Application of the optimal homotopy asymptotic method for the solution of the Korteweg-de Vries equation. (English) Zbl 1255.35194
Summary: The Optimal Homotopy Asymptotic Method (OHAM), a semi-analytic approximate technique for the treatment of time-dependent partial differential equations, has been used in this presentation. To see the effectiveness of the method, we consider Korteweg-de Vries (KdV) equation with different initial conditions. It provides us with a convenient way to control the convergence of approximate solutions. The obtained solutions show that the OHAM is more effective, simpler and easier than other methods. The results reveal that the method is explicit.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35A35 | Theoretical approximation in context of PDEs |
Keywords:
asymptotic method; initial value and time-dependent partial differential equations; perturbationReferences:
| [1] | Nayfeh, A. H., Introduction to Perturbation Techniques (1981), John Wiley: John Wiley New York · Zbl 0449.34001 |
| [2] | Nayfeh, A. H., Perturbation Methods (2000), John Wiley & Sons: John Wiley & Sons New York · Zbl 0375.35005 |
| [3] | Marinca, V.; Herisanu, N.; Nemes, I., Optimal homotopy asymptotic method with application to thin film flow, Central European Journal of Physics, 6, 648-653 (2008) |
| [4] | Marinca, V.; Herisanu, N., Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, International Communication in Heat and Mass Transfer, 08, 35, 710-715 (2008) |
| [5] | Marinca, V.; Herisanu, N., An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Applied Mathematics Letters, 22, 245-251 (2009) · Zbl 1163.76318 |
| [6] | Herişanu, N.; Marinca, V.; Dordea, T.; Madescu, G., A new analytical approach to nonlinear vibration of an electric machine, Proceedings of the Romanian Academy. Series A, 9, 3, 229-236 (2008) |
| [7] | V. Marinca, N. Herisanu, Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method, Journal of Sound and Vibration, doi:10.1016/j.jsv.2009.11.005; V. Marinca, N. Herisanu, Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method, Journal of Sound and Vibration, doi:10.1016/j.jsv.2009.11.005 · Zbl 1202.34072 |
| [8] | Marinca, V.; Herisanu, N., Optimal homotopy perturbation method for strongly nonlinear differential equations, Nonlinear Science Letters A, 1, 3, 273-280 (2010) · Zbl 1222.65089 |
| [9] | Islam, S.; Ali Shah, Rehan; Ali, Ishtiaq, Optimal homotopy asymptotic solutions of Couette and Poiseuille flows of a third grade fluid with heat transfer analysis, International Journal of Nonlinear Sciences and Numerical Simulations, 11, 6, 389-400 (2010) |
| [10] | Ali Shah, Rehan; Islam, S.; Tariq Hussain, Gulzaman, M., Solution of stagnation point flow with heat transfer by optimal homotopy asymptotic method, Proceedings of the Romanian Academy of Science Series, 11, 4, 312-321 (2010) · Zbl 1324.76076 |
| [11] | Idrees, M.; Islam, S.; Haq, Sirajul; Islam, Sirajul, Application of optimal homotopy asymptotic method to squeezing flow, Computers and Mathematics with Applications, 59, 11, 3858-3866 (2010) · Zbl 1198.76095 |
| [12] | Ali, Javed; Islam, S.; Islam, Sirajul; Zaman, Gul, The solution of multipoint boundary value problems by the optimal homotopy asymptotic method, Computers & Mathematics with Applications, 59, 2000-2006 (2010) · Zbl 1189.65154 |
| [13] | Iqbal, S.; Idrees, M.; Siddiqui, A. M.; Ansari, A. R., Some solutions of the linear and non-linear Klien-Gorden equations using the optimal homotopy asymptotic method, Applied Mathematics and Computation, 216, 2898-2909 (2010) · Zbl 1193.35190 |
| [14] | Herisanu, N.; Marinca, V., Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method, Computers and Mathematics with Applications, 60, 1607-1615 (2010) · Zbl 1202.34072 |
| [15] | Herisanu, N.; Marinca, V., Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia, Meccanica, 45, 847-855 (2010) · Zbl 1258.74102 |
| [16] | Idrees, M.; Islam, S.; Haq, Sirajul, Application of optimal homotopy asymptotic method to special sixth order boundary value problems, World Applied Science Journal, 9, 2, 138-143 (2010) · Zbl 1198.76095 |
| [17] | Idrees, M.; Islam, S.; Haq, Sirajul; Zaman, G., Application of optimal homotopy asymptotic method to fourth order boundary value problems, World Applied Science Journal, 9, 2, 131-137 (2010) |
| [18] | Liao, S. J., An optimal homotopy-analysis approach for strongly nonlinear differential equations, Communication in Nonlinear Science and Numerical Simulation (2009), doi:10.1016/j.cnsns.2009.09.002 (Online) · Zbl 1222.65088 |
| [19] | He, J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-linear Mechanics, 35, 1, 37-43 (2000) · Zbl 1068.74618 |
| [20] | He, J. H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 3-4, 257-262 (1999) · Zbl 0956.70017 |
| [21] | He, J.-H., Homotopy perturbation method: a new nonlinear technique, Applied Mathematics and Computation, 135, 73-79 (2003) · Zbl 1030.34013 |
| [22] | He, J. H., Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20, 10, 1141-1199 (2006) · Zbl 1102.34039 |
| [23] | Siddiqui, A. M.; Mahmood, R.; Ghori, Q. K., Thin film flow of a third grade fluid on a moving belt by He’s homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 7-14 (2006) · Zbl 1187.76622 |
| [24] | Siddiqui, A. M.; Ahmed, M.; Ghori, Q. K., Couette and Poiseuille flows for non-Newtonian fluids, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 15-26 (2006) · Zbl 1401.76018 |
| [25] | Islam, S.; Ali, I.; Shah, A.; Ran, X. J.; Siddiqui, A. M., Effect of couple stresses on flow of third grade fluids between parallel plates using homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 1, 97-110 (2009) |
| [26] | Kaya, Dogan; Aassila, Mohammed, Application for a generalized KdV equation by the decomposition method, Physics Letters A, 299, 201-206 (2002) · Zbl 0996.35061 |
| [27] | Drazin, P. G.; Johnson, R. S., Solutions: An Introduction (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0661.35001 |
| [28] | Saucez, P.; Wouwer, A. V.; Schiesser, W. E., An adaptive method of lines solution of the Korteweg-de Vries equation, Computers & Mathematics with Applications, 35, 12, 13-25 (1998) · Zbl 0999.65098 |
| [29] | T.A. Abassy, Magdy A. El-Tawil, H. El-Zoheiry, Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms andthe Pad’e technique, Computers and Mathematics with Applications, doi:10.1016/j.camwa.2006.12.067; T.A. Abassy, Magdy A. El-Tawil, H. El-Zoheiry, Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms andthe Pad’e technique, Computers and Mathematics with Applications, doi:10.1016/j.camwa.2006.12.067 · Zbl 1141.65382 |
| [30] | F. Kangalgil, F. Ayaz, Solitary wave solutions for the KdV and KdV equations by differential transform method, Chaos, Solitons and Fractals, doi:10.1016/j.chaos.2008.02.009; F. Kangalgil, F. Ayaz, Solitary wave solutions for the KdV and KdV equations by differential transform method, Chaos, Solitons and Fractals, doi:10.1016/j.chaos.2008.02.009 · Zbl 1198.35222 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
