Variation of parameters method with an auxiliary parameter for initial value problems

Ain Shams Engineering Journal xxx (2017) xxx–xxx

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Variation of parameters method with an auxiliary parameter for initial value problems Waseem Sikandar a, Umar Khan b, Naveed Ahmed a, Syed Tauseef Mohyud-Din a,⇑ a b

Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan

a r t i c l e

i n f o

Article history: Received 3 June 2016 Revised 1 September 2016 Accepted 23 September 2016 Available online xxxx Keywords: Variation of Parameter Method (VPM) Auxiliary parameter Evolution and RLW equations

a b s t r a c t In this paper, an unknown auxiliary parameter is inserted in Variation of Parameter Method to solve Evolution and RLW equations. The proposed algorithm is very effective and is practically well suited for use in these problems. Three practical examples are given to verify the accuracy and efficiency of the developed technique. Comparison with the standard VPM is made to show the reliability and efficiency of the proposed algorithm. Ó 2017 Faculty of Engineering, Ain Shams University 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/).

1. Introduction Several analytical and numerical technique including Perturbation [1,2], Homotopy Perturbation Method [3–6], Differential Transform Method [7–9], Adomian’s Decomposition Method [10– 13], Polynomial Spline [14], Exp-function [15,16], Variational Iteration Method [17–22] and Sink-Galerkin [23,24] have been developed for solving initial and boundary value problems arise in Mathematics, Physics, Engineering and applied Sciences. Unfortunately, this entire algorithm has some inbuilt deficiencies including huge amount of computational work, lengthy calculation, limited convergence, assumption of small parameter, calculation of Adomian’s polynomials, divergent results and even in certain cases they are unreliable with the physical nature of the problem. Ma et al. [25–27] used VPM for solving differential equations of the diversified physical nature and constructing an approximate solution coupled with a number of other new solutions including solitons, rational, interaction and complexions solutions of the KdV equations. It has been observed that Ma’s VPM [25–27] is much better than the above mentioned algorithms. The suggested algorithm finds the solution without any linearization, discretization, perturbation, restrictive assumptions and is free from calculation of Adomian’s polynomials. The results obtained are completely reliable with the proposed algorithm and are very encouraging. Recently, Mohyud-Din et al. [28–31] applied the Peer review under responsibility of Ain Shams University. ⇑ Corresponding author. E-mail address: [email protected] (S.T. Mohyud-Din).

method to Helmholtz and Boussinesq equations with great success and pointed out that VPM is very reliable and promising tool for solving these problems. Ghaneai and Hosseini [35] inserted an auxiliary parameter in variational iteration algorithm to obtain solution of wave-like and heat-like equations and concluded that an auxiliary parameter provides a simple way to control and adjust the convergence region of approximate solution in a large domain. Semary and Hassan [36] used variational iteration method coupled with an auxiliary parameter to predict the multiple solutions of nonlinear boundary value problems with great success. In this study, we use Variation of Parameters Method (VPM) coupling with an unknown auxiliary parameter h for Evolution and RLW equations. Numerical results obtained by the proposed algorithm are very effective, reliable and encouraging as compared with the standard VPM. 2. Variation of Parameters Method (VPM) Consider the following general nonlinear ordinary differential equation in operator form:

Lf ðnÞ þ Rf ðnÞ þ Nf ðnÞ þ gðnÞ ¼ 0:

ð1Þ

where L is a higher order linear operator that is assumed to be easily invertible, R is a linear operator of order less then L, N is a nonlinear operator, and g is an inhomogeneous term. The variation of parameters method [25–31] provides the following iterative scheme for Eq. (1):

http://dx.doi.org/10.1016/j.asej.2016.09.014 2090-4479/Ó 2017 Faculty of Engineering, Ain Shams University 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/).

Please cite this article in press as: Sikandar W et al. Variation of parameters method with an auxiliary parameter for initial value problems. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.09.014

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8 k X > > ni f i ð0Þ < f ðnÞ ¼ is an initial approximation; 0 i! i¼0 > > Rn : f nþ1 ðnÞ ¼ f 0 ðnÞ þ 0 kðn; gÞðRfn ðgÞ  Nfn ðgÞ  gðgÞÞdg; n P 0: ð2Þ where k is the order of given differential equation and kðn; gÞ is multiplier which can be determined with the help of Wronskian’s technique

kðn; gÞ ¼

k X

g n ð1Þ

i¼1

ði  1Þ!ðk  iÞ!

i1

i1 ki

ð3Þ

Consequently, an exact solution can be obtained when n approaches to infinity:

f ðnÞ ¼ lim f n ðnÞ:

ð4Þ

n!1

3. Insertion of auxiliary parameter in VPM

(

f 0 ðn; tÞ ¼ f ðn; 0Þ; f nþ1 ðn; tÞ ¼ f 0 ðn; tÞ þ

Rt n 0

@

4

o

f n ðn;sÞ @n4

ds:

ð9Þ

Fig. 1 shows the absolute error for 10th order approximation by standard VPM. The result obtained by standard VPM is not valid for large values of n and t. Now, using the iterative scheme (6), we have

f 0 ðn; tÞ ¼ f ðn; 0Þ ¼ sinðnÞ;

ð10Þ

f 1 ðn; t; hÞ ¼ f 0 ðn; tÞ !) Z t( @f 0 ðn; sÞ @f 0 ðn; sÞ @ 4 f 0 ðn; sÞ þ þh þ ds: @s @s @n4 0

ð11Þ

Generally,

f nþ1 ðn; t; hÞ ¼ f 0 ðn; tÞ !) Z t( @f n ðn; s; hÞ @f n ðn; s; hÞ @ 4 f n ðn; s; hÞ þ þh þ ds: n P 1 @s @s @n4 0

An unknown auxiliary parameter h can be inserted into the Variation of Parameter Method. Eq. (1) can be easily written in the following form:

First few terms of solution are

Lf ðnÞ ¼ Lf ðnÞ þ hðLf ðnÞ þ Rf ðnÞ þ Nf ðnÞ þ gðnÞÞ:

f 0 ðn; tÞ ¼ sinðnÞ;

ð13Þ

f 1 ðn; t; hÞ ¼ sinðnÞ þ ht sinðnÞ;

ð14Þ

1 2 2 f 2 ðn; t; hÞ ¼ sinðnÞ þ 2ht sinðnÞ þ h t sinðnÞ þ h t 2 sinðnÞ; 2

ð15Þ

ð5Þ

According to variation of parameters method, we can construct the following iterative scheme for Eq. (5):

8 k X > > ni f i ð0Þ > > f 0 ðnÞ ¼ is an initial approximation; > i! > > i¼0 > > > Rn < f 1 ðn; hÞ ¼ f 0 ðnÞ þ 0 kðn; gÞðLf0 ðgÞ þ hðLf0 ðgÞ þ Rf0 ðgÞ þ Nf0 ðgÞ > > þgðgÞÞÞdg; > > > Rn > > f > nþ1 ðn; hÞ ¼ f 0 ðnÞ þ 0 kðn; gÞðLfn ðg; hÞ þ hðLfn ðg; hÞ þ Rfn ðg; hÞ > > : þNfn ðg; hÞ þ gðgÞÞÞdg; n P 1:

ð12Þ

.. .: The auxiliary parameter h is used to control and adjust the convergence of approximate solution. In order to obtain the proper value

ð6Þ f n ðn; hÞ; n P 1 contain an auxiliary parameter h, which is used to control and adjust the convergence of approximate solution over the domain of the problem. In general, the optimal value of h can be determined by means of the so-called h curve [36–38]. In fact, the suggested algorithm is simple, reliable, effective and is accurately approximate the solution over a large domain.

of h, we plot the so called h curve of @f ðn;t;hÞ for 10th order approxi@t mation when n ¼ 5 and t ¼ 5 as shown in Fig. 2. According to this h curve, it can be easily determine the valid region of h, which corresponds to the line segments nearly parallel to the horizontal axis. It can be seen that admissible range is 0:9 6 h 6 0:25. From Fig. 2, we choose h ¼ 0:55 and absolute error for 10th order approximation by the developed technique is shown in Fig. 3. 4.2. Example 4.2

4. Numerical examples

(

Consider the following evolution eq. [32] @f @t

3

t > 0;

n

0 6 n 6 10:

@f þ @n ¼ 2 @n@2 f@t ;

In this section, we apply Variation of Parameters Method (VPM) coupled with an unknown auxiliary parameter developed in Section 3 for solving initial value problems. Numerical results obtained by the proposed algorithm are very encouraging.

The exact solution for this problem is

4.1. Example 4.1

f ðn; tÞ ¼ ent :

f ðn; 0Þ ¼ e ;

0 6 n 6 10;

ð16Þ

ð17Þ

An ordinary or partial differential equation can be regarded as evolution equation on an infinite dimensional state space. The linear heat equation or linear wave equation describing heat conduction or vibration of string is two simple examples of evolution equations. Consider the following evolution eq. [32]

(

@f @t

4

@ f þ @n 4 ¼ 0;

f ðn; 0Þ ¼ sinðnÞ;

0 6 n 6 10; t > 0; 0 6 n 6 10:

ð7Þ

The closed form solution for this problem is

f ðn; tÞ ¼ et sinðnÞ:

ð8Þ

According to standard VPM, we have the following iterative scheme (2):

Fig. 1. Absolute residual error for Eq. (7) for 10th order approximation by the standard VPM.

Please cite this article in press as: Sikandar W et al. Variation of parameters method with an auxiliary parameter for initial value problems. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.09.014

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W. Sikandar et al. / Ain Shams Engineering Journal xxx (2017) xxx–xxx

f 0 ðn; tÞ ¼ en ;

ð21Þ n

f 1 ðn; t; hÞ ¼ en  hte ; n

f 2 ðn; t; hÞ ¼ en  2hte

ð22Þ 1 2 2 þ h ten þ h t2 en ; 2

ð23Þ

other terms of the solution can also be obtained in a similar way. From Fig. 5, we select h ¼ 0:5. The accuracy of absolute error is remarkably improved for 10th order approximation, as illustrated in Fig. 6. 4.3. Example 4.3 Fig. 2. h-curve of @f ðn;t;hÞ for 10th order approximation given by Eq. (12) when n ¼ 5 @t and t ¼ 5:

(

Consider the following RLW eq. [32] @f @s

3

 @n@2 @f s þ f

@f @n

¼ 0;

f ðn; 0Þ ¼ n;

0 6 n 6 10; t > 0; 0 6 n 6 10:

ð24Þ

The closed form solution is

f ðn; tÞ ¼

Fig. 3. Absolute residual error for Eq. (7) for 10th-order approximation by the present technique when h ¼ 0:55.

n : 1þt

The RLW equation was proposed by Peregrine [33], which plays a very important role in the study of nonlinear dispersive waves [33,34] because of its description in a large number of physical phenomena, such as ion acoustic plasma waves and shallow water waves. Adopting the similar procedure as in the previous examples, Fig. 7 shows the absolute error of f 6 ðn; tÞ; which confirm that the obtained result by standard VPM is not valid for large value of n and t. Now by the iterative scheme (6), we have

f 0 ðn; tÞ ¼ f ðn; 0Þ ¼ n; According to standard VPM, the absolute error of f 10 ðn; tÞ is shown in Fig. 4 which confirm that the result is not valid for large value of n and t. Now, using the iterative scheme (6), we have

f 0 ðn; tÞ ¼ f ðn; 0Þ ¼ en ;

ð18Þ

ð25Þ

ð26Þ

f 1 ðn;t;hÞ ¼ f 0 ðn;tÞ !) Z t( @f 0 ðn; sÞ @f 0 ðn; sÞ @ 3 f 0 ðn; sÞ @f 0 ðn; sÞ ds; þ þh  þ f 0 ðn; sÞ @s @s @n @n2 @ s 0 ð27Þ

f 1 ðn; t; hÞ ¼ f 0 ðn; tÞ !) Z t( @f 0 ðn; sÞ @f 0 ðn; sÞ @f 0 ðn; sÞ @ 3 f 0 ðn; sÞ ds; þh þ 2 þ @s @s @n @n2 @ s 0 ð19Þ generally,

 Z t @f n ðn; s; hÞ @f n ðn; s; hÞ @f n ðn; s; hÞ f nþ1 ðn; t; hÞ ¼ f 0 ðn; tÞ þ þh þ @s @s @n 0 !) @ 3 f n ðn; s; hÞ 2 ds: n P 1 ð20Þ @n2 @ s First few terms of solution are

Fig. 5. h-curve of @f ðn;t;hÞ for 10th order approximation given by Eq. (20) when n ¼ 5 @t and t ¼ 5.

Fig. 4. Absolute residual error for Eq. (16) for 10th-order approximation by the standard VPM.

Fig. 6. Absolute residual error for Eq. (16) for 10th order approximation by the present technique when h ¼ 0:5.

Please cite this article in press as: Sikandar W et al. Variation of parameters method with an auxiliary parameter for initial value problems. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.09.014

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5. Conclusion In this article, we have used Variation of Parameter Method coupled with an auxiliary parameter for Evolution and RLW equations. Numerical results obtained by the suggested technique are very effective, completely reliable and powerful in obtaining analytical solutions for initial value problems. It would be quite interesting that the proposed algorithm involving an auxiliary parameter is very feasible and effective for nonlinear physical problem in a large domain. Moreover, the iterative scheme (6) reduces to standard VPM for h ¼ 1: Fig. 7. Absolute residual error for Eq. (24) for 6th order approximation by the standard VPM.

generally,

Z t( @f n ðn; s; hÞ @f n ðn; s; hÞ @ 3 f n ðn; s; hÞ f nþ1 ðn; t; hÞ ¼ f 0 ðn; tÞ þ þh  @s @s @n2 @ s 0  @f ðn; s; hÞ ds: n P 1 þ f n ðn; s; hÞ n ð28Þ @n First few terms of solution are

f 0 ðn; tÞ ¼ n;

ð29Þ

f 1 ðn; t; hÞ ¼ n þ htn;

ð30Þ

1 3 2 2 f 2 ðn; t; hÞ ¼ n þ 2htn þ h tn þ h t2 n þ h t 3 n; 3

ð31Þ

In a similar fashion, other terms of the solution can be calculated. From Fig. 8, we select h ¼ 0:7, and absolute error of f ðn; tÞ for 6th order approximation by the present technique for ðn; tÞ 2 ½0; 10  ½0; 4 is shown in Fig. 9.

Fig. 8. h-curve of and t ¼ 1.

@f ðn;t;hÞ @t

for 6th order approximation given by Eq. (28) when n ¼ 1

Fig. 9. Absolute residual error for Eq. (24) for 6th order approximation by the present technique when h ¼ 0:7.

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Waseem Sikandar completed his master degree from COMSATS Institute of Information Technology, Abbottabad, Pakistan. He is pursuing his Ph.D. in the field of Applied Mathematics at HITEC University Taxila Pakistan. He has a teaching experience of 5 years and a research experience of 5 years.

5

Naveed Ahmed completed his Ph.D. in the field of Applied Mathematics at HITEC University Taxila Pakistan. He is now serving in the same department as an Assistant Professor, He has a teaching experience of 12 years and a research experience of 8 years. His fields of interest include fluid mechanics, analytical and numerical methods for solving differential equations, and computational fluid dynamics.

Syed Tauseef Mohyud-Din completed his Ph.D. degree from COMSATS Institute of Information Technology, Islamabad, Pakistan. He is currently working as Dean Faculty of Sciences at HITEC University Taxila Cantt, Pakistan. He has won Presidents Pride of Performance in 2011. Also, he won best research paper award from Higher Education Commission Pakistan in 2010. His field of interest includes Analytical and numerical solutions of initial and boundary value problems related to the applied/engineering sources using Variational iteration, Homotopy Perturbation, Variation of Parameters, Modified Variation of Parameters, Variational Iteration using He’s polynomials, Variational Iteration using Adomain’s polynomials, exp function, iterative, decomposition and finite difference schemes. He has a teaching experience of 21 years and a research experience of 11 years.

Umar Khan completed his Ph. D in Allied Mathematics from HITEC University Taxila Cantt, Pakistan. He is currently working as Assistant Professor at COMSATS Institute of Information Technology, Abbottabad, Pakistan. He has teaching experience of 4 years and research experience of 6 years. His field of research includes, fluid mechanics and heat transfer, analytical and numerical schemes for boundary value problems, and nanofluid dynamics.

Please cite this article in press as: Sikandar W et al. Variation of parameters method with an auxiliary parameter for initial value problems. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.09.014