he populaion of he ne generaion. The selecion operaor process can be epressed: ( ) if ( ) ( ) ( ) if ( ) > ( ) ui F ui F vi vi ( + 1) = vi F ui F vi where ui () and vi () are he rial and arge vecors, respecively, and F is he objecive funcion. The process is considered o be convergen if he bes funcion values, in he new and old populaions, wih he new bes poin and he old bes poin, have less difference han he olerance level. Thus, in DE algorih, he soluions are easily obained by jus specifying he populaion se, rial soluion and he objecive funcion. For he governing probles, he objecive funcion is defined by using he error funcions, E, for he conforable fracional operaor: E = TZ+ Z Z + N+ G (10) where do and prie represen he parial derivaives of he funcions wih respec o τ and, respecively, which can be epressed in ari for: (, τ ; ) (, τ ; ) (, τ ; ) Z = zˆ ˆ ˆ 1 1 Φ z Φ z Φ (, τ ; ) (, τ ; ) (, τ ; ) Z = zˆ ˆ ˆ 1 1 Φ z Φ z Φ Z zˆ( 1, 1; ) zˆ(, ; ) zˆ = τ Φ τ Φ (, τ ; Φ) { ˆ ( ) ˆ ( ) ˆ N z 1, τ1; N z, τ; N z(, τ ; ) } N = Φ Φ Φ (, τ ; ) (, τ ; ) (, τ ; ) G = G 1 1 Φ G Φ G Φ τ τ 0 T = 0 τ (9)

5 Khan, N. A., e al.: A Heurisic Opiizaion Mehod of Fracional Convecion Reacion... S47 And he selecion operaor process is epressed: 1 in ( E ) 10 η for he populaion se (0, ], where η is any posiive ineger. The algorih Accordingly, he algorihic process of TOM for iediae ipleenaion on convecion advecion eq. (1) is oulined: (, τ) = (, τ) + + τ + (, τ) z z v w AF 1 1 Sep 1. (i) Se N. (ii) Find w0, w1, v 1by using z 0 = q 0 ( ), za = s0( τ), zb = s1( τ), and (iii) Consruc he rial soluion: ( τ ) ( τ) ( ) zˆ,, Φ = z AF 0, AF,0 + rial 0 + z b z a AF ( 0, τ) AF ( b, τ) + AF (,0) + AF (, τ; Φ) b Sep. (i) Se 0< λ 1and (ii) copue all he coponens of given advecion, i. e. T Z (, τ, Φ ) = τ D zˆ (, τ, Φ ), Z = D zˆ (, τ, Φ ) Z = D zˆ (, τ, Φ) λ rial τ rial rial rial Sep 3. (i) Se sapling poins: ( b ai ) i = τ i =, i = 0,1,, for N and (ii) Subsiue copued coponen and he rial soluion in error funcion, defined in eq. (10): in E ( ) ˆrial (, ; ) ˆrial (, ; ) (, τ ; ) ˆ (, τ ; ) (, τ ) 1 τi Dτ z i τi Φ + Dz i τi Φ Φ = i= 1 D ˆ zrial i i Φ + N zrial i i Φ + G i i Sep 4. Inpu: Fi he error funcions in he DE algorih. On soe anipulaion, effecive values of each ers of Φ along wih a global iniu value of ean square of error funcions are aained by using MATHEMATICA sofware. Oupu: Global iniu value of E and he values of all unknown ers in Φ. Sep 5. Inpu: Subsiue he values of Φ in rial soluions zˆ rial (, τ ). Oupu: Approiae soluions of zˆ rial (, τ ). Tesing he algorih Tes proble 1. Consider he iniial-boundary proble of ie-fracional convecion-diffusion equaion [15, 16] defined in eq. (1) wih: λ ( ) ( ) N z = 0, A=, G, τ = τ + +, 0 < < 1, 0 < τ < 1 (11) and iniial-boundary condiions: Γ ( λ + 1) (,0) =, z( 0, τ ) = ( ) λ, z( τ ) Γ λ + 1 z ( λ 1) ( ) λ 1 Γ + 1, = 1 + λ (1) Γ + Here, Γ () is he gaa funcion. The eac soluion of eq. (11) is given:

6 S48 Khan, N. A., e al.: A Heurisic Opiizaion Mehod of Fracional Convecion Reacion... ( λ 1) ( λ 1) z(, τ ) = + Γ + λ Γ + Taking N = 6 rial soluion of eqs. (11) and (1), can be copued by seing: ( λ ) ( λ ) ( λ ) ( λ ) Γ + 1 Γ + 1 z0 =, z =, z = 1+ Γ + 1 Γ + 1 λ λ a b The consruced rial soluion of eqs. (1) and (13) can be wrien: rial ( τ ) ( τ ) ( ) ( τ ) zˆ,, Φ = AF 0, ; Φ AF,0; Φ + AF, ; Φ AF ( 0, τ; Φ) AF ( b, τ; Φ ) + AF (,0; Φ) b Equaion (14) helps us o copue all coponens of eq. (6). Using definiion given in secion Conforable fracional derivaive, and subsiue all of hese in he residual error funcion: ˆrial (, ; ) ˆrial (, ; ) (, τ ; ) ˆ (, τ ; ) (, τ ) 1 τi Dτ z i τi Φ + Dz i τi Φ in E( i, τ i, Φ ) = (15) i= 1 D ˆ zrial i i Φ + N zrial i i Φ + G i i where τ i, i [0,1]. Now, on ipleening DE algorih he ean square error in eq. (15) is globally iniized and approiae soluion is obained. Here, we consider si ers of Taylor s series epansion, i. e. N = 6, and populaion size NP = 0, fro he populaion se [0,1], o acquire he graphical and abulaed soluions of zˆ rial (, τ ) a various values of λ. Sequenially, fig. 1(a) displays he coparison of he eac soluion wih an approiae soluion of zˆ rial (, τ ) wih conforable fracional operaor λ = 1. The absolue error of he funcion zˆ rial (, τ ) is ploed in fig. 3(a). Addiionally, coparaive eplanaion of he proposed algorih wih he previous ehod [15] is ehibied in ab. 1 for differen values of λ. Moreover, soe nuerical approiaions are also ploed in fig. (a) for differen values of λ, in order o deonsrae he effecs of fracional operaor on he soluion. (13) (14) z(,τ) z(,τ) τ TOM a λ = 1 Eac τ TOM a λ = 1 Eac (a) (b) Figure 1. Soluion of zτ (, ) (a) es proble 1 and (b) es proble (for color iage see journal web sie) Tes proble. We consider he hoogeneous fracional convecion-diffusion equaion [16,18] wih he following values of he funcions and he paraeers:

7 Khan, N. A., e al.: A Heurisic Opiizaion Mehod of Fracional Convecion Reacion... S z(,0.5) λ = 1 λ = 0.9 λ = 0.7 z(,1) λ = 1 λ = (a) (b) Figure. Diffusion behaviors a τ = 0.5 (a) es proble 1 and (b) es proble Table 1. Coparison of zˆ(,1) and he ehod in [11] for es proble 1 λ = 1 λ = 0.9 λ = 0.7 TOM [11] TOM [11] TOM [11] Absolue error Absolue error (a) (b) Figure 3. Absolue error a λ = 1 and τ = 0.5 (a) es proble 1 and (b) es proble ( ) ( ) ( ) N z = z z z, Α= 1, G, τ = +Γ λ + 1, 0 < < 1, 0 < τ < 1 (16) wih iniial and he boundary condiions: λ (,0) =, z ( 0, τ) = τ, z ( τ) z λ 1, = 1 + τ (17) λ The eac soluion of he proble is z (, τ)= + τ. Taking N = 6 he rial soluion of eqs. (16) and (17), can be copued by seing ( ) =, ( ) = λ, ( ) = 1 + λ z, 0 za τ zb τ and we have he sae rial soluion in eq. (14). Afer subsiuing all coponens of eq. (6) he residual error funcion:

8 S50 Khan, N. A., e al.: A Heurisic Opiizaion Mehod of Fracional Convecion Reacion... ˆrial (, ; ) ˆrial (, ; ) (, τ ; ) ˆ (, τ ; ) (, τ ) 1 τi Dτ z i τi Φ + Dz i τi Φ in E( i, τ i, Φ ) = (18) i= 1 D ˆ zrial i i Φ + N zrial i i Φ + G i i The ie-fracional convecion-diffusion equaion enioned in eq. (14) have also been sudied in [18], where fracional order Legendre funcion ehod is uilized o aain he approiae soluion. Here, we use TOM o aain he opiu soluions a differen values of λ. Figure 3(a) reveal he significan precision of approiae soluion wih he eac soluion. Following he scheaic algorih, he easured nuerical soluions of zˆ rial (, τ ), a differen values of λ are shown in fig. 3(b). Addiionally, coparaive eplanaion of he proposed algorih wih he previous ehods [16, 18] is ehibied in ab.. Table. Coparisons of zˆ(,0.5) and he ehods in [10, 1] for es proble λ = 1 λ = 0.7 λ = 0.5 TOM [1] [10] TOM [10] TOM [1] [10] Conclusions In his work, ie-fracional convecion diffusion equaions were analyzed wih conforable fracional derivaives. We deliberaed he opiized soluions by eans of TOM. The anifesaion given in secion Taylor opiizaion ehod and he ascerained values of E 10 k for posiive inegers k = 1,...7 in ab. 3 signify he worh enioning accuracy of he proposed approach. Thus, fro he facs and figures, i is possible o conclude as follows. y Modeling wih conforable fracional derivaive suppored he physical eaning of he Table 3. Glob al opiu error values for NP = 0 and N = 6 governing odel and provided a new purse for odelling any probles of applied sciences. λ Tes proble 1 Tes proble y Jus depending on he basic lii definiion of he derivaive, conforable fracional derivaive is siple and can be easily used o eecue fracional behaviors of he funcions. y To calculae he unknown ers of Taylor s series epansion, he opiizing algorihs give effecive resuls by siply opiizing he error funcions. y As shown in ab. 3, he populaion based opiizing algorih DE, gives he global opiu values of ean square errors of he es probles a differen fracional values in a coninuous doain, which verifies he appropriaeness of approiaed funcions of he governing odels. Noenclaure AF(,τ) runcaed Taylor s series E(Φ) he residual error N(z) real nubers poenial energy

9 Khan, N. A., e al.: A Heurisic Opiizaion Mehod of Fracional Convecion Reacion... S51 v k, w k weigh vecor i, τ i sapling poins zˆ (, τ ) rial soluion rial Greek sybols η posiive ineger λ fracional order derivaive T α conforable fracional operaor Φ vecor References [1] Miller, K. S., An Inroducion o Fracional Calculus and Fracional Differenial Equaions, John Wiley and Sons, New York, USA, 1993 [] He, J. H., A New Fracal Derivaion, Theral Science, 15 (011), Suppl. 1, pp. S145-S147 [3] Aangana, A., Baleanu, D., New Fracional Derivaives wih Nonlocal and Non-Singular Kernel: Theory and Applicaion o Hea Transfer Model, Theral Science, 0 (016),, pp [4] Capuo, M., Fabrizio, M., A New Definiion of Fracional Derivaive wihou Singular Kernel, Progress in Fracional Differeniaion and Applicaion, 1 (015),, pp [5] Khalil, R., e al., A New Definiion of Fracional Derivaive, Journal of Copuaional and Applied Maheaics, 64 (014), July, pp [6] Hosseini, K., e al., New Eac Soluions of he Conforable Tie-Fracional Cahn-Allen and Cahn-Hilliard Equaions using he Modified Kudryashov Mehod, Opik Inernaional Journal for Ligh and Elecron Opics, 13 (017), Mar., pp [7] Wang, K, L., Liu., S. Y., He s Fracional Derivaive for Non-Linear Fracional Hea Transfer Equaion, Theral Science, 0 (016), 3, pp [8] Wang, K., Liu, S. Y., A New Soluion Procedure for Nonlinear Fracional Porous Media Equaion Based on a New Fracional Derivaive, Nonlinear Science. Leers A, 7 (016), 4, pp [9] Sayevand, K., Pichaghchi, K., Analysis of Nonlinear Fracional KdV Equaion Based on He s Fracional Derivaive, Nonlinear Science. Leers A, 7 (016), 3, pp [10] Eslai, M., Rezazadeh, H., The Firs Inegral Mehod for Wu-Zhang Syse wih Conforable Tie-Fracional Derivaive, Calcolo 53 (016), 3, pp [11] Jarad., F., e al., On a New Class of Fracional Operaors, Advances in Difference Equaions, 47 (017), Dec., pp [1] Goufo, E. F. D., Applicaion of The Capuo-Fabrizio Fracional Derivaive wihou Singular Kernel o Koreweg-de Vries-Bergers Equaion, Mahaical Modelling and Analysis 1 (016),, pp [13] Aangana, A., Koca, I., Chaos in a Siple Nonlinear Syse wih Aangana-Baleanu Derivaives wih Fracional Order, Chaos Solions & Fracals. 89 (016), Aug., pp [14] Wu, G. C., e al., Lyapunov Funcions for Rieann-Liouville-like Fracional Difference Equaions, Applied Maheaics and Copuaion, 314 (017), Dec., pp [15] Odiba, Z., Moani, S., A Generalized Differenial Transfor Mehod for Linear Parial, Differenial Equaions of Fracional Order, Applied Maheaics Leers, 1 (008),, pp [16] Abolhasani, M., e al., A New Variaional Ieraion Mehod for a Class of Fracional Convecion-Diffusion Equaions in Large Doains, Maheaics, 5 (017), 6, pp [17] Chen, Y., e al., Wavele Mehod for a Class of Fracional Convecion-Diffusion Equaion wih Variable Coefficiens, Journal of Copuaional Science, 1 (010), 3, pp [18] Abbasbandya, S., e al., Applicaion of he Operaional Mari of Fracional-Order Legendre Funcions for Solving he Tie-Fracional Convecion-Diffusion Equaion, Applied Maheaics and Copuaion, 66 (015), Sep., pp [19] Hariharan, G., Kannan, K., Review of Wavele Mehods for he Soluion of Reacion-Diffusion Probles in Science and Engineering, Applied Maheaical Modelling, 38 (014), 3, pp [0] Abdusala, H. A., Analyic and Approiae Soluions for Naguo Telegraph Reacion Diffusion Equaion, Applied Maheaics and Copuaion, 157 (004),, pp [1] Wu, G. C., e al., Analysis of Fracional Non-Linear Diffusion Behaviors Based on Adoian Polynoial, Theral Science, 1 (017),, pp [] Saruhan, H., Differenial Evoluion and Siulaed Annealing Algorihs For Mechanical Syses Design, Engineering Science and Technology, 17 (014), 3, pp [3] Sorn, R., Price, K, Differenial Evoluion: A Siple and Efficien Adapive Schee for Global Opiizaion over Coninuous Spaces, Inernaional Copuer Science Insiue, 1 (1995), Mar., pp. 1-16

10 S5 Khan, N. A., e al.: A Heurisic Opiizaion Mehod of Fracional Convecion Reacion... [4] Sorn, R., e al., Differenial Evoluion A Pracical Approach o Global Opiizaion, Springer Naural Copuing Series, Vanderplaas, Novi, Mich., USA, 005 [5] Bulbul, B., Sezer, M., Nuerical Soluion of Duffing Equaion by Using an Iproved Taylor Mari Mehod, Journal of Applied Maheaics, 013 (013), ID [6] Aslan, B. B., e al., A Taylor Mari-Collocaion Mehod Based on Residual Error for Solving Lane-Eden Type Differenial Equaions, New Trends Mah. Sci., 4 (015),, pp Paper subied: July 17, 017 Paper revised: Noveber 15, 017 Paper acceped: Noveber 30, Sociey of Theral Engineers of Serbia Published by he Vinča Insiue of Nuclear Sciences, Belgrade, Serbia. This is an open access aricle disribued under he CC BY-NC-ND 4.0 ers and condiions