- Email: [email protected]
A short remark on fractional variational iteration method
Physics Letters A 375 (2011) 3362–3364
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
A short remark on fractional variational iteration method Ji-Huan He ∗ National Engineering Laboratory for Modern Silk, College of Textile and Engineering, Soochow University, 199 Ren-ai Road, Suzhou 215123, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 18 June 2011 Accepted 18 July 2011 Available online 28 July 2011 Communicated by R. Wu
This Letter compares the classical variational iteration method with the fractional variational iteration method. The fractional complex transform is introduced to convert a fractional differential equation to its differential partner, so that its variational iteration algorithm can be simply constructed. © 2011 Elsevier B.V. All rights reserved.
Keywords: Modified Riemann–Liouville derivative Fractional complex transform Variational iteration method
1. Introduction The variational iteration method was originally proposed to solve ordinary differential equations and partial differential equations [1–6]. In 1998, the method was first adopted to solve fractional differential equations [7,8], and the solution procedure was routinely used to solve various fractional differential equations [9–14] before the fractional variational iteration method [15–17] and the fractional complex transform [18–22] were appeared in literature. In this Letter we will first compare the classical variational iteration algorithm with the fractional variational iteration algorithm, and then introduce the fractional complex transform to construct a simpler variational iteration algorithm for fractional calculus.
where u is a continuous (but not necessarily differentiable) function. In order to solve fractional differential equation by the variational iteration method, we have to re-write Eq. (1) in the form
∂u + F (u ) = 0 ∂t
(3)
where
F (u ) = −
∂u + D tα u + f (u ) ∂t
(4)
The correction functional is constructed as follows
un+1 (t ) = un (t ) +
t ∂ un λ + F˜ (un ) ds ∂s
(5)
t0
2. Classical variational iteration method vs fractional variational iteration method
(1)
where D tα u = D α u / Dt α denotes Jumarie’s fractional derivation [23–26], which is a modified Riemann–Liouville derivative defined as
D α u (t ) = t
*
1
d
(1 − α ) dt
α D t un + f (un ) ds
(6)
t0
0<α1
t
t un+1 (t ) = un (t ) −
Consider the following fractional differential equation
D tα u + f (u ) = 0,
After identification of the Lagrange multiplier, we have
−α
(t − ξ )
u (ξ ) − u (0) dξ
0
Tel.: +86 512 6588 4633. E-mail address: [email protected].
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.07.033
(2)
The Lagrange multiplier above was approximately identified under the frame of the integer differential mode. Recently Wu and Lee [15] suggested a more accurate way to identify the Lagrange multiplier by constructing a correction functional in the form
un+1 (t ) = un (t ) +
1
(1 + α )
t
λ D tα un + f (un ) (ds)α
(7)
t0
The functional contains fractional integral with respect to (ds)α which is elusive to nonmathematicians, furthermore, the identification of the Lagrange multiplier requires the full knowledge of variational theory for fractional differential equations [27].
J.-H. He / Physics Letters A 375 (2011) 3362–3364
3. Fractional complex transform
Consider an example [15]
∂ α u 1 2 ∂ 2u − x =0 ∂tα 2 ∂ x2
(8)
The classical variational iteration algorithm reads
t α ∂ un (x, s) 1 2 ∂ 2 un (x, s) un+1 (x, t ) = un (x, t ) − − ds x ∂ sα 2 ∂ x2
(9)
0
t ×
The fractional complex transform [18–22] can convert a fractional differential equation into its differential partner, so that the variational iteration method can be effectively used. By the fractional complex transform
T=
tα
∂u ∂αu ∂u ∂α T = · = ∂tα ∂t ∂tα ∂T
1
(1 + α )
∂ α un (x, s) 1 2 ∂ 2 un (x, s) (ds)α − x ∂ sα 2 ∂ x2
(10)
If we begin with u 0 = x2 , both algorithms lead to series solutions converging to the exact solution. Consider another example [15]
∂u 1 ∂ 2u − x2 2 = 0 ∂T 2 ∂x
T u n +1 ( T ) = u n ( T ) −
∂ 2u + N (u ) = 0 ∂ x2
X=
∂ 2 u ∂ u ∂ 2α u + + 2α ∂t ∂ x2 ∂x
xα
(14)
∂ 2u ∂u + =0 ∂t ∂ X2
X
After identification of the Lagrange multiplier, we have the following iteration formulation
un+1 (x, t ) = un (x, t ) +
(22)
Its variational iteration formulation can be easily constructed, which is
0
x
(21)
(1 + α )
Eq. (11) becomes
(13)
The correction functional is constructed as
x 2 ∂ un ˜ (un ) ds un+1 (x, t ) = un (x, t ) + λ + N ∂ s2
(20)
Comparing with Eq. (10), we find that the fractional integral is excluded in Eq. (20), so it is accessible to those who know little fractional calculus. For Eq. (11), we can use the following fractional complex transform
(12)
where
∂ un 1 2 ∂ 2 un − x ds ∂s 2 ∂ x2
0
(11)
The classical variational iteration method begins with a modification of Eq. (11) in the form
(19)
The variational iteration algorithm is
2α
∂u ∂ u + 2α = 0 ∂t ∂x
(18)
Eq. (8) becomes an ordinary differential equation, which reads
0
N (u ) = −
(17)
(1 + α )
and using Jumarie’s chain rule [23–26], we have
while Wu–Lee’s fractional variational iteration algorithm is
un+1 (x, t ) = un (x, t ) −
3363
2 ∂ u n (s, t ) (s − x) + N u ( s , t ) ds n ∂ s2
u n +1 ( X , t ) = u n ( X , t ) +
∂ u n (s, t ) ∂ 2 u n (s, t ) (s − X ) + ds ∂t ∂ s2
0
(23) 4. Conclusions
0
x = un (x, t ) +
∂ u n ( s , t ) ∂ 2α u n ( s , t ) (s − x) + ds ∂t ∂ s 2α
0
(15) while Wu and Lee’s iteration formulation is
un+1 (x, t ) = un (x, t ) +
1
(1 + α ) x (s − x)α ∂ 2α un (s, t ) ∂ un (s, t ) (ds)α (16) + × (1 + α ) ∂t ∂ s 2α 0
If we begin with u 0 (t ) = et , both iteration formulae result in the same result as given in Ref. [15].
This Letter compares the classical variational iteration method with the fractional variational iteration method, the former is simpler to identify the Lagrange multiplier; while the later has a rigorousness in mathematical derivation of the Lagrange multiplier using the variational theory for fractional calculus. Both cases lead to solutions converging to the exact one. The fractional complex transform can easily convert a fractional differential equation to its differential partner, so that its variational iteration algorithm can be simply constructed, and the solution procedure excludes fractional operation. Acknowledgement The work is supported by PAPD (A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions).
3364
J.-H. He / Physics Letters A 375 (2011) 3362–3364
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
J.H. He, Int. J. Non-Linear Mech. 34 (1999) 699. J.H. He, J. Comput. Appl. Math. 207 (2007) 3. J.H. He, X.H. Wu, Comput. Math. Applicat. 54 (2007) 881. J.H. He, Non-perturbative Methods for Strongly Nonlinear Problems, Verlag im Internet GmbH, Berlin, 2006, http://www.dissertation.de. J.H. He, Int. J. Mod. Phys. B 20 (2006) 1141. J.H. He, Int. J. Mod. Phys. B 22 (2008) 3487. J.H. He, Comput. Meth. Appl. Mech. Eng. 167 (1998) 57. J.H. He, G.C. Wu, F. Austin, Nonlinear Sci. Lett. A 1 (2010) 1. S. Momani, Z. Odibat, Chaos Solitons Fractals 31 (2007) 1248. S. Momani, Z. Odibat, Int. J. Nonlinear Sci. Num. 7 (2006) 27. S. Momani, Z. Odibat, Phys. Lett. A 355 (2006) 271. G.E. Draganescu, J. Math. Phys. 47 (2006) 082902. D.D. Ganji, Seyed H. Hashemi Kachapi, Analysis of Nonlinear Equations in Fluids, Progress in Nonlinear Science, vol. 2, 2011, pp. 1–293.
[14] D.D. Ganji, Seyed H. Hashemi Kachapi, Analytical and Numerical Methods in Engineering and Applied Sciences, Progress in Nonlinear Science, vol. 3, 2011, pp. 1–579. [15] G. Wu, E.W.M. Lee, Phys. Lett. A 374 (2010) 2506. [16] S. Guo, L. Mei, Phys. Lett. A 375 (2011) 309. [17] G.C. Wu, Comput. Math. Applicat. 61 (2011) 2186. [18] Z.B. Li, J.H. He, Math. Comput. Applicat. 15 (2010) 970. [19] Z.B. Li, Int. J. Nonlinear Sci. Num. 11 (2010) 335. [20] Z.B. Li, J.H. He, Nonlinear Sci. Lett. A 2 (2011) 121. [21] J.H. He, Thermal Science 15 (2011) S1. [22] J.H. He, Thermal Science 15 (2011) S145. [23] G. Jumarie, Appl. Math. Lett. 23 (2010) 1444. [24] G. Jumarie, Internat. J. Systems Sci. 6 (1993) 1113. [25] G. Jumarie, J. Appl. Math. Comput. 24 (2007) 31. [26] G. Jumarie, Appl. Math. Lett. 22 (2009) 1659. [27] G.C. Wu, J.H. He, Nonlinear Sci. Lett. A 1 (2010) 281.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
A short remark on fractional variational iteration method Ji-Huan He ∗ National Engineering Laboratory for Modern Silk, College of Textile and Engineering, Soochow University, 199 Ren-ai Road, Suzhou 215123, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 18 June 2011 Accepted 18 July 2011 Available online 28 July 2011 Communicated by R. Wu
This Letter compares the classical variational iteration method with the fractional variational iteration method. The fractional complex transform is introduced to convert a fractional differential equation to its differential partner, so that its variational iteration algorithm can be simply constructed. © 2011 Elsevier B.V. All rights reserved.
Keywords: Modified Riemann–Liouville derivative Fractional complex transform Variational iteration method
1. Introduction The variational iteration method was originally proposed to solve ordinary differential equations and partial differential equations [1–6]. In 1998, the method was first adopted to solve fractional differential equations [7,8], and the solution procedure was routinely used to solve various fractional differential equations [9–14] before the fractional variational iteration method [15–17] and the fractional complex transform [18–22] were appeared in literature. In this Letter we will first compare the classical variational iteration algorithm with the fractional variational iteration algorithm, and then introduce the fractional complex transform to construct a simpler variational iteration algorithm for fractional calculus.
where u is a continuous (but not necessarily differentiable) function. In order to solve fractional differential equation by the variational iteration method, we have to re-write Eq. (1) in the form
∂u + F (u ) = 0 ∂t
(3)
where
F (u ) = −
∂u + D tα u + f (u ) ∂t
(4)
The correction functional is constructed as follows
un+1 (t ) = un (t ) +
t ∂ un λ + F˜ (un ) ds ∂s
(5)
t0
2. Classical variational iteration method vs fractional variational iteration method
(1)
where D tα u = D α u / Dt α denotes Jumarie’s fractional derivation [23–26], which is a modified Riemann–Liouville derivative defined as
D α u (t ) = t
*
1
d
(1 − α ) dt
α D t un + f (un ) ds
(6)
t0
0<α1
t
t un+1 (t ) = un (t ) −
Consider the following fractional differential equation
D tα u + f (u ) = 0,
After identification of the Lagrange multiplier, we have
−α
(t − ξ )
u (ξ ) − u (0) dξ
0
Tel.: +86 512 6588 4633. E-mail address: [email protected].
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.07.033
(2)
The Lagrange multiplier above was approximately identified under the frame of the integer differential mode. Recently Wu and Lee [15] suggested a more accurate way to identify the Lagrange multiplier by constructing a correction functional in the form
un+1 (t ) = un (t ) +
1
(1 + α )
t
λ D tα un + f (un ) (ds)α
(7)
t0
The functional contains fractional integral with respect to (ds)α which is elusive to nonmathematicians, furthermore, the identification of the Lagrange multiplier requires the full knowledge of variational theory for fractional differential equations [27].
J.-H. He / Physics Letters A 375 (2011) 3362–3364
3. Fractional complex transform
Consider an example [15]
∂ α u 1 2 ∂ 2u − x =0 ∂tα 2 ∂ x2
(8)
The classical variational iteration algorithm reads
t α ∂ un (x, s) 1 2 ∂ 2 un (x, s) un+1 (x, t ) = un (x, t ) − − ds x ∂ sα 2 ∂ x2
(9)
0
t ×
The fractional complex transform [18–22] can convert a fractional differential equation into its differential partner, so that the variational iteration method can be effectively used. By the fractional complex transform
T=
tα
∂u ∂αu ∂u ∂α T = · = ∂tα ∂t ∂tα ∂T
1
(1 + α )
∂ α un (x, s) 1 2 ∂ 2 un (x, s) (ds)α − x ∂ sα 2 ∂ x2
(10)
If we begin with u 0 = x2 , both algorithms lead to series solutions converging to the exact solution. Consider another example [15]
∂u 1 ∂ 2u − x2 2 = 0 ∂T 2 ∂x
T u n +1 ( T ) = u n ( T ) −
∂ 2u + N (u ) = 0 ∂ x2
X=
∂ 2 u ∂ u ∂ 2α u + + 2α ∂t ∂ x2 ∂x
xα
(14)
∂ 2u ∂u + =0 ∂t ∂ X2
X
After identification of the Lagrange multiplier, we have the following iteration formulation
un+1 (x, t ) = un (x, t ) +
(22)
Its variational iteration formulation can be easily constructed, which is
0
x
(21)
(1 + α )
Eq. (11) becomes
(13)
The correction functional is constructed as
x 2 ∂ un ˜ (un ) ds un+1 (x, t ) = un (x, t ) + λ + N ∂ s2
(20)
Comparing with Eq. (10), we find that the fractional integral is excluded in Eq. (20), so it is accessible to those who know little fractional calculus. For Eq. (11), we can use the following fractional complex transform
(12)
where
∂ un 1 2 ∂ 2 un − x ds ∂s 2 ∂ x2
0
(11)
The classical variational iteration method begins with a modification of Eq. (11) in the form
(19)
The variational iteration algorithm is
2α
∂u ∂ u + 2α = 0 ∂t ∂x
(18)
Eq. (8) becomes an ordinary differential equation, which reads
0
N (u ) = −
(17)
(1 + α )
and using Jumarie’s chain rule [23–26], we have
while Wu–Lee’s fractional variational iteration algorithm is
un+1 (x, t ) = un (x, t ) −
3363
2 ∂ u n (s, t ) (s − x) + N u ( s , t ) ds n ∂ s2
u n +1 ( X , t ) = u n ( X , t ) +
∂ u n (s, t ) ∂ 2 u n (s, t ) (s − X ) + ds ∂t ∂ s2
0
(23) 4. Conclusions
0
x = un (x, t ) +
∂ u n ( s , t ) ∂ 2α u n ( s , t ) (s − x) + ds ∂t ∂ s 2α
0
(15) while Wu and Lee’s iteration formulation is
un+1 (x, t ) = un (x, t ) +
1
(1 + α ) x (s − x)α ∂ 2α un (s, t ) ∂ un (s, t ) (ds)α (16) + × (1 + α ) ∂t ∂ s 2α 0
If we begin with u 0 (t ) = et , both iteration formulae result in the same result as given in Ref. [15].
This Letter compares the classical variational iteration method with the fractional variational iteration method, the former is simpler to identify the Lagrange multiplier; while the later has a rigorousness in mathematical derivation of the Lagrange multiplier using the variational theory for fractional calculus. Both cases lead to solutions converging to the exact one. The fractional complex transform can easily convert a fractional differential equation to its differential partner, so that its variational iteration algorithm can be simply constructed, and the solution procedure excludes fractional operation. Acknowledgement The work is supported by PAPD (A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions).
3364
J.-H. He / Physics Letters A 375 (2011) 3362–3364
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
J.H. He, Int. J. Non-Linear Mech. 34 (1999) 699. J.H. He, J. Comput. Appl. Math. 207 (2007) 3. J.H. He, X.H. Wu, Comput. Math. Applicat. 54 (2007) 881. J.H. He, Non-perturbative Methods for Strongly Nonlinear Problems, Verlag im Internet GmbH, Berlin, 2006, http://www.dissertation.de. J.H. He, Int. J. Mod. Phys. B 20 (2006) 1141. J.H. He, Int. J. Mod. Phys. B 22 (2008) 3487. J.H. He, Comput. Meth. Appl. Mech. Eng. 167 (1998) 57. J.H. He, G.C. Wu, F. Austin, Nonlinear Sci. Lett. A 1 (2010) 1. S. Momani, Z. Odibat, Chaos Solitons Fractals 31 (2007) 1248. S. Momani, Z. Odibat, Int. J. Nonlinear Sci. Num. 7 (2006) 27. S. Momani, Z. Odibat, Phys. Lett. A 355 (2006) 271. G.E. Draganescu, J. Math. Phys. 47 (2006) 082902. D.D. Ganji, Seyed H. Hashemi Kachapi, Analysis of Nonlinear Equations in Fluids, Progress in Nonlinear Science, vol. 2, 2011, pp. 1–293.
[14] D.D. Ganji, Seyed H. Hashemi Kachapi, Analytical and Numerical Methods in Engineering and Applied Sciences, Progress in Nonlinear Science, vol. 3, 2011, pp. 1–579. [15] G. Wu, E.W.M. Lee, Phys. Lett. A 374 (2010) 2506. [16] S. Guo, L. Mei, Phys. Lett. A 375 (2011) 309. [17] G.C. Wu, Comput. Math. Applicat. 61 (2011) 2186. [18] Z.B. Li, J.H. He, Math. Comput. Applicat. 15 (2010) 970. [19] Z.B. Li, Int. J. Nonlinear Sci. Num. 11 (2010) 335. [20] Z.B. Li, J.H. He, Nonlinear Sci. Lett. A 2 (2011) 121. [21] J.H. He, Thermal Science 15 (2011) S1. [22] J.H. He, Thermal Science 15 (2011) S145. [23] G. Jumarie, Appl. Math. Lett. 23 (2010) 1444. [24] G. Jumarie, Internat. J. Systems Sci. 6 (1993) 1113. [25] G. Jumarie, J. Appl. Math. Comput. 24 (2007) 31. [26] G. Jumarie, Appl. Math. Lett. 22 (2009) 1659. [27] G.C. Wu, J.H. He, Nonlinear Sci. Lett. A 1 (2010) 281.