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Optimal variational iteration method for nonlinear problems
Journal of the Association of Arab Universities for Basic and Applied Sciences (2016) xxx, xxx–xxx
University of Bahrain
Journal of the Association of Arab Universities for Basic and Applied Sciences www.elsevier.com/locate/jaaubas www.sciencedirect.com
Optimal variational iteration method for nonlinear problems Syed Tauseef Mohyud-Din *, Waseem Sikander, Umar Khan, Naveed Ahmed Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan Received 22 April 2016; revised 8 August 2016; accepted 21 September 2016
KEYWORDS Auxiliary parameter; Variational iteration method; Boundary value problems; Error estimate
Abstract This paper focuses on the study of boundary value problems using well-known He’s variational iteration method which is coupled with an auxiliary parameter. Three examples are given to show the efficiency and importance of the proposed algorithm. The reliability and accuracy has been proved by comparing our results with the solution obtained by standard variational iteration method. Ó 2016 University of Bahrain. Publishing services 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/).
1. Introduction Inokuti et al. (1978) proposed a general Lagrange multiplier method to solve nonlinear problems especially in the field of quantum mechanics. Later on, He (1999, 2000, 2007) modified the method to a new kind of an analytical technique for nonlinear problems and named it as variational iteration method (VIM), which is effectively and easily used to obtain solution of nonlinear equations accurately. For example, Belgacem et al. (2015) and Baskonus et al. (2015) obtain solutions of nonlinear fractional differential equations systems (NFDES) through implementation of VIM and concluded that VIM remains a valuable tool for the treatments of NFDES. Bulut and Baskonus (2009) obtain an exact solution of dispersive equation. Wazwaz (2007a, 2007b, 2007c, 2008) applied the method to nonlinear differential equations and pointed out that VIM is a very effective and reliable analytical tool for solving these equations. Saberi and Tamamgar (2008) concluded that the method is highly reliable for * Corresponding author. E-mail address: [email protected] (S.T. Mohyud-Din). Peer review under responsibility of University of Bahrain.
integro-differential equations. Goh et al. (2009) applied the method to hyperchaotic system with great success. Uremen and Yildirim (2009) and Sadighi and Ganji (2007) obtain exact solutions of poisson equation and nonlinear diffusion equations respectively. With the passage of time, several modifications were made in He’s VIM, which have further improved the efficiency and accuracy of the iterative algorithm to a tangible level. Moreover, with the passage of time many analytical techniques are developed to solve nonlinear problems. Liao (1992) came up with a new idea, he developed a nonlinear analytical technique called homotopy analysis method (HAM), which is free from assumption of small parameters and can be used to obtain approximate solution of nonlinear problems. In this method, Liao inserted an auxiliary parameter h, which is used to control the convergence of an approximate solution over the domain of the problem. Liao (2003), further generalized the method so called optimal homotopy analysis method (OHAM) for strongly nonlinear differential equations by inserting multiple parameters, which are used to control the convergence of approximate solutions. The optimal value of auxiliary parameters is obtained by minimizing the absolute residual error, which is a reliable, effective and accurate
http://dx.doi.org/10.1016/j.jaubas.2016.09.004 1815-3852 Ó 2016 University of Bahrain. Publishing services 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/). Please cite this article in press as: Mohyud-Din, S.T. et al., Optimal variational iteration method for nonlinear problems. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016), http://dx.doi.org/10.1016/j.jaubas.2016.09.004
2
S.T. Mohyud-Din et al.
method even for higher order of approximation. Different available studies that used the OHAM to solve various nonlinear equations can be seen in the literature (Xu et al., 2015; Nawaz et al., 2015; Ellahi et al., 2015a, 2015b; Zeeshan et al., 2014, 2016). In this paper, an auxiliary parameter h is inserted into the correctional functional of VIM for boundary value problems. We consume all of the boundary conditions to establish an integral equation before constructing an iterative algorithm to obtain an approximate solution. Thus we establish a modified iterative algorithm that does not contain undetermined coefficients, whereas most previous iterative methods do incorporate undetermined coefficients. It is observed that the coupling algorithm provides a convenient way to control and adjust the convergence region of approximate solution over the domain of the problem. Three examples are given to explicitly reveal the performance and reliability of the suggested algorithm. 2. Variational iteration method (VIM) To illustrate the steps of variational iteration method, we consider the following general nonlinear ordinary differential equation. LfðnÞ þ NfðnÞ þ gðnÞ ¼ 0:
ð1Þ
where L and N are linear and nonlinear operator respectively and gðnÞ illustrates an inhomogeneous term. According to VIM (He, 1999, 2000, 2007; Noor and Mohyud-Din, 2008), we can construct the correction functional as follows Z n fnþ1 ðnÞ ¼ fn ðnÞ þ kðLfn ðsÞ þ Nf~n ðsÞ þ gðsÞÞds; ð2Þ 0
Making the correction functional stationary, the Lagrange multiplier is identified as k ¼ s n; (Noor and Mohyud-Din, 2008; Xu, 2009), we get the following iterative formula Z n fnþ1 ðnÞ ¼ fn ðnÞ þ ðs nÞðLfn ðsÞ þ Nf~n ðsÞ þ gðsÞÞds: ð7Þ 0
An unknown auxiliary parameter h can be inserted into the iterative formula (7), for n ¼ 0, Eq. (7), becomes Z n f1 ðnÞ ¼ fð0Þ þ nf 0 ð0Þ þ h ðs nÞ Lf0 ðsÞ þ Nf~0 ðsÞ þ gðsÞ ds: ð8Þ 0
for optimal variational iteration method, we will proceed as follows, From Eq. (8), Z n fðnÞ ¼ fð0Þ þ nf 0 ð0Þ þ h ðs nÞðLfðsÞ þ NfðsÞ 0
þ gðsÞÞds:
ð9Þ
Substituting n ¼ a and n ¼ b in Eq. (9) and solve for fð0Þ, f 0 ð0Þ we get Z a ab h ðs aÞðLfðsÞ þ NfðsÞ þ gðsÞÞds fð0Þ ¼ a a ab 0 Z a ah ðs aÞðLfðsÞ þ NfðsÞ þ gðsÞÞds þ ab 0 Z b ðs bÞðLfðsÞ þ NfðsÞ þ gðsÞÞds ; 0
Z a ab h ðs aÞðLfðsÞ þ NfðsÞ þ gðsÞÞds ab ab 0 Z b þ ðs bÞðLfðsÞ þ NfðsÞ þ gðsÞÞds :
f 0 ð0Þ ¼
0
where k is a Lagrange multiplier (He, 1999, 2000, 2007; Noor and Mohyud-Din, 2008), which can be identified optimally via variational theory, fn is the nth approximate solution, and f~n is consider as a restricted variation, i.e. df~n ¼ 0. After identification of Lagrange multiplier, the successive approximations fnþ1 ðnÞ; n P 0, of the solution f can be readily obtained. Consequently, the exact solution will be of the form: fðnÞ ¼ lim fn ðnÞ:
ð3Þ
n!1
3. Optimal variational iteration method (OVIM)
Substituting the value of fð0Þ and f0 ð0Þ in Eq. (9) yields ab ab þn fðnÞ ¼ a a ab ab Z n þh ðs nÞðLfðsÞ þ NfðsÞ þ gðsÞÞds 0 Z a h ðs aÞðLfðsÞ þ NfðsÞ þ gðsÞÞds þ ða nÞ ab 0 Z b h ðs bÞðLfðsÞ þ NfðsÞ þ gðsÞÞds ðn aÞ þ ab 0 Z a ðs aÞðLfðsÞ þ NfðsÞ þ gðsÞÞds; ð10Þ h 0
To illustrate the steps of optimal variational iteration method, we consider the following second order nonlinear ordinary differential equation. LfðnÞ þ NfðnÞ þ gðnÞ ¼ 0;
a 6 n 6 b;
ð4Þ
subject to the boundary conditions fðaÞ ¼ a;
fðbÞ ¼ b; 2
ð5Þ
where L ¼ dnd 2 is the linear differential operator, N represents the nonlinear operator and gðnÞ illustrate an inhomogeneous term. According to standard VIM, the correction functional is given as Z n fnþ1 ðnÞ ¼ fn ðnÞ þ k Lfn ðsÞ þ Nf~n ðsÞ þ gðsÞ ds: ð6Þ
which can be solved by the modified iterative algorithm as ab ab f0 ðnÞ ¼ a a þn ; ab ab Z n f1 ðn; hÞ ¼ f0 ðnÞ þ h ðs nÞðLf0 ðsÞ þ Nf0 ðsÞ þ gðsÞÞds 0 Z a h ðs aÞðLf0 ðsÞ þ Nf0 ðsÞ þ gðsÞÞds ða nÞ þ ab 0 Z b h ðs bÞðLf0 ðsÞ þ Nf0 ðsÞ þ gðsÞÞds þ ðn aÞ ab 0 Z a ðs aÞðLf0 ðsÞ þ Nf0 ðsÞ þ gðsÞÞds h 0
0
Please cite this article in press as: Mohyud-Din, S.T. et al., Optimal variational iteration method for nonlinear problems. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016), http://dx.doi.org/10.1016/j.jaubas.2016.09.004
Optimal variational iteration method for nonlinear problems Z fnþ1 ðn; hÞ ¼ fn ðnÞ þ h
1 2 1 3 An þ Bn 2! 3! 1 2 1 3 f1 ðnÞ ¼ 1 þ n þ An þ Bn 2! 3! 1 1 c c þ þ A þ A n4 24 24 24 24 c 1 c c c 6 c þ B þ n5 þ A n Bn7 ; 120 120 120 720 720 5040 .. .
n
f0 ðnÞ ¼ 1 þ n þ
ðs nÞðLfn ðs; hÞ þ Nfn ðs; hÞ 0 Z a h ðs aÞðLfn ðs; hÞ ða nÞ þ gðsÞÞds þ ab 0 þ Nfn ðs; hÞ þ gðsÞÞds Z b h ðn aÞ ðs bÞðLfn ðs; hÞ þ Nfn ðs; hÞ þ ab 0 Z a ðs aÞðLfn ðs; hÞ þ gðsÞÞds h 0
þ Nfn ðs; hÞ þ gðsÞÞds;
n P 1:
3
ð11Þ
The Iterative algorithm (11) does not contain undetermined coefficients. fn ðn; hÞ; n P 1un ðx; hÞ; n P 1 contain only an auxiliary parameter h, which is used to control the convergence of approximate solution that can be determined optimally by minimizing the norm 2 of the residual function over the domain of given problem (Hosseini et al., 2010a, 2010b, 2012).
where A and B can be calculated using the boundary conditions corresponding to different values of parameter c. According to OVIM, we will proceed as follows, Z n 1 1 A3 n2 þ A4 n3 þ h 2! 3! 0 ðs nÞ3 ðivÞ 1 00 f ðsÞ ð1 þ cÞf ðsÞ þ cfðsÞ cs2 þ 1 ds; 3! 2
fðnÞ ¼ A1 þ A2 n þ
ð16Þ
using the boundary conditions, we get 4. Numerical examples In this section, we use Optimal Variational Iteration Method (OVIM) to find an approximate solution of boundary value problems. Numerical results obtained by the developed iterative algorithm are very encouraging. Example 4.1. Consider the following fourth order boundary value problem (Mohyud-Din et al., 2010). 1 f ðivÞ ðnÞ ð1 þ cÞf 00 ðnÞ þ cfðnÞ cn2 þ 1 ¼ 0; 2
ð12Þ
subject to the boundary conditions fð0Þ ¼ 1;
f 0 ð0Þ ¼ 1;
fð1Þ ¼ 1:5 þ sinhð1Þ;
0
f ð1Þ ¼ 1 þ coshð1Þ: The exact solution for this problem is 1 fðnÞ ¼ 1 þ n2 þ sinhðnÞ: 2
ð13Þ
According to standard VIM, the correction functional is given as Z
k fnðivÞ ðsÞ ð1 þ cÞf~00n ðsÞ 0 1 þcf~n ðsÞ cs2 þ 1 ds; n P 0: 2
fnþ1 ðnÞ ¼ fn ðnÞ þ
n
A1 ¼ 1; A2 ¼ 1; A3 ¼ 3 þ 6 sinhð1Þ 2 coshð1Þ Z 1 2 s 1 þ 2h þs 2 2 0 1 ðivÞ f ðsÞ ð1 þ cÞf 00 ðsÞ þ cfðsÞ cs2 þ 1 ds 2 Z 1 3 s s2 s 1 þ 6h 6 2 2 6 0 1 f ðivÞ ðsÞ ð1 þ cÞf 00 ðsÞ þ cfðsÞ cs2 þ 1 ds; 2 Z 1 2 s 1 þs A4 ¼ 6 þ 6 coshð1Þ 12 sinhð1Þ 6h 2 2 0 1 f ðivÞ ðsÞ ð1 þ cÞf 00 ðsÞ þ cfðsÞ cs2 þ 1 ds 2 Z 1 3 2 s s s 1 þ þ 12h 6 2 2 6 0 1 2 ðivÞ 00 f ðsÞ ð1 þ cÞf ðsÞ þ cfðsÞ cs þ 1 ds: 2 Consequently, we have the following iterative algorithm 2
f0 ðnÞ ¼ 1 þ n þ n2 ð3 þ 6 sinhð1Þ 2coshð1ÞÞ 3
þ n6 ð6 þ 6 coshð1Þ 12 sinhð1ÞÞ;
ð14Þ
Making the correction functional stationary, the Lagrange
f1 ðn;hÞ ¼ f0 ðnÞ þ h
R n s3 0
6
2
2
3
s2n þ sn2 n6
ðivÞ f0 ðsÞ ð1 þ cÞf000 ðsÞ þ cf0 ðsÞ 12 cs2 þ 1 ds
3
multiplier is identified as k ¼ ðsnÞ ; (Noor and Mohyud-Din, 3! 2008; Xu, 2009), we get the following iterative formula Z n 3 ðs nÞ fnþ1 ðnÞ ¼ fn ðnÞ þ 3! 0 1 fnðivÞ ðsÞ ð1 þ cÞf~n00 ðsÞ þ cf~n ðsÞ cs2 þ 1 ds: 2
ð15Þ
Consuming the initial conditions, we have the following approximants
0
1 R1 3 2 ðivÞ 3 0 s6 s2 þ 2s 16 f0 ðsÞ ð1 þ cÞf000 ðsÞ þ cf0 ðsÞ 12 cs2 þ 1 ds B C B C þhn2 B C @ R1 A 2 ðivÞ 00 s 1 1 2 þ 0 2 þ s 2 f0 ðsÞ ð1 þ cÞf0 ðsÞ þ cf0 ðsÞ 2 cs þ 1 ds 0 R 1 2 1 s3 ðivÞ 2 s þ s 1 f0 ðsÞ ð1 þ cÞf000 ðsÞ þ cf0 ðsÞ 12 cs2 þ 1 ds B 0 6 2 2 6 C B C þhn3 B C; @ R1 A 2 ðivÞ 00 s 1 1 2 0 2 þ s 2 f0 ðsÞ ð1 þ cÞf0 ðsÞ þ cf0 ðsÞ 2 cs þ 1 ds
Please cite this article in press as: Mohyud-Din, S.T. et al., Optimal variational iteration method for nonlinear problems. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016), http://dx.doi.org/10.1016/j.jaubas.2016.09.004
4
S.T. Mohyud-Din et al. Error estimates: error = Abs (exact solution approximate solution).
Table 1
c ¼ 1000; h ¼ 0:0549
Error (VIM)
n ;
Exact
Approximate
Error (OVIM)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1 1.105170 1.221340 1.349520 1.490750 1.646100 1.816650 2.003580 2.208110 2.431520 2.675200
1 1.105170 1.221340 1.349520 1.490750 1.646090 1.816650 2.003580 2.208100 2.431510 2.675200
0 9.75E7 6.58E7 5.70E7 6.85E7 8.01E7 8.60E7 7.90E7 1.08E6 1.98E6 2.6E12
fnþ1 ðn;hÞ ¼ fn ðn;hÞ þ h
0 7.91E7 0.000020 0.000485 0.011451 0.270513 6.382930 149.2260 3368.650 70385.10 1,307,730
Z n 3 2 s s n sn2 n3 þ 2 6 2 6 0
Error (OVIM)
1 1.105170 1.221340 1.349520 1.490750 1.646100 1.816650 2.003580 2.208110 2.431520 2.675200
1 1.105170 1.221340 1.349520 1.490750 1.646090 1.816650 2.003580 2.208100 2.431510 2.675200
0 1.27E6 5.61E7 1.44E7 4.64E8 1.04E7 3.01E7 4.63E7 1.14E6 2.70E6 5.7E12
0 2.51E7 0.000015 0.001280 0.111747 9.337910 628.7880 27217.50 634860.0 5,764,110 0.046360
and the error of norm 2 of residual function (18) is, " 2 #12 30 1 X i e15 ðhÞ ¼ fr15 ;h g : 30 i¼0 30
1 R1 3 2 3 0 s6 s2 þ 2s 16 fnðivÞ ðs;hÞ ð1 þ cÞfn00 ðs;hÞ þ cfn ðs;hÞ 12 cs2 þ 1 ds B C C þ hn2 B @ R A 1 ðivÞ 00 s2 1 1 2 þ 0 2 þ s 2 fn ðs;hÞ ð1 þ cÞfn ðs;hÞ þ cfn ðs;hÞ 2 cs þ 1 ds 0
1 0 R1 3 2 s 2 s þ s 1 fnðivÞ ðs;hÞ ð1 þ cÞfn00 ðs;hÞ þ cfn ðs;hÞ 12 cs2 þ 1 ds B 0 6 2 2 6 C C þ hn @ A; n P 1; R1 2 0 s2 þ s 12 fnðivÞ ðs;hÞ ð1 þ cÞfn00 ðs;hÞ þ cfn ðs;hÞ 12 cs2 þ 1 ds 3B
ð17Þ
we obtain the following approximations f0 ðnÞ ¼ 1 þ n þ 0:482523n2 þ 0:192678n3 ; f1 ðn; hÞ ¼ 1 þ n þ ð0:482523 þ 0:0178114h
ð19Þ
The above residual function can be used to approximate e15 ðhÞ ; and the optimal value of h can be determined by minimizing the residual error given in Eq. (19). Table 1 exhibits the errors obtained using the optimal variational iteration method (OVIM) and variational iteration method (VIM) corresponding to different values of parameter c : It is obvious that the results of OVIM are more reliable and accurate than VIM. Example 4.2. Consider the following second order nonlinear differential equation, the temperature distribution equation in a uniformly thick rectangular fin radiation to free space with non-linearity of high order (Marinca and Herisanu, 2008). f 00 ðnÞ ef 4 ðnÞ ¼ 0; ð20Þ
þ 0:0003728chÞn2 þ ð0:192678 0:0259889h 0:0000361chÞn3 ð0:0014564h þ 0:0014564chÞn4 þ ð0:0096339h þ 0:0013005chÞn5 þ 0:0000485chn6 0:0002293chn7 ; f2 ðn; hÞ ¼ 1 þ n þ ð0:482523 þ 0:0356228h
subject to the boundary conditions fð1Þ ¼ 1; f 0 ð0Þ ¼ 0:
þ 0:00074572ch 0:0181541h2 0:0007641ch2
According to standard VIM, the correction functional is given as Z n fnþ1 ðnÞ ¼ fn ðnÞ þ k fn00 ðsÞ ef~n4 ðsÞ ds; n P 0: ð21Þ
0:0000101c2 h2 Þn2 þ ð0:192678 0:0519778h 0:0000723ch þ 0:025966h2 þ 0:0000733ch2 þ 1:29 106 c2 h2 Þn3 ð0:0029128h þ 0:0029128ch
0
Making the correction functional stationary, the Lagrange multiplier is identified as k ¼ s n; (Noor and Mohyud-Din, 2008; Xu, 2009), we get the following iterative formula Z n fnþ1 ðnÞ ¼ fn ðnÞ þ ðs nÞ fn00 ðsÞ ef~n4 ðsÞ ds; n P 0: ð22Þ
0:0029407h2 0:0029717ch2 0:0000310c2 h2 Þn4 þ ð0:019267h þ 0:002601ch 0:0109334h2 0:0026018ch2 1:80 106 c2 h2 Þn5 þ ð0:0000970ch 0:0000485h2 0:0001951ch2
0
0:0000495c h Þn þ ð0:0004587ch þ 0:0002293h 6
Error (VIM)
Approximate
1 ðivÞ r15 ðn; hÞ ¼ f15 ðn; hÞ ð1 þ cÞf1500 ðn; hÞ þ cf15 ðn; hÞ cn2 þ 1; 2 ð18Þ
1 fnðivÞ ðs;hÞ ð1 þ cÞfn00 ðs;hÞ þ cfn ðs;hÞ cs2 þ 1 ds 2
2 2
c ¼ 2000; h ¼ 0:0293 Exact
2
þ 0:0005206ch2 þ 0:0000310c2 h2 Þn7 þ ð1:73 106 ch2 þ 1:73 106 c2 h2 Þn8 þ ; .. . we define the following residual function, to obtain the proper value of h for the series solution (17),
Consuming the initial condition, we have the following approximants f0 ðnÞ ¼ A; 1 f1 ðnÞ ¼ A þ A4 en2 ; 2 1 1 1 1 13 4 8 f2 ðnÞ ¼ A þ A4 en2 þ A7 e2 n4 þ A10 e3 n6 þ A en 2 6 20 112 1 A16 e5 n10 ; þ 1440 .. .
Please cite this article in press as: Mohyud-Din, S.T. et al., Optimal variational iteration method for nonlinear problems. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016), http://dx.doi.org/10.1016/j.jaubas.2016.09.004
Optimal variational iteration method for nonlinear problems
5
Error estimates: error = Abs (numerical solution approximate solution).
Table 2
e ¼ 0:1; h ¼ 0:9223
Error (VIM)
n #
Numerical
Approximate
Error (OVIM)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.956820 0.957240 0.958499 0.960602 0.963557 0.967375 0.972068 0.977655 0.984155 0.991594 1.000000
0.956821 0.957240 0.958499 0.960603 0.963558 0.967375 0.972068 0.977655 0.984155 0.991594 1.000000
4.66E7 4.56E7 4.36E7 4.16E7 3.90E7 3.51E7 3.03E7 2.45E7 1.79E7 9.60E8 1.03E8
0.677522 0.680675 0.690197 0.706274 0.729233 0.759557 0.797918 0.845227 0.902696 0.971948 1.055160
where A can be calculated using the boundary condition fð1Þ ¼ 1; corresponding to different values of parameter e. According to OVIM, we will proceed as follows, Z n ðs nÞ f00 ðsÞ ef4 ðsÞ ds; ð23Þ fðnÞ ¼ A1 þ A2 n þ h 0
using the boundary conditions, we get Z 1 A1 ¼ 1 h ðs 1Þðf 00 ðsÞ ef 4 ðsÞÞds; 0
A2 ¼ 0: Consequently, we have the following iterative algorithm f0 ðnÞ ¼ 1; f1 ðn; hÞ ¼ f0 ðnÞ þ h h
R1 0
Rn
ðs
0
ðs nÞðf000 ðsÞ ef04 ðsÞÞds
1Þðf000 ðsÞ
fnþ1 ðn; hÞ ¼ fn ðn; hÞ þ h h
R1 0
Rn 0
ef04 ðsÞÞds;
ð24Þ
ðs nÞðfn00 ðs; hÞ efn4 ðs; hÞÞds
ðs 1Þ fn00 ðs; hÞ efn4 ðs; hÞ ds;
n P 1;
we obtain the following approximations
e ¼ 0:5; h ¼ 0:7557
Error (VIM)
Numerical
Approximate
Error (OVIM)
0.852329 0.853650 0.857629 0.864315 0.873795 0.886194 0.901681 0.920478 0.942871 0.969224 1.000000
0.852460 0.853780 0.857756 0.864438 0.873912 0.886301 0.901777 0.920559 0.942934 0.969261 1.000000
1.31E4 1.30E4 1.28E4 1.23E4 1.16E4 1.07E4 9.58E5 8.12E5 6.21E5 3.64E5 6.35E8
0.202374 0.204153 0.209547 0.218732 0.232020 0.249886 0.273022 0.302408 0.339434 0.386100 0.445335
we define the following residual function, r4 ðn; hÞ ¼ f400 ðn; hÞ ef44 ðn; hÞ; and the error of norm 2 of residual function (25) is, " 2 #12 30 1 X i ;h e4 ðhÞ ¼ r4 : 30 i¼0 30
ð25Þ
ð26Þ
The optimal value of h can be determined by minimizing the residual error given in Eq. (26). Table 2 exhibits the errors obtained using the optimal variational iteration method (OVIM) and variational iteration method (VIM) corresponding to different values of parameter e : It is obvious that the results of OVIM are better and more accurate than VIM. Example 4.3. Consider non-dimensional steady state reactive transport model, so-called nonlinear reaction diffusion model in porous catalysts which has been used to study porous catalyst pellets, which is governed by (Semary and Hassan, 2015). f 00 ðnÞ Pf 0 ðnÞ
AfðnÞ ¼ 0; B þ fðnÞ
ð27Þ
subject to the boundary conditions
f0 ðnÞ ¼ 1; 1 1 f1 ðn; hÞ ¼ 1 he þ hen2 ; 2 2 1 5 11 93 4 4 193 5 5 f2 ðn; hÞ ¼ 1 he þ h2 e þ h2 e2 h3 e3 þ he he 2 6 20 560 10080 1 3 1 1 þ he h2 e h2 e2 þ h3 e3 h4 e4 þ h5 e5 n2 2 4 4 32 1 1 1 1 þ h2 e2 h3 e3 þ h4 e4 h5 e5 n4 6 4 8 48 1 3 3 1 4 4 1 5 5 6 he he þ he n þ 20 20 80 1 4 4 1 5 5 8 1 5 5 10 þ he he n þ hen ; 112 224 1440 .. .
fð1Þ ¼ 1;
f 0 ð0Þ ¼ 0:
According to standard VIM, we have the following iterative formula, when P ¼ 0; A ¼ 12 and B ¼ 15 Z
n
fnþ1 ðnÞ ¼ fn ðnÞ þ 0
1 1 ðs nÞ fn00 ðsÞ þ fn ðsÞfn00 ðsÞ fn ðsÞ ds; n P 0: 5 2 ð28Þ
Consuming the initial condition, we have the following approximants f0 ðnÞ ¼ C; 1 f1 ðnÞ ¼ C þ Cn2 ; 4 11 1 1 1 C C2 n2 þ C C2 n4 ; f2 ðnÞ ¼ C þ 20 4 96 96 .. .
Please cite this article in press as: Mohyud-Din, S.T. et al., Optimal variational iteration method for nonlinear problems. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016), http://dx.doi.org/10.1016/j.jaubas.2016.09.004
6
S.T. Mohyud-Din et al.
where C can be calculated using the boundary condition fð1Þ ¼ 1: According to OVIM, we will proceed as follows, Z n fðnÞ ¼ C1 þ C2 n þ h ðs nÞ 0 1 00 1 00 f ðsÞ þ fðsÞf ðsÞ fðsÞ ds; 5 2
ð29Þ
using the boundary conditions, we get Z
1
C1 ¼ 1 h 0
1 1 ðs 1Þ f 00 ðsÞ þ fðsÞf 00 ðsÞ fðsÞ ds; 5 2
C2 ¼ 0: Consequently, we have the following iterative algorithm f0 ðnÞ ¼ 1;
1 1 f1 ðn; hÞ ¼ f0 ðnÞ þ h ðs nÞ f000 ðsÞ þ f0 ðsÞf000 ðsÞ f0 ðsÞ ds 5 2 0 Z 1 1 1 h ðs 1Þ f000 ðsÞ þ f0 ðsÞf000 ðsÞ f0 ðsÞ ds; 5 2 0 fnþ1 ðn; hÞ ¼ fn ðn; hÞ Z n 1 1 ðs nÞ fn00 ðs; hÞ þ fn ðs; hÞfn00 ðs; hÞ fn ðs; hÞ ds þh 5 2 0 Z 1 1 1 h ðs 1Þ fn00 ðs; hÞ þ fn ðs; hÞfn00 ðs; hÞ fn ðs; hÞ ds; 5 2 0 Z
"
2 #12 30 1 X i e5 ðhÞ ¼ ;h r5 : 30 i¼0 30
n
ð32Þ
The minimum point of e5 ðhÞ; as h ¼ 1:4149, is obtained. By substituting h ¼ 1:4149; in r5 ðn; hÞ, the absolute error at 5th order approximation by OVIM reduces remarkably (see Figs. 1 and 2). n P 1; ð30Þ
5. Conclusion
we obtain the following approximations f0 ðnÞ ¼ 1; 1 1 f1 ðn; hÞ ¼ 1 h þ hn2 ; 4 4 1 121 2 5 1 21 1 h h3 þ h h2 þ h3 n2 f2 ðn; hÞ ¼ 1 h þ 2 480 96 2 80 16 1 2 1 3 4 h h n; þ 96 96 .. .
The proposed study is based on a variational iteration method coupled with an auxiliary parameter to obtain an approximate solution of the ordinary differential equations with boundary conditions. An optimal auxiliary parameter can be determined by minimizing the residual function over the domain of the problem. Graphical representation and numerical results demonstrate the complete reliability and efficiency of the developed algorithm. References
we define the following residual function, 1 1 r5 ðn; hÞ ¼ f500 ðn; hÞ þ f5 ðn; hÞf500 ðn; hÞ f5 ðn; hÞ; 5 2
Figure 2 Absolute residual error for the problem (27) at 5th order approximation by OVIM, when h ¼ 1:4149.
ð31Þ
and the error of norm 2 of residual function (31) is,
Figure 1 Absolute residual error for the problem (27) at 5th order approximation by standard VIM.
Baskonus, H.M., Belgacem, F.B.M., Bulut, H., 2015. Solutions of nonlinear fractional differential equations systems through an implementation of the variational iteration method. Fra. Dyn. De Gruyter., 333–342 Belgacem, F.B.M., Baskonus, H.M., Bulut, H., 2015. Variational iteration method for hyperchaotic nonlinear fractional differential equations systems. Adv. Math. Stat. Sci., 445–453 Bulut, H., Baskonus, H.M., 2009. A comparison among homotopy perturbation method and the decomposition method with the variational iteration method for dispersive equation. Int. J. Basic Appl. Sci. 9 (10), 32–42. Ellahi, R., Hassan, M., Zeeshan, A., 2015a. Shape effects of nanosize particles in Cu–H2 O nanofluid on entropy generation. Int. J. Heat Mass Transfer 81, 449–456. Ellahi, R., Hassan, M., Zeeshan, A., 2015b. Study of natural convection MHD nanofluid by means of single and multi-walled carbon nanotubes suspended in a salt water solutions. Nanotechnol. IEEE Trans. 14 (4), 726–734. Goh, S.M., Noorani, M.S.N., Hashim, I., 2009. Variational iteration method as a reliable treatment for the hyperchaotic Rossler system. Int. J. Nonlinear Sci. Numer. 10, 363–371. He, J.H., 1999. Variational iteration method-a kind of non-linear analytical technique: some examples. Int. J. Nonlinear Mech. 34 (4), 699–708.
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Optimal variational iteration method for nonlinear problems He, J.H., 2000. Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114 (2–3), 115–123. He, J.H., 2007. Variational iteration method-some recent results and new interpretations. J. Comput. Appl. Math. 207, 3–17. Hosseini, M.M., Mohyud-Din, S.T., Ghaneai, H., Usman, M., 2010a. Auxiliary parameter in the variational iteration method and its optimal determination. Int. J. Nonlinear Sci. Numer. Simul. 11 (7), 495–502. Hosseini, M.M., Mohyud-Din, S.T., Ghaneai, H., 2010b. Variational iteration method for nonlinear age-structured population models using auxiliary parameter. Z. Nat. Forsch. A. 65 (12), 1137. Hosseini, M.M., Mohyud-Din, S.T., Ghaneai, H., 2012. Variational iteration method for Hirota-Satsuma coupled KdV equation using auxiliary parameter. Int. J. Numer. Methods Heat Fluid Flow 22 (3), 277–286. Inokuti, M., Sekine, H., Mura, T., 1978. General use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Naseer, S. (Ed.), Variational Method in the Mechanics of Solids. Pergamon press, New York, pp. 156–162. Liao, S.J., 1992. The proposed homotopy analysis technique for the solution of nonlinear problems (PhD thesis). Shanghai Jiao Tong University. PhD thesis. Liao, S.J., 2003. Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman and Hall. CRC Press, Boca Raton. Marinca, V., Herisanu, N., 2008. Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int. Commun. Heat Mass Transfer 35, 710–715. Mohyud-Din, S.T., Noor, M.A., Waheed, A., 2010. Variation of parameters method for initial and boundary value problems. World Appl. Sci. J. 11 (5), 622–639. Nawaz, M., Zeeshan, A., Ellahi, R., Abbasbandy, S., Rashidi, S., 2015. Joules and Newtonian heating effects on stagnation point flow over a stretching surface by means of genetic algorithm and NelderMead method. Int. J. Numer. Methods Heat Fluid Flow 25 (3), 665–684. Noor, M.A., Mohyud-Din, S.T., 2008. Solution of twelfth-order boundary value problems by variational iteration technique. J. Appl. Math. Comput. 28, 123–131.
7 Saberi, N.J., Tamamgar, M., 2008. The variational iteration method: a highly promising method for solving the system of integrodifferential equations. Comput. Math. Appl. 56, 346–351. Sadighi, A., Ganji, D.D., 2007. Exact solutions of nonlinear diffusion equations by variational iteration method. Comput. Math. Appl. 54, 1112–1121. Semary, M.S., Hassan, H.N., 2015. A new approach for a class of nonlinear boundary value problems with multiple solutions. J. Assoc. Arab Univ. Basic Appl. Sci. 17, 27–35. Uremen, S., Yildirim, A., 2009. Exact solutions of Poisson equation for electrostatic potential problems by means of the variational iteration method. Int. J. Nonlinear Sci. Numer. 10, 867–871. Wazwaz, A.M., 2007a. The variational iteration method for solving linear and nonlinear systems of PDEs. Comput. Math. Appl. 54, 895–902. Wazwaz, A.M., 2007b. The variational iteration method: a reliable analytic tool for solving linear and nonlinear wave equations. Comput. Math. Appl. 54, 926–932. Wazwaz, A.M., 2007c. The variational iteration method: a powerful scheme for handling linear and nonlinear diffusion equations. Comput. Math. Appl. 54, 933–939. Wazwaz, A.M., 2008. A study on linear and nonlinear Schrodinger equations by the variational iteration method. Chaos Solitons Fractals 37, 1136–1142. Xu, L., 2009. The variational iteration method for fourth-order boundary value problems. Chaos. Solitons Fract. 39 (3), 1386– 1394. Xu, D., Cui, J., Liao, S., Alsaedi, A., 2015. A HAM-based analytic approach for physical models with an infinite number of singularities. Numer. Algor. 69 (1), 59–74. Zeeshan, A., Baig, M., Ellahi, R., Hayat, T., 2014. Flow of viscous nanofluid between the concentric cylinders. J. Comput. Theor. Nanosci. 11 (3), 646–654. Zeeshan, A., Hassan, M., Ellahi, R., Nawaz, M., 2016. Shape effect of nanosize particles in unsteady mixed convection flow of nanofluid over disk with entropy generation. In: Proc. IMechE. Part E: J. Process Mechanical Engineering. 0954408916646139.
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University of Bahrain
Journal of the Association of Arab Universities for Basic and Applied Sciences www.elsevier.com/locate/jaaubas www.sciencedirect.com
Optimal variational iteration method for nonlinear problems Syed Tauseef Mohyud-Din *, Waseem Sikander, Umar Khan, Naveed Ahmed Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan Received 22 April 2016; revised 8 August 2016; accepted 21 September 2016
KEYWORDS Auxiliary parameter; Variational iteration method; Boundary value problems; Error estimate
Abstract This paper focuses on the study of boundary value problems using well-known He’s variational iteration method which is coupled with an auxiliary parameter. Three examples are given to show the efficiency and importance of the proposed algorithm. The reliability and accuracy has been proved by comparing our results with the solution obtained by standard variational iteration method. Ó 2016 University of Bahrain. Publishing services 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/).
1. Introduction Inokuti et al. (1978) proposed a general Lagrange multiplier method to solve nonlinear problems especially in the field of quantum mechanics. Later on, He (1999, 2000, 2007) modified the method to a new kind of an analytical technique for nonlinear problems and named it as variational iteration method (VIM), which is effectively and easily used to obtain solution of nonlinear equations accurately. For example, Belgacem et al. (2015) and Baskonus et al. (2015) obtain solutions of nonlinear fractional differential equations systems (NFDES) through implementation of VIM and concluded that VIM remains a valuable tool for the treatments of NFDES. Bulut and Baskonus (2009) obtain an exact solution of dispersive equation. Wazwaz (2007a, 2007b, 2007c, 2008) applied the method to nonlinear differential equations and pointed out that VIM is a very effective and reliable analytical tool for solving these equations. Saberi and Tamamgar (2008) concluded that the method is highly reliable for * Corresponding author. E-mail address: [email protected] (S.T. Mohyud-Din). Peer review under responsibility of University of Bahrain.
integro-differential equations. Goh et al. (2009) applied the method to hyperchaotic system with great success. Uremen and Yildirim (2009) and Sadighi and Ganji (2007) obtain exact solutions of poisson equation and nonlinear diffusion equations respectively. With the passage of time, several modifications were made in He’s VIM, which have further improved the efficiency and accuracy of the iterative algorithm to a tangible level. Moreover, with the passage of time many analytical techniques are developed to solve nonlinear problems. Liao (1992) came up with a new idea, he developed a nonlinear analytical technique called homotopy analysis method (HAM), which is free from assumption of small parameters and can be used to obtain approximate solution of nonlinear problems. In this method, Liao inserted an auxiliary parameter h, which is used to control the convergence of an approximate solution over the domain of the problem. Liao (2003), further generalized the method so called optimal homotopy analysis method (OHAM) for strongly nonlinear differential equations by inserting multiple parameters, which are used to control the convergence of approximate solutions. The optimal value of auxiliary parameters is obtained by minimizing the absolute residual error, which is a reliable, effective and accurate
http://dx.doi.org/10.1016/j.jaubas.2016.09.004 1815-3852 Ó 2016 University of Bahrain. Publishing services 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/). Please cite this article in press as: Mohyud-Din, S.T. et al., Optimal variational iteration method for nonlinear problems. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016), http://dx.doi.org/10.1016/j.jaubas.2016.09.004
2
S.T. Mohyud-Din et al.
method even for higher order of approximation. Different available studies that used the OHAM to solve various nonlinear equations can be seen in the literature (Xu et al., 2015; Nawaz et al., 2015; Ellahi et al., 2015a, 2015b; Zeeshan et al., 2014, 2016). In this paper, an auxiliary parameter h is inserted into the correctional functional of VIM for boundary value problems. We consume all of the boundary conditions to establish an integral equation before constructing an iterative algorithm to obtain an approximate solution. Thus we establish a modified iterative algorithm that does not contain undetermined coefficients, whereas most previous iterative methods do incorporate undetermined coefficients. It is observed that the coupling algorithm provides a convenient way to control and adjust the convergence region of approximate solution over the domain of the problem. Three examples are given to explicitly reveal the performance and reliability of the suggested algorithm. 2. Variational iteration method (VIM) To illustrate the steps of variational iteration method, we consider the following general nonlinear ordinary differential equation. LfðnÞ þ NfðnÞ þ gðnÞ ¼ 0:
ð1Þ
where L and N are linear and nonlinear operator respectively and gðnÞ illustrates an inhomogeneous term. According to VIM (He, 1999, 2000, 2007; Noor and Mohyud-Din, 2008), we can construct the correction functional as follows Z n fnþ1 ðnÞ ¼ fn ðnÞ þ kðLfn ðsÞ þ Nf~n ðsÞ þ gðsÞÞds; ð2Þ 0
Making the correction functional stationary, the Lagrange multiplier is identified as k ¼ s n; (Noor and Mohyud-Din, 2008; Xu, 2009), we get the following iterative formula Z n fnþ1 ðnÞ ¼ fn ðnÞ þ ðs nÞðLfn ðsÞ þ Nf~n ðsÞ þ gðsÞÞds: ð7Þ 0
An unknown auxiliary parameter h can be inserted into the iterative formula (7), for n ¼ 0, Eq. (7), becomes Z n f1 ðnÞ ¼ fð0Þ þ nf 0 ð0Þ þ h ðs nÞ Lf0 ðsÞ þ Nf~0 ðsÞ þ gðsÞ ds: ð8Þ 0
for optimal variational iteration method, we will proceed as follows, From Eq. (8), Z n fðnÞ ¼ fð0Þ þ nf 0 ð0Þ þ h ðs nÞðLfðsÞ þ NfðsÞ 0
þ gðsÞÞds:
ð9Þ
Substituting n ¼ a and n ¼ b in Eq. (9) and solve for fð0Þ, f 0 ð0Þ we get Z a ab h ðs aÞðLfðsÞ þ NfðsÞ þ gðsÞÞds fð0Þ ¼ a a ab 0 Z a ah ðs aÞðLfðsÞ þ NfðsÞ þ gðsÞÞds þ ab 0 Z b ðs bÞðLfðsÞ þ NfðsÞ þ gðsÞÞds ; 0
Z a ab h ðs aÞðLfðsÞ þ NfðsÞ þ gðsÞÞds ab ab 0 Z b þ ðs bÞðLfðsÞ þ NfðsÞ þ gðsÞÞds :
f 0 ð0Þ ¼
0
where k is a Lagrange multiplier (He, 1999, 2000, 2007; Noor and Mohyud-Din, 2008), which can be identified optimally via variational theory, fn is the nth approximate solution, and f~n is consider as a restricted variation, i.e. df~n ¼ 0. After identification of Lagrange multiplier, the successive approximations fnþ1 ðnÞ; n P 0, of the solution f can be readily obtained. Consequently, the exact solution will be of the form: fðnÞ ¼ lim fn ðnÞ:
ð3Þ
n!1
3. Optimal variational iteration method (OVIM)
Substituting the value of fð0Þ and f0 ð0Þ in Eq. (9) yields ab ab þn fðnÞ ¼ a a ab ab Z n þh ðs nÞðLfðsÞ þ NfðsÞ þ gðsÞÞds 0 Z a h ðs aÞðLfðsÞ þ NfðsÞ þ gðsÞÞds þ ða nÞ ab 0 Z b h ðs bÞðLfðsÞ þ NfðsÞ þ gðsÞÞds ðn aÞ þ ab 0 Z a ðs aÞðLfðsÞ þ NfðsÞ þ gðsÞÞds; ð10Þ h 0
To illustrate the steps of optimal variational iteration method, we consider the following second order nonlinear ordinary differential equation. LfðnÞ þ NfðnÞ þ gðnÞ ¼ 0;
a 6 n 6 b;
ð4Þ
subject to the boundary conditions fðaÞ ¼ a;
fðbÞ ¼ b; 2
ð5Þ
where L ¼ dnd 2 is the linear differential operator, N represents the nonlinear operator and gðnÞ illustrate an inhomogeneous term. According to standard VIM, the correction functional is given as Z n fnþ1 ðnÞ ¼ fn ðnÞ þ k Lfn ðsÞ þ Nf~n ðsÞ þ gðsÞ ds: ð6Þ
which can be solved by the modified iterative algorithm as ab ab f0 ðnÞ ¼ a a þn ; ab ab Z n f1 ðn; hÞ ¼ f0 ðnÞ þ h ðs nÞðLf0 ðsÞ þ Nf0 ðsÞ þ gðsÞÞds 0 Z a h ðs aÞðLf0 ðsÞ þ Nf0 ðsÞ þ gðsÞÞds ða nÞ þ ab 0 Z b h ðs bÞðLf0 ðsÞ þ Nf0 ðsÞ þ gðsÞÞds þ ðn aÞ ab 0 Z a ðs aÞðLf0 ðsÞ þ Nf0 ðsÞ þ gðsÞÞds h 0
0
Please cite this article in press as: Mohyud-Din, S.T. et al., Optimal variational iteration method for nonlinear problems. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016), http://dx.doi.org/10.1016/j.jaubas.2016.09.004
Optimal variational iteration method for nonlinear problems Z fnþ1 ðn; hÞ ¼ fn ðnÞ þ h
1 2 1 3 An þ Bn 2! 3! 1 2 1 3 f1 ðnÞ ¼ 1 þ n þ An þ Bn 2! 3! 1 1 c c þ þ A þ A n4 24 24 24 24 c 1 c c c 6 c þ B þ n5 þ A n Bn7 ; 120 120 120 720 720 5040 .. .
n
f0 ðnÞ ¼ 1 þ n þ
ðs nÞðLfn ðs; hÞ þ Nfn ðs; hÞ 0 Z a h ðs aÞðLfn ðs; hÞ ða nÞ þ gðsÞÞds þ ab 0 þ Nfn ðs; hÞ þ gðsÞÞds Z b h ðn aÞ ðs bÞðLfn ðs; hÞ þ Nfn ðs; hÞ þ ab 0 Z a ðs aÞðLfn ðs; hÞ þ gðsÞÞds h 0
þ Nfn ðs; hÞ þ gðsÞÞds;
n P 1:
3
ð11Þ
The Iterative algorithm (11) does not contain undetermined coefficients. fn ðn; hÞ; n P 1un ðx; hÞ; n P 1 contain only an auxiliary parameter h, which is used to control the convergence of approximate solution that can be determined optimally by minimizing the norm 2 of the residual function over the domain of given problem (Hosseini et al., 2010a, 2010b, 2012).
where A and B can be calculated using the boundary conditions corresponding to different values of parameter c. According to OVIM, we will proceed as follows, Z n 1 1 A3 n2 þ A4 n3 þ h 2! 3! 0 ðs nÞ3 ðivÞ 1 00 f ðsÞ ð1 þ cÞf ðsÞ þ cfðsÞ cs2 þ 1 ds; 3! 2
fðnÞ ¼ A1 þ A2 n þ
ð16Þ
using the boundary conditions, we get 4. Numerical examples In this section, we use Optimal Variational Iteration Method (OVIM) to find an approximate solution of boundary value problems. Numerical results obtained by the developed iterative algorithm are very encouraging. Example 4.1. Consider the following fourth order boundary value problem (Mohyud-Din et al., 2010). 1 f ðivÞ ðnÞ ð1 þ cÞf 00 ðnÞ þ cfðnÞ cn2 þ 1 ¼ 0; 2
ð12Þ
subject to the boundary conditions fð0Þ ¼ 1;
f 0 ð0Þ ¼ 1;
fð1Þ ¼ 1:5 þ sinhð1Þ;
0
f ð1Þ ¼ 1 þ coshð1Þ: The exact solution for this problem is 1 fðnÞ ¼ 1 þ n2 þ sinhðnÞ: 2
ð13Þ
According to standard VIM, the correction functional is given as Z
k fnðivÞ ðsÞ ð1 þ cÞf~00n ðsÞ 0 1 þcf~n ðsÞ cs2 þ 1 ds; n P 0: 2
fnþ1 ðnÞ ¼ fn ðnÞ þ
n
A1 ¼ 1; A2 ¼ 1; A3 ¼ 3 þ 6 sinhð1Þ 2 coshð1Þ Z 1 2 s 1 þ 2h þs 2 2 0 1 ðivÞ f ðsÞ ð1 þ cÞf 00 ðsÞ þ cfðsÞ cs2 þ 1 ds 2 Z 1 3 s s2 s 1 þ 6h 6 2 2 6 0 1 f ðivÞ ðsÞ ð1 þ cÞf 00 ðsÞ þ cfðsÞ cs2 þ 1 ds; 2 Z 1 2 s 1 þs A4 ¼ 6 þ 6 coshð1Þ 12 sinhð1Þ 6h 2 2 0 1 f ðivÞ ðsÞ ð1 þ cÞf 00 ðsÞ þ cfðsÞ cs2 þ 1 ds 2 Z 1 3 2 s s s 1 þ þ 12h 6 2 2 6 0 1 2 ðivÞ 00 f ðsÞ ð1 þ cÞf ðsÞ þ cfðsÞ cs þ 1 ds: 2 Consequently, we have the following iterative algorithm 2
f0 ðnÞ ¼ 1 þ n þ n2 ð3 þ 6 sinhð1Þ 2coshð1ÞÞ 3
þ n6 ð6 þ 6 coshð1Þ 12 sinhð1ÞÞ;
ð14Þ
Making the correction functional stationary, the Lagrange
f1 ðn;hÞ ¼ f0 ðnÞ þ h
R n s3 0
6
2
2
3
s2n þ sn2 n6
ðivÞ f0 ðsÞ ð1 þ cÞf000 ðsÞ þ cf0 ðsÞ 12 cs2 þ 1 ds
3
multiplier is identified as k ¼ ðsnÞ ; (Noor and Mohyud-Din, 3! 2008; Xu, 2009), we get the following iterative formula Z n 3 ðs nÞ fnþ1 ðnÞ ¼ fn ðnÞ þ 3! 0 1 fnðivÞ ðsÞ ð1 þ cÞf~n00 ðsÞ þ cf~n ðsÞ cs2 þ 1 ds: 2
ð15Þ
Consuming the initial conditions, we have the following approximants
0
1 R1 3 2 ðivÞ 3 0 s6 s2 þ 2s 16 f0 ðsÞ ð1 þ cÞf000 ðsÞ þ cf0 ðsÞ 12 cs2 þ 1 ds B C B C þhn2 B C @ R1 A 2 ðivÞ 00 s 1 1 2 þ 0 2 þ s 2 f0 ðsÞ ð1 þ cÞf0 ðsÞ þ cf0 ðsÞ 2 cs þ 1 ds 0 R 1 2 1 s3 ðivÞ 2 s þ s 1 f0 ðsÞ ð1 þ cÞf000 ðsÞ þ cf0 ðsÞ 12 cs2 þ 1 ds B 0 6 2 2 6 C B C þhn3 B C; @ R1 A 2 ðivÞ 00 s 1 1 2 0 2 þ s 2 f0 ðsÞ ð1 þ cÞf0 ðsÞ þ cf0 ðsÞ 2 cs þ 1 ds
Please cite this article in press as: Mohyud-Din, S.T. et al., Optimal variational iteration method for nonlinear problems. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016), http://dx.doi.org/10.1016/j.jaubas.2016.09.004
4
S.T. Mohyud-Din et al. Error estimates: error = Abs (exact solution approximate solution).
Table 1
c ¼ 1000; h ¼ 0:0549
Error (VIM)
n ;
Exact
Approximate
Error (OVIM)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1 1.105170 1.221340 1.349520 1.490750 1.646100 1.816650 2.003580 2.208110 2.431520 2.675200
1 1.105170 1.221340 1.349520 1.490750 1.646090 1.816650 2.003580 2.208100 2.431510 2.675200
0 9.75E7 6.58E7 5.70E7 6.85E7 8.01E7 8.60E7 7.90E7 1.08E6 1.98E6 2.6E12
fnþ1 ðn;hÞ ¼ fn ðn;hÞ þ h
0 7.91E7 0.000020 0.000485 0.011451 0.270513 6.382930 149.2260 3368.650 70385.10 1,307,730
Z n 3 2 s s n sn2 n3 þ 2 6 2 6 0
Error (OVIM)
1 1.105170 1.221340 1.349520 1.490750 1.646100 1.816650 2.003580 2.208110 2.431520 2.675200
1 1.105170 1.221340 1.349520 1.490750 1.646090 1.816650 2.003580 2.208100 2.431510 2.675200
0 1.27E6 5.61E7 1.44E7 4.64E8 1.04E7 3.01E7 4.63E7 1.14E6 2.70E6 5.7E12
0 2.51E7 0.000015 0.001280 0.111747 9.337910 628.7880 27217.50 634860.0 5,764,110 0.046360
and the error of norm 2 of residual function (18) is, " 2 #12 30 1 X i e15 ðhÞ ¼ fr15 ;h g : 30 i¼0 30
1 R1 3 2 3 0 s6 s2 þ 2s 16 fnðivÞ ðs;hÞ ð1 þ cÞfn00 ðs;hÞ þ cfn ðs;hÞ 12 cs2 þ 1 ds B C C þ hn2 B @ R A 1 ðivÞ 00 s2 1 1 2 þ 0 2 þ s 2 fn ðs;hÞ ð1 þ cÞfn ðs;hÞ þ cfn ðs;hÞ 2 cs þ 1 ds 0
1 0 R1 3 2 s 2 s þ s 1 fnðivÞ ðs;hÞ ð1 þ cÞfn00 ðs;hÞ þ cfn ðs;hÞ 12 cs2 þ 1 ds B 0 6 2 2 6 C C þ hn @ A; n P 1; R1 2 0 s2 þ s 12 fnðivÞ ðs;hÞ ð1 þ cÞfn00 ðs;hÞ þ cfn ðs;hÞ 12 cs2 þ 1 ds 3B
ð17Þ
we obtain the following approximations f0 ðnÞ ¼ 1 þ n þ 0:482523n2 þ 0:192678n3 ; f1 ðn; hÞ ¼ 1 þ n þ ð0:482523 þ 0:0178114h
ð19Þ
The above residual function can be used to approximate e15 ðhÞ ; and the optimal value of h can be determined by minimizing the residual error given in Eq. (19). Table 1 exhibits the errors obtained using the optimal variational iteration method (OVIM) and variational iteration method (VIM) corresponding to different values of parameter c : It is obvious that the results of OVIM are more reliable and accurate than VIM. Example 4.2. Consider the following second order nonlinear differential equation, the temperature distribution equation in a uniformly thick rectangular fin radiation to free space with non-linearity of high order (Marinca and Herisanu, 2008). f 00 ðnÞ ef 4 ðnÞ ¼ 0; ð20Þ
þ 0:0003728chÞn2 þ ð0:192678 0:0259889h 0:0000361chÞn3 ð0:0014564h þ 0:0014564chÞn4 þ ð0:0096339h þ 0:0013005chÞn5 þ 0:0000485chn6 0:0002293chn7 ; f2 ðn; hÞ ¼ 1 þ n þ ð0:482523 þ 0:0356228h
subject to the boundary conditions fð1Þ ¼ 1; f 0 ð0Þ ¼ 0:
þ 0:00074572ch 0:0181541h2 0:0007641ch2
According to standard VIM, the correction functional is given as Z n fnþ1 ðnÞ ¼ fn ðnÞ þ k fn00 ðsÞ ef~n4 ðsÞ ds; n P 0: ð21Þ
0:0000101c2 h2 Þn2 þ ð0:192678 0:0519778h 0:0000723ch þ 0:025966h2 þ 0:0000733ch2 þ 1:29 106 c2 h2 Þn3 ð0:0029128h þ 0:0029128ch
0
Making the correction functional stationary, the Lagrange multiplier is identified as k ¼ s n; (Noor and Mohyud-Din, 2008; Xu, 2009), we get the following iterative formula Z n fnþ1 ðnÞ ¼ fn ðnÞ þ ðs nÞ fn00 ðsÞ ef~n4 ðsÞ ds; n P 0: ð22Þ
0:0029407h2 0:0029717ch2 0:0000310c2 h2 Þn4 þ ð0:019267h þ 0:002601ch 0:0109334h2 0:0026018ch2 1:80 106 c2 h2 Þn5 þ ð0:0000970ch 0:0000485h2 0:0001951ch2
0
0:0000495c h Þn þ ð0:0004587ch þ 0:0002293h 6
Error (VIM)
Approximate
1 ðivÞ r15 ðn; hÞ ¼ f15 ðn; hÞ ð1 þ cÞf1500 ðn; hÞ þ cf15 ðn; hÞ cn2 þ 1; 2 ð18Þ
1 fnðivÞ ðs;hÞ ð1 þ cÞfn00 ðs;hÞ þ cfn ðs;hÞ cs2 þ 1 ds 2
2 2
c ¼ 2000; h ¼ 0:0293 Exact
2
þ 0:0005206ch2 þ 0:0000310c2 h2 Þn7 þ ð1:73 106 ch2 þ 1:73 106 c2 h2 Þn8 þ ; .. . we define the following residual function, to obtain the proper value of h for the series solution (17),
Consuming the initial condition, we have the following approximants f0 ðnÞ ¼ A; 1 f1 ðnÞ ¼ A þ A4 en2 ; 2 1 1 1 1 13 4 8 f2 ðnÞ ¼ A þ A4 en2 þ A7 e2 n4 þ A10 e3 n6 þ A en 2 6 20 112 1 A16 e5 n10 ; þ 1440 .. .
Please cite this article in press as: Mohyud-Din, S.T. et al., Optimal variational iteration method for nonlinear problems. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016), http://dx.doi.org/10.1016/j.jaubas.2016.09.004
Optimal variational iteration method for nonlinear problems
5
Error estimates: error = Abs (numerical solution approximate solution).
Table 2
e ¼ 0:1; h ¼ 0:9223
Error (VIM)
n #
Numerical
Approximate
Error (OVIM)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.956820 0.957240 0.958499 0.960602 0.963557 0.967375 0.972068 0.977655 0.984155 0.991594 1.000000
0.956821 0.957240 0.958499 0.960603 0.963558 0.967375 0.972068 0.977655 0.984155 0.991594 1.000000
4.66E7 4.56E7 4.36E7 4.16E7 3.90E7 3.51E7 3.03E7 2.45E7 1.79E7 9.60E8 1.03E8
0.677522 0.680675 0.690197 0.706274 0.729233 0.759557 0.797918 0.845227 0.902696 0.971948 1.055160
where A can be calculated using the boundary condition fð1Þ ¼ 1; corresponding to different values of parameter e. According to OVIM, we will proceed as follows, Z n ðs nÞ f00 ðsÞ ef4 ðsÞ ds; ð23Þ fðnÞ ¼ A1 þ A2 n þ h 0
using the boundary conditions, we get Z 1 A1 ¼ 1 h ðs 1Þðf 00 ðsÞ ef 4 ðsÞÞds; 0
A2 ¼ 0: Consequently, we have the following iterative algorithm f0 ðnÞ ¼ 1; f1 ðn; hÞ ¼ f0 ðnÞ þ h h
R1 0
Rn
ðs
0
ðs nÞðf000 ðsÞ ef04 ðsÞÞds
1Þðf000 ðsÞ
fnþ1 ðn; hÞ ¼ fn ðn; hÞ þ h h
R1 0
Rn 0
ef04 ðsÞÞds;
ð24Þ
ðs nÞðfn00 ðs; hÞ efn4 ðs; hÞÞds
ðs 1Þ fn00 ðs; hÞ efn4 ðs; hÞ ds;
n P 1;
we obtain the following approximations
e ¼ 0:5; h ¼ 0:7557
Error (VIM)
Numerical
Approximate
Error (OVIM)
0.852329 0.853650 0.857629 0.864315 0.873795 0.886194 0.901681 0.920478 0.942871 0.969224 1.000000
0.852460 0.853780 0.857756 0.864438 0.873912 0.886301 0.901777 0.920559 0.942934 0.969261 1.000000
1.31E4 1.30E4 1.28E4 1.23E4 1.16E4 1.07E4 9.58E5 8.12E5 6.21E5 3.64E5 6.35E8
0.202374 0.204153 0.209547 0.218732 0.232020 0.249886 0.273022 0.302408 0.339434 0.386100 0.445335
we define the following residual function, r4 ðn; hÞ ¼ f400 ðn; hÞ ef44 ðn; hÞ; and the error of norm 2 of residual function (25) is, " 2 #12 30 1 X i ;h e4 ðhÞ ¼ r4 : 30 i¼0 30
ð25Þ
ð26Þ
The optimal value of h can be determined by minimizing the residual error given in Eq. (26). Table 2 exhibits the errors obtained using the optimal variational iteration method (OVIM) and variational iteration method (VIM) corresponding to different values of parameter e : It is obvious that the results of OVIM are better and more accurate than VIM. Example 4.3. Consider non-dimensional steady state reactive transport model, so-called nonlinear reaction diffusion model in porous catalysts which has been used to study porous catalyst pellets, which is governed by (Semary and Hassan, 2015). f 00 ðnÞ Pf 0 ðnÞ
AfðnÞ ¼ 0; B þ fðnÞ
ð27Þ
subject to the boundary conditions
f0 ðnÞ ¼ 1; 1 1 f1 ðn; hÞ ¼ 1 he þ hen2 ; 2 2 1 5 11 93 4 4 193 5 5 f2 ðn; hÞ ¼ 1 he þ h2 e þ h2 e2 h3 e3 þ he he 2 6 20 560 10080 1 3 1 1 þ he h2 e h2 e2 þ h3 e3 h4 e4 þ h5 e5 n2 2 4 4 32 1 1 1 1 þ h2 e2 h3 e3 þ h4 e4 h5 e5 n4 6 4 8 48 1 3 3 1 4 4 1 5 5 6 he he þ he n þ 20 20 80 1 4 4 1 5 5 8 1 5 5 10 þ he he n þ hen ; 112 224 1440 .. .
fð1Þ ¼ 1;
f 0 ð0Þ ¼ 0:
According to standard VIM, we have the following iterative formula, when P ¼ 0; A ¼ 12 and B ¼ 15 Z
n
fnþ1 ðnÞ ¼ fn ðnÞ þ 0
1 1 ðs nÞ fn00 ðsÞ þ fn ðsÞfn00 ðsÞ fn ðsÞ ds; n P 0: 5 2 ð28Þ
Consuming the initial condition, we have the following approximants f0 ðnÞ ¼ C; 1 f1 ðnÞ ¼ C þ Cn2 ; 4 11 1 1 1 C C2 n2 þ C C2 n4 ; f2 ðnÞ ¼ C þ 20 4 96 96 .. .
Please cite this article in press as: Mohyud-Din, S.T. et al., Optimal variational iteration method for nonlinear problems. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016), http://dx.doi.org/10.1016/j.jaubas.2016.09.004
6
S.T. Mohyud-Din et al.
where C can be calculated using the boundary condition fð1Þ ¼ 1: According to OVIM, we will proceed as follows, Z n fðnÞ ¼ C1 þ C2 n þ h ðs nÞ 0 1 00 1 00 f ðsÞ þ fðsÞf ðsÞ fðsÞ ds; 5 2
ð29Þ
using the boundary conditions, we get Z
1
C1 ¼ 1 h 0
1 1 ðs 1Þ f 00 ðsÞ þ fðsÞf 00 ðsÞ fðsÞ ds; 5 2
C2 ¼ 0: Consequently, we have the following iterative algorithm f0 ðnÞ ¼ 1;
1 1 f1 ðn; hÞ ¼ f0 ðnÞ þ h ðs nÞ f000 ðsÞ þ f0 ðsÞf000 ðsÞ f0 ðsÞ ds 5 2 0 Z 1 1 1 h ðs 1Þ f000 ðsÞ þ f0 ðsÞf000 ðsÞ f0 ðsÞ ds; 5 2 0 fnþ1 ðn; hÞ ¼ fn ðn; hÞ Z n 1 1 ðs nÞ fn00 ðs; hÞ þ fn ðs; hÞfn00 ðs; hÞ fn ðs; hÞ ds þh 5 2 0 Z 1 1 1 h ðs 1Þ fn00 ðs; hÞ þ fn ðs; hÞfn00 ðs; hÞ fn ðs; hÞ ds; 5 2 0 Z
"
2 #12 30 1 X i e5 ðhÞ ¼ ;h r5 : 30 i¼0 30
n
ð32Þ
The minimum point of e5 ðhÞ; as h ¼ 1:4149, is obtained. By substituting h ¼ 1:4149; in r5 ðn; hÞ, the absolute error at 5th order approximation by OVIM reduces remarkably (see Figs. 1 and 2). n P 1; ð30Þ
5. Conclusion
we obtain the following approximations f0 ðnÞ ¼ 1; 1 1 f1 ðn; hÞ ¼ 1 h þ hn2 ; 4 4 1 121 2 5 1 21 1 h h3 þ h h2 þ h3 n2 f2 ðn; hÞ ¼ 1 h þ 2 480 96 2 80 16 1 2 1 3 4 h h n; þ 96 96 .. .
The proposed study is based on a variational iteration method coupled with an auxiliary parameter to obtain an approximate solution of the ordinary differential equations with boundary conditions. An optimal auxiliary parameter can be determined by minimizing the residual function over the domain of the problem. Graphical representation and numerical results demonstrate the complete reliability and efficiency of the developed algorithm. References
we define the following residual function, 1 1 r5 ðn; hÞ ¼ f500 ðn; hÞ þ f5 ðn; hÞf500 ðn; hÞ f5 ðn; hÞ; 5 2
Figure 2 Absolute residual error for the problem (27) at 5th order approximation by OVIM, when h ¼ 1:4149.
ð31Þ
and the error of norm 2 of residual function (31) is,
Figure 1 Absolute residual error for the problem (27) at 5th order approximation by standard VIM.
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Please cite this article in press as: Mohyud-Din, S.T. et al., Optimal variational iteration method for nonlinear problems. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016), http://dx.doi.org/10.1016/j.jaubas.2016.09.004