10.1186/2251-7456-6-32

Effective approximate methods for strongly nonlinear differential equations with oscillations

HTML PDF
  • Marwan Alquran*1,
  • Kamel Al-Khaled1,
  1. Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, 22110, JO
Cover Image

Published in Issue 2012-09-05

How to Cite

Alquran, M., & Al-Khaled, K. (2012). Effective approximate methods for strongly nonlinear differential equations with oscillations. Mathematical Sciences, 6(1). https://doi.org/10.1186/2251-7456-6-32
Crossref
Scopus
Google Scholar
Europe PMC

HTML views: 19

PDF views: 74

Abstract

Abstract Purpose This paper proposes the use of different analytical methods in obtaining approximate solutions for nonlinear differential equations with oscillations. Methods Three methods are considered in this paper: Lindstedt-Poincare method, the Krylov-Bogoliubov first approximate method, and the differential transform method. Results Figures that are given in this paper give a strong evidence that the proposed methods are effective in handling nonlinear differential equations with oscillations. Conclusions This study reveals that the differential transform method provides a remarkable precision compared with other perturbation methods.

Keywords

  • Lindstedt-Poincare method,
  • Krylov-Bogoliubov method,
  • Differential transform method,
  • Nonlinear oscillations
HTML PDF

References

  1. Logan (2006) Wiley
  2. He (2006) Some asymptotic methods for strongly nonlinear equations 20(10) (pp. 1141-1199) https://doi.org/10.1142/S0217979206033796
  3. Cveticanin (2006) Homotopy-perturbation method for pure nonlinear differential equation (pp. 1221-1230) https://doi.org/10.1016/j.chaos.2005.08.180
  4. He (2000) A new perturbation technique which is also valid for large parameters (pp. 1257-1263) https://doi.org/10.1006/jsvi.1999.2509
  5. Hu (2004) A classical perturbation technique which is valid for large parameters (pp. 409-412) https://doi.org/10.1016/S0022-460X(03)00318-3
  6. Al-Khaled and Ali Hajji (2008) On the existence of solutions for strongly nonlinear differential equations (pp. 1165-1175) https://doi.org/10.1016/j.jmaa.2008.03.022
  7. Guo and Leung (2010) The iterative homotopy harmonic balance method for conservative Helmholtz-Duffing oscillators. Appl (pp. 3163-3169)
  8. Chen and Liu (2007) A new method based on the harmonic balance method for nonlinear oscillators. Phys. Lett (pp. 371-378)
  9. Liu et al. (2007) A novel harmonic balance analysis for the Van Der Pol oscillator (pp. 2-12) https://doi.org/10.1016/j.ijnonlinmec.2006.09.004
  10. Gottlieb (2004) Harmonic balance approach to periodic solutions of nonlinear jerk equations (pp. 671-683) https://doi.org/10.1016/S0022-460X(03)00299-2
  11. Hu (2008) Perturbation method for periodic solutions of nonlinear jerk equations (pp. 4205-4209) https://doi.org/10.1016/j.physleta.2008.03.027
  12. Alquran and Al-Khaled (2010) Approximate solutions to nonlinear partial integro-differential equations with applications in heat flow 3(2) (pp. 93-116)
  13. Farzaneh and Tootoonchi (2010) Global Error Minimization method for solving strongly nonlinear oscillator differential equations (pp. 2887-2895) https://doi.org/10.1016/j.camwa.2010.02.006
  14. Alquran and Al-Khaled (2010) Approximations of Sturm-Liouville Eigenvalues using Sinc-Galerkin and differential transform methods 5(1) (pp. 128-147)
  15. Ayaz (2004) Solution of the system of differential equations by differential transform method (pp. 547-567) https://doi.org/10.1016/S0096-3003(02)00794-4
  16. He (2002) Modified Lindstedt-Poincare methods for strongly non-linear oscillations: part I: expansion of a constant (pp. 309-314) https://doi.org/10.1016/S0020-7462(00)00116-5
  17. He (2002) Modified Lindstedt-Poincare methods for strongly nonlinear oscillations: part II: a new transformation (pp. 315-320) https://doi.org/10.1016/S0020-7462(00)00117-7
  18. Cheung et al. (1991) A modified Lindstedt-Poincare method for certain strongly non-linear oscillators (pp. 367-378) https://doi.org/10.1016/0020-7462(91)90066-3
  19. Ross (1964) Blaisdel