NEW FRACTIONAL DERIVATIVES WITH NON-LOCAL AND NON-SINGULAR KERNEL Theory and Application to Heat Transfer Model

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1 Angn, A., e l.: New Frcionl Derivives wih Non-Locl nd THERMAL SCIENCE, Yer 216, Vol. 2, No. 2, pp NEW FRACTIONAL DERIVATIVES WITH NON-LOCAL AND NON-SINGULAR KERNEL Theory nd Applicion o He Trnsfer Model by Abdon ATANGANA * nd Dumiru BALEANU b, c Insiue for Groundwer Sudies, Fculy of Nurl nd Agriculurl Sciences, Universiy of he Free Se, Bloemfonein, Souh Afric b Deprmen of Mhemics nd Compuer Sciences, Cny Universiy, Anr, Turey c Insiue of Spce Sciences, Mgurele-Buchres, Romni Inroducion Originl scienific pper DOI: /TSCI A In his pper new frcionl derivive wih non-locl nd no-singulr ernel is proposed. Some useful properies of he new derivive re presened nd pplied o solve he frcionl he rnsfer model. Key words: frcionl derivive, non-locl ernel, non-singulr ernel, generlized Mig-Leffler funcion, frcionl he rnsfer model A new derivive ws recenly lunched by Cpuo nd Fbrizio [1] nd i ws followed by some reled heoreicl nd pplied resuls (for exmple [2-4] nd he references herein). We recll h he exising frcionl derivives hve been used in mny rel world problems wih gre success (for exmple [5-12] nd he references herein) bu sill here re mny hins o be done in his direcion. Definiion 1: [1] Le f H 1 ( b, ), < b, [,1] hen, he definiion of he new Cpuo frcionl derivive is: ' (1) M( ) x D [ f( )] = f ( x)exp dx 1 1 where M() denoes normlizion funcion obeying M() = M(1) = 1. However, if he funcion does no belong o H 1 (,b) hen, he derivive hs he form: M( ) x D [ f( )] = [ f( ) f( x)] exp dx 1 1 (2) If σ = (1 )/ [, ], = 1/(1 + σ) [, 1], hen eq. (2) ssumes he form: * Corresponding uhor; e-mil: bdonngn@yhoo.fr

2 Angn, A., e l.: New Frcionl Derivives wih Non-Locl nd 764 THERMAL SCIENCE, Yer 216, Vol. 2, No. 2, pp σ N( σ ) ' x D [ f( )] = f ( x)exp d x, N() N( ) 1 σ = = σ The im of [1] ws o inroduce of new derivive wih exponenil ernel. Is niderivive ws repored in [2] nd i ws found o be he verge of given funcion. The derivive inroduced in [1] cnno produce he originl funcion when = 1. However, his issue ws, so fr, independenly solved in [13, 14], respecively. We believe h he min messge presened in [1] ws o find wy o describe even beer he dynmics of sysems wih memory effec. For given d we s he following quesion: wh is he mos ccure ernel which beer describe i? We sugges possible nswer in he following secions. New derivives wih non-locl ernel We recll h he Mig-Leffler funcion is he soluion of he following frcionl ordinry differenil equion [12, 15, 16]: d y = y, < < 1 (3) dx The Mig-Leffler funcion nd is generlized versions re herefore considered s non-locl funcions. Le us consider he following generlized Mig-Leffler funcion: = ( ) E ( ) = (4) Γ( + 1) The Tylor series of exp [ ( y)] he poin is given by: [ ( y)] exp[ ( y)] = (5)! If we chose = /(1 ) nd replce expression (5) ino Cpuo-Fbrizio derivive we conclude h: = = M( ) ( ) d f( y) D [ f( )] = [( y)] dy 1! dy (6) To solve he problem of non-locliy, we derive he following expression. In eq. (6), we replce! by Γ( + 1) lso ( y) is replced by ( y) o obin: M( ) ( ) d f( y) D [ f( )] = [( y)] dy 1 Γ( + 1) dy = Thus, he following derivive is proposed. Definiion 2: Le f H 1 ( b, ), < b, [,1] hen, he definiion of he new frcionl derivive is given: ' (7) B( ) ( x) D [ f( )] = f( xe ) dx 1 1

3 Angn, A., e l.: New Frcionl Derivives wih Non-Locl nd THERMAL SCIENCE, Yer 216, Vol. 2, No. 2, pp Of course, B() hs he sme properies s in Cpuo nd Fbrizio cse. The previous definiion will be helpful o discuss rel world problems nd i lso will hve gre dvnge when using he Lplce rnsform o solve some physicl problem wih iniil condiion. However, when = we do no recover he originl funcion excep when he origin he funcion vnishes. To void his issue, we propose definiion 3. Definiion 3: Le f H 1 ( b, ), < b, [,1] hen, he definiion of he new frcionl derivive is given: B( ) d ( x) D [ f( )] = f( xe ) dx 1 d 1 (8) Equions (7) nd (8) hve non-locl ernel. Also in eq. (7) when he funcion is consn we ge zero. Properies of he new derivives In his secion, we sr by presening he relion beween boh derivives wih Lplce rnsform. By simple clculion we conclude h: nd B( ) p A{ f( )}( p) A { D [ f( )]}( p) = (9) 1 p B( ) p A{ f( )}( p) p f() A { D [ f( )]}( p) = (1) 1 p + 1 respecively. The following heorem cn herefore be esblished. Theorem 1: Le f H 1 ( b, ), < b, [,1] hen, he following relion is obined: D [ f()] = D [ f()] + H() (11) Proof: By using he definiion (11) nd he Lplce rnsform pplied on boh sides we obin esily he resul: B( ) p A{ f( )}( p) p f() B( ) A { D [ f( )]}( p) = (12) 1 p + p Following eq. (9) we hve: 1 1 A p f() B( ) { D [ f( )]}( p) = A { D [ f( )]}( p) p + 1 (13) 1 Applying he inverse Lplce on boh sides of eq. (13) we obin:

4 Angn, A., e l.: New Frcionl Derivives wih Non-Locl nd 766 THERMAL SCIENCE, Yer 216, Vol. 2, No. 2, pp B( ) 1 1 D [ f( )] = D [ f( )] f() E This complees he proof. Theorem 2: Le f be coninuous funcion on closed inervl [, b]. Then he following inequliy is obined on [, b]: Proof: B( ) D [ f()] < K, h () = mx h () 1 b B( ) d ( x) B( ) d B( ) D [ f( )] = f( x) E d x < f( x)d x = f( x) 1 d 1 1 d 1 Then ing K o be f( x) he proof is compleed. Theorem 3: The A.B. derivive in Riemnn nd Cpuo sense possess he Lipschiz condiion, h is o sy, for given couple funcion f nd h, he following inequliies cn be esblished: nd lso: (14) (15) D [ f()] D [ h ()] H f() h () (16) D [ f()] D [ h ()] H f() h () (17) We presen he proof of (16) s he proof of (17) nd i cn be obined similrly. Proof: B( ) d ( x) D [ f( )] D [ h ( )] = f( xe ) dx 1 d 1 B( ) d ( x) hxe ( ) dx 1 d 1 Using he Lipschiz condiion of he firs order derivive, we cn find smll posiive consn such h: B( ) 1 D [ f( )] D [ f( )] < θ E f( x)d x h( x)dx 1 1 nd hen he following resul is obined: (18) B( ) 1 D [ f( )] D [ f( )] < θ E f( x) hx ( ) H f( x) hx ( ) 1 1 = which produces he requesed resul.

5 Angn, A., e l.: New Frcionl Derivives wih Non-Locl nd THERMAL SCIENCE, Yer 216, Vol. 2, No. 2, pp Le f be n n-imes differenible wih nurl number nd f () () =, = 1, 2, 3, n, hen by inspecion we obin: n n d f( x) d n n D = { D [ f( )]} (19) d d Now, we cn esily prove by ing he inverse Lplce rnsform nd using he convoluion heorem h he following ime frcionl ordinry differenil equion: hs unique soluion, nmely: D [ f()] = u () (2) 1 1 f() = u () + uy ( )( y) dy B( ) B( )Γ( ) Definiion 4: The frcionl inegrl ssocie o he new frcionl derivive wih non-locl ernel is defined: When = we recover he iniil funcion, nd if = 1, we obin he ordinry inegrl. 1 I { f( )} = f( ) + f( y)( y) dy B( ) B( )Γ( ) AB 1 (21) Applicion o herml science: A new he rnsfer model The nlyses of he experimenl d coming from he invesigion of dynmics of complex sysems is sill n ineresing open problem. Therefore, new mehods nd echniques re sill o be discovered o nd pply wih more success o hve n even beer descripion of he dynmics of rel world problems. Priculrly, finding new derivives suggesed in his mnuscrip is becuse of he necessiy of employing beer model porrying he behviour of orhodox viscoelsic merils, herml medium, nd oher. The new pproch is ble o porry meril heerogeneiies nd some srucure or medi wih differen scles. The nonlocliy of he new ernel llows beer descripion of he memory wihin srucure nd medi wih differen scles. In ddiion of his, we lso rely h his new derivive cn ply specific role in he sudy of mcroscopic behvior of some merils, reled wih non-locl exchnges, which re predominn in defining he properies of he meril [1]. Thus, his new derivive will be very useful in describing mny complex problems in herml science. We recll here h noher derivive ws inroduced in [17, 18] wih he im of solving some problems wihin he scope of herml science. To describe he ime re of he rnsfer hrough meril wih differen scle or heerogeneous, we propose new lw of he conducion which will be refereed s frcionl Fourier s lw. The frcionl Fourier lw ses h ime re of he rnsfer vi meril wih differen scle is proporionl o negive grdien in emperure nd re righ ngles o h grdien vi which he he flows nd is given: D Q = T d d A (22) S

6 Angn, A., e l.: New Frcionl Derivives wih Non-Locl nd 768 THERMAL SCIENCE, Yer 216, Vol. 2, No. 2, pp where D Qr (,) is he moun of he rnsferred wihin meril wih differen scle per uni ime. Now, for heerogeneous meril of 1-D geomery beween wo endpoins consn emperure, produced new he flow re: dq = A Dr T (23) d For insnce, in cylindricl heerogeneous shells lie pipes, he he conducion vi n heerogeneous shell will be deermined from inernl rdius, r 1, nd he exernl rdius, r 2, he lengh, LL, nd he difference beween inner nd ouer wll we hve: dq = 2πlrA (24) d Rerrnging nd pply he Lplce rnsform on boh sides, hen pplying he inverse Lplce rnsform, we obin: Q + 1 r1 T1 T2 = 1+ r2 Hyper geomeric PFQ {1, 1, 1 }, {2, 2}, r1 2πl 1 r2 1 Hrmonic number [ ] ln r + r r2 where K is he consn, L [m] he lengh, R [m] he rdius, nd T [ C] he emperure. Conclusions The im of his pper ws o sugges new derivives wih non-locl nd nonsingulr ernel. To chieve his gol, we me use he generlized Mig-Leffler funcion o build he non-locl ernel. One derivive is bsed upon he Cpuo viewpoin nd he second on he Riemnn-Liouville pproch. We derive he frcionl inegrl ssocie using he Lplce rnsform operor. The new derivive ws used o model he flow of he in meril wih differen scle nd lso hose wih heerogeneous medi. References [1] Cpuo, M., Fbrizio M., A New Definiion of Frcionl Derivive wihou Singulr Kernel, Progress in Frcionl Differeniion nd Applicions, 1 (215), 2, pp [2] Losd, J., Nieo, J. J., Properies of New Frcionl Derivive wihou Singulr Kernel, Progress in Frcionl Differeniion nd Applicions, 1 (215), 2, pp [3] Angn, A., On he New Frcionl Derivive nd Applicion o Nonliner Fisher s Recion- Diffusion Equion, Applied Mhemics nd Compuion, 273 (216), Jn., pp [4] Angn, A., Nieo, J. J., Numericl Soluion for he Model of RLC Circui Vi he Frcionl Derivive wihou Singulr Kernel, Advnces in Mechnicl Engineering, 7 (215), 1, pp. 1-7 [5] Benson, D., e l., Applicion of Frcionl Advecion-Dispersion Equion, Wer Resources Reserch, 36 (2), 6, pp [6] Cpuo, M., Liner Model of Dissipion whose Q is Almos Frequency Independen-II, Geophysicl Journl Royl Asronomicl Sociey, 13 (1967), 5, pp [7] Whecrf, S. W., Meerscher, M. M., Frcionl Conservion of Mss, Advnces in Wer Resources, 31 (28), 1, pp [8] Nsholm, S. P., Holm, S., Lining Muliple Relxion, Power-Lw Aenuion, nd Frcionl Wve Equions, Journl of he Acousicl Sociey of Americ, 13 (211), 5, pp (25)

7 Angn, A., e l.: New Frcionl Derivives wih Non-Locl nd THERMAL SCIENCE, Yer 216, Vol. 2, No. 2, pp [9] Hrisov, J., Double Inegrl-Blnce Mehod o he Frcionl Subdiffusion Equion: Approxime Soluions, Opimizion Problems o be Resolved nd Numericl Simulions, Journl of Vibrion nd Conrol, doi: / [1] Pedro, H. T. C., e l., Vrible Order Modeling of Diffusive-Convecive Effecs on he Oscillory Flow Ps Sphere, Journl of Vibrion nd Conrol, 14 (28), 9-1, pp [11] Wu, G. C., Blenu, D., Jcobin Mrix Algorihm for Lypunov Exponens of he Discree Frcionl Mps, Communicions in Nonliner Science nd Numericl Simulion, 22 (215), 1-3, pp [12] Kilbs, A. A., e l., Theory nd Applicions of Frcionl Differenil Equions, Elsevier, Amserdm, The Neherlnd, 26 [13] Cpuo, M., Fbrizio, M., Applicions of New Time nd Spil Frcionl Derivives wih Exponenil Kernels, Progress in Frcionl Differeniion nd Applicions, 2 (216), 1, pp [14] Doungmo Goufo, E. M., Applicion of he Cpuo-Fbrizio Frcionl Derivive wihou Singulr Kernel o Koreweg-de Vries-Bergers Equion, Mhemicl Modelling nd Anlysis, 21 (215), 2, pp [15] Hrisov, J., Diffusion Models wih Wely Singulr Kernels in he Fding Memories: How he Inegrl- Blnce Mehod Cn be Applied?, Therml Science, 19 (215), 3, pp [16] Hrisov, J., Approxime Soluions o Time-Frcionl Models by Inegrl Blnce Approch, in: Frcionl Dynmics, Chper 5, (Ed. C. Cni, H. M. Srivsv, X. J. Yng), De Gruyer Open, Wrsw, 215, pp [17] Liu, F. J., e l., He's Frcionl Derivive for He Conducion in Frcl Medium Arising in Silworm Cocoon Hierrchy, Therml Science, 19 (215), 4, pp [18] He, J. H., A New Frcl Derivion, Therml Science, 15 (211), Suppl. 1, pp. S145-S147 Pper submied: Jnury 11, 216 Pper revised: Jnury 17, 216 Pper cceped: Jnury 19, 216