A New Analytical Method for Solving Linear and Nonlinear Fractional Partial Differential Equations

Transcription

1 Progr. Fract. Differ. Appl. 2, No. 4, ) 247 Progress in Fractional Differentiation and Applications An International Jornal A New Analytical Method for Solving Linear and Nonlinear Fractional Partial Differential Eqations Sheh Maitama* and Irahim Adllahi Department of Mathematics, Faclty of Science, Northwest University, Kano, Nigeria Received: 2 May 201, Revised: 15 Jl. 201, Accepted: 18 Jl. 201 Plished online: 1 Oct. 201 Astract: In this paper, a new analytical method called the Natral Homotopy Pertration Method NHPM) for solving linear and the nonlinear fractional partial differential eqation is introdced. The proposed analytical method is an elegant comination of a wellknown method, Homotopy Pertration Method HPM) and the Natral Transform Method NTM). In this new analytical method, the fractional derivative is compted in Capto sense and the nonlinear terms are calclated sing He s polynomials. Exact soltion of linear and nonlinear fractional partial differential eqations are sccessflly otained sing the new analytical method, and the reslt is compared with the reslt of the existing methods. Keywords: Natral homotopy pertration method, He s polynomials, Mittag-Leffler fnction, linear and nonlinear fractional partial differential eqations. 1 Introdction In recent years, there is a rapid development in the concept of fractional calcls and its applications [1,2,3,4. The fractional calcls which deals with derivatives and integrals of aritrary orders [5 plays a vital role in many field of applied science and engineering. The linear and nonlinear fractional partial differential eqations have wide applications in flid mechanics, acostic, electromagnetism, signal processing, analytical chemistry, iology and many other areas of physical science and engineering [. In last few decades, many analytical and nmerical methods has een developed and sccessflly applied to solve linear and nonlinear fractional partial differential eqations sch as, Adomian Decomposition Method [7, 8, 9, 10, 11, G /G-Expansion Method [12, Homotopy Analysis Method [13, Jacoi Spectral Collocation Method [14, Laplace Decomposition Method [15, and New Spectral Algorithm [1. Moreover, Homotopy Pertration Method [17, 18, Yang Laplace Transform [19, Local Fractional Variational Iteration Method [20, 21, Cylindrical-Coordinate Method [22, Modified Laplace Decomposition Method [15, Spectral Legendre-Gass-Loatto Collocation Method [23, 24, Homotopy Pertration Smd Transform Method [25, 2, and Fractional Complex Transform Method [27 are applied to linear and nonlinear fractional partial differential eqations. However, despite the potential of the nmerical methods, they can not e considered as niversal methods for solving linear and nonlinear fractional partial differential eqations ecase of many deficiencies and some comptational difficlties sch as nnecessary linearization, discretization of variales, transformation or taking some restrictive assmptions. In this paper, a new analytical method called the Natral Homotopy Pertration Method NHPM) for solving linear and nonlinear fractional partial differential eqations withot the aove-mentioned deficiencies is introdced. The constrctive analytical method is applied directly to linear and nonlinear fractional partial differential eqations. The proposed analytical method gives a series soltion which converges rapidly to an exact or approximate soltion with elegant comptational terms. In this new analytical method, the nonlinear terms are compted sing He s Polynomials [28, 29, 30. Exact soltion of linear and nonlinear fractional partial differential eqation are sccessflly otained sing the new analytical method. Ths, the proposed analytical method is a powerfl mathematical method for solving linear and nonlinear fractional partial differential eqations and is a refinement of the existing methods. Corresponding athor sheh.maitama@yahoo.com Natral Sciences Plishing Cor.

2 248 S. Maitama, I. Adllahi : A new analytical method for solving linear and nonlinear... 2 Natral transform In this section, the asic definition and properties of the Natral transform are presented. Definition: { The Natral Transform of the fnction vt) for t 0, } ) is defined over the set of fnctions, t τ A= vt) : M, τ 1, τ 2 > 0, vt) <Me j, i f t 1) j [0,), y the following integral: N + [vt)= Vs,)= 1 Here s and are the Natral transform variales [31,32. The asic properties of the Natral transform method are given elow. 0 e st vt)dt; s > 0, >0. 1) Property 1: If Vs,) is the Natral transform and Fs) is the Laplace transform of the fnction ft) A, thenn + [ ft)= Vs,)= 1 st 0 e ft)dt = 1 F ) s. Property 2: If Vs,) is the Natral transform and G) is the Smd transform of the fnction vt) A, thenn + [vt)= Vs,)= 1 s 0 e t v ) t s dt = 1 s G ) s. Property 3: If N + [vt)= Vs,), thenn + [vαt)= 1 α Vs,). Property 4: If N + [vt)= Vs,), thenn + [v t)= s v0) Vs,). Property 5: If N + [vt)= Vs,), thenn + [v t)= s2 2 Vs,) s 2 v0) v 0). Remark: The Natral transform is a linear operator. That is, if α and β are non zero constants, then N + [α ft)±β gt)=αn + [ ft)±βn + [gt)=αf + s,)±β G + s,). Moreover, F + s,) and G + s,) are the Natral transforms of ft) and gt), respectively [31. Tale 1. List of some special Natral transforms Fnctional Form Natral transform Form t n s t s 2 e at s a 1 n 1 s n s n 1)!,n=1,2,... sint) 3 Basic Definitions of Fractional Calcls In this section, the asic definitions of fractional calcls are presented. Definition 1: A fnction fx), x>0 is said to e in the space C m α, m N {0}, if f m) C α. Definition 2: A real fnction fx), x>0 is said to e in the apace C α α R if there exist a real nmer p > α) sch that fx)=x p f 1 x) where f 1 x) C[0,). Clearly C α C β if β α. Definition 3: The left sided Riemann-Lioville fractional integral operator of order µ > 0, of a fnction ft) C α, and α 1 is define as [10,33. Dt µ ft)= 1 t Γµ) 0 fτ) dτ, µ > 0, t > 0, 2) t τ) 1 µ D 0 ft)= ft). 3) Definition 4: The left sided) Capto fractional derivative f, f ε C m 1, mε N {0}, is defined as [4,5. Note that [, 10, 4, 5: D µ t ft)= D µ t γ = { [ µ m D m ft) t t, m m 1< µ < m, m N, m ft) t m, µ = m. Γγ+ 1) Γγ+ µ+ 1) tγ+µ, µ > 0, γ > 1, t > 0. 5) 4) Natral Sciences Plishing Cor.

3 Progr. Fract. Differ. Appl. 2, No. 4, ) / D µ D t µ m 1 ft)= ft) v k) 0+) tk, m 1< µ m, m N. ) k! Definition 5: The Natral transform of the Capto fractional derivative is defined as: k=0 N + [Dt α m 1 sα vt)= α Vs,) k=0 s α k+1) α k v k) 0+), 7) m 1<α m). Definition : Mittag-Leffler fnction E α with α > 0 is defined y the following series representation, valid in the whole complex plane [5: E α z)= k=0 z k, α > 0, z C. 8) Γαk+1) 4 Analysis of the Method In this section, the asic idea of the Natral Homotopy Pertration Method is clearly illstrated y the standard nonlinear fractional partial differential eqation of the form: sject to the initial condition D α t vx,t)+mvx,t))+fvx,t))=gx,t), 9) vx,0)= fx). 10) where Fvx,t)) represent the nonlinear terms, Dt α t α is the Capto fractional derivative of the fnction vt), Mvx,t)) is the linear differential operator, and gx,t) is a sorce term. = α Applying the Natral transform to Eq. 9) sject to the given initial condition we get: Vx,s,)= 1 s fx)+ α s α N+ [gx,t) α s α N+ [Mvx,t))+Fvx,t)). 11) Taking the inverse Natral transform of Eq. 11), we have: vx,t)=gx,t) N 1 α s α N+ [Mvx,t))+Fvx,t)), 12) where Gx,t) is a term arising from the sorce term and the prescried initial condition. Now we apply the Homotopy Pertration Method. The nonlinear terms Fvx, t)) is decomposed as: vx,t)= Fvx,t))= p n v n x,t). 13) where H n v) is the He s polynomial and e compted sing the following formla: n H n v 1,v 2,,v n )= 1 n! p n [ F p n H n v), 14) n j=0 p j v j ) p=0,,1,2, 15) Natral Sciences Plishing Cor.

4 250 S. Maitama, I. Adllahi : A new analytical method for solving linear and nonlinear... Sstitting Eq. 13) and Eq. 14) into Eq. 12), we get: [ p n v n x,t)=gx,t) p N 1 α s α N+[ p n Mvx,t))+ p n H n v) Using the coefficient of the likes powers of p in Eq.1), the following approximations are otained: Hence, the series soltion of Eq. 9) is given y: p 0 : v 0 x,t)=gx,t), p 1 : v 1 x,t)= N 1 α s α N+ [Mvx,t))+H 0 v), p 2 : v 2 x,t)= N 1 α s α N+ [Mvx,t))+H 1 v), [ p 3 : v 3 x,t)= N 1 α s α N+ [Mvx,t))+H 2 v),., vx,t)= lim N N ). 1) v n x,t). 17) 5 Applications In this section, the application of the Natral Homotopy Pertration Method to linear and nonlinear fractional partial differential eqations are clearly demonstrated to show its simplicity and high accracy. Example 1Consider the following linear fractional partial differential eqation of the form: sject to the initial condition D α t v 2v xx 2v yy = 0, <x,y<, t > 0, 18) vx,y,0)=sinx)siny), v t x,y,0)=0, α ε 1,2). 19) Applying the Natral transform to Eq. 18) sject to the given initial condition, we get: Vx,y,s,)= sinx)siny) s + α s α N+ [2v xx + 2v yy. 20) Taking the inverse Natral transform of Eq. 20), we get: vx,y,t)=sinx)siny)+n 1 α s α N+ [2v xx + 2v yy. 21) Now we apply the Homotopy Pertration Method. vx,y,t)= Then Eq. 21) will ecome: [ p n v n x,y,t)=sinx)siny)+ p N 1 α p n v n x,y,t). 22) s α N+ [ 2 p n v nxx + 2 p n v nyy ). 23) Natral Sciences Plishing Cor.

5 Progr. Fract. Differ. Appl. 2, No. 4, ) / Using the coefficients of the like powers of p in Eq.23), the following approximations are otained: Then, the series soltion of Eq. 18) is given y: vx,y,t)= lim N N p 0 : v 0 x,y,t)= sinx)siny), [ p 1 : v 1 x,y,t)=n 1 α s α N+ [2v 0xx + 2v 0yy 4t α ) = sinx)siny) Γα+ 1), p 2 : v 2 x,y,t)=n 1 α s α N+ [2v 1xx + 2v 1yy 4t α ) 2 = sinx)siny) Γ2α+ 1), p 3 : v 3 x,y,t)=n 1 α s α N+ [2v 2xx + 2v 2yy 4t α ) 3 = sinx)siny) Γ3α+ 1), p 4 : v 4 x,y,t)=n 1 α s α N+ [2v 3xx + 2v 3yy. 4t α ) 4 = sinx)siny) Γ4α+ 1), v n x,y,t) 24) = v 0 x,y,t)+v 1 x,y,t)+v 2 x,y,t)+v 3 x,y,t)+ = sinx)siny) 1 4tα ) Γα+ 1) + 4tα ) 2 Γ2α+ 1) 4tα ) 3 Γ3α+ 1) + 4tα ) 4 ) Γ4α+ 1) + = sinx)siny) m=0 4t α ) m Γmα+ 1) = sinx)siny)e α 4t α ). When α = 1, we otained the following reslt: vx,y,t)= lim N N v n x,y,t) 25) = v 0 x,y,t)+v 1 x,y,t)+v 2 x,y,t)+v 3 x,y,t)+ = sinx)siny) 1 4t) ) 1! + 4t)2 4t)3 + 4t)4 + 2! 3! 4! = sinx)siny)e 4t, which is the exact soltion of Eq. 18). The exact soltion is in close agreement with the reslt otained y HPSTM) [2. Natral Sciences Plishing Cor.

6 252 S. Maitama, I. Adllahi : A new analytical method for solving linear and nonlinear... Example 2Consider the following nonlinear fractional partial differential eqation of the form: D α t v vv x + v xxx = 0, 2) sject to the initial condition vx,0)= 1 x 1). 27) Applying the Natral transform on oth sides of Eq. 2), we get: Vx,s,)= 1 α x 1)+ s s α N+ [vv x v xxx 28) Taking the inverse Natral transform of Eq.28), we get: vx,t)= 1 α x 1)+N 1 s α N+ [vv x v xxx. 29) Now we apply the Homotopy Pertration Method. Then, Eq. 29), will ecome: vx,t)= [ p n v n x,t)= 1 x 1)+ p N 1 α s α N+ p n v n x,t). 30) [ where H n v) is a He s Polynomial which represent the nonlinear term, vv x. Some few components of the He s Polynomials are given elow: p n H n v) H 0 v)=v 0 v 0x, H 1 v)=v 1 v 0x + v 0 v 1x, H 2 v)=v 0x v 2 + v 1x v 1 + v 2x v 0, H 3 v)=v 0x v 3 + v 1x v 2 + v 2x v 1 + v 3x v 0,., p n v nxxx x,t) Using the coefficient of the like powers of p, in Eq.31), we otained the following approximations: p 0 : v 0 x,t)= 1 x 1), p 1 : v 1 x,t)=n 1 α s α N+ [H 0 v) v 0xxx =N 1[ s N+ [v 0 v 0x v 0xxx = x 1) t α ), Γα+ 1) p 2 : v 2 x,t)=n 1[ s N+ [H 1 v) v 1xxx =N 1[ = x 1) s N+ [v 1 v 0x + v 0 v 1x ) v 1xxx t 2α ), Γ2α+ 1) ), 31) Natral Sciences Plishing Cor.

7 Progr. Fract. Differ. Appl. 2, No. 4, ) / p 3 : v 3 x,t)=n 1[ s N+ [H 2 v) v 2xxx =N 1[.., = x 1) Therefore, the series soltion of Eq. 2) is given y: vx,t)= lim N N v n x,t) s N+ [v 0x v 2 + v 1x v 1 + v 2x v 0 ) v 2xxx t 3α ), Γ3α+ 1) = v 0 x,t)+v 1 x,t)+v 2 x,t)+v 3 x,t)+ = x 1) t α 1+ Γα+ 1) + t 2α Γ2α+ 1) + t 3α ) Γ3α+ 1) + = x 1) t α ) m Γmα+ 1) m=0 = x 1) E α t α ). When α = 1, we otained the exact soltion of Eq.2) as: vx,t)= 1 x 1 1 t ), t <1. 32) The exact soltion is in close agreement with the reslt otained y ADM) [11 and NDM) [34. Example 3Consider the following nonlinear time-fractional Harry Dym eqation of the form: sject to the initial condition D α t vx,t) v3 x,t)d x vx,t)=0, 0<α 1, 33) vx,0)= Applying the Natral transform to Eq.33) sject to the given initial condition, we get: Vx,s,)= 1 s Taking the inverse Natral transform of Eq.35), we get: Now we apply the Homotopy Pertration Method. Then Eq.3) will ecome: ) 2 2 x. 34) ) 2 2 x + α s α N+[ v 3 x,t)d x vx,t). 35) ) 2 vx,t)= [ 2 x +N 1 α s α N+[ v 3 x,t)d x vx,t). 3) vx,t)= p n v n x,t). 37) ) 2 [ p n v n x,t)= 2 x + pn 1 ) α s α N+[ p n H n v), 38) Natral Sciences Plishing Cor.

8 254 S. Maitama, I. Adllahi : A new analytical method for solving linear and nonlinear... where H n v) is the He s polynomials which represent the nonlinear term v 3 x,t)d x vx,t). Some few components of the He s polynomials are given elow: H 0 v)=v 3 0 D xv 0, H 1 v)=v 3 0 D xv 1 + 3v 1 v 2 0 D xv 0, H 2 v)=v 3 0 D xv 2 + 3v 1 v 2 0D x v 1 + 3v 0 v v 2 0v 2 ) D 3 x v 0,.., Using the coefficients of the like powers of p in Eq.38), we otained the following approximations: ) 2 p 0 : v 0 x,t)= 2 x, p 1 : v 1 x,t)=n 1 α s α N+ [H 0 v) =N 1 α s α N+[ v 3 0 D xv 0 = 2 3 ) 1 2 x t α Γα+ 1), p 2 : v 2 x,t)=n 1 α s α N+ [H 1 v) [ =N 1 α s α N+[ v 3 0 D xv 1 + 3v 1 v 2 0D x v 0 = 3 2 ) 4 2 x t 2α Γ2α+ 1), p 3 : v 3 x,t)=n 1 α s α N+ [H 2 v) [ =N 1 α s α N+[ v 3 0 D xv 2 + 3v 1 v 2 0D x v 1 + 3v 0 v v 2 ) 0v 2 D 3 x v 0., = 9 2 a 3 2 x ) Γ2α+ 1) 2Γα+ 1)) 2 1 ) t 3α Γ3α+ 1), Then, the series soltion of Eq.33) is given y: vx,t) = v n x,t) = v 0 x,t)+v 1 x,t)+v 2 x,t)+v 3 x,t)+ ) 2 ) = 1 2 x x t α Γα+ 1) ) 4 ) x t 2α 7 Γ2α+ 1) x Γ2α+ 1) 2 2Γα+ 1)) 2 1 ) t 3α Γ3α+ 1) 39) Natral Sciences Plishing Cor.

9 Progr. Fract. Differ. Appl. 2, No. 4, ) / When α = 1, we otained the exact soltion of Eq.33) as: vx,t)= The exact soltion is in close agreement with the reslt otained y ADM) [25. ) 2 2 x+t). 40) Conclsion In this paper, Natral transform method NTM) and Homotopy Pertration Method HPM) are sccessflly comined to form a powerfl analytical method called the Natral Homotopy Pertration Method NHPM) for solving linear and nonlinear fractional partial differential eqations. The new analytical method gives a series soltion which converges rapidly to an exact or approximate soltion with elegant comptational terms. In this new analytical method, the fractional derivative are handle in Capto sense and the nonlinear term are compted sing He s Polynomials. The new analytical method is applied sccessflly and otained an exact soltion of linear and nonlinear fractional partial differential eqations. The simplicity and high accracy of the new analytical method is clearly illstrated. Ths, the Natral Homotopy Pertration Method is a powerfl analytical method for solving linear and nonlinear fractional partial differential eqations. Acknowledgment The athors are highly gratefl to the editor and the anonymos referees for their sefl comments and sggestions in this paper. References [1 K. S. Miller and B. Ross, An introdction to the fractional calcls and fractional differential eqations, New York: Johan Willey and Sons, Inc [2 K. B. Oldham and J. Spanier, The fractional calcls, New York: Academic Press [3 K. Diethelm and N. J. Ford, Analysis of fractional differential eqations, J. Math. Appl., ). [4 S. G. Samko, A. A. Kilas and O. I. Marichev, Fractional integrals and derivatives: theory and applications, Yverdon: Gordon and Breach [5 I. Podlny, Fractional differential eqations, Academic Press, San Diego [ B. J. West, M. Bologna and P. Grigolini, Physics of fractal operators, New York: Springer [7 S. S. Ray, Analytical soltion for the space fractional diffsion eqation y two-step Adomian decomposition method, Commn. Nonlinear Sci. Siml. 14, ). [8 K. Aai and Y. Cherralt, New ideas for proving convergence of Adomian decomposition methods, Compt. Math. Appl. 32, ). [9 S. Momani and Z. Odiat, Analytical soltion of time-fractional Navier-Stoke eqation y Adomian decomposition method, Appl. Math. Compt. 177, ). [10 H. Jafari and V. Daftardar-Gejji, Solving linear and nonlinear fractional diffsion and wave eqations y Adomian decomposition, Appl. Math. Compt. 180, ). [11 A. M. Wazwaz, Partial differential eqations and solitary waves theory, Springer Verlag, Heidelerg [12 A. H. Bhrawy, M. A. Adelkawy, A. Alshaery and E. M. Hilal, Symolic comptation of some nonlinear fractional differential eqations, Rom. J. Phys. 595), ). [13 M. Ganjiana, Soltion of nonlinear fractional differential eqations sing homotopy analysis method, Appl. Math. Model Siml. 3, ). [14 A. H. Bhrawy, A Jacoi spectral collocation method for solving mlti-dimensional nonlinear fractional s-diffsion eqations, Nmer. Algoritm. DOI: /s ). [15 H. Jafari, C. M. Khaliqe and M. Nazari, Application of the Laplace decomposition method for solving linear and nonlinear fractional diffsion-wave eqations, Appl. Math. Lett. 24, ). [1 A. H. Bhrawy, A new spectral algorithm for time-space fractional partial differential eqations with sdiffsion and sperdiffsion, Proc. Rom. Acad. A. 17, ). [17 P. K. Gpta and M. Singh, Homotopy pertration method for fractional Fornerg-Whitham eqation, Compt. Math. Appl. 1, ). Natral Sciences Plishing Cor.

10 25 S. Maitama, I. Adllahi : A new analytical method for solving linear and nonlinear... [18 S. Momani and Z. Odiat, homotopy pertration method for nonlinear partial differential eqations of fractional order, Phys. Lett. A. 35, ). [19 Y. Z. Zhang, A. M. Yang and Y. Long, initial ondary vale prolem for fractal heat eqation in the semi-infinite region y Yang-Laplace transform, Thermal Sci. 18, ). [20 X. J. Yang and D. Balean, Fractional heat condction prolem solved y local fractional variation iteration method, Thermal Sci. 17, ). [21 X. J. Yang, D. Balean, Y. Khan and S. T. Mohyd-Din, local fractional variation iteration method for diffsion and wave eqations on Cantor sets, Rom. J. Phys. 59, ). [22 X. J. Yang, H. M. Srivastava, J. H. He and D. Balean, cantor-type cylindrical-coordinate fractional derivatives, Proc. Rom. Acad. Series A. 14, ). [23 A. H. Bhrawy and D. Balean, A spectral Legendre-Gass-Loatto collocation method for a space-fractional advection eqations with variale coefficients, Rep. Math. Phys. 72, ). [24 A. H. Bhrawy, A new Legendre collocation method for solving a two-dimensional diffsion eqations, Astr. Appl. Anal. 2014, Article ID: ). [25 D. Kmar, J. Singh and A. Kilicman, An efficient approach for fractional Harry Dym eqation y sing Smd transform. Astr. Appl. Anal Article ID 08943, ). [2 A. Atangana and A. Kilicman, The se of Smd transform for solving certain nonlinear fractional heat-like eqations, Astr. Appl. Anal. 2013, Article ID: ). [27 B. Ghazanfari and A. G. Ghazanfari, Solving fractional nonlinear Schrodinger eqation y fractional complex transform method, Int. J. Math. Model. Compt. 2, ). [28 J. H. He, Homotopy pertration techniqe, Compt. Meth. Appl. Mech. Eng. 178, ). [29 J. H. He, Recent development of the homotopy pertration method, Topolog. Meth. Nonlinear Anal. 31, ). [30 J. H. He, The homotopy pertration method for nonlinear oscillators with discontinities, Appl. Math. Compt. 151, ). [31 F. B. M. Belgacem and R. Silamarasan, Theory of natral transform, Mathematics in Engineering, Science and Aerospace 3, ). [32 F. B. M. Belgacem and R. Silamarasan, Advances in the natral transform, AIP Conference Proceedings; 1493 Janary 2012; USA: American Institte of Physics ). [33 K. Snil, K. Deepak, A. Saied and M. M. Rashidi, Analytical soltion of fractional Navier-Stoke eqation y sing modified Laplace decomposition method, Ain Shams Eng. J. 5, ). [34 M. S. Rawashdeh and S. Maitama, Solving PDEs sing the natral decomposition method, Nonlinear Std. 23, ). Natral Sciences Plishing Cor.