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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 ,
Khalida Inayat Noor ,
Asif Waheed  and
Eisa Al-Said
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
*
Corresponding author. Email: [email protected]
Email: [email protected]
Email: [email protected]
Email: [email protected]
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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.

Keywords

65N9965R2065K15Variation of parameters methodoceanographysystem of nonlinear Volterra integro-differential equationserror estimates
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 2 , April 2012 , pp. 190 - 204
Copyright
Copyright © Global-Science Press 2012

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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. 1253–1258.Google 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. 223–228.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. 105–112.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. 163–168.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. 538–548.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. 745–751.Google Scholar
[7] He, J. H., Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering’ 98, Dalian, China, 1998, pp. 288–291.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. 1753–1778.Google Scholar
[9] Ma, W. X. and You, Y., Rational solutions of the Toda lattice equation in casoratian form, Chaos. Solit. Fract., 22 (2004), pp. 395–406.Google 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. 4245–4258.Google Scholar
[11] Ma, W. X. and Fan, E., Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl., 61 (2011), pp. 950–959.Google 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.Google 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. 1356–1363.Google 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. 17–328.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. 1139–1146.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. 605–615.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. 346–351.Google Scholar
[19] Noor, M. A., Iterative methods for nonlinear equations using homotopy perturbation method, Appl. Math. Inf. Sci., 4(2) (2010), pp. 2227–2235.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. 201–226.Google Scholar
[21] Noor, M. A., Some developments in general variaitonal inequalities, Appl. Math. Comput., 152 (2004), pp. 199–277.Google Scholar
[22] Noor, M. A., Extended general variational inequalities, Appl. Math. Lett., 22 (2009), pp. 182–185.Google Scholar
[23] Noor, M. A., Noor, K. I. and TH. Rassias, M., Some aspects of variational inequalities, J. Comput. Appl. Math., 47 (1993), pp. 285–312.Google 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. 135–141.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. 2426–2431.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. 128–135.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. 39–69.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. 211–222.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. 171–178.Google Scholar
[31] Wang, S. Q. and He, J. H., Variational iteration method for solving integro-differential equations, Phys. Lett. A., 367 (2007), pp. 188–191.Google 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. 327–342.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. 191–202.Google Scholar
[34] Yusufoglu, E., An efficient algorithm for solving integro-differential equations system, Appl. Math. Comput., 192(2) (2007), pp. 51–55.Google Scholar