Iterative Laplace Transform Method for Solving Fractional Heat and Wave- Like Equations

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1 Research Journal of Mahemaical and Saisical Sciences ISSN Vol. 3(), 4-9, February (5) Res. J. Mahemaical and Saisical Sci. Ieraive aplace Transform Mehod for Solving Fracional Hea and Wave- ike Euaions Absrac S.C. Sharma and R.K. Bairwa Deparmen of Mahemaics, Universiy of Rajashan, Jaipur-34, Rajashan, INDIA Available online a: Received 8 h December 4, revised s January 5, acceped h February 5 In his paper, we derive he closed form soluions of he fracional hea and wave like euaions in erms of Miag-effler funcions by he use of ieraive aplace ransform mehod. In he process he ime-fracional derivaives are considered in Capuo sense for he said problem. Keywords: aplace ransform, Ieraive aplace ransform mehod, hea and wave-like euaions, Capuo fracional derivaive, Miag-effler funcion, fracional differenial euaion. MSC (): 6A33, 33E, 35R, 44A. Inroducion Fracional calculus has been aracing he aenion of scieniss and engineers from long ime ago, resuling in he developmen of many applicaions -3. Various mehods for he soluion of fracional differenial euaions are available in lieraure, including fracional subeuaion mehod 4, fracional wavele mehod 5-8, fracional aplace Adomian decomposiion mehod 9,, fracional operaional marix mehod,, fracional variaional ieraion mehod 3,4, fracional improved homoopy perurbaion mehod 5,6, fracional differenial ransform mehod 7 and fracional complex ransform mehod 8. The ieraive mehod was inroduced in 6 by Dafardar-Gejji and Jafari o solve numerically he nonlinear funcional euaions 9,. By now, he ieraive mehod has been used o solve many ineger and fracional boundary value problem,. Jafari e al. firsly solved he fracional parial differenial euaions by he use of ieraive aplace ransform mehod (ITM) 3. More recenly, Fracional Fokker-Planck euaions are solved by he ITM 4. For he presen problem, we considered he fracional hea and wave-like euaions wih variable coefficiens in he following form: u u u D u = f ( x, y, z) + g( x, y, z) + h( x, y, z), () and iniial condiions: u( x, y, z,) = h( x, y, z), u ( x, y, z,) = l ( x, y, z), () where ( < ) denoes he fracional derivaive. In he case, when <, and < ; hen euaion. () leads o a fracional hea-like and wave-like euaions wih variable coefficiens, respecively. Preliminaries and Noaions In his secion, we give some basic definiions and properies of fracional calculus and aplace ransform heory, which shall be used in his paper: Definiion: The Capuo fracional derivaive of funcion u( x, ) is defined as 5 D u x u x d N (3) ( ) (, ) = ( υ) (, υ) υ, <,, ( ) = J D u x,. d here D and J sands for he Riemann-iouville d fracional inegral operaor of order > defined as 6 J u x, = u x, d, >, <, N υ ( υ) υ υ ( ) Γ( ) (4) Definiion: The aplace ransform of a funcion φ, > is defined as s [ φ] = Φ ( s) = e φ d. (5) Definiion: aplace ransform of D u( x, ) is given as 7 [ k k (, )] [ (, )] (,),, k= o D u x = u x u x s < N (6) u x is he k-order derivaive of u( x, ) a =. k where (, ) Definiion: The Miag-effler funcion which is a generalizaion of exponenial funcion is defined as 8 z E ( z ) = C > (, Re ). (7) = Γ + Inernaional Science Congress Associaion 4

2 Research Journal of Mahemaical and Saisical Sciences ISSN Vol. 3(), 4-9, February (5) Res. J. Mahemaical and Saisical Sci. a furher generalizaion of (7) is given in he form 9 z E z = C R > R > ;(,,, ). (8) = Γ ( + β ), β β β Basic Idea of Ieraive aplace Transform Mehod To illusrae he basic idea of his mehod, we consider a general fracional nonlinear nonhomogeneous parial differenial euaion wih he iniial condiions of he form: D u( x, ) + Ru( x, ) + Nu( x, ) = g( x, ), <, N (9) k u ( x,) = h ( x), k =,,..., () where D u(x, ) k is he Capuo fracional derivaive of he funcion u(x, ), R is he linear differenial operaor, N represens he general nonlinear differenial operaor and g(x, ) is he source erm. Applying he aplace ransform (denoed by hroughou he presen paper) in Euaion (9), we ge [ D u( x, )] + [ R u( x, ) + Nu( x, )] = [ g( x, )]. () Using Euaion (6), we have k k [ u( x, )] = s u ( x,) [ g( x, )] [ Ru( x, ) N u( x, )]. + + () s s s k= Taking inverse aplace ransform of Euaion () implies k k u( x, ) = s u ( x,) [ g( x, )] [ Ru( x, ) Nu ( x, )], + + s (3) k= s Now we apply he Ieraive mehod, u( x, ) = u ( x, ) (4) i= i Since R is a linear operaor, R ui ( x, ) = R( ui ( x, ) i = i = and he nonlinear operaor N is decomposed as i i N ui ( x, ) = N( u( x, )) + N( uk ( x, )) N( uk ( x, )) i= i= k= k= (5) (6) Subsiuing (4), (5) and (6) in (3), we ge k k u i( x, ) = s u ( x,) [ g( x, )] + i= s k= (7) i i R( ui ( x, ) ) N( u ( x, )) N( uk ( x, )) N( uk ( x, )), + + s i= i= k= k= We define he recurrence relaions as k k u ( x, ) = s u ( x,) [ g( x, )] + s k= u + ( x, ) = R( u ( x, ) ) N( uk ( x, )) N( uk ( x, )), s k= k= u ( x, ) = R ( u ( x, ) ) + N ( u ( x, ) ) s (8) Therefore he-erm approximae soluion of (9) - () in series form is given by u( x, ) u ( x, ) + u ( x, ) + u ( x, ) u ( x, ), =,,... (9) Applicaions In his secion, he fracional hea and wave-like euaions wih variable coefficiens are solved by ITM. Example: Consider he following one-dimensional fracional hea-like euaion: u D u( x, ), <, () x subjec o he iniial condiion u( x, ) () Applying he aplace ransform in Euaion () and making use of () we ge x u [ u( x, ) ] = + x () s s x Taking inverse aplace ransform of Euaion () implies u u( x, ) + x (3) s x Now, applying he Ieraive mehod, Subsiuing (4) - (6) ino (3) and applying (8), we obain he componens of he soluion as follows: u ( x, ) = u( x,) (4) u u ( x, ) = x s x Γ ( (5) u + u u u ( x, ) = x x s x s x + x Γ ( Γ ( Γ ( (6) Γ ( Therefore, he series form soluion is given by Inernaional Science Congress Associaion 5

3 Research Journal of Mahemaical and Saisical Sciences ISSN Vol. 3(), 4-9, February (5) Res. J. Mahemaical and Saisical Sci. u( x, ) = u ( x, ) + u ( x, ) + u ( x, ) +... u( x, ) E ( ) Γ ( Γ ( (7) where E ( ) is he Miag-effler funcion, defined by Euaion(7) Remark: Seing=, Euaion () reduced o onedimensional hea-like euaion: u u, x wih soluion u ( x, ) e. (8) Example: Consider he following Two-dimensional fracional hea-like euaion: u u D u( x, y, ) = +, <, x y (9) subjec o he iniial condiions u( x, y,) = sin xsin y, (3) Applying he aplace ransform in Euaion (9) and making use of (3) we ge sin x sin y u u [ u( x, y, ) ] = + + s s x y (3) Taking inverse aplace ransform of Euaion (3) implies u u u( x, ) = sin xsin y + + (3) s x y Now, applying he Ieraive mehod, Subsiuing (4) - (6) ino (3) and applying (8), we obain he componens of he soluion as follows: u ( x, y, ) = u ( x, y,) = sin x sin y (33) u u s x y (,, ) = + u x y = sin x sin y (34) Γ ( ` ( u + u ) ( u + u ) u u u ( x, y, ) = v + + s x y s x y ( ) = sin x sin y + sin x sin y Γ ( Γ ( Γ ( = ( ) sin x sin y (35) Γ ( Therefore, he series form soluion is given by u( x, y, ) = u ( x, y, ) + u ( x, y, ) + u ( x, y, ) +... ( ) ( ) u( x, y, ) = sin xsin y = sin xsin ye ( ) (36) Γ ( Γ ( Remark.Seing=, Euaion (9) reduced o Twodimensional hea-like euaion: u u u = +, x y wih soluion u x, y, = e sin x sin y (37) Example 3.Consider he following Three-dimensional fracional hea-like euaion: u u u D u( x, y, z, ) y z + x + y + z,, < (38) 36 Subjec o he iniial condiion u( x, y, z,) =, (39) Applying he aplace ransform in Euaion (38) and making use of (39) we ge x y z [ u( x, y, z, ) ] = ( x y z ) uxx uyy u zz s s 36 s 36 s (4) Taking inverse aplace ransform of Euaion (4) implies x y z u( x, y, z, ) = ( x y z ) + uxx uyy u + + zz s 36 s 36 s 36 s (4) Now we apply he Ieraive mehod, Subsiuing (4) - (6) ino (4) and applying (8), we obain he componens of he soluion as follows: (,,, ) u x y z y z Γ ( (4) u u u u ( x, y, z, ) = x y z s x 36s y 36s z = x y z Γ ( (43) ( u u ) u + u ( u u ) + + u ( x, y, z, ) = x y z s x 36 s y 36 s z u u u x y z s x 36 s y 36 s z = x y z + x y z x y z Γ ( Γ (3 Γ ( y z (44) Γ (3 Inernaional Science Congress Associaion 6

4 Research Journal of Mahemaical and Saisical Sciences ISSN Vol. 3(), 4-9, February (5) Res. J. Mahemaical and Saisical Sci. Therefore, he series form soluion is given by u( x, y, z, ) = u ( x, y, z, ) + u ( x, y, z, ) + u ( x, y, z, ) +... u x y z x y z x y z E (,,, ) = =. Γ ( Γ ( 3 Γ (3 (45) Remark: Seing=, Euaion (38) reduced o Threedimensional hea-like euaion: u u u u x y z = + x + y + z, 36 wih soluion u ( x, y, z, ) y z ( e ) (46) Example: Consider he following one-dimensional fracional wave-like euaion: u D u( x, ), <, x (47) subjec o he iniial condiions u( x,), u ( x, ) (48) Applying he aplace ransform in Euaion (47) and making use of (48) we ge x x x u [ u( x, ) ] = + + s s s x (49) Taking inverse aplace ransform of Euaion (49) implies u u( x, ) + x + (5) s x Now, applying he Ieraive mehod, Subsiuing (4) - (6) ino (5) and applying (8), we obain he componens of he soluion as follows: u ( x, ) + x (5) u u ( x, ) x + = s x Γ ( + ) (5) ( u + u ) u u ( x, ) = x x s x s x x (53) Γ ( + ) Γ ( + ) Γ ( + ) Γ ( + ) Therefore, he series form soluion is given by u( x, ) = u ( x, ) + u ( x, ) + u ( x, ) u( x, ) + x x E,( ) Γ ( + ) Γ ( + ) (54) where E, ( ) β is generalizaion form of Miag-effler funcion, defined by Euaion(8). = Remark 4.Seing, Euaion (47) reduced o onedimensional wave-like euaion: u u, x wih soluion u ( x, ) + x sinh (55) Example 5.Consider he following Two-dimensional fracional wave-like euaion: u u D u( x, y, ) = x + y,, < x y (56) Subjec o he iniial condiions 4 4 u( x, y,), u ( x, y,) = y (57) Applying he aplace ransform in Euaion (56) and making use of (57) we ge 4 4 x y x u y u [ u( x, y, ) ] = + + s s s + x s y (58) Taking inverse aplace ransform of Euaion (58) implies 4 4 x u y u u( x, y, ) + y + + (59) s x s y Now, applying he Ieraive mehod, Subsiuing (4) - (6) ino (59) and applying (8), we obain he componens of he soluion as follows: u ( x, y, ) 4 + y 4 (6) x u y u s x s y (,, ) = + u x y u y Γ ( Γ ( + ) ( u u ) y ( u u ) + + s x s y x ( x, y, ) = + (6) x u y u + (6) s x s y y + x + y, ( ( ( + ) Γ ( + ) ( ( + ) y, Γ ( Γ ( + ) Therefore, he series form soluion is given by u( x, y, ) = u ( x, y, ) + u ( x, y, ) + u ( x, y, ) +... (63) Inernaional Science Congress Associaion 7

5 Research Journal of Mahemaical and Saisical Sciences ISSN Vol. 3(), 4-9, February (5) Res. J. Mahemaical and Saisical Sci ux (, y, ) y Γ ( Γ ( Γ ( + ) Γ ( + ) = +., (64) x 4 E y 4 E = Remark 5 Seing, Euaion (56) reduced o Twodimensional wave-like euaion: u u x y u = +, x y wih soluion 4 4 u ( x, y, ) cosh + y sinh. (65) Example 6.Consider he following Three-dimensional fracional wave-like euaion: u u u D u( x, y, z, ) + y + z + x + y + z,, < (66) subjec o he iniial condiions u( x, y, z,) =, u ( x, y, z,) + y z. (67) Applying he aplace ransform in Euaion (66) and making use of (67) we ge x + y z x + y + z x u y u z u [ ux (, y, z, )] = + s s + s + + x s y s z (68) Taking inverse aplace ransform of Euaion (68) implies x u y u z u ux (, y, z, ) = ( x + y z ) + ( x + y + z ) (69) s s x s y s z Now, applying he Ieraive mehod, Subsiuing (4) - (6) ino (69) and applying (8), we obain he componens of he soluion as follows: u ( x, y, z, ) = ( x + y z ) + ( x + y + z ) s = x + y z + x + y + z Γ ( (,,, ) = + + u x y z (7) x u y u z u s x s y s z + = ( x + y z ) + ( x + y + z ) Γ ( + ) Γ ( (7) ( u u ) ( u u ) ( u u ) s x s y s z x y z (,,, ) = + + u x y z x u y u z u + + s x s y s z + 3 = ( x + y z ) + ( x + y + z ) Γ ( + ) Γ (3 (7) Therefore, he series form soluion is given by u( x, y, z, ) = u ( x, y, z, ) + u ( x, y, z, ) + u ( x, y, z, ) u( x, y, z, ) = ( x + y z ) ( x + y + z ) Γ ( + ) Γ ( + ) Γ ( Γ ( Γ (3 = x + y z E, + x + y + z E ( ). = (73) Remark 6.Seing, Euaion (66) reduced o Threedimensional wave-like euaion: u u u u x y z = x + y + z, wih soluion u x, y, z, + y e + z e + x + y z. (74) Conclusion The soluions of he fracional hea and wave like euaions in erms of Miag-effler funcions by he use of ieraive aplace ransform mehod were derived. The soluions are obained in series form ha rapidly converges in a closed exac formula wih simply compuable erms. The calculaions are simple and sraighforward. The mehod was esed on six examples on differen siuaions. References. Baleanu D., Diehelm K., Scalas E. and Trujillo J.J., Fracional Calculus, vol. 3 of Series on Complexiy, Nonlineariy and Chaos, World Scienific, Singapore, (). Origueira M.D., Fracional calculus for scieniss and engineers, Springer, () 3. Sabaier J., Agrawal O.P. and Tenreiro Machado J.A., Advances in Fracional Calculus: Theoreical Developmens and Applicaions in Physics and Engineering, Springer,(7) 4. Zhang S. and Zhang H.Q., Fracional sub-euaion mehod and is applicaions o nonlinear fracional PDEs, Physics eers A, 375(7), () 5. epik U., Solving fracional inegral euaions by he Haar wavele mehod, Applied Mahemaics and Compuaion,4(), (9) 6. i Y., Solving a nonlinear fracional differenial euaion using Chebyshev waveles, Communicaions in Nonlinear Science and Numerical Simulaion, 5(9), 84 9, () 7. Rehman M. and Ali Khan R., The egendre wavele mehod for solving fracional differenial euaions, Communicaions in Nonlinear Science and Numerical Simulaion,6 (), () Inernaional Science Congress Associaion 8

6 Research Journal of Mahemaical and Saisical Sciences ISSN Vol. 3(), 4-9, February (5) Res. J. Mahemaical and Saisical Sci. 8. Wu J.., A wavele operaional mehod for solving fracional parial differenial euaions numerically, Applied Mahemaics and Compuaion, 4(), 3 4 (9) 9. Jafari H., Khaliue C.M., and Nazari M., Applicaion of he aplace decomposiion mehod for solving linear and nonlinear fracional diffusion-wave euaions, Applied Mahemaics eers, 4(), (). Ongun M.Y., The aplace adomian decomposiion mehod for solving a model for HIV infecion of CD4+ cells, Mahemaical and Compuer Modelling, 53( 5-6), (). iu Y. and Sun N, Numerical soluion of fracional differenial euaions using he generalized block pulse operaional marix, Compuers and Mahemaics wih Applicaions, 6(3), (). Saadamandi A. and Dehghan M., A new operaional marix for solving fracional-order differenial euaions, Compuers and Mahemaics wih Applicaions, 59(3), () 3. Das S., Analyical soluion of a fracional diffusion euaion by variaional ieraion mehod, Compuers and Mahemaics wih Applicaions,57(3), (9) 4. Sweilam N.H., Khader M.M. and Al-Bar R.F., Numerical sudies for a muli-order fracional differenial euaion, Physics eers A, 37(-), 6 33 (7) 5. iu Y., Approximae soluions of fracional nonlinear euaions using homoopy perurbaion ransformaion mehod, Absrac and Applied Analysis,, Aricle ID 75869, 4 ( ) 6. iu Y., Variaional homoopy perurbaion mehod for solving fracional iniial boundary value problems, Absrac and Applied Analysis,, Aricle ID 773, () 7. Erurk V.S. and Momani S., Solving sysems of fracional differenial euaions using differenial ransform mehod, Journal of Compuaional and Applied Mahemaics, 5(), 4 5 (8) 8. Ibrahim R.W., Fracional complex ransforms for fracional differenial euaions, Advances in Difference Euaions, 9 () 9. Dafardar-Gejji V. and Jafari H., An ieraive mehod for solving nonlinear funcional euaions, Journal of Mahemaical Analysis and Applicaions, 36(), (6). Jafari H., Ieraive Mehods for solving sysem of fracional differenial euaions [Ph.D. hesis], Pune Universiy, (6). Bhalekar S. and Dafardar-Gejji V., Solving evoluion euaions using a new ieraive mehod, Numerical Mehods for Parial Differenial Euaions, 6(4), (). Dafardar-Gejji V. and Bhalekar S., Solving fracional boundary value problems wih Dirichle boundary condiions using a new ieraive mehod, Compuers andmahemaics wih Applicaions, 59(5), 8 89 () 3. Jafari H., Nazari M., Baleanu D. and Khaliue C.M., A new approach for solving a sysem of fracional parial differenial euaions, Compuers and Mahemaics wih Applicaions, 66(5), ( 3) 4. Yan., Numerical Soluions of Fracional Fokker-Planck Euaions Using Ieraive aplace Transform Mehod, Absrac and Applied Analysis,3, Aricle ID 4656, 7 pages, (3) 5. Capuo M, Elasicia e Dissipazione, Zani-Chelli, Bologna,( 969) 6. Podlubny I., Fracional Differenial Euaions, vol. 98, Academic Press, New York, NY, USA, (999) 7. Kilbas A.A., Srivasava H.M. and Trujillo J.J., Theory and Applicaions of Fracional Differenial Euaions, Elsevier, Amserdam,(6) 8. Miller K.S. and Ross B., An Inroducion o he Fracional Calculus and Fracional Differenial Euaions, John Wiley and Sons, New York,USA,(993) 9. Wiman A., Uber de fundamenal saz in der heorie der funkionen E ( x), Aca Mah., 9, 9-, (95) Inernaional Science Congress Associaion 9