Skip to main content Accessibility help
×
Home
Hostname: page-component-cf9d5c678-9z9qw Total loading time: 0.387 Render date: 2021-07-27T19:52:25.321Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Variation of Parameters Method for Solving System of Nonlinear Volterra Integro-Differential Equations

Published online by Cambridge University Press:  03 June 2015

Muhammad Aslam Noor
Affiliation:
Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan
Khalida Inayat Noor
Affiliation:
Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan
Asif Waheed
Affiliation:
Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan
Eisa Al-Said
Affiliation:
Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia
Get access

Abstract

It is well known that nonlinear integro-differential equations play vital role in modeling of many physical processes, such as nano-hydrodynamics, drop wise condensation, oceanography, earthquake and wind ripple in desert. Inspired and motivated by these facts, we use the variation of parameters method for solving system of nonlinear Volterra integro-differential equations. The proposed technique is applied without any discretization, perturbation, transformation, restrictive assumptions and is free from Adomian’s polynomials. Several examples are given to verify the reliability and efficiency of the proposed technique.

Type
Research Article
Copyright
Copyright © Global-Science Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Biazar, J., Ghazvini, H. and Eslami, M., He’s homotopy perturbation method for system of integro-differential equations, Chaos. Solit. Fract., 39 (2009), pp. 12531258.CrossRefGoogle Scholar
[2] Bo, T. L., Xie, L., X, and Zheng, J., Numerical approach to wind ripple in dessert, Int. J. Nonl. Sci. Num. Sim., 8(2) (2007), pp. 223228.Google Scholar
[3] El-Sayed, S. M., Kaya, D. and Zarea, S., The decomposition method applied to solve high order linear Voltera Fredholm integro-differential equations, Int. J. Nonl. Sci. Num. Sim., 5(2) (2004), pp. 105112.Google Scholar
[4] El-Shahed, M., Application of He’s homotopy perturbation method to Voltera’s integro-differential equation, Int. J. Nonl. Sci. Num. Sim., 6(2) (2005), pp. 163168.Google Scholar
[5] Ghasemi, M., Kajani, M. T. and Babolian, E., Application of He’s homotopy perturbation method to nonlinear integro differential equations, Appl. Math. Comput., 188 (2007), pp. 538548.Google Scholar
[6] Goghary, S., Javadi, H. S. and Babolian, E., Restarted Adomian method for system of nonlinear Volterra integral equations, Appl. Math. Comput., 161 (2005), pp. 745751.Google Scholar
[7] He, J. H., Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering98, Dalian, China, 1998, pp. 288291.Google Scholar
[8] Ma, W. X. and You, Y., Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions, Trans. Am. Math. Soc., 357 (2004), pp. 17531778.CrossRefGoogle Scholar
[9] Ma, W. X. and You, Y., Rational solutions of the Toda lattice equation in casoratian form, Chaos. Solit. Fract., 22 (2004), pp. 395406.CrossRefGoogle Scholar
[10] Ma, W. X., Li, C. X. and He, J. S., A second Wronskian formulation of the Boussinesq equation, Nonl. Anal., 70(12) (2008), pp. 42454258.CrossRefGoogle Scholar
[11] Ma, W. X. and Fan, E., Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl., 61 (2011), pp. 950959.CrossRefGoogle Scholar
[12] Ma, W. X., Huang, T. and Zhang, Y., A multiple exp-function method for nonlinear differential equations and its applications, Phys. Scr., 82 (2010), 065003.CrossRefGoogle Scholar
[13] Ma, W. X., A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation, Chaos. Solit. Fract., 42 (2009), pp. 13561363.CrossRefGoogle Scholar
[14] Maleknejad, K., Mirzaee, F. and Abbasbandy, S., Solving linear integro-differential equations system by using rationalized Haar functions method, Appl. Math. Comput., 155 (2004), pp. 17328.Google Scholar
[15] Mohyud-Din, S. T., Noor, M. A., Noor, K. I. and Waheed, A., Modified variation of parameters method for solving nonlinear boundary value problems, Int. J. Mod. Phys. B., (2011), in press.Google Scholar
[16] Mohyud-Din, S. T., Noor, M. A. and Noor, K. I., Modified variation of parameters method for second-order integro-differential equations and coupled systems, World. Appl. Sci. J., 6(8) (2009), pp. 11391146.Google Scholar
[17] Mohyud-Din, S. T., Noor, M. A. and Waheed, A., Variation of parameters method for solving sixth-order boundary value problems, Commun. Korean. Math. Soc., 24(4) (2009), pp. 605615.CrossRefGoogle Scholar
[18] Nadjafi, J. S. and Tamamgar, M., The variational iteration method, a highly promising method for system of integro-differential equations, Comput. Math. Appl., 56 (2008), pp. 346351.CrossRefGoogle Scholar
[19] Noor, M. A., Iterative methods for nonlinear equations using homotopy perturbation method, Appl. Math. Inf. Sci., 4(2) (2010), pp. 22272235.Google Scholar
[20] Noor, M. A., Variational inequalities in physical oceanography, in: Ocean Wave Engineering, (edit. Rahman, M.), Computational Mechanics Publications, Southampton, UK, 1994, pp. 201226.Google Scholar
[21] Noor, M. A., Some developments in general variaitonal inequalities, Appl. Math. Comput., 152 (2004), pp. 199277.Google Scholar
[22] Noor, M. A., Extended general variational inequalities, Appl. Math. Lett., 22 (2009), pp. 182185.CrossRefGoogle Scholar
[23] Noor, M. A., Noor, K. I. and TH. Rassias, M., Some aspects of variational inequalities, J. Comput. Appl. Math., 47 (1993), pp. 285312.CrossRefGoogle Scholar
[24] Noor, M. A., Mohyud-Din, S. T. AND Waheed, A., Variation of parameters method for solving fifth-order boundary value problems, Appl. Math. Inf. Sci., 2 (2008), pp. 135141.Google Scholar
[25] Noor, M. A., Noor, K. I., Waheed, A. and Al-Said, E., Modified variation of parameters method for solving a system of second-order nonlinear boundary value problems, Int. J. Phys. Sci., 5(16) (2010), pp. 24262431.Google Scholar
[26] Noor, M. A., Noor, K. I., Waheed, A. and Al, E.-SAID, On computation methods for solving systems of fourth-order nonlinear boundary value problems, Int. J. Phys. Sci,. 6(1) (2011), pp. 128135.Google Scholar
[27] Noor, M. A., Noor, K. I., Waheed, A. and Al, E.-SAID, Variation of parameters method for solving a class of eight-order boundary value problems, Int. J. Comput. Methods., 8 (2011), in press.Google Scholar
[28] Ramos, J. I., On the variational iteration method and other iterative techniques for nonlinear differential equations, Appl. Math. Comput., 199 (2008), pp. 3969.Google Scholar
[29] Sun, F. Z., Gao, M. and Lei, S. H., The fractal dimension of fractal model of dropwise condensation and its experimental study, Int. J. Nonl. Sci. Num. Sim., 8(2) (2007), pp. 211222.Google Scholar
[30] Wang, H., Fu, H. M. and Zhang, H. F., A practical thermodynamic method to calculate the best glass-forming composition for bulk metallic glasses, Int. J. Nonl. Sci. Num. Sim., 8(2) (2007), pp. 171178.Google Scholar
[31] Wang, S. Q. and He, J. H., Variational iteration method for solving integro-differential equations, Phys. Lett. A., 367 (2007), pp. 188191.CrossRefGoogle Scholar
[32] Wazwaz, A. M., A reliable algorithm for solving boundary value problems for higher-order integro-differential equations, Appl. Math. Comput., 118 (2001), pp. 327342.Google Scholar
[33] Xu, L., He, J. H. and Liu, Y., Electrospun nanoporous spheres with Chinese drug, Int. J. Nonl. Sci. Num. Sim., 8(2) (2007), pp. 191202.Google Scholar
[34] Yusufoglu, E., An efficient algorithm for solving integro-differential equations system, Appl. Math. Comput., 192(2) (2007), pp. 5155.Google Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Variation of Parameters Method for Solving System of Nonlinear Volterra Integro-Differential Equations
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Variation of Parameters Method for Solving System of Nonlinear Volterra Integro-Differential Equations
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Variation of Parameters Method for Solving System of Nonlinear Volterra Integro-Differential Equations
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *