Results in Physics 9 (2018) 321–327
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Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics
On the comparison of perturbation-iteration algorithm and residual power series method to solve fractional Zakharov-Kuznetsov equation Mehmet Sß enol a,⇑, Marwan Alquran b, Hamed Daei Kasmaei c a
Nevsßehir Hacı Bektasß Veli University, Department of Mathematics, Nevsßehir, Turkey Jordan University of Science and Technology, Department of Mathematics and Statistics, Irbid, Jordan c Islamic Azad University, Central Tehran Branch, Department of Mathematics and Statistics, Tehran, Iran b
a r t i c l e
i n f o
Article history: Received 26 January 2018 Received in revised form 21 February 2018 Accepted 23 February 2018 Available online 2 March 2018 Keywords: Fractional partial differential equations Caputo fractional derivative Perturbation-iteration algorithm Residual power series method
a b s t r a c t In this paper, we present analytic-approximate solution of time-fractional Zakharov-Kuznetsov equation. This model demonstrates the behavior of weakly nonlinear ion acoustic waves in a plasma bearing cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Basic definitions of fractional derivatives are described in the Caputo sense. Perturbation-iteration algorithm (PIA) and residual power series method (RPSM) are applied to solve this equation with success. The convergence analysis is also presented for both methods. Numerical results are given and then they are compared with the exact solutions. Comparison of the results reveal that both methods are competitive, powerful, reliable, simple to use and ready to apply to wide range of fractional partial differential equations. Ó 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction Fractional calculus has been studied by increasingly interest in recent years and has been extensively investigated and applied for many real problems which are modeled in various areas of science and engineering. Besides it has been the focus of many studies due to their frequent appearance in various applications such as fluid mechanics, thermodynamics, biology, electrical circuits and control theory. Therefore, a comprehensive attention has been noted to obtain numerical and analyticapproximate solutions of FDEs. These methods that have been used for different types of non-fractional and fractional differential equations include, Adomian decomposition method (ADM) [1,2] for fractional diffusion equations and for linear and nonlinear systems of fractional differential equations, homotopy perturbation method (HPM) [3–5] for Klein-Gordon equation, fractional IVPs and coupled system of partial fractional differential equations, homotopy analysis method [6–9] for fractional Hirota equations, time fractional PDEs, Sturm-Liouville problems and systems of nonlinear fractional differential equations, variational iteration method [10–12] for seepage flow problems, time-fractional partial differential equations and ZakharovKuznetsov equation, perturbation methods [13–15] for nonlinear ⇑ Corresponding author. E-mail addresses:
[email protected] (M. Sßenol),
[email protected] (M. Alquran).
problems in science and engineering and perturbation-iteration algorithm [16–22] for Bratu-type equations, nonlinear heat transfer equations, first-order differential equations, Fredholm and Volterra type integral equations and ordinary fractional differential equations. In this paper, firstly, we proposed fractional ZakharovKuznetsov equation, then we described perturbation-iteration algorithm and in the next part, we investigated convergence analysis of PIA. Then we explained residual power series method (RPSM) in order to implement on Zakharov-Kuznetsov equation. After introducing RPSM, we described its convergence analysis and then we presented one example that shows reliability and efficiency of two methods in order to compare their numerical results. At last, we discussed about obtained results as a section for conclusion. In this paper, the fractional Zakharov-Kuznetsov equation considered is of the form:
Dat u þ aðup Þx þ bðuq Þxxx þ bður Þttx ¼ 0
ð1Þ
where u ¼ uðx; y; tÞ, a is order of the fractional derivative ð0 < a 6 1Þ, a and b are arbitrary constants and p; q and r are integers and p; q; r – 0 that demonstrate the behavior of weakly nonlinear ion acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field [23,24]. Firstly, the Zakharov-Kuznetsov equation was formulated for describing weakly nonlinear ion-acoustic waves in strongly
https://doi.org/10.1016/j.rinp.2018.02.056 2211-3797/Ó 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
M. Sßenol et al. / Results in Physics 9 (2018) 321–327
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magnetized lossless plasma in two dimensions studied in [25]. The FZK equations have been solved by means of VIM in [26] and HPM in [27]. Basic definitions Definition 1. A real function f ðtÞ, t > 0 is said to be in the space C l , ðl > 0Þ if there exists a real number pð> l) such that f ðtÞ ¼ tp f 1 ðtÞ where f 1 2 C½0; 1Þ and it is said to be in the space C m l if f
ðmÞ
2 C l ; m 2 @ [28].
1 CðaÞ
Z
t
0
ðt sÞa1 f ðsÞds;
a; t > 0
ð2Þ
and J0 f ðtÞ ¼ f ðtÞ, where C is the well-known gamma function. Then the following properties hold for f 2 C l , l P 1, a, b P 0 and k > 1. J a J b f ðtÞ ¼ J aþb f ðtÞ J a J b f ðtÞ ¼ J b J a f ðtÞ ðkþ1Þ kþa . J a tk ¼ CCðkþ1þ aÞ t
Definition 3. The Caputo fractional derivative of a function f of order a, f 2 C m 1 , m 2 @ [ f0g, is defined as [30]:
Da f ðtÞ ¼ J ma f ¼
ðmÞ
1
ðtÞ Z
Cðm aÞ
t 0
ðt sÞma1 f
ðmÞ
ðsÞds;
a; t > 0
ð3Þ
F
þ ðucðaÞ Þt n F uðaÞ þ ðuc Þx n F ux þ ðuc Þxxx n F uxxx þ ðuc Þttx n F uttx þ F e ¼ 0 t
ð7Þ
where F uðaÞ ¼ t
F uttx ¼
@F @uxxx
and F e ¼
@F . @e
F e þ Fe F uk uk ðtÞ ¼ F u0k F u0k
ð8Þ
F ukþ1 F e þ Fe ukþ1 ðtÞ ¼ F u0kþ1 F u0kþ1
ð9Þ
Then, by categorizing the elements F, F u , F ut and F in PIAð1; 1Þ based on Eq. (1), we have:
F ¼ ut ;
In this study, we first develop PIA approach in order to solve fractional Zakharov-Kuznetsov equation. To obtain the approximate solution of this model, we introduce and present perturbation-iteration algorithm PIA(m,n) by taking m ¼ 1 as one correction term in the perturbation expansion and n ¼ 1 as correction terms of just first derivatives in the Taylor series expansion. Now, we consider the following initial value problem as follows:
ð4Þ
F u ¼ 0;
F ut ¼ 1;
F e ¼ ut þ aupx þ buxxx þ buttx þ uDat
ð5Þ
Substituting Eq. (5) into Eq. (4) and writing in the Taylor Series expansion for first order derivatives in the neighborhood of e ¼ 0 gives
r
u0k ðx; y; tÞ ¼ uk ðx; y; tÞ þ Dat uk ðx; y; tÞ þ aðuk ðx; y; tÞÞpx þ bðuk ðx; y; tÞÞqxxx þ bðuk ðx; y; tÞÞrttx þ
uk ðx; y; tÞ
ð10Þ
e
Thus, we can write it in k-th and k þ 1-th steps as follows:
u0k ðx; y; tÞ ¼
uk ðx; y; tÞ Dat uk ðx; y; tÞ aðuk ðx; y; tÞÞpx
bðuk ðx; y; tÞÞqxxx bðuk ðx; y; tÞÞrttx
where u ¼ uðx; y; tÞ and e is a small perturbation parameter. The perturbation expansions with only one correction term is
q
So, by substituting these elements in PIA formula, we have:
uðx; y; 0Þ ¼ c
unþ1 ¼ un þ eðuc Þn
F uxxx ¼
@F , @uxxx
Proof. The general iteration formula of PIA(m; n) for m ¼ 1 and n ¼ 1 according to do some computations on Eq. (7) in k-th and k þ 1-th step results as follows:
u0kþ1 ðtÞ þ
Overview and convergence of the PIA(1,1)
ða Þ F ut ; ux ; uxxx ; uttx ; e ¼ 0
F ux ¼
@F , @ux
Theorem 1. PIA(m; n) converges to Eq. (1) for m ¼ 1 and n ¼ 1 when kukþ1 uk k 6 e0 and e0 ! 0.
a
Da ðaf ðtÞ þ bgðtÞÞ ¼ aDa f ðtÞ þ bD gðtÞ, a; b 2 R, Da J a f ðtÞ ¼ f ðtÞ, Pk1 ðjÞ j J a Da f ðtÞ ¼ f ðtÞ j¼0 f ð0Þ tj! , t > 0.
@F ðaÞ , @ut
All derivatives in the expansion are evaluated at e ¼ 0. For this reason in the calculation procedure, each term is obtained when e tends to zero. Beginning with an initial function u0 ðx; y; tÞ, first ðuc Þ0 ðx; y; tÞ is determined by the help of Eq. (7). Then using Eq. (5), n þ 1-th iteration solution is found. Iteration process is repeated using Eqs. (7) and (5) until obtaining an acceptable result. After presenting the convergence of PIA method, an example is given to show reliability and effectiveness of the method. The aim of this paper is to obtain approximate solutions of the fractional Zakharov-Kuznetsov equation by PIA and RPSM and to determine series solutions with high accuracy. Regards the convergence of PIA, we present the following theorem.
u0k ðtÞ þ
where m 1 < a < m with the following properties;
ð6Þ
or
e
Definition 2. The Riemann-Liouville fractional integral operator ðJ a Þ of order a P 0 of a function f 2 C l , l P 1 is defined as [29]:
J a f ðtÞ ¼
F ðuðnaÞ Þt ; ðun Þx ; ðun Þxxx ; ðun Þttx ; 0 þ F uðaÞ ðuðnaÞ Þt ; ðun Þx ; ðun Þxxx ; ðun Þttx ; 0 e ðuðcaÞ Þt n t þ F ux ðuðnaÞ Þt ; ðun Þx ; ðun Þxxx ; ðun Þttx ; 0 e ðuc Þx n ða Þ þ F uxxx ðun Þt ; ðun Þx ; ðun Þxxx ; ðun Þttx ; 0 e ðuc Þxxx n þ F uttx ðuðnaÞ Þt ; ðun Þx ; ðun Þxxx ; ðun Þttx ; 0 e ðuc Þttx n ðaÞ þ F e ðun Þt ; ðun Þx ; ðun Þxxx ; ðun Þttx ; 0 e ¼ 0
uk ðx; y; tÞ
ð11Þ
e
and
u0kþ1 ðx; y; tÞ ¼ ukþ1 ðx; y; tÞ Dat ukþ1 ðx; y; tÞ aðukþ1 ðx; y; tÞÞpx bðukþ1 ðx; y; tÞÞqxxx bðukþ1 ðx; y; tÞÞrttx
ukþ1 ðx; y; tÞ
e
ð12Þ
M. Sßenol et al. / Results in Physics 9 (2018) 321–327
Now, we need to make kukþ1 uk k from u0kþ1 u0k . Therefore, it needs some mathematical computations about algebra of inequalities. Meanwhile, we use the obtained inequalities from 0 u and u0 . So, we rewrite inequalities to obtain them with k kþ1 respect to u0k and u0kþ1 respectively. Then, we need to form u0kþ1 u0k as follows:
0 u ðx; y; tÞ 6 kuk ðx; y; tÞk þ Da uk ðx; y; tÞ þ aðuk ðx; y; tÞÞp t k x 1 þ bðuk ðx; y; tÞÞqxxx þ bðuk ðx; y; tÞÞrttx þ
e
ð13Þ
xxx
1 þ bðukþ1 ðx; y; tÞÞrttx þ kukþ1 ðx; y; tÞk
e
ð14Þ
Now, we rewrite these inequalities as follows:
1 1þ kuk ðx; y; tÞk P u0k ðx; y; tÞ Dat uk ðx; y; tÞ e aðuk ðx; y; tÞÞpx bðuk ðx; y; tÞÞqxxx bðuk ðx; y; tÞÞr ttx
Then, we have: Assuming
u0kþ1 ¼ L½ukþ1
ð21Þ
where L is a linear operator that is defined as follows:
L¼
d ½: dðÞ
ð22Þ
L¼
@ ½: @ðÞ
ð23Þ
0 u ¼ kL½uk k 6 N1 k
ð24Þ
0 u ¼ kL½ukþ1 k 6 N2 kþ1
ð25Þ
kukþ1 ðtÞ uk ðtÞk P 2ðkukþ1 ðtÞk kuk ðtÞkÞ P N2 N1 M 1 þ M 2 aM 3 þ aM 4 bM5 ð15Þ
þ bM6 bM7 þ bM8
ð26Þ
If we want
kukþ1 uk k ! 0
ð27Þ
then, we have to consider
N2 N1 M 1 þ M 2 aM 3 þ aM 4 bM5 þ bM 6 bM7 þ bM 8 ! 0 ð16Þ
From magnitude rules in calculus and optional values of c and d, we have:
kc dk P kck kdk
ð20Þ
Therefore, we obtain:
and
1 1þ kukþ1 ðx; y; tÞk P u0kþ1 ðx; y; tÞ Dat ukþ1 ðx; y; tÞ e aðukþ1 ðx; y; tÞÞpx bðukþ1 ðx; y; tÞÞqxxx bðukþ1 ðx; y; tÞÞrttx
u0k ¼ L½uk
Meanwhile, we know that any linear operator is bounded in theory of operators from pure mathematics. Then, we can define:
0 u ðx; y; tÞ 6 kukþ1 ðx; y; tÞk þ Da ukþ1 ðx; y; tÞ t kþ1 þ aðukþ1 ðx; y; tÞÞp þ bðukþ1 ðx; y; tÞÞq x
323
ð17Þ
e ¼ 1, we have:
kukþ1 ðx; y; tÞ uk ðx; y; tÞk P 2ðkukþ1 ðx; y; tÞk kuk ðx; y; tÞkÞ P u0kþ1 ðx; y; tÞ u0k ðx; y; tÞ Dat ukþ1 ðx; y; tÞ a þ D uk ðx; y; tÞ
e0 ! 0
ð28Þ
If the above expression tends to zero, it means that we have a Cauchy sequences of our approximate solutions that converges to our exact solution uðx; y; tÞ. The proof is complete. The proof is done in similar manner for PIAð1; 2Þ, PIAð2; 2Þ and so on. Therefore, we can find such these conditions for PIAð1; 2Þ, PIAð2; 2Þ states and so on. In fact, we have found the condition of stop process for PIA method by all involved expressions in the iteration algorithm. Therefore, all the governing conditions needs to be imposed on PIA method to become convergent just in a few of computational iterations. h
t
akukþ1 ðx; y; tÞkpx
Algorithm and convergence of RPSM
þ akuk ðx; y; tÞkpx bkukþ1 ðx; y; tÞkqx þ bkukþ1 ðx; y; tÞkqxxx bkukþ1 ðx; y; tÞkrttx þ bkuk ðx; y; tÞkrttx
ð18Þ
Also, we know from a theorem in mathematical analysis too that any continuous and differentiable function based on defined space are bounded. On the other hand, according to first definitions of Caputo derivatives and basic definitions in fractional calculus, all norms of functions with respect to u are bounded. Therefore, there exists constants such that we have:
kukþ1 ðx; y; tÞ uk ðx; y; tÞk P 2ðkukþ1 ðx; y; tÞk kuk ðx; y; tÞkÞ P u0 ðx; y; tÞ u0 ðx; y; tÞ kþ1
uðx; y; tÞ ¼
1 X f n ðx; yÞ n¼0
x 2 ½a; b;
uk ðx; y; tÞ ¼
ð19Þ
and now just we need to prove that u0k and u0kþ1 are bounded. Therefore we consider:
0 < a 6 1;
06t
