Variational Iteration Method and He’s Polynomials for Time-Fractional Partial Differential Equations

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  Abstract: In this work, we have applied the variational iteration method and He’s polynomials to solve partial differential equation (PDEs) with time-fractional derivative. The variational homotopy perturbation iteration method (VHPIM) is presented
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  Progr. Fract. Differ. Appl.  1 , No. 1, 47-55 (2015) 47 Progress in Fractional Differentiation and Applications  An International Journal http://dx.doi.org/10.12785/pfda/010105 Variational Iteration Method and He’s Polynomials forTime-Fractional Partial Differential Equations  Abdolali Neamaty 1 , ∗  , Bahram Agheli 1 and Rahmat Darzi 2 1 Department of Mathematics, University of Mazandaran, Babolsar, Iran 2 Department of Mathematics, Neka Branch, Islamic Azad University, Neka, IranReceived: 12 Nov. 2014, Revised: 16 Dec. 2014, Accepted: 19 Dec. 2014Published online: 1 Jan. 2015 Abstract:  In this work, we have applied the variational iteration method and He’s polynomials to solve partial differential equation(PDEs) with time-fractional derivative. The variational homotopy perturbation iteration method (VHPIM) is presented in two steps.Some illustrative examples are given in order to show the ability and simplicity of the approach. All numerical calculations in thismanuscript were performed on a PC applying some programs written in  Maple 18. Keywords:  Variational-iteration method (VIM), Homotopy-perturbation method (HPM), fractional partial differential equation(FPDEs). 1 Introduction In recent years, fractional area has been one of the most interesting issues that has attracted many scientists, speciallyin the fields of mathematics and engineering sciences. Many natural phenomena can be presented by boundary valueproblems of fractional differential equations.Many authors in various fields such as chemical physics, fluid flows, electrical networks, viscoelasticity, have triedto present a model of these phenomena by fractional differential equations [1,2]. In order to achieve extra information in fractional calculus, interested readers can refer to more valuable books that are written by other authors [3,4,5]. Most fractional differential equations do not have accurate analytical solutions. Therefore,direct and iterative estimate methodsare used. In the recent years, several methods have been used to solve FDEs and FPDEs as ADM [6,7,8], VIM [9,10,11, 12], HPM[13,14,15,16], HAM[17,18] and so on[19,20,21]. In this research work, combining two different methods VIM and HPM is purposed for solving the time-fractionalpartial differential equation.The purpose behind this research work is to use He’s homotopy offered by He [22,23] and extend its application in order to solve FPDEs. First using an algorithm, after combining VIM and HPM methods, in two steps, we arrive to thefollowing equation: ∞ ∑ n = 0  p n u n (  x , t  ) =  u 0 (  x , t  )+  p   ∞ ∑ n = 1  p n u n (  x , t  )+  I  β   λ  ( t  )( ∞ ∑ n = 0  p n  D α  u n (  x , t  ) − ∞ ∑ n = 0  p n ℜ ( ˇ u n , ( ˇ u n )  x , ( ˇ u n )  xx ) −  f  (  x , t  ))  . Lagrange multiplier  λ   is calculated with VIM method. Eventually using HPM method for  p → 1, we get an estimate of the solution, that is: u (  x , t  ) = ∞ ∑ n = 0 u n (  x , t  ) . The present research work is arranged in five sections. In Section 2, the preliminaries and a reliable algorithm of VHPIM are given. In Section 3, we describe the estimate of solution. In Section 4, as the applications of this method, ∗ Corresponding author e-mail: namaty@umz.ac.ir c  2015 NSPNatural Sciences Publishing Cor.  48 A. Neamaty et. al. : Variational Iteration Method and He’s Polynomials... time-fractionaladvection,hyperbolicandFisherequationshavebeensensiblysolved.A conclusionis presentedin Section5. 2 Preliminaries and a reliable algorithm of VHPIM In this part of the paper, we present and define Riemann-Liouville fractional integral and Caputo’s fractional derivative[5]. Then VHIPM method is introduced and explained in detail. Definition 1.  A real function  f  (  x ) ,  x  >  0, is purposed to be in the space C  ν  , ( ν  ∈  R ), if there exists a real number  n ( >  ν  ) ,so that  f  (  x ) =  x n  f  1 (  x ) , where  f  1 (  x ) ∈ C  [ 0 , ∞ ) , and  f   ∈ C  k  ν   iff   f  ( k  ) ∈ C  ν  ,  k  ∈  N  . Definition 2.  The Riemann-Liouville fractional integral operator of order of   α   >  0,  f   ∈ C  ν  ,  ν  ≥− 1, is given:  I  α  a  f  (  x ) =  1 Γ  ( α  )    xa (  x − r  ) α  − 1  f  ( r  ) dr  ,  I  α   f  (  x ) =  I  α  0  f  (  x ) ,  I  0  f  (  x ) =  f  (  x ) . Definition 3.  The Caputo’s fractional derivative of   f   is defined:  D α   f  (  x ) =  I  k  − α   D k   f  (  x ) =  1 Γ  ( k  − α  )    x 0 (  x − r  ) k  − α  − 1  f  ( k  ) ( r  ) dr  ,  x  >  0 . where,  f   ∈ C  k  − 1 ,  k  − 1  <  α  ≤ k   and  k  ∈ N .  Property 1.  For  k  − 1 <  α  ≤ k  ,  k  ∈ N ,  f   ∈ C  k  ν  ,  ν  ≥− 1 and  x  >  0 ,  the following properties satisfyi)  D α  a  I  α  a  f  (  x ) =  f  (  x ) .ii)  I  α  a  D α  a  f  (  x ) =  f  (  x ) − k  − 1 ∑  j = 0  f  (  j ) ( a + ) (  x − a )  j  j !  .First of all, we purpose the following nonlinear problem, with two variables  x  and  t  :  D α  t   u (  x , t  ) = R u (  x , t  )+  f  (  x , t  ) ,  (1)where  D α  t   is the fractional Caputo derivative with respect to  t  ,  α   >  0, R is an operator in  x , and  t   which might includederivatives with respect to ”  x ”,  u (  x , t  )  is an uncertain function, and  f  (  x , t  )  is the srcin in homogeneous sentence.Following this, VHPM is introduced and explained in two steps. Step 1.  Pursuant to VIM, we create the revision functional on Eq.(1): u n + 1 (  x , t  ) =  u n (  x , t  )+  I  β   λ  ( t  )   D α  u n (  x , t  ) − R  ˇ u n , ( ˇ u n )  x , ( ˇ u n )  xx  −  f  (  x , t  )  =  u n (  x , t  )+  1 Γ  ( β  )    t  0 ( t  − τ  ) β  − 1 λ  ( τ  )   D α  u n (  x , τ  ) − R  ˇ u n , ( ˇ u n )  x , ( ˇ u n )  xx  −  f  (  x , τ  )  d  τ  ,  (2)where  I  β  is the Riemann-Liouville fractional integral operator of under  β   =  α  −  floor  ( α  ) , that is  β   =  α   + 1 − m , and  λ  is a common Lagrange multiplier, which may be discerned through variational principle.The function ˇ u k   is regarded on a confined variation, that is  δ   ˇ u k   =  0. Step 2.  By applying the VIM and HPM, we create the following repetition equation: ∞ ∑ n = 0  p n u n (  x , t  ) =  u 0 (  x , t  )+  p   ∞ ∑ n = 1  p n u n (  x , t  )+  I  β   λ  ( t  )( ∞ ∑ n = 0  p n  D α  u n (  x , t  ) − ∞ ∑ n = 0  p n ℜ ( ˇ u n , ( ˇ u n )  x , ( ˇ u n )  xx ) −  f  (  x , t  ))  .  (3)which is named as VHPIM.In the Eq.(3),  u 0  is primary estimation of Eq.(1), and  p ∈ [ 0 , 1 ]  is embedded parameter.Equating the sentences with identical powers of   p  in the Eq.(3), we can acquire  u i  ( i  =  0 , 1 , 2 ,... ) .Finally, pursuant to HPM, for  p → 1, we estimate the solution: u (  x , t  ) = ∞ ∑ n = 0 u n (  x , t  ) .  (4) c  2015 NSPNatural Sciences Publishing Cor.  Progr. Fract. Differ. Appl.  1 , No. 1, 47-55 (2015) /  www.naturalspublishing.com/Journals.asp 49 3 Estimate of solution Pursuant to VIM, the revision functional Eq.(2) may be estimable: u n + 1 (  x , t  ) =  u n (  x , t  )+    t  0 λ  ( τ  )   d  m dt  m u (  x , τ  ) − R  ˇ u n , ( ˇ u n )  x , ( ˇ u n )  xx  −  f  (  x , τ  )  d  τ  , where ˇ u n  is a revision functional, but ˇ u n  is regarded on a confined variation, i.e  δ   ˇ u n (  x ,  t  ) =  0.Next, by making functional still: δ  u n + 1 (  x , t  ) =  δ  u n (  x , t  ) + δ     t  0 λ  ( τ  )   d  m dt  m u (  x , τ  ) −  f  (  x , τ  )  d  τ  , we acquire the Lagrange multiplier as1)  m  =  1 :  λ   = − 1,2)  m  =  2 :  λ   =  τ  − t  . Case 1.  If   m  =  1, i.e in case 0  <  α  ≤ 1, substituting  λ   = − 1 into Eq.(2.2), we acquire the repetition formula: u n + 1 (  x , t  ) =  u n (  x , t  ) −  I  α    D α  u n (  x , t  ) − R u n (  x , t  ) −  f  (  x , t  )  . Using Eq.(3), we may construct the repetition formula as follows: ∞ ∑ n = 0  p n u n (  x , t  ) =  u 0 (  x , t  )+  p   ∞ ∑ n = 1  p n u n (  x , t  ) −  I  α   ∞ ∑ n = 0  p n  D α  u n (  x , t  ) − ∞ ∑ n = 0  p n R  ˇ u n , ( ˇ u n )  x , ( ˇ u n )  xx  −  f  (  x , t  )  .  (5)contrasting the sentences with identical powers of   p  in the Eq.(5), we can acquire  u i (  x , t  ) ,  ( i  =  0 , 1 ,  2 ,  3 ,... ) .Pursuant to HPM, we have: u (  x , t  ) =  u 0 (  x , t  )+ u 1 (  x , t  )+ u 2 (  x , t  )+ ··· , which is an estimate solution for Eq.(1). Case 2.  For  m  =  2, i.e in the case 1  <  α  ≤ 2, we substitute  λ   =  τ  − t   into functional Eq.(2), to acquire: u n + 1 (  x , t  ) =  u n (  x , t  ) − α  − 1 Γ  ( α  )    t  0 ( t  − τ  ) α  − 1 (  D α  u n (  x , τ  ) − R u n (  x , τ  ) −  f  (  x , τ  )) d  τ  . Therefore, we have: u n + 1 (  x , t  ) =  u n (  x , t  ) − β   I  α  [  D α  u n (  x , t  ) − R u n (  x , t  ) −  f  (  x , t  )] . where  β   =  α  − 1.Using Eq.(3), we have the following repetition formula: ∞ ∑ n = 0  p n u n (  x , t  ) =  u 0 (  x , t  )+  p   ∞ ∑ n = 1  p n u n (  x , t  ) − β   I  α   ∞ ∑ n = 0  p n  D α  u n (  x , t  ) − ∞ ∑ n = 0  p n R  ˇ u n , ( ˇ u n )  x , ( ˇ u n )  xx  −  f  (  x , t  )  .  (6) 4 Applications and results This part, we applyVIM and He’s polynomialsto solve nonlineartime-fractionaladvectionpartial differentialequation,time-fractionalhyperbolicequation,time-fractionalFisher’s equation. All of the plots and computationsfor this equationshave been done on a PC applying some programs written in  Maple 18. Example 1.  We purpose the time-fractional advection partial differential equation: d  α  dt  α   u (  x , t  )+ u (  x , t  )  u  x (  x , t  ) =  x ( 1 + t  2 ) ,  t   >  0 ,  x ∈ R ,  0  <  α  ≤ 1 ,  (7)with the primary condition: u (  x , 0 ) =  0 .  (8) c  2015 NSPNatural Sciences Publishing Cor.  50 A. Neamaty et. al. : Variational Iteration Method and He’s Polynomials... Now if we substitute the primary amount  u (  x , 0 )  into the repetition formulation Eq. (5), the result will be:  p 0 :  u 0 (  x , t  ) =  0  p 1 :  u 1 (  x , t  ) =  xt  α  Γ  ( α   + 1 ) +  2!  xt  α  + 2 Γ  ( α   + 3 ) ,  p 2 :  u 2 (  x , t  ) = −  x Γ  ( 2 α   + 1 ) t  3 α  Γ  ( 3 α   + 1 )  − 4  x Γ  ( 2 α   + 3 ) t  3 α  + 2 Γ  ( α   + 1 ) Γ  ( α   + 3 ) Γ  ( 3 α   + 3 ) − 4 Γ  ( 2 α   + 5 ) t  3 α  + 4 Γ  ( α   + 3 ) Γ  ( α   + 3 ) Γ  ( 3 α   + 5 ) ,  p 3 :  u 3 (  x , t  ) = −  x  Γ  ( 2 α   + 1 ) Γ  ( 3 α   + 1 )  2 Γ  ( 6 α   + 1 ) Γ  ( 7 α   + 1 ) t  7 α  +  8 Γ  ( 2 α   + 1 ) Γ  ( 2 α   + 3 ) Γ  ( 6 α   + 3 ) Γ  ( α   + 1 ) Γ  ( α   + 3 ) Γ  ( 3 α   + 1 ) Γ  ( 3 α   + 3 ) Γ  ( 7 α   + 3 ) t  7 α  + 2 +  8 Γ  ( 2 α   + 1 ) Γ  ( 2 α   + 5 ) Γ  ( 6 α   + 5 )[ Γ  ( α   + 3 )] 2 Γ  ( 3 α   + 1 ) Γ  ( 3 α   + 5 ) Γ  ( 7 α   + 5 ) t  7 α  + 4 +  16 [ Γ  ( 2 α   + 3 )] 2 Γ  ( 6 α   + 5 )[ Γ  ( α   + 1 ) Γ  ( α   + 3 ) Γ  ( 3 α   + 3 )] 2 Γ  ( 7 α   + 5 ) t  7 α  + 4 +  32 Γ  ( 2 α   + 3 ) Γ  ( 2 α   + 5 ) Γ  ( 6 α   + 7 )[ Γ  ( α   + 3 )] 3 Γ  ( α   + 1 ) Γ  ( 3 α   + 3 ) Γ  ( 3 α   + 5 ) Γ  ( 7 α   + 7 ) t  7 α  + 6 +  16 [ Γ  ( 2 α   + 5 )] 2 Γ  ( 6 α   + 9 )[ Γ  ( α   + 3 )] 4 [ Γ  ( 3 α   + 5 )] 2 Γ  ( 7 α   + 9 ) t  7 α  + 8  ,... . Then, with purpose the first three sentence as estimate of solution for Eq.(7) is: u (  x , t  ) =  xt  α  Γ  ( α   + 1 ) +  2!  xt  α  + 2 Γ  ( α   + 3 ) −  x Γ  ( 2 α   + 1 ) t  3 α  Γ  ( 3 α   + 1 )  − 4  x Γ  ( 2 α   + 3 ) t  3 α  + 2 Γ  ( α   + 1 ) Γ  ( α   + 3 ) Γ  ( 3 α   + 3 ) − 4 Γ  ( 2 α   + 5 ) t  3 α  + 4 Γ  ( α   + 3 ) Γ  ( α   + 3 ) Γ  ( 3 α   + 5 ) . The solution that we have found is match to the accurate solution  u (  x , t  ) =  xt  , which is the same third order sentenceestimate solution for Eq. (7)-(8) acquired from [25] applying VIM. We can also solve the advection partial differential equation with time-fractional derivative Eq.(7) in [25] through applying ADM. Accordingly, the third order estimate of the decomposition series solution is presented in [25] on Eq.(7): u (  x , t  ) =  x   t  α  Γ  ( α   + 1 ) +  2 t  α  + 2 Γ  ( α   + 3 ) − Γ  ( 2 α   + 1 ) t  3 α  Γ  ( α   + 1 ) 2 Γ  ( 3 α   + 1 ) − 4 Γ  ( 2 α   + 3 ) t  3 α  + 2 Γ  ( α   + 1 ) Γ  ( α   + 3 ) Γ  ( 3 α   + 3 )  . In Table 1, we can see the estimate solutions toward  α   = 1 which is derivedfor variousvalues of   x  and t   applyingVHPIM,VIM, HPM and ADM. c  2015 NSPNatural Sciences Publishing Cor.
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