Research Article Analytical Solutions of the One-Dimensional Heat Equations Arising in Fractal Transient Conduction with Local Fractional Derivative

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1 Absrac and Applied Analysis Volume 3, Aricle ID 535, 5 pages hp://d.doi.org/.55/3/535 Research Aricle Analyical Soluions of he One-Dimensional Hea Equaions Arising in Fracal Transien Conducion wih Local Fracional Derivaive Ai-Ming Yang,, Carlo Caani, 3 Hossein Jafari,,5 and Xiao-Jun Yang College of Science, Hebei Unied Universiy, Tangshan 39, China College of Mechanical Engineering, Yanshan Universiy, Qinhuangdao, China 3 Deparmen of Mahemaics, Universiy of Salerno, Via Pone don Melillo, Fisciano, Salerno, Ialy Deparmen of Mahemaics, Faculy of Mahemaical Sciences, Universiy of Mazandaran, Babolsar, Iran 5 Inernaional Insiue for Symmery Analysis and Mahemaical Modelling, Deparmen of Mahemaical Sciences, Norh-Wes Universiy, Mafikeng Campus, Privae Bag X, Mmabaho 735, Souh Africa Deparmen of Mahemaics and Mechanics, China Universiy of Mining and Technology, Xuzhou, Jiangsu, China Correspondence should be addressed o Ai-Ming Yang; aimin heu@3.com Received Sepember 3; Acceped 7 Ocober 3 Academic Edior: Abdon Aangana Copyrigh 3 Ai-Ming Yang e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. The one-dimensional hea equaions wih he hea generaion arising in fracal ransien conducion associaed wih local fracional derivaive operaors are invesigaed. Analyical soluions are obained by using he local fracional Adomian decomposiion mehod via local fracional calculus heory. The mehod in general is easy o implemen and yields good resuls. Illusraive eamples are included o demonsrae he validiy and applicabiliy of he new echnique.. Inroducion The Adomian decomposiion mehod [ 3] was applied o process linear and nonlinear problems in he fields of science and engineering. Taari and Dehghan [] applied Adomian decomposiion mehod o process he mulipoin boundary value problem. Wazwaz [5] used Adomian decomposiion mehod o deal wih he Brau-ype equaions. Dafardar- Gejji and Jafari [] considered Adomian decomposiion mehod o analyze he Bagley Torvik equaion. Larsson [7] presened he soluion for Helmholz equaion by using he Adomain decomposiion mehod. Taari and coworkers [] invesigaed soluion for he Fokker-Planck equaion by Adomain decomposiion mehod. Fracional calculus [9 ]was applied o model he physical and engineering problems for epressions of sress-srain consiuive relaions of differen viscoelasic fracional order properies of maerials, diffusion processes wih fracional order properies, fracional order flows, analyical mechanics of fracional order discree sysem vibraions [3 5], and so on. Recenly, he applicaion of Adomian decomposiion mehod for solving he linear and nonlinear fracional parial differenialequaionsinhefieldsofhephysicsandengineering had been esablished in [, 7]. Adomian decomposiion mehod was applied o handle he ime-fracional Navier- Sokes equaion [], fracional space diffusion equaion [9], fracional KdV-Burgers equaion [], linear and nonlinear fracional diffusion and wave equaions [], KdV-Burgers- Kuramoo equaion [], fracional Burgers equaion [3], and so on. For more deails on some mehods for solving fracional differenial equaions, see [ ]. Recenly, local fracional calculus heory was applied o model some nondiffereniable problems for mahemaical physics (see [9 3] and he references herein). The Adomian decomposiion mehod, as one of efficien ools for solving he linear and nonlinear differenial equaions, was eended o find he soluions for local fracional differenial equaions [37 ] and nondiffereniable soluions were obained.

2 Absrac and Applied Analysis The parial differenial equaionfs describing hermal process of fracal hea conducion were suggesed in [3, 3] in he following form: The iniial and boundary condiions are α =. () u (, ) =f(), α u (, ) α =g(), where he operaor is he local fracional differenial operaor [9, 3, 3, 37, 3], which is applied o model he hea conducion problems in fracal media, fracal maerials, fracal fracure mechanics, fracal wave behavior, Navier- Sokes equaions on Canor ses, Schrödinger equaion wih local fracional derivaive, and diffusion equaions on canor space-ime. The one-dimensional hea equaions wih he hea generaion arising in fracal ransien conducion were considered in [3]as () α =φ(, ), (3) where φ(, ) is he hea generaion erm. We use iniial and boundary condiions as follows: u (, ) =f(), α u (, ) α =g(). The aim of his paper is o invesigae he one-dimensional hea equaions wih he hea generaion arising in fracal ransien conducion by using he local fracional Adomian decomposiion mehod. This paper is srucured as follows. In Secion, wegive he basic noaions and definiions of local fracional operaors. In Secion 3, local fracional Adomian decomposiion mehod for hea generaion arising in fracal ransien conducion is presened. Three eamples are shown in Secion. Finally, Secion5 presens conclusions.. Preliminaries In his secion we presen some basic definiions and noaions of he local fracional operaors which are used furher hrough he paper. Le us denoe local fracional coninuiy of f() as () f () C α (a, b). (5) Definiion. Local fracional derivaive operaor of f() a he poin is given by [9, 3, 3 3]: f (α) ( )= dα f () d α = = lim Δ α (f () f( )) ( ) α, () where Δ α (f() f( )) Γ(+α)Δ(f() f( )) and f() C α (a, b). Local fracional derivaive of high order and local fracional parial derivaive of high order are wrien in he form [9, 3, 3] k imes f (kα) () = D (α) D(α) f (), (7) k imes kα α f(,y)= kα α α α f(,y), () respecively. As inverse of local fracional differenial operaor, he local fracional inegral operaor of f() in he inerval [a, b] is defined as [9, 3, 3 3] ai (α) b b f () = Γ (+α) = a f ()(d) α j=n Γ (+α) lim Δ j= f( j )(Δ j ) α, where a pariion of he inerval [a, b] is denoed as Δ j = j+ j, Δ = ma{δ,δ,δ j,...} and j =,...,N, =a,and N =b. The properies are only presened as follows [9, 3, 37]: D (α) [f () g ()] =(D (α) ai (α) f () g(α) () f ())g() +f() (D(α) g ()), =[f() g ()] a ai (α) f(α) () g (), D (α) I (α) b kα Γ (+kα) = kα Γ (+kα) = 3. Analysis of he Mehod (k )α Γ [+(k ) α], (k+)α Γ [+(k+) α]. (9) () Le us rewrie he hea equaions wih he hea generaion arising in fracal ransien conducion in he form L (α) u L (α) u=φ, () subjec o he iniial and boundary condiions u (, ) =f(), α u (, ) α =g(), and L (α),respec- where α / and α / α symbolize L (α) ively. ()

3 Absrac and Applied Analysis 3 By defining he wofold local fracional inegral operaor as L ( α),wehave so ha u=l ( α) Hence, we ge L ( α) L (α) u L ( α) φ+ u φ]=l ( α) L (α) u, (3) α g () +f(). () Γ (+α) u (, ) =u (, ) +L ( α) u (, )], (5) where u (, ) = L ( α) α φ+ g () +f(). () Γ (+α) So,from (5) we have ieraive formula as follows: u n+ (, ) =L ( α) u n (, )], n, (7) where u (, ) = L ( α) φ+( α /Γ( + α))g() + f(). Finally,heeacsoluioncanbeconsrucedasfollows:. Illusraive Eamples n u (, ) = lim u i (, ). () n i= Eample. In view of (3), we consider φ(, ) =, f() = α /Γ( + α),andg() = α /Γ( + α). We have subjec o he iniial value condiion α =, (9) α u (, ) = Γ (+α) + α α Γ (+α) Γ (+α) + α Γ (+α). () From (9) we have he following recursive relaions: u n+ (, ) =L ( α) u n (, )]. () In view of (), he firs few erms of he decomposiion series read α u (, ) = Γ (+α) + α α Γ (+α) Γ (+α) + α Γ (+α), From (5)wege u (, ) = 3α Γ (+3α) + α Γ (+α). () u (, ) =u 3 (, ) = =u n (, ) =. (3) u(, ) Figure : Soluion for he one-dimensional hea equaions wih a fied value α=ln / ln 3. Therefore, he eac soluion of (9) can be wrien as u (, ) = 3α Γ (+3α) + α α Γ (+α) Γ (+α) + Γ (+α). () The value of he fracal-dimension order α=ln / ln 3 of he behavior of he soluion is shown in Figure. Eample. When φ(, ) =, f() = α /Γ( + α),andg() =,wege α α =. (5) We give he iniial value condiion as follows: α u (, ) = Γ (+α) + α Γ (+α). () From (9)wehavehefollowingrecursiverelaions: u n+ (, ) =L ( α) u n (, )]. (7) From (7), we have he firs few erms of he decomposiion series as follows: Hence, we ge α u (, ) = Γ (+α) + α Γ (+α), u (, ) = α Γ (+α). () u (, ) =u 3 (, ) = =u n (, ) =. (9) So, he eac soluion of (9)reads u (, ) = α Γ (+α). (3) The soluion wih fracal-dimension order α = ln / ln 3 is shown in Figure.

4 Absrac and Applied Analysis u(, ) Figure : Soluion for he one-dimensional hea equaions wih a fied value α=ln / ln 3. Figure 3: The surface shows he eac soluion u(, ) wih a fied value α=ln / ln 3. Eample 3. When φ(, ) =, f() =,andg() = α /Γ( + α),wege α =. (3) The iniial value condiion is presened as follows: α u (, ) = Γ (+α) + Γ (+α) Γ (+α). (3) From (9)herecursiverelaionsfollow α u n+ (, ) =L ( α) u n (, )]. (33) In view of (7), we ge he few erms of he series; namely, α α u (, ) = Γ (+α) + Γ (+α) Γ (+α), Hence, we ge u (, ) = 3α Γ (+3α). α α (3) u (, ) =u 3 (, ) = =u n (, ) =. (35) So, he eac soluion of (9)reads 3α u (, ) = Γ (+3α) α Γ (+α) + α α Γ (+α) Γ (+α). (3) Figure 3 shows he eac soluion when α=ln / ln Conclusions In his work, analyical soluions for he one-dimensional hea equaions wih he hea generaion arising in fracal ransien conducion associaed wih local fracional derivaive operaors were discussed. The obained soluions are nondiffereniable funcions, which are Canor funcions and hey disconinuously depend on he local fracional derivaive. I is shown ha he local fracional Adomian decomposiion mehod is an efficien and simple ool for solving local fracional differenial equaions. Conflic of Ineress The auhors declare ha here is no conflic of ineress regarding he publicaion of his paper. Acknowledgmens This paper was suppored by he Naional Scienific and Technological Suppor Projecs (no. BAE9B), he Naional Naural Science Foundaion of China (no. 3 and no. 737), and he Naional Naural Science Foundaion of he Hebei Province (no. E395). References [] G. Adomian, Nonlinear Sochasic Operaor Equaions, Academic Press, Orlando, Fla, USA, 9. [] G. Adomian, Solving Fronier Problems of Physics: The Decomposiion Mehod, vol.offundamenal Theories of Physics, Kluwer Academic Publishers, Dordrech, The Neherlands, 99. [3] G. Adomian, A review of he decomposiion mehod in applied mahemaics, Mahemaical Analysis and Applicaions, vol. 35, no., pp. 5 5, 9. [] M. Taari and M. Dehghan, The use of he Adomian decomposiion mehod for solving mulipoin boundary value problems, Physica Scripa,vol.73,no.,pp.7 7,. [5] A.-M. Wazwaz, Adomian decomposiion mehod for a reliable reamen of he Brau-ype equaions, Applied Mahemaics and Compuaion,vol.,no.3,pp.5 3,5. [] V. Dafardar-Gejji and H. Jafari, Adomian decomposiion: a ool for solving a sysem of fracional differenial equaions, Mahemaical Analysis and Applicaions,vol.3,no.,pp.5 5,5.

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