PAMM · Proc. Appl. Math. Mech. 7, 2050017–2050018 (2007) / DOI 10.1002/pamm.200701001
An Iterative method for solving fractional differential equations Varsha Daftardar-Gejji* and Sachin Bhalekar** Department of Mathematics, University of Pune, Pune 411007, India In the present paper non-linear, time fractional advection partial differential equation has been solved using the new iterative method presented by Daftardar-Gejji and Jafari [1]. The results are compared with those obtained by Adomian decomposition and Homotopy perturbation methods. It is demonstrated that the new iterative method gives the best approximation among these. © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction Fractional differential equations are gaining considerable importance recently due to their wide range of applications in the fields of physics, engineering, chemistry and biology [2]. Several techniques such as Adomian decomposition method (ADM) [3], Homotopy perturbation method (HPM) [5], and a new iterative method [1] have been developed for solving non linear functional equations in general and solving fractional differential equations in particular. In the present paper we solve non-linear time-fractional advection partial differential equation. Daftardar-Gejji and Jafari in [1] have given a new iterative method to solve the general functional equation u( x )= N(u( x )) + f( x ), x = (x1,x2,…,xn), where N is a nonlinear operator from a Banach space B → B and f a known function. A solution u of the above equation having the series form ∞
u( x ) = ¦ ui ( x )
is constructed as follows.
i =0
The nonlinear operator N can be decomposed as
§
·
∞
¦ u ¸¹ = N (u ©
N¨
i
i =0
0
∞ i i −1 ½ ) + ¦ ® N (¦ u j ) − N (¦ u j ) ¾ i =1 ¯ j =0 j =0 ¿
(1.1)
Thus the functional equation becomes ∞
¦ ui = f + N (u 0 ) + i =1
i i −1 ½ − N ( u ) N ( u j )¾ . ® ¦ j ¦ ¦ i =1 ¯ j =0 j =0 ¿ ∞
(1.2)
We define the recurrence relation:
u 0 = f , u1 = N (u 0 ), u m +1 = N (u 0 + + u m ) − N (u 0 + + u m −1 ), m = 1,2,.
(1.3)
∞
Then,
u = f + ¦ u i , in view of (1.1)—(1.3). i =1
2 Numerical Example For the notations and basic definitions regarding fractional calculus, we refer to [2]. Consider the non-linear time-fractional advection partial differential equation [4]: ____________________ *
Corresponding author: e-mail
[email protected], Phone: +91 022 25601272, e-mail
[email protected], Phone: +91 022 25601272
**
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
2050018
ICIAM07 Contributed Papers
Dtα u ( x, t ) + u ( x, t )u x ( x, t ) = x + xt 2 , u ( x,0) = 0 , t > 0, − ∞ < x < ∞ , 0 < α ≤ 1 ,
(2.1)
Dtα denote Caputo fractional derivative of order Į [2]. Applying Dt−α on both sides of (2.1), we get
where
u=
xt α 2 xt 2+α + − Dt−α (uu x ) = f + N (u ) . Γ(1 + α ) Γ(3 + α )
(2.2)
From the algorithm (1.1) (with the help of Mathematica) we obtain
u0 =
2 xt 2+α xt α + , Γ(1 + α ) Γ(3 + α )
u1 = N (u 0 ) = −
§ Γ(1 + 2α ) 1 §¨ 3α § 4t 2 Γ(3 + 2α ) 4t 4 Γ(5 + 2α ) · · ·¸ ¸¸ ¸ xt ¨¨ + + Γ(α )¨¨ 2 2 ¸ α α α Γ(α ) ¨© Γ + Γ + ( 3 ) ( 3 3 ) α α Γ + α Γ + α Γ + Γ + ( 1 ) ( 1 3 ) ( 3 ) ( 5 3 ) © ¹ ¹ ¸¹ ©
and so on. In the following table we compare the third order approximate solution ( u
= u 0 + u1 + u 2 ) of Eq. (2.1) obtained by the
Adomian decomposition method and homotopy perturbation method [4] with the new iterative method (IM): t
0.2
0.4
0.6
α = 0.5
x
u ADM
u HPM
0.25
0.112844
0.104573
0.50
0.225688
1.0
α = 0.75 u IM
α = 1.0
u ADM
u HPM
u IM
0.1124
0.078787
0.078306
0.209146
0.2248
0.157574
0.451375
0.418293
0.449601
0.25
0.164004
0.177229
0.50
0.328008
1.0
u IM
u Exact
u ADM
u HPM
0.078781
0.050000
0.049989
0.0500001
0.05
0.156612
0.157562
0.100000
0.099978
0.1
0.1
0.315148
0.313225
0.315124
0.200001
0.199957
0.2
0.2
0.15824
0.128941
0.136806
0.12869
0.100023
0.099645
0.100016
0.1
0.354458
0.316481
0.257881
0.273612
0.25738
0.200046
0.199290
0.200032
0.2
0.656015
0.708915
0.632962
0.515762
0.547225
0.51476
0.400092
0.398580
0.400063
0.4
0.25
0.243862
0.230500
0.214187
0.177238
0.185146
0.17478
0.150411
0.147158
0.150274
0.15
0.50
0.487721
0.461000
0.428374
0.354477
0.370292
0.34956
0.300823
0.294317
0.300548
0.30
1.0
0.975441
0.921999
0.856749
0.7089541
0.740583
0.69912
0.601646
0.588634
0.601096
0.60
3 Conclusion The new iterative method gives better approximation than by ADM and HPM. Acknowledgements
Varsha Daftardar-Gejji is grateful to National Board for Higher Mathematics, India for the financial support.
References [1] V. Daftardar-Gejji, H. Jafari, Journal of Math Anal and Appl 316, 753–763 (2006). [2] I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999). [3] G. Adomian, Solving Frontier Problems of Physics: The decomposition method (Kluwer Academic Publishers, Boston, 1994). [4] Z. Odibat, S. Momani, Physics Letters A 365, 345–350 (2007). [5] J. H. He, Comput Meth Appl Mech Eng 178, 257–62 (1999).
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
