Research Article Multivariate Padé Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order

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1 Absrac and Applied Analysis Volume 23, Aricle ID 7464, 2 pages hp://ddoiorg/55/23/7464 Research Aricle Mulivariae Padé Approimaion for Solving Nonlinear Parial Differenial Equaions of Fracional Order Veyis Turu and Nuran Güzel 2 Deparmen of Mahemaics, Faculy of Ars and Sciences, Baman Universiy, Baman, Turkey 2 Deparmen of Mahemaics, Faculy of Ars and Sciences, Yıldız Technical Universiy, İsanbul, Turkey Correspondence should be addressed o Nuran Güzel; nguzel@yildizedur Received 27 November 22; Revised 4 January 23; Acceped 4 January 23 Academic Edior: Hassan Elayeb Copyrigh 23 V Turu and N Güzel 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 Two echniques were implemened, he Adomian decomposiion mehod (ADM) and mulivariae Padé approimaion (MPA), for solving nonlinear parial differenial equaions of fracional order The fracional derivaives are described in Capuo sense Firs, he fracional differenial equaion has been solved and convered o power series by Adomian decomposiion mehod (ADM), hen power series soluion of fracional differenial equaion was pu ino mulivariae Padé series Finally, numerical resuls were compared and presened in ables and figures Inroducion Recenly, differenial equaions of fracional order have gained much ineres in engineering, physics, chemisry, and oher sciences I can be said ha he fracional derivaive has drawn much aenion due o is wide applicaion in engineering physics [ 9] Some approimaions and numerical echniques have been used o provide an analyical approimaion o linear and nonlinear differenial equaions and fracional differenial equaions Among hem, he variaional ieraion mehod, homoopy perurbaion mehod [, ], and he Adomian decomposiion mehod are relaively new approaches [5 9, 2, 3] The decomposiion mehod has been used o obain approimae soluions of a large class of linear or nonlinear differenialequaions [2, 3] Recenly, he applicaion of he mehod is eended for fracional differenial equaions [6 9, 4] Many definiions and heorems have been developed for mulivariae Padé approimaions MPA (see [5] for a survey on mulivariae Padé approimaion)the mulivariae Padé Approimaionhasbeenusedoobainapproimae soluionsoflinearornonlineardifferenialequaions[6 9] Recenly, he applicaion of he unvariae Padé approimaion is eended for fracional differenial equaions [2, 2] The objecive of he presen paper is o provide approimae soluions for iniial value problems of nonlinear parial differenial equaions of fracional order by using mulivariae Padéapproimaion 2 Definiions For he concep of fracional derivaive, we will adop Capuo s definiion, which is a modificaion of he Riemann- Liouville definiion and has he advanage of dealing properly wih iniial value problems in which he iniial condiions are given in erms of he field variables and heir ineger order, which is he case in mos physical processes The definiions canbeseenin[3, 4, 22, 23] 3 Decomposiion Mehod [24] Consider D α u (, ) =f(u, u,u ) +g(, ), m < α m ()

2 2 Absrac and Applied Analysis The decomposiion mehod requires ha a nonlinear fracional differenial equaion () is epressed in erms of operaor form as D α u (, ) +Lu(, ) +Nu(, ) =g(, ), >, (2) where L is a linear operaor which migh include oher fracional derivaives of order less han α, N is a nonlinear operaor which also migh include oher fracional derivaives of order less han α, D α = α / α is he Capuo fracional derivaive of order α, andg(, ) ishesourcefuncion[24] Applying he operaor J α [3, 4, 22, 23], he inverse of he operaor D α, o boh sides of (5) OdibaandMomani[24] obained n= u (, ) = u n (, ) = m k u k= (, k + ) k k! +Jα g (, ) J α [Lu (, ) +Nu(, )], m k u k= (, k + ) k k! +Jα g (, ) J α [L ( u n (, ))+ n= A n ] From his, he ieraes are deermined in [24] by he following recursive way: n= m k u u (, ) = k= (, k + ) k k! +Jα g (, ), u (, ) = J α (Lu +A ), u 2 (, ) = J α (Lu +A ), u n+ (, ) = J α (Lu n +A n ) 4 Mulivariae Padé Aproimaion [25] Consider he bivariae funcion f(, y) wih Taylor series developmen f(,y)= i,j= (3) (4) c ij i y j (5) around he origin We know ha a soluion of unvariae Padé approimaion problem for f () = i= c i i (6) is given by m c i i i= m c i i nm n c i i i= i= p () = c m+ c m c m+ n, (7) c m+n c m+n c m q () = n c m+ c m c m+ n (8) c m+n c m+n c m Le us now muliply jh row in p() and q() by j+m (j = 2,, n + ) and aferwards divide jh column in p() and q() by j (j=2,,n+) This resuls in a muliplicaion of numeraor and denominaor by mn Havingdoneso,we ge p () q () = m i= c i i m i= c i i m n i= c i i c m+ m+ c m m c m+ n m+ n c m+n m+n c m+n m+n c m m c m+ m+ c m m c m+ n m+ n c m+n m+n c m+n m+n c m m if (D = de D m,n =) This quoen of deerminans can also immediaely be wrien down for a bivariae funcion f(, y) The sum k i= c i i shallbereplacedbykh parial sum of he Taylor series developmen of f(, y) and he epression c k k by an epression ha conains all he erms of degree k in f(, y) Here a bivariae erm c ij i y j is said o be of degree i+jifwe define p(,y) = m c ij i y j i+j= c ij i y j i+j=m+ c ij i y j i+j=m+n m c ij i y j i+j= m n c ij i y j i+j= c ij i y j c ij i y j i+j=m i+j=m+ n, c ij i y j c ij i y j i+j=m+n i+j=m (9)

3 Absrac and Applied Analysis 3 q(,y) = c ij i y j i+j=m+ c ij i y j i+j=m+n c ij i y j c ij i y j i+j=m i+j=m+ n c ij i y j c ij i y j i+j=m+n i+j=m () Then i is easy o see ha p(, y) and q(, y) are of he form p(,y)= q(,y)= mn+m a ij i y j, i+j=mn mn+n b ij i y j i+j=mn () We know ha p(, y) and q(, y) arecalledpadéequaions [25] So he mulivariae Padé approiman of order (m, n) for f(, y) is defined as 5 Numerical Eperimens r m,n (, y) = p(,y) q(,y) (2) In his secion, wo mehods, ADM and MPA, shall be illusraed by wo eamples All he resuls are calculaed byusinghesofwaremaple2thefulladmsoluionsof eamples can be seen from Odiba and Momani [24] Eample Consider he nonlinear ime-fracional advecion parial differenial equaion [24] D α u (, ) +u(, ) u (, ) =+ 2, subjec o he iniial condiion >, R, < α, (3) u (, ) = (4) Odiba and Momani [24] solved he problem using he decomposiion mehod, and hey obained he following recurrence relaion [24]: where A j are he Adomian polynomials for he nonlinear funcion N=uu Inviewof(5), he firs few componens of he decomposiion series are derived in [24] as follows: α u (, ) =( Γ (α+) + 2α+2 Γ (α+3) ), Γ (2α + ) 3α u (, ) = ( Γ(α+) 2 Γ (3α + ) + 4Γ (2α + 3) 3α+2 Γ (α+) Γ (α+3) Γ (3α + 3) 4Γ (2α + 5) 3α+4 + Γ(α+3) 2 Γ (3α + 5) ), Γ (2α + ) Γ (4α + ) 5α u 2 (, ) =2( Γ(α+) 3 Γ (3α + ) Γ (5α + ) 8Γ (2α + 5) Γ (4α + 7) 5α+6 + Γ(α+) 3 Γ (3α + 5) Γ (5α + 7) + ), (6) and so on; in his manner, he res of componens of he decomposiion series can be obained [24] The firs hree erms of he decomposiion series are given by [24] u (, ) α =( Γ (α+) + 2α+2 Γ (α+3) Γ (2α + ) 3α Γ(α+) 2 Γ (3α + ) 4Γ (2α + 3) 3α+2 Γ (α+) Γ (α+3) Γ (3α + 3) + ) (7) For α=(6)is u (, ) = (8) Now, le us calculae he approimae soluion of (8)for m= 4 and n = 2 by using Mulivariae Padé approimaionto obain mulivariae Padé equaions of(8)for m=4and n= 2,weuse() By using (), we obain p (, ) u (, ) =u(, ) +J α (+ 2 ) α =( Γ (α+) + 2α+2 Γ (α+3) ), u j+ (, ) = J α (A j ), j, (5) = = ( ) 3 7,

4 4 Absrac and Applied Analysis q (, ) = = ( ) 2 6 (9) So, he mulivariae Padé approimaion of order (4, 2) for (7), ha is, [4, 2] (,) = ( ) (2) For α = 5 (7)is u (, ) = For simpliciy, le /2 =a;hen u (, a) = a a a a 7 (2) (22) Using () o calculae he mulivariae Padé equaions for (22)wege p (, a) a a a a a a a 3 = a a a 5 = ( a a ) 3 a, q (, a) = a 5 = ( a 2 ) 3 a 75686a a 5 (23) recalling ha /2 =a,wegemulivariaepadéapproimaion of order (6, 2) for (2), ha is, [6, 2] (,) = ( For α = 75 (7)is ) ( ) u (, ) = (24) (25) Using () ocalculaehemulivariaepadéequaionsand hen recalling ha /2 = a, we ge mulivariae Padé approimaion of order (49, 2) for (25), ha is, [49, 2] (,) = /2 ( ) ( ) (27) Table, Figures(a), (b), (c), 2(a), 2(b), 2(c), and2(d) shows he approimae soluions for (3) obained for differen values of α using he decomposiion mehod (ADM) and he mulivariae Padé approimaion (MPA) The value of α= is for he eac soluion u(, ) = [24] For simpliciy, le /2 =a;hen u (, a) = a a a a 49 (26) Eample 2 Consider he nonlinear ime-fracional hyperbolic equaion [24] D α u (, ) u (, ) = (u (, ) ), (28) >, R, < α 2, subjec o he iniial condiion u (, ) = 2, u (, ) = 2 2 (29)

5 Absrac and Applied Analysis (a) (b) (c) Figure : (a) Eac soluion of Eample for α=(b) ADM soluion of Eample for α=(c) Mulivariae PadéapproimaionofADM soluion for α=in Eample Odiba and Momani [24] solved he problem using he decomposiion mehod, and hey obained he following recurrence relaion in [24]: u (, ) =u(, ) +u (, ) = 2 ( 2), u j+ (, ) =J α (A j ), j, (3) where A j are he Adomian polynomials for he nonlinear funcion N=uu Inviewof(3), he firs few componens of he decomposiion series are derived in [24] as follows: u (, ) = 2 ( 2), α u (, ) =6 2 ( Γ (α+) 4α+ Γ (α+2) + 8α+2 Γ (α+3) ), u 2 (, ) = α ( Γ (2α + ) 42α+ Γ (2α + 2) + 82α+2 Γ (2α + 3) 2Γ (α+2) 2α+ Γ (α+) Γ (2α + 2) ) Γ (α+3) 2α+2 ( Γ (α+2) Γ (2α + 3) 6Γ (α+4) 2α+3 Γ (α+3) Γ (2α + 4) ), (3)

6 6 Absrac and Applied Analysis (a) (b) 6e 5 4e 5 2e 5 2e 5 4e 5 6e 5 6e 5 4e 5 2e 5 2e 5 4e 5 6e (c) (d) Figure 2: (a) ADM soluion of Eample for α = 5 (b) Mulivariae Padé approimaion of ADM soluion for α = 5 in Eample (c) ADM soluion of Eample for α = 75 (d) Mulivariae Padé approimaion of ADM soluion for α = 75 in Eample and so on; in his manner he res of componens of he decomposiion series can be obained The firs hree erms of he decomposiion series (7) are given [24]by For α=2(43)is u (, ) = 2 ( 2) u (, ) = 2 ( 2) ( ) (33) α ( Γ (α+) 4α+ Γ (α+2) + 8α+2 Γ (α+3) ) (32) α ( Γ (2α + ) + ) Now, le us calculae he approimae soluion of (33)for m= 4 and n = 2 by using mulivariae Padé approimaionto

7 Absrac and Applied Analysis (a) (b) (c) Figure 3: (a) Eac soluion of Eample 2 for α = 2 (b) ADM soluion of Eample 2 for α = 2 (c) Mulivariae Padé approimaion of ADM soluion for α = 2 in Eample 2 obain mulivariae Padé equaions of(33)for m=4and n= 2,weuse() By using (), we obain So, he mulivariae Padé approimaion of order (4, 2) for (33), ha is, p (, ) 2 ( 2) ( 2) 2 = = 2 4 ( ( ) 6, q (, ) = = 2 4 ( ) 4 (34) [4, 2] (,) = ( ) 2 ( ) (35)

8 8 Absrac and Applied Analysis (a) (b) (c) (d) Figure 4: (a) ADM soluion of Eample 2 for α = 5 (b) Mulivariae Padé approimaion of ADM soluion for α = 5 in Eample 2 (c) ADM soluion of Eample 2 for α = 75 (d) Mulivariae Padé approimaion of ADM soluion for α = 75 in Eample 2 For α = 5 (32)is u (, ) = 2 ( 2) +6 2 ( ) (36) ( a a ) a 6 = 2 2 2a a a a 7 For simpliciy, le /2 =a;hen a 6 (37) u (, a) = 2 ( 2a 2 )+6 2 Using () o calculae mulivariae Padé equaions of(37) for m=7and n=2,weuse() By using (), we obain

9 Absrac and Applied Analysis 9 Table : Numerical values when α = 5, α = 75,andα = for (3) α = 5 α = 75 α = u ADM u MPA u ADM u MPA u ADM u MPA u Eac

10 Absrac and Applied Analysis Table 2: Numerical values when α = 5, α = 75,andα = 2 for (28) α = 5 α = 75 α = 2 u ADM u MPA u ADM u MPA u ADM u MPA u Eac

11 Absrac and Applied Analysis 2 2 2a a a a a a a 3 p (, a) = a a a a = ( a a a a a ) 6 a, q (, a) = 2 2 a a a a a = ( a a 2 ) 4 a, (38) recalling ha /2 =a,wegemulivariaepadéapproimaion of order (7, 2) for (36), ha is, [7, 2] (,) = ( / For α = 75 (32)is / ) 2 ( ) (39) u (, ) = 2 ( 2) +6 2 ( ) (4) For simpliciy, le /4 =a;hen u (, a) = 2 ( 2a 4 ) +6 2 ( a a a 5 ) a 4 = a a a a a 4 (4) Using () o calculae mulivariae Padé equaions of(4) for m=5and n=2,weuse() By using (), we obain p (, a) a a a a a a a a a = a a a 4 = a 28 ( + 2a a a ), q (, a) = a 4 = a 28, a 4 (42) recalling ha /4 =a,wegemulivariaepadéapproimaion of order (5, 2) for (4), ha is, [5, 2] (,) = ( / /4 ) ( ) (43) Table 2, Figures 3(a), 3(b), 3(c), 4(a), 4(b), 4(c), and 4(d) show he approimae soluions for (28) obained for differen values of α using he decomposiion mehod (ADM) and he mulivariae Padé approimaion (MPA) The value of α=2 is for he eac soluion u(, ) = (/ + ) 2 [24] 6 Concluding Remarks The fundamenal goal of his paper has been o consruc an approimae soluion of nonlinear parial differenial

12 2 Absrac and Applied Analysis equaions of fracional order by using mulivariae Padé approimaion The goal has been achieved by using he mulivariae Padé approimaion and comparing wih he Adomian decomposiion mehod The presen work shows he validiy and grea poenial of he mulivariae Padé approimaion for solving nonlinear parial differenial equaions of fracional order from he numerical resuls Numerical resuls obained using he mulivariae Padéapproimaionandhe Adomian decomposiion mehod are in agreemen wih eac soluions References [] J H He, Nonlinear oscillaion wih fracional derivaive and is applicaions, in Proceedings of Inernaional Conference on Vibraing Engineering, pp , Dalian, China, 998 [2] J H He, Some applicaions of nonlinear fracional differenial equaions and heir approimaions, Bullein of Science and Technology,vol5,no2,pp86 9,999 [3] Y Luchko and R Gorenflo, The Iniial Value Problem for Some Fracional Differenial Equaions wih he Capuo Derivaive, Series A8-98, Fachbreich Mahemaik und Informaik, Freic Universia Berlin, Berlin, Germany, 998 [4] I Podlubny, Fracional Differenial Equaions,vol98ofMahemaics in Science and Engineering, Academic Press, San Diego, Calif, USA, 999 [5] S Momani, Non-perurbaive analyical soluions of he spaceand ime-fracional Burgers equaions, Chaos, Solions & Fracals,vol28,no4,pp93 937,26 [6] Z M Odiba and S Momani, Applicaion of variaional ieraion mehod o nonlinear differenial equaions of fracional order, Inernaional Nonlinear Sciences and Numerical Simulaion,vol7,no,pp27 34,26 [7] S Momani and Z Odiba, Analyical soluion of a imefracional Navier-Sokes equaion by Adomian decomposiion mehod, Applied Mahemaics and Compuaion, vol 77, no2, pp , 26 [8] S Momani and Z Odiba, Numerical comparison of mehods for solving linear differenial equaions of fracional order, Chaos, Solions & Fracals,vol3,no5,pp ,27 [9] Z M Odiba and S Momani, Approimae soluions for boundary value problems of ime-fracional wave equaion, Applied Mahemaics and Compuaion, vol8,no,pp , 26 [] G Domairry and N Nadim, Assessmen of homoopy analysis mehod and homoopy perurbaion mehod in non-linear hea ransfer equaion, Inernaional Communicaions in Hea and Mass Transfer,vol35,no,pp93 2,28 [] G Domairry, M Ahangari, and M Jamshidi, Eac and analyical soluion for nonlinear dispersive K(m, p) equaions using homoopy perurbaion mehod, Physics Leers A, vol 368, no 3-4, pp , 27 [2] G Adomian, A review of he decomposiion mehod in applied mahemaics, Mahemaical Analysis and Applicaions, vol 35, no 2, pp 5 544, 988 [3] G Adomian, Solving Fronier Problems of Physics: The Decomposiion Mehod, vol6offundamenal Theories of Physics, Kluwer Academic Publishers, Dordrech, The Neherlands, 994 [4] S Momani, An eplici and numerical soluions of he fracional KdV equaion, Mahemaics and Compuers in Simulaion, vol 7, no 2, pp 8, 25 [5] Ph Guillaume and A Huard, Mulivariae Padé approimaion, Compuaional and Applied Mahemaics, vol 2, no -2, pp 97 29, 2 [6] V Turu, E Çelik, and M Yiğider, Mulivariae Padé approimaion for solving parial differenial equaions (PDE), Inernaional Journal for Numerical Mehods in Fluids,vol66,no9, pp 59 73, 2 [7] V Turu and N Güzel, Comparing numerical mehods for solving ime-fracional reacion-diffusion equaions, ISRN Mahemaical Analysis, vol22,aricleid73726,28pages, 22 [8] V Turu, Applicaion of Mulivariae padé approimaion for parial differenial equaions, Baman Universiy Life Sciences,vol2,no,pp7 28,22 [9] V Turu and N Güzel, On solving parial differenial eqauaions of fracional order by using he variaional ieraion mehod and mulivariae padé approimaion, European Journal of Pure and Applied Mahemaics Acceped [2] S Momani and R Qaralleh, Numerical approimaions and Padé approimans for a fracional populaion growh model, Applied Mahemaical Modelling, vol3,no9,pp97 94, 27 [2] S Momani and N Shawagfeh, Decomposiion mehod for solving fracional Riccai differenial equaions, Applied Mahemaics and Compuaion,vol82,no2,pp83 92,26 [22] K B Oldham and J Spanier, The Fracional Calculus, Academic Press, New York, NY, USA, 974 [23] M Capuo, Linear models of dissipaion whose Q is almos frequency independen Par II, he Royal Asronomical Sociey,vol3,no5,pp ,967 [24] Z Odiba and S Momani, Numerical mehods for nonlinear parial differenial equaions of fracional order, Applied Mahemaical Modelling,vol32,no,pp28 39,28 [25] A Cuy and L Wuyack, Nonlinear Mehods in Numerical Analysis,vol36ofNorh-Holland Mahemaics Sudies,Norh- Holland Publishing, Amserdam, The Neherlands, 987

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