Numerical method for the wave and nonlinear diffusion equations with the homotopy perturbation method. (English) Zbl 1186.65138
Summary: The homotopy perturbation method and a modified homotopy perturbation method are used for analytical treatment of the wave equation and some nonlinear diffusion equations, respectively. Some examples are given to illustrate that a suitable choice of an initial solution can lead to the exact solution, this revealing the reliability and effectiveness of the method.
MSC:
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
Keywords:
homotopy perturbation method; He’s polynomials; wave equation; d’alembert formula; nonlinear diffusion equations; approximate analytical solution; exact solutionsReferences:
| [1] | He, J. H., Homotopy perturbation technique, Comput. Methods. Appl. Mech. Engrg., 178, 257-262 (1999) · Zbl 0956.70017 |
| [2] | He, J. H., A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Internat. J. Non-Linear Mech., 35, 1, 37-43 (2000) · Zbl 1068.74618 |
| [3] | He, J. H., Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput., 135, 1, 73-79 (2003) · Zbl 1030.34013 |
| [4] | He, J. H., New interpretation of homotopy perturbation method, Internat. J. Modern Phys. B, 20, 2561-2568 (2006) |
| [5] | J.H. He, Recent development of the homotopy perturbation method, The full text is available at: http://works.bepress.com/ji_huan_he/37; J.H. He, Recent development of the homotopy perturbation method, The full text is available at: http://works.bepress.com/ji_huan_he/37 |
| [6] | He, J. H., Some asymptotic methods for strongly nonlinear equation, Internat. J. Modern Phys. B, 20, 10, 1144-1199 (2006) · Zbl 1102.34039 |
| [7] | J.H. He, An elementary introduction to recently developed asymptotic methods and nanomechanics intextile engineering, The full paper can be downloaded from: http://www.worldscinet.com/cgi-bin/details.cgi?id=jsname:ijmpb&type=current; J.H. He, An elementary introduction to recently developed asymptotic methods and nanomechanics intextile engineering, The full paper can be downloaded from: http://www.worldscinet.com/cgi-bin/details.cgi?id=jsname:ijmpb&type=current |
| [8] | He, J. H., The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput., 151, 1, 287-292 (2004) · Zbl 1039.65052 |
| [9] | He, J. H., Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals, 26, 3, 695-700 (2005) · Zbl 1072.35502 |
| [10] | He, J. H., Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul., 6, 2, 207-208 (2005) · Zbl 1401.65085 |
| [11] | He, J. H., Periodic solutions and bifurcations of delay-differential equations, Phys. Lett. A, 347, 4-6, 228-230 (2005) · Zbl 1195.34116 |
| [12] | He, J. H., Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals, 26, 3, 827-833 (2005) · Zbl 1093.34520 |
| [13] | He, J. H., Determination of limit cycles for strongly nonlinear oscillators, Phys. Rev. Lett., 90, 17 (2003), Art. No. 174301 |
| [14] | He, J. H., Homotopy perturbation method for solving boundary value problems, Phys. Lett. A, 350, 1-2, 87-88 (2006) · Zbl 1195.65207 |
| [15] | He, J. H., Asymptotology by homotopy perturbation method, Appl. Math. Comput., 156, 3, 591-596 (2004) · Zbl 1061.65040 |
| [16] | El-Shahed, M., Application of He’s homotopy perturbation method to Volterra’s integro-differential equation, Int. J. Nonlinear Sci. Numer. Simul., 6, 2, 163-168 (2005) · Zbl 1401.65150 |
| [17] | 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, Int. J. Nonlinear Sci. Numer. Simul., 7, 1, 7-14 (2006) · Zbl 1187.76622 |
| [18] | Siddiqui, A. M.; Mahmood, R.; Ghori, Q. K., Couette and Poiseuille flows for non-Newtonian fluids, Int. J. Nonlinear Sci. Numer. Simul., 7, 1, 15-26 (2006) · Zbl 1187.76622 |
| [19] | Cveticanin, L., Homotopy perturbation method for pure nonlinear differential equation, Chaos Solitons Fractals, 30, 5, 1221-1230 (2006) · Zbl 1142.65418 |
| [20] | Cai, X. C.; Wu, W. Y.; Li, M. S., Approximate period solution for a kind of nonlinear oscillator by He’s perturbation method, Int. J. Nonlinear Sci. Numer. Simul., 7, 1, 109-112 (2006) |
| [21] | Boyce, W.; DiPrima, R., Elementary Differential Equations (1991), Wiley: Wiley New York · Zbl 0178.09001 |
| [22] | Wazwaz, A., A reliable technique for solving the wave equation in an infinite one-dimensional medium, Appl. Math. Comput., 92, 1-7 (1998) · Zbl 0942.65107 |
| [23] | Abbasbandy, S., Numerical method for non-linear wave and diffusion equations by the variational iteration method, Int. J. Numer. Mech. Engrg., 73, 12, 1836-1843 (2008) · Zbl 1159.76372 |
| [24] | Ghorbani, A.; Nadjfi, J. S., He’s homotopy perturbation method for calculating Adomian polynomialsl, Int. J. Nonlinear Sci. Numer. Simul., 8, 2, 229-332 (2007) · Zbl 1401.65056 |
| [25] | Ghorbani, A., Beyond Adomian polynomials: He polynomial, Chaos Solitons Fractals (2007) |
| [26] | Noor, M. A.; Mohyud-Din, S. T., Variational iteration method for solving higher-order nonlinear boundary value problems using He’s polynomials, Int. J. Nonlinear Sci. Numer. Simul., 9, 2, 141-156 (2008) · Zbl 1145.65050 |
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.
