LOCAL FRACTIONAL LAPLACE SERIES EXPANSION METHOD FOR DIFFUSION EQUATION ARISING IN FRACTAL HEAT TRANSFER
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1 Yn, S.-P.: Locl Frctonl Lplce Seres Expnson Method for Dffuson THERMAL SCIENCE, Yer 25, Vol. 9, Suppl., pp. S3-S35 S3 LOCAL FRACTIONAL LAPLACE SERIES EXPANSION METHOD FOR DIFFUSION EQUATION ARISING IN FRACTAL HEAT TRANSFER b Sheng-Png YAN School of Mechncs nd Cvl Engneerng, Chn Unverst of Mnng nd Technolog, Xuzhou, Jngsu, Chn Orgnl scentfc pper DOI:.2298/TSCI463Y In ths pper, we frst propose the locl frctonl Lplce seres expnson method, whch s couplng method of seres expnson method nd Lplce trnsform v locl frctonl dfferentl opertor. An llustrtve exmple for hndlng the dffuson equton rsng n frctl het trnsfer s gven. Ke words: nltcl soluton, dffuson equton, het trnsfer, Lplce seres expnson method, Lplce trnsform Introducton Locl frctonl ntegrl trnsforms hve potentl pplctons for scence nd engneerng [-4]. The ws utlzed to fnd the solutons for dfferentl equtons n the mthemtcl modelng of complex sstems n engneerng to cpture the reltons n spce nd tme wth the kernels wthn non-dfferentblt nd rregulr sets lke frctls [5-]. The locl frctonl Lplce trnsform (LFLT ws ppled to couple other methods, such s decomposton method (DM [5] nd vrtonl terton method (VIM [-9]. Recentl, the locl frctonl seres expnson method (LFSEM ws suggested n [2] nd developed to solve the dfferentl equtons wthn locl frctonl dervtves (LFD [2, 22]. However, the couplng scheme of LFSEM wth LFLT s not consdered. The trget of ths pper s to present the locl frctonl Lplce seres expnson method to del wth the dffuson equton rsng n frctl het trnsfer [23-25]. Fundmentls The locl frctonl ntegrl opertor of ω ( x s defned s [-5, 23]: b j= N ( I ω ( = ω ( (d = lm ω ( ( Γ ( + ( Γ ( + b j = where t = tj+ tj, j =,..., N, t =, tn = b. As the nverse opertor of eq. (, the locl frctonl dervtve of Ω ( s defned s [-5, 23-25]: ( d Ω( [ Ω( Ω( ] Ω ( = lm = = (2 d ( wth [ Ω( Ω( ] Γ ( + Ω [ ( Ω ( ]. Author s e-ml: spn@cumt.edu.cn
2 S32 Yn, S.-P.: Locl Frctonl Lplce Seres Expnson Method for Dffuson THERMAL SCIENCE, Yer 25, Vol. 9, Suppl., pp. S3-S35 The LFLT of Ω ( s defned s [2-5]: Y, Y E ( { Ω ( } =Ω ( = ( Ω ( (d, < Γ + (3 The nverse LFLT of Ω ( s defned s [2-5]: β + Y, Y, Y E (2π β Ω ( = { Ω ( } = ( Ω ( (d (4 where = β +, nd Re( = β. Some propertes whch re ppled to ths mnuscrpt re [, 4]: Y { Ω ( + bω ( } = Y { Ω ( } + by { Ω ( } (5 2 2 n ( n n ( k ( n k Y{ Ω ( } = Y[ Ω( ] Ω ( (6 k = Y { E ( x } = (7 k Y = ( k + ( k Γ + (8 Anlss of the method We consder gven dfferentl equton n locl form: ( ψ = Κ ψ (9 where ( ψ ( x, /d = ψ nd K s lner locl opertor wth respect to x. We consder mult-term seprted functons of ndependent vrbles t nd x, nmel: ψ( x, = σ( ω ( x ( where σ ( nd ω ( x re two locl frctonl contnuous functons. Settng σ ( = / Γ ( +, we hve: = = Tkng the LFLT of eq. (, we obtn: ψ( x, = ω ( x ( Γ ( + Hence, we obtn: (, = ω ( (2 Γ ( + =
3 Yn, S.-P.: Locl Frctonl Lplce Seres Expnson Method for Dffuson THERMAL SCIENCE, Yer 25, Vol. 9, Suppl., pp. S3-S35 S33 Y ( { ( x t, } { ( } ( ( Y x ( t t t = ω + = ω + Γ + Γ + (3 = = Y { K (, } K ( ( ( ( = ω = ( K ω = Γ + = Γ + (4 Mkng use of eqs. (3 nd (4, from eq. (9 one obtn: ω+ = = = ( ( ( ( t t ( K ω Γ + Γ + (5 whch leds to the recurson: ω+ ( = ( Kω ( (6 Adoptng the recurson formul (6, we hve: where the convergent condton reds: (, = ω ( (7 Γ ( + = lm ω ( = Γ ( + Hence, the soluton of eq. (9 s determned b: (8 (, = Y { (, } = Y { ω ( } (9 Γ ( + An nltcl soluton for dffuson equton rsng n frctl het trnsfer = We now consder the dffuson equton rsng n frctl het trnsfer [23-25]: ( ( 2 ψ ( x, t ψ ( x, t =, < (2 We present ntl vlues s follows: Adoptng (6, we hve: such tht the recurrence terms re wrtten s: t x ψ ( x, = E ( x (2 ω+ ( = ( Kω( = ω( Y { ( x,} Y { E ( x } = = = (22
4 S34 Yn, S.-P.: Locl Frctonl Lplce Seres Expnson Method for Dffuson THERMAL SCIENCE, Yer 25, Vol. 9, Suppl., pp. S3-S35 ω ( = (23 2 ω ( = (24 3 ω ( = (25 nd so on. Hence, we get: Tkng nverse LFLT, the non-dfferentble soluton of dffuson equton rsng n frctl het trnsfer cn be wrtten s: ψ( x, = E ( x = Γ ( + = = E ( x E ( (27 nd ts grph s gven n fg.. Conclusons In ths work, we frst hd proposed the couplng scheme of LFSEM wth LFLT, whch clled locl frctonl Lplce seres expnson method (LFLSEM. Bsed on t, (, = (26 Γ ( + = we fnd the non-dfferentble soluton of dffuson equton rsng n frctl het trnsfer. The obtned result shows tht the presented technolog s es, smple, effcent nd ccurte. Fgure. The non-dfferentble soluton of dffuson equton rsng n frctl het trnsfer when the frctl dmenson s equl to ln2/ln3 Nomenclture x spce co-ordntes, [m] Y [ Ω ( ] LFLT of Ω (, [ ] Y, Y [ Ω ( ] nverse LFLT of Y, Ω (, [ ] Greek smbols tme frctl dmensonl order, [ ] tme, [s] ψ(x, concentrton, [ ] References [] Yng, X. J., Locl Frctonl Functonl Anlss & Its Applctons, Asn Acdemc Publsher Lmted, Hong Kong, 2 [2] Yng, X.-J., et l., Locl Frctonl Integrl Trnsforms nd Applctons, Elsever, 25 [3] Srvstv, H. M., et l., Specl Functons n Frctonl Clculus nd Relted Frctonl Dfferntegrl Equtons, World Scentfc, Sngpore, 25 [4] Yng, X. J. Locl Frctonl Integrl Trnsforms, Progress n Nonlner Scence, 4 (2,, pp [5] Cttn, C., et l., Frctonl Dnmcs, Emergng Scence Publshers, 25
5 Yn, S.-P.: Locl Frctonl Lplce Seres Expnson Method for Dffuson THERMAL SCIENCE, Yer 25, Vol. 9, Suppl., pp. S3-S35 S35 [6] Zhong, W. P., et l., Applctons of Yng-Fourer Trnsform to Locl Frctonl Equtons wth Locl Frctonl Dervtve nd Locl Frctonl Integrl, Advnced Mterls Reserch, 46 (22, Mrch, pp [7] Yng, A. M., et l., The Yng-Fourer Trnsforms to Het-Conducton n Sem-Infnte Frctl Br, Therml Scence, 7 (23, 3, pp [8] Yng, X. J., et l., A Novel Approch to Processng Frctl Sgnls Usng the Yng-Fourer Trnsforms, Proced Engneerng, 29 (22, Feb., pp [9] Yng, X. J., et l., Mthemtcl Aspects of the Hesenberg Uncertnt Prncple wthn Locl Frctonl Fourer Anlss, Boundr Vlue Problems, 23 (23,, pp. -6 [] Wng, S. Q., et l., Locl Frctonl Functon Decomposton Method for Solvng Inhomogeneous Wve Equtons wth Locl Frctonl Dervtve, Abstrct nd Appled Anlss, 24 (24, ID [] Yng, X. J., Locl Frctonl Prtl Dfferentl Equtons wth Frctl Boundr Problems, Advnces n Computtonl Mthemtcs nd ts Applctons, (22,, pp [2] He, J.-H., A Tutorl Revew on Frctl Spcetme nd Frctonl Clculus, Interntonl Journl of Theoretcl Phscs, 53 (24,, pp [3] Zhng, Y. Z., et l., Intl Boundr Vlue Problem for Frctl Het Equton n the Sem-Infnte Regon b Yng-Lplce Trnsform, Therml Scence, 8 (24, 2, pp [4] Zho, Y., et l., Mppngs for Specl Functons on Cntor Sets nd Specl Integrl Trnsforms v Locl Frctonl Opertors, Abstrct nd Appled Anlss, 23 (23, ID [5] Zho, C. G., et l., The Yng-Lplce Trnsform for Solvng the IVPs wth Locl Frctonl Dervtve, Abstrct nd Appled Anlss, 24 (24, ID [6] Lu, C. F., et l., Reconstructve Schemes for Vrtonl Iterton Method wthn Yng-Lplce Trnsform wth Applcton to Frctl Het Conducton Problem, Therml Scence, 7 (23, 3, pp [7] Yng, A. M., et l., Locl Frctonl Lplce Vrtonl Iterton Method for Solvng Lner Prtl Dfferentl Equtons wth Locl Frctonl Dervtve, Dscrete Dnmcs n Nture nd Socet, 24 (24, ID [8] L, Y., et l., Locl Frctonl Lplce Vrtonl Iterton Method for Frctl Vehculr Trffc Flow, Advnces n Mthemtcl Phscs, 24 (24, ID [9] Xu, S., et l., Locl Frctonl Lplce Vrtonl Iterton Method for Nonhomogeneous Het Equtons Arsng n Frctl Het Flow, Mthemtcl Problems n Engneerng, 24 (24, ID [2] Yng, A. M., et l., Locl Frctonl Seres Expnson Method for Solvng Wve nd Dffuson Equtons on Cntor Sets, Abstrct nd Appled Anlss, 23 (23, ID 3557 [2] Zho, Y., et l., Approxmton Solutons for Locl Frctonl Schrödnger Equton n the One-Dmensonl Cntorn Sstem, Advnces n Mthemtcl Phscs, 23 (23, ID [22] Yng, A. M., et l., Applcton of Locl Frctonl Seres Expnson Method to Solve Klen-Gordon Equtons on Cntor Sets, Abstrct nd Appled Anlss, 24 (24, ID [23] Yng, X. J., Advnced Locl Frctonl Clculus nd ts Applctons, World Scence, New York, USA, 22 [24] Yng, X. J., et l., Approxmte Solutons for Dffuson Equtons on Cntor Spce-Tme, Proceedngs of the Romnn Acdem, Seres A, 4 (23, 2, pp [25] Ho, Y. J., et l., Helmholtz nd Dffuson Equtons Assocted wth Locl Frctonl Dervtve Opertors Involvng the Cntorn nd Cntor-Tpe Clndrcl Coordntes, Advnces n Mthemtcl Phscs, 23 (23, ID Pper submtted: October, 24 Pper revsed: Jnur 2, 25 Pper ccepted: Februr 3, 25
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