Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T00:39:19.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Application of Homotopy Analysis Method for Solving Systems of Volterra Integral Equations

Published online by Cambridge University Press:  03 June 2015

M. Matinfar ,
M. Saeidy  and
J. Vahidi
M. Matinfar*
Affiliation:
Department of Mathematics and Computer Science, University of Mazandaran, Babolsar, Iran
M. Saeidy*
Affiliation:
Department of Mathematics and Computer Science, University of Mazandaran, Babolsar, Iran
J. Vahidi*
Affiliation:
Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran
*
Corresponding author. Email: [email protected]
Email: [email protected]
Email: [email protected]
Get access

Abstract

In this paper, we prove the convergence of homotopy analysis method (HAM). We also apply the homotopy analysis method to obtain approximate analytical solutions of systems of the second kind Volterra integral equations. The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method.

Keywords

54A4026E50Homotopy analysis methodhomotopy perturbation methodsystems of Volterra integral equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 1 , February 2012 , pp. 36 - 45
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. A. and Ghazvini, H., He’s homotopy perturbation method for solving systems of Volterra integral equations of the second kind, Chaos. Soliton. Fract., 39(2) (2009), pp. 770–777.CrossRefGoogle Scholar
[2] Chena, X. M. and Wang, L. J., The variational iteration method for solving a neutral functional-differential equation with proportional delays, Comput. Math. Appl., 59 (2010), pp. 2696–2702.CrossRefGoogle Scholar
[3] He, J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear. Mech., 35(1) (2000), pp. 37–43.CrossRefGoogle Scholar
[4] He, J. H., Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B., 20(10) (2006), pp. 1141–1199.CrossRefGoogle Scholar
[5] Jafari, H., Saeidy, M. and Vahidi, J., The homotopy analysis method for solving fuzzy system of linear equations, Int. J. Fuzzy. Syst., 11(4) (2009), pp. 308–313.Google Scholar
[6] Jafari, H., Saeidy, M. and Firoozjaee, M. A., The Homotopy Analysis Method for Solving Higher Dimensional Initial Boundary Value Problems of Variable Coefficients, 2009, doi 10.1002/num.20471.CrossRefGoogle Scholar
[7] Jafari, H., Zabihi, M. and Saidy, M., Application of homotopy perturbation method for solving gas dynamic equation, Appl. Math. Sci., 2 (2008), pp. 2393–2396.Google Scholar
[8] Khan, Y. and Faraz, N., Application of modified Laplace decomposition method for solving boundary layer equation, Journal of King Saud University (Science), 23 (2011), pp. 115–119.CrossRefGoogle Scholar
[9] Liao, S. J., The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.Google Scholar
[10] Liao, S. J., An approximate solution technique which does not depend upon small parameters (part 2): an application in fluid mechanics, Int. J. Nonlinear. Mech., 32(5) (1997), pp. 815–822.CrossRefGoogle Scholar
[11] Liao, S. J., Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, Boca Raton: Chapman & Hall, 2003.CrossRefGoogle Scholar
[12] Li, J. L., Adomian’s decomposition method and homotopy perturbation method in solving nonlinear equations, J. Comput. Appl. Math., 228 (2009), pp. 168–173.CrossRefGoogle Scholar
[13] Matinfar, M. and Saeidy, M., The homotopy perturbation method for solving higher dimensional initial boundary value problems of variable coefficients, World. J. Model. Sim., 4 (2009), pp. 72–80.Google Scholar
[14] Ma, W. X. and Fan, E. G., Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl., 61 (2011), pp. 950–959.CrossRefGoogle Scholar
[15] Ma, W. X., Gu, X. and Gao, L., A note on exact solutions to linear differential equations by the matrix exponential, Adv. Appl. Math. Mech., 1 (2009), pp. 573–580.CrossRefGoogle Scholar
[16] Ma, W. X., Huang, T. W. and Zhang, Y., A multiple exp-function method for nonlinear differential equations and its application, Phys. Scr., 82 (2010), 065003.CrossRefGoogle Scholar
[17] Noor, M. A., KH. Noor, I. and S. T. MOHYUD-DIN, Variational iteration method for solving sixth-order boundary value problems, Commun. Nonlinear. Sci. Numer. Sim., 14 (2009), pp. 2571–2580.CrossRefGoogle Scholar
[18] Noor, M. A. and Mohyud-Din, S. T., Variational Iteration Decomposition Method for Solving Eighth-Order Boundary Value Problems, Hindawi Publishing Corporation-Differential Equations and Nonlinear Mechanics, 19529, 16 pages.Google Scholar
[19] Noor, M. A. and Mohyud-Din, S. T., Homotopy perturbation method for solving sixth-order boundary value problems, Comput. Math. Appl., 55 (2008), pp. 2953–2972.CrossRefGoogle Scholar
[20] Noor, M. A., KH. Noor, I. and Mohyud-Din, S. T., Variational iteration method for solving sixth-order boundary value problems, Commun. Nonlinear. Sci. Numer. Sim., 14 (2009), pp. 2571–2580.CrossRefGoogle Scholar
[21] Noor, M. A. and Mohyud-Din, S. T., A reliable approach for higher-order integro-differential equations, Appl. Appl. Math., 3(2) (2008), pp. 188–199.Google Scholar
[22] Shen, Sh., Application of homotopy perturbation method to the fifth-order boundary value problems, Int. J. Contemp. Math. Sci., 2 (2007), pp. 1227–1236.Google Scholar