REMARKS ON A GREEN FUNCTIONS APPROACH TO DIFFUSION MODELS WITH SINGULAR KERNELS IN FADING MEMORIES

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1 Alkahani, B. S. T., e al.: Remark on a Green Funcion Approach o Diffuion Model... THERMAL SCIENCE: Year 7, Vol., No., pp REMARKS ON A GREEN FUNCTIONS APPROACH TO DIFFUSION MODELS WITH SINGULAR KERNELS IN FADING MEMORIES by Badr Saad T. ALKAHTANI a and Abdon ATANGANA b* a Deparmen of Mahemaic, College of Science, King Saud Univeriy, Riyadh, Saudi Arabia b Iniue for Groundwaer Sudie, Faculy of Naural and Agriculural Science, Univeriy of he Free Sae, Bloemfonein, Souh Africa Original cienific paper hp://doi.org/.98/tsci6434a Diffuion problem wih ingular kernel in fading memorie are very inereing phyical problem ha have araced aenion of many reearcher. In hi paper, we aim o provide eac oluion of hee problem uing he green funcion mehod wih ome inegral ranform operaor. The ingular kernel ued in hi paper i baed upon he power law funcion, which i ued o conruc he well-known Riemann-Liouville derivaive wih fracional order. Key word: diffuion model, ingular kernel, green funcion, Laplace ranform Inroducion The pread of molecule from an area of high-piched of ineniy o an area of lile concenraion ha been a ubec of many inveigaion wihin he pa decade, due o he compleiie of hi movemen wihin variou ype of media [-3]. Deprived of reervaion, hi heory of diffuion i eenively employed acro many field including bu no limied o finance wih he pread of mankind, alo wih he concep of price value: in ociology, biology, phyic, and chemiry [4-6]. In all hee field, he main idea i preading ou from an area of prohibiive concenraion of ha obec. I i imporan o menion ha, here ei wo differen way of iniiaing he concep of diffuion including, a phenomenological mehodology which of coure ar wih Fick law of preading ogeher wih heir mahemaical reul and an aomiic approach which conider he random walk of he diffuion paricle. We hall recall ha in he cope of ime, he heory underpinning he diffuion in olid wa long employed before ha of diffuion ielf. For inance, one can remember he preading of colour of ained gla alo he Chinee ceramic. Alo in he conemporary cholarhip, many reearcher have devoed heir focu in enhancing model for differen ype of diffuion. We hall ar by quoing he eample of Thoma Graham inveigaion ha conied in diffuion of gae. Due o he wide applicabiliy of hi concep, many reearcher have uggeed new analyical mehod o olve diffuion model, for inance: Hriov [7] applied he inegral balance mehod o olve model wih elaic effec and ha wih memory kernel including a vicou damping, and many oher reul can be found in he following inveigaion [9-]. * Correponding auhor, abdonaangana@yahoo.fr

2 Alkahani, B. S. T., e al.: Remark on a Green Funcion Approach o Diffuion Model THERMAL SCIENCE: Year 7, Vol., No., pp In hi ecion, we ar he udy of a imple parabolic mahemaical equaion able o porray he rigid hea conducor and relaing vicoelaiciy wih fading memory. The mahemaical equaion under inveigaion here i given: C u (,) C η [(,)] u = D í a ý, î þ () - η -η d D [ f( ) ] = ( -y) f( y)dy Γ( η) dy where η i he fracional order in [, ]. Here we aume he following iniial-boundary condiion: u (,) = f( ), u(, ) = g ( ), u(,) = u (,) = () The inegral operaor conidered ued in hi udy i given: L[ f( )]( ) = f( )ep( -) d (3) Therefore, applying he Laplace ranform on boh ide of eq. (), we ge: α u (, ) u (, )- u (,) = a The ne operaor ha will be ued here i given: Fy [ ( )] = y ( )ep(- πiw)d (5) - Then applying he Fourier ranform on boh ide of eq. (4), we ge: uw uw awuw uw (,) uw (, ) = - α w a uw (,) uw (, ) = α (, )- (,) =- (, ) α -α - wa Thu, an applicaion of he invere Laplace ranform on boh ide of eq. (6) produce: é - - uw (,) uw (, ) = L [ uw (, )] = L (7) ê α -α - wa Then uing he convoluion heorem for Laplace ranform, we obain: - uw (,) α- α- α- uw (, ) = L [ uw (, )] = ( - ) Eα-, α- ( aw ) d Γ( α ) (8) The previou oluion i he eac oluion in Fourier pace, now o obain he oluion in real pace we apply he invere Fourier ranform in boh ide of eq. (8) o obain: (4) (6)

3 Alkahani, B. S. T., e al.: Remark on a Green Funcion Approach o Diffuion Model... THERMAL SCIENCE: Year 7, Vol., No., pp é uw (,) u(, ) F { L [ u( w, )]} F α- α- α- = = ( ) Eα-, α- ( aw )d - Γ( α) (9) ê However, he invere can no be obained raighforward. The generalized Miag-Leffler can be reformulaed in erie: α- ( aw ) α- ( aw ) Eα-, α- = å () l Γ é = l( α- ) + α- Replacing eq. () ino eq. (9) yield: = F α- l α- ( ) (,) aw - uw u (,) F α- α- = í ( - ) dý = Γ( α ) å ( ) l Γ é = l α- + α- î þ l l uw (,) aw - α ( α ) l α F = ( ) d í - ý = å Γ( α ) l= Γ él( α- ) + α- î þ l l uw (,) aw l α α l a α í ý = åγ él( α- ) + α- å l= l= Γ él( α- ) + α- î þ - + ( -) - -l --l l l l l + ( α-) l- i Γ( + l){[( -i) - i ]co( l)gn( ) + i[( -i) + i ]in( l) } f( - v) d Equaion () i he eac oluion of he imple cae of diffuion wih elaic influence. Diffuion model wih memory kernel including a vicou damping. In hi ecion, we eamine he more general equaion where he relaaion can be hown in erm of fracional derivaive. The ranien flow of econd grade vicoelaic non-olidified can be moulded by mean of he leading mahemaical formulaion, which in a preading form i ariculaed: u (,) u (,) é C u (,) = v + D p, < ê () and u (,) = f( ), u(, ) = f ( ), u (,) = u (, ) = Uing he inegral balance mehod, Hriov preened an approimae oluion of he eq. () [7]. Many oher cholar in hi field have alo eamined he previou equaion when he ime fracional derivaive i replaced by he local derivaive, which i a very eay cae o handle. However, here i eac oluion of hi equaion in cae of fracional order derivaive. In hi ecion, we hall ue he Laplace ranform ogeher wih Fourier ranform o derive he eac oluion of eq. (). Analyical oluion via Laplace Fourier mehod Applying he Laplace ranform in ime componen in eq. (), we obain: ()

4 Alkahani, B. S. T., e al.: Remark on a Green Funcion Approach o Diffuion Model THERMAL SCIENCE: Year 7, Vol., No., pp u (, ) u (, ) u (, ) u (,) v é - = + p ê Thu applying he Fourier ranform operaor on boh ide of eq. (3) yield: uw uw vwuw pwuw (, )- (,) =- (, )- (, ) ( ) uw (, ) + vw + pw = uw (,) uw (,) uw (,) uw (, ) = = ( + vw + p w ) + w Therefore, an applicaion of invere Fourier ranform on boh ide of eq. (4) provide: u (, ) é ê uw (,) - = F ê + w êv + p Applying he convoluion heorem for Fourier ranform, we ge: e - ay é ( e ay) gn( ){ gn[re( a )]} ê a u (, ) = f( -y) dy í ay v p æe HeaviideThea{ - y gn[re( a)]} + ö ý + + HeaviideThea{ y gn[re( a)]}gn[re( a)] î è + ø þ a = The eac oluion i hen obained only by applying he invere Laplace ranform on boh ide o obain: u (,) = e ay a - é ( e ay ) gn( ){ gn[re( a )]} ê a - L f( y) = í - í a y v p æe HeaviideThea{ - y gn[re( a)]} + ö ý + ýdy + HeaviideThea{ y gn[re( a)]}gn[re( a)] è+ ø î î þ þ For impliciy, we le: h (,) = e ay a - é ( e ay) gn( ){ gn[re( a )]} ê a - L f( y) = í - í æ a y e HeaviideThea{ - y gn[re( a)]} + ö ý ýdy + HeaviideThea{ y gn[re( a)]})gn[re( a)] î î è + ø þþ (3) (4) (5) (6)

5 Alkahani, B. S. T., e al.: Remark on a Green Funcion Approach o Diffuion Model... THERMAL SCIENCE: Year 7, Vol., No., pp However, he invere Laplace ranform of /(ν + p ) i given: æ ö æ ö æ v ö è ø p è + ø p è ø L = p L = p E, - v By convoluion heorem for Laplace ranform, we obain he eac oluion of he equaion under inveigaion: - - v u (, ) = h (, -k) p k E, í- k ýdk p î þ (8) (7) Analyical oluion via double-laplace ranform mehod In hi ecion, we employ he double-laplace ranform operaor o obain an equivalen oluion of eq. (). Thu, applying he double-laplace ranform on pace and ime componen, we ge: U l U l vl U p l uu p (, )- (,) = (, ) + (, ) { } U ( p, ) -vl - p l = U ( l,) Uw (,) Ul (,) Ul (, ) = = = { -vl -p l } =- l Ul (,) - -l Applying he invere Laplace in pace componen we obain: æ ö in y è ø u (, ) =- f( - y) dy= i æ ö - è ø i+ = f( - y) y dy í ý = å i= (i + )! î þ i æ ö v p v p + è + ø = f( -y) í i+ å ýy dy= (i + )! i= î þ (9)

6 Alkahani, B. S. T., e al.: Remark on a Green Funcion Approach o Diffuion Model THERMAL SCIENCE: Year 7, Vol., No., pp i i- é æ ö i+ = f( - y) å y dy= ê i (i )! v p = + è + ø i i- é æ ö i+ = å f( - yy ) dy= ê i= (i + )! è ø i- æ ö i i- é æpö = å ê i (i )! v v = + è ø + p è ø i+ f( -y) y dy () Bu we have he following relaion: é i- æ ö æ ( i ) ( i ) v ö L E, = - v ( i )! p - + è ø p è ø ê () alo -- i - i L { } = Γ( - i) () By convoluion heorem we have: i i- é --i æpö ( - ) æ ( i ) ( i ) v ö i+ u (,) = E, - d f( -yy ) dy ê í ý i= (i+ )! è vø Γ( -i)( i-)! è p ø î þ The previou equaion i he eac oluion of eq. (). Concluion Parial differenial equaion conruced wih he concep of fracional differeniaion are uiable o model real world problem. However, hee equaion are omeime very difficul o handle analyically. For hoe phyical problem ha can be decribed wih linear fracional parial differenial equaion, an aemp o find an eac oluion i done uing ome echnique like Laplace, Fourier, Sumudu, and Mellin ranform mehod. The ue of hee inegral ranform commonly lead o obaining he green funcion. In hi paper uing boh Laplace and Fourier ranform wih he green funcion echnique, we conruced he eac oluion of ome diffuion problem wih ingular kernel. Acknowledgemen The auhor eend heir incere appreciaion o he Deanhip of Science Reearch a King Saud Univeriy for funding hi prolific reearch group PRG Reference [] Ferreira, J. A., de Oliveira, P., Qualiaive Analyi of a Delayed Non-Fickian Model, Applicable Analyi, 87 (8), 8, pp

7 Alkahani, B. S. T., e al.: Remark on a Green Funcion Approach o Diffuion Model... THERMAL SCIENCE: Year 7, Vol., No., pp [] Caaneo, C., On he Conducion of Hea (in Ialian), Ai Sem. Ma. Fi. Univeri a Modena, 3 (948),, pp. 83- [3] Curin, M. E, Pipkin, A. C., A General Theory of Hea Conducion wih Finie Wave Speed, Archive of Raional Mahemaical Analyi, 3 (968),, pp [4] Olmead, W. E., Bifurcaion wih Memory, SIAM J. Appl. Mah., 46 (986),, pp [5] Langford, D., The Hea Balance Inegral Mehod, Inernaional Journal of Hea and Ma Tranfer, 6 (973),, pp [6] Royon, L., e al., Inveigaion of Hea Tranfer in a Polymeric Phae Change Maerial for Low Level Hea Sorage, Energy Conver. Mgm., 38 (997), 6, pp [7] Hriov, J., Subdiffuion Model wih Time-Dependen Diffuion Coefficien: Inegral-Balance Soluion and Analyi, Thermal Science, on-line fir, doi:.98/tsci64747h [8] Myer, T. G., e al., A Cubic Hea Balance Inegral Mehod for One-Dimenional Meling of a Finie Thickne Layer, Inernaional Journal of Hea and Ma Tranfer, 5 (7), 5-6, pp [9] Anic, A., Hill, J. M., The Double-Diffuiviy Hea Tranfer Model for Grain Sore Incorporaing Microwave Heaing, Applied Mahemaical Modelling, 7 (3), 8, pp [] Thambynayagam, R., K., M., The Diffuion Handbook: Applied Soluion for Engineer, McGraw-Hill, New York, USA, [] Hriov, J., Diffuion Model wih Weakly Singular Kernel in he Fading Memorie: How he Inegral Balance Mehod Can be Applied, Thermal. Science, (5), 3, pp [] Hriov, J., Approimae Soluion o Time-Fracional Model by Inegral Balance Approach, in: Fracional Dynamic (Ed. C. Caani, H. M. Srivaava, Xia-Jun Yang), De Gruyer Open, Waraw, 5, Chaper 5, pp Paper ubmied: April, 6 Paper revied: May 3, 6 Paper acceped: June 5, 6 7 Sociey of Thermal Engineer of Serbia Publihed by he Vinča Iniue of Nuclear Science, Belgrade, Serbia. Thi i an open acce aricle diribued under he CC BY-NC-ND 4. erm and condiion

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