A FINITE-DIFFERENCE SCHEME FOR SOLUTION OF A FRACTIONAL HEAT DIFFUSION-WAVE EQUATION WITHOUT INITIAL CONDITIONS

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1 Beilev, V. D., e l.: A Finie-Difference Scheme for Soluion of... THERMAL SCIENCE: Yer 5, Vol. 9, No., A FINITE-DIFFERENCE SCHEME FOR SOLUTION OF A FRACTIONAL HEAT DIFFUSION-WAVE EQUATION WITHOUT INITIAL CONDITIONS y Velugin D. BEIBALAEV nd Mumin R. SHABANOVA Dgesn Se Insiue of Nionl Economy, Dgesn Se Universiy, Mhchkl, Dgesn, Russi Originl scienific er DOI:.98/TSCI4848B Efficien finie-difference scheme o solve frcionl diffusion-wve equions wihou iniil condiions hs een develoed. The efficien roximion of he Riesz frcionl derivives is demonsred nd efficienly exemlified y wo simle rolems wih/wihou source erms. Key words: frcionl diffusion-wve equions, Riesz derivive, finie-difference scheme Inroducion Nowdys he frcionl differenil equions [, ] widely encounered in licions o rnsien rheology [3-5], he [6-8] nd mss rnsfer [9, ], non-liner diffusion in orous [, ] nd grnulr medi [], nd Sefn rolem [-4] re ower ools for efficien soluions of comlex engineering rolems. The frcionl differenil equions cn e solved eiher nlyiclly y he homooy erurion mehod [5-7], he vriionl ierion mehod [], he He-Blnce Inegrl Mehod [5,, 8, 9], he ex-funcion mehod [], nd ohers [, ]. The numericl soluions re oriened minly o finie-difference roximions [3-5], mrix mehod [6], oundry elemen mehod [7], generlized differenil rnsform mehod [8], nd ohers [9, 3]. The rolems wihou iniil soluions re rciclly imorn nd descrie commonly he diffusion rocesses fr wy from he iniil sr-u momen when he iniil condiions do no ffec he emerure disriuions. A clssicl exmle is he emerure disriuion in he soil ccouning oh he dily nd nnul flucuions of he emerure he erh surfce. The soluions of such rolems llow idenifying he hermohysicl chrcerisics of vrious merils nd comosiion srucures forming he grounds hving frcl srucures wih resec o oh he ime nd sce co-ordines. This work ddresses numericl finie-difference soluion he frcionl diffusion-wve equion wih Riesz derivives looking for soluion u(x, y)c (D), in he domin D ={<x < ;< < T} of oundry rolem descried y non-locl he diffusion equion wih Riesz frcionl derivives, nmely: * Corresonding uhor; e-mil: ksij_3@mil.ru

2 Beilev, V. D., e l.: A Finie-Difference Scheme for Soluion of THERMAL SCIENCE: Yer 5, Vol. 9, No., u D u f (,) x () x wih oundry condiions: u(, ) m( ),,, f ( x, ), m ( ) C( D) (, c, d) The Riesz derivives re defined s []: u ux (, s) uu (, s) s ds () G( )cos u ( ) ux ( s, ) ux (, ) ux ( s, ) ds () s G( )cos Soluion Numericl roximion of he frcionl derivives The rolem soluion is considered in he domin D * ={x;t}. From he definiion of he Riesz derivive [] in he ime inervl [ n, n+ ], we ge: u n G( )cos n ux (, n s) ux (, n s) d s s Exressing u(x, + s) nd u(x, s) in Tylor series wih resec o he exonen s nd hen using finie differences for u (x, )in[ n, n+ ]s(du/d) n [u(x, n+ ) u(x, n )/] we ge finie-difference roximion of he frcionl Riesz derivive of order in he inervl [ n, n+ ]: u n [ u( x, n) u( x, n)] ~ n ( ) G( )cos (3) Similrly, develoing u(x, + s) nd u(x s,) s Tylor series nd reresening u xx (x, ) in he rnge [x, x m+ ] in finie-difference form (d u/dx ) m [u(x m+, ) u(x m, )+u(x m, )]/h we ge he finie-difference roximion of he frcionl derivive of order s: (3) u m m m m m x u x u x u x ~ ( ) [ (, ) (, ) ( ( ) G( )cos h, )] (4) Furher, develoing he funcions u(x, n + ), u(x, n ), u(x m + h, ), nd u(x m h, )s Tylor series (wih resec o nd h we ge: n [ u( x, n) u( x, n)] u ( ) G( )cos n O( ) (5)

3 Beilev, V. D., e l.: A Finie-Difference Scheme for Soluion of... THERMAL SCIENCE: Yer 5, Vol. 9, No., ( ) xm [ u( xm, ) u( xm, ) u( xm, )] u ( ) G( )cos h x m Oh ( ) Therefore, he finie-difference roximions wih resec o he ime nd he sce co-ordine hve orders of, nd, resecively. The numericl soluion D * needs mesh W ={(x m, n ): n =,,..., N; m =,,..., M}, where x m = mh, n = n, h = /M, nd = T/N. Then, using exressions (3), (4), nd () we ge weighed finie-difference scheme: n ( um n um n ) ( ) G( )cos ( ) xm [ sl u m n ( s) um n ] fm n (6) ( ) G( )cos h where um n u( xm, n ), um n u( xm, n), u u x m n ( m, n), nd u u x m n ( m, n) (6) For s = we ge n exlici finie-difference: u u x n ( m n m n ) ( ) m ( u u u ) f G( )cos G( 3 )cos h m n m n m n m n Finding he soluion on he zero-h lyer y he Euler mehod (8,, c): un m( n ), u u hf m M m m m,,,...,, (8,,c) we cn develo he soluions in ll oher knos of he mesh y lying exression (6) s i is demonsred in he nex exmle. Exmles Exmle Consider rolem wih: u u D e x, x u(, ) cos( 7, ) (9,) in he domin D * ={ x ; 5} y hel of exression (6) nd lying exressions (6) nd (8) we ge: u C h C h h u u u C C x x C m n m n m n m n un n cos( 7. C C n),, G( )cos e xm n xm x x (5) (7) ( ) () (,c,d) G( 3 )cos

4 Beilev, V. D., e l.: A Finie-Difference Scheme for Soluion of THERMAL SCIENCE: Yer 5, Vol. 9, No.,. 53- Figure. Soluions of he rolem ; () for =, () for =4 This soluion is resen grhiclly in fig. for differen frcionl orders nd, nd imes. Exmle Le us solve he rolem: Figure. Sce disriuion of he dimensionless emerure u(x, ) for vrious vlues of he frcionl order ; solid line =, dshed line =.5, doed line =.8 u D u, u(, ) cos( 7. ) () x in he domin D ={ x 4, }. Alying he sme echniques s hose o Exmle we ge he resuls illusred in fig. <. Plos clerly show h chrcerisic enuion lengh increses while he sme ime he chrcerisic dely ime decreses which re resonle hysiclly sound resuls.

5 Beilev, V. D., e l.: A Finie-Difference Scheme for Soluion of... THERMAL SCIENCE: Yer 5, Vol. 9, No., Conclusions The frcionl clculus llows develoing new roch o he heory of non-locl differenil equions. Unlike he clssicl roch where he ccoun of he non-locl effecs re reresened y differenil/inegrl oerors re sere erms of he governing equions, he frcionl clculus ermis hose effecs o e incorored in he models y frcionl derivives/inegrls. I is quie imorn o noice h here lrge moun of differenil models sed on he coninuum roch nd corresonding ses of corresonding mulirmeric fundmenl soluions. In his conex, he frcionl clculus formules new rolems, such hose h he soluion of frcionl differenil equions cree funcionl sce, rolem sill undeveloed in mhemics. The finie-difference scheme develoed in his work nd he soluions of he exmles sed on i show he efficiency of he roch nd forms sis o deermine he diffusiviies of heerogeneous medi. The resen ricle demonsres n efficien finie-difference scheme o solve frcionl diffusion-wve equions wihou iniil condiions. The efficien roximion of he Riesz frcionl derivives is demonsred nd efficienly exemlified y wo simle rolems. The frcionl he diffusion rolems considered in he resen work llow ccouning he ime non-locliy in nurl wy. Such differenil equions form lrge clss of soluions deending on he frcionl rmeer (order). For we ge he clssicl soluion while in ll oher cse wih non-ineger vlue of he new clss of soluions hs symoic ehviour differing from hose of he well-known clssicl counerrs. I is very imorn h he funcionl forms of he new soluions re chrcerized y he frcionl order hus defining new rmeer in he modelling of exerimenl d. Nomenclure domin oundry m couner of he nodes in he finie-difference scheme, [ ] n couner of he nodes in he finie-difference scheme, [ ] T ime limi ime, [s] u(x, ) dimensionless emerure u(x, y) dimensionless soluion x sce co-ordine y sce co-ordine Greek symols frcionl order, [ ] frcionl order, [ ] G Gmm funcion References [] Oldhm, K. B., Snier, J., The Frcionl Clculus, Acdemic Press, New York, USA, 974 [] Podluny, I., Frcionl Differenil Equions, Acdemic Press, Sn Diego, Cl., USA, 999 [3] Siddique, I., Vieru, D., Sokes Flows of Newonin Fluid wih Frcionl Derivives nd Sli he Wll, In. Rev. Chem. Eng., 3 (), 6, [4] Qi, H., Xu, M., Some Unsedy Unidirecionl Flows of Generlized Oldroyd-B Fluid wih Frcionl Derivive, Al. Mh. Model., 33 (9),, [5] Hrisov, J., Trnsien Flow of A Generlized Second Grde Fluid Due o Consn Surfce Sher Sress: An Aroxime Inegrl-Blnce Soluion, In. Rev. Chem. Eng., 3 (), 6, [6] Agrwl, O. P., Alicion of Frcionl Derivives in Therml Anlysis of Disk Brkes, Nonliner Dynmics, 38 ( 4),,. 9-6 [7] Hrisov, J., He-Blnce Inegrl o Frcionl (Hlf-Time) He Diffusion Su-Model, Therml Science, 4 (),, [8] Kulish, V. V., Lge, J. L., Frcionl-Diffusion Soluions for Trnsien Locl Temerure nd He Flux, J. He Trnsfer, (),,

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