##
**Couple of the variational iteration method and Legendre wavelets for nonlinear partial differential equations.**
*(English)*
Zbl 1266.65218

Summary: This paper develops a modified variational iteration method coupled with the Legendre wavelets, which can be used for the efficient numerical solution of nonlinear partial differential equations (PDEs). The approximate solutions of PDEs are calculated in the form of a series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The main advantage of the new method is that it can avoid solving the nonlinear algebraic system and symbolic computation. Furthermore, the developed vector-matrix form makes it computationally efficient. The results show that the proposed method is very effective and easy to implement.

### MSC:

65T60 | Numerical methods for wavelets |

PDF
BibTeX
XML
Cite

\textit{F. Yin} et al., J. Appl. Math. 2013, Article ID 157956, 11 p. (2013; Zbl 1266.65218)

Full Text:
DOI

### References:

[1] | G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. · Zbl 0843.34026 |

[2] | G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501-544, 1988. · Zbl 0671.34053 |

[3] | G. Adomian, “Solutions of nonlinear P. D. E,” Applied Mathematics Letters, vol. 11, no. 3, pp. 121-123, 1998. · Zbl 0933.65121 |

[4] | Q. Esmaili, A. Ramiar, E. Alizadeh, and D. D. Ganji, “An approximation of the analytical solution of the Jeffery-Hamel flow by decomposition method,” Physics Letters A, vol. 372, no. 19, pp. 3434-3439, 2008. · Zbl 1220.76035 |

[5] | A.-M. Wazwaz, “A new algorithm for calculating Adomian polynomials for nonlinear operators,” Applied Mathematics and Computation, vol. 111, no. 1, pp. 53-69, 2000. · Zbl 1023.65108 |

[6] | S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488-494, 2006. · Zbl 1096.65131 |

[7] | J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999. · Zbl 0956.70017 |

[8] | J.-H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37-43, 2000. · Zbl 1068.74618 |

[9] | J.-H. He, “The homotopy perturbation method nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287-292, 2004. · Zbl 1039.65052 |

[10] | S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving fourth-order boundary value problems,” Mathematical Problems in Engineering, vol. 2007, Article ID 98602, 15 pages, 2007. · Zbl 1144.65311 |

[11] | S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving partial differential equations,” Zeitschrift für Naturforschung A, vol. 64, no. 3-4, pp. 157-170, 2009. · Zbl 1185.35005 |

[12] | S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method and Padé approximants for solving Flierl-Petviashivili equation,” Applications and Applied Mathematics, vol. 3, no. 2, pp. 224-234, 2008. · Zbl 1177.65112 |

[13] | S. J. Liao, “An approximate solution technique not depending on small parameters: a special example,” International Journal of Non-Linear Mechanics, vol. 30, no. 3, pp. 371-380, 1995. · Zbl 0837.76073 |

[14] | S. J. Liao, “Boundary element method for general nonlinear differential operators,” Engineering Analysis with Boundary Elements, vol. 20, no. 2, pp. 91-99, 1997. |

[15] | J. H. He, “Variational iteration method-a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699-708, 1999. · Zbl 1342.34005 |

[16] | J.-H. He, “Variational iteration method for autonomous ordinary differential systems,” Applied Mathematics and Computation, vol. 114, no. 2-3, pp. 115-123, 2000. · Zbl 1027.34009 |

[17] | J.-H. He, “Variational principles for some nonlinear partial differential equations with variable coefficients,” Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 847-851, 2004. · Zbl 1135.35303 |

[18] | J.-H. He and X.-H. Wu, “Construction of solitary solution and compacton-like solution by variational iteration method,” Chaos, Solitons & Fractals, vol. 29, no. 1, pp. 108-113, 2006. · Zbl 1147.35338 |

[19] | J.-H. He and X.-H. Wu, “Variational iteration method: new development and applications,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 881-894, 2007. · Zbl 1141.65372 |

[20] | J.-H. He, “Variational iteration method-some recent results and new interpretations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3-17, 2007. · Zbl 1119.65049 |

[21] | J. H. He, G. C. Wu, and F. Austin, “The variational iterational method which should be follow,” Nonlinear Science Letters A, vol. 1, no. 1, pp. 1-30, 2010. |

[22] | J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141-1199, 2006. · Zbl 1102.34039 |

[23] | J. H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012. · Zbl 1257.35158 |

[24] | G. Hariharan, K. Kannan, and K. R. Sharma, “Haar wavelet method for solving Fisher’s equation,” Applied Mathematics and Computation, vol. 211, no. 2, pp. 284-292, 2009. · Zbl 1162.65394 |

[25] | Ü. Lepik, “Numerical solution of evolution equations by the Haar wavelet method,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 695-704, 2007. · Zbl 1110.65097 |

[26] | S. G. Venkatesh, S. K. Ayyaswamy, and S. Raja Balachandar, “The Legendre wavelet method for solving initial value problems of Bratu-type,” Computers & Mathematics with Applications, vol. 63, no. 8, pp. 1287-1295, 2012. · Zbl 1247.65180 |

[27] | S. A. Yousefi, “Legendre wavelets method for solving differential equations of Lane-Emden type,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1417-1422, 2006. · Zbl 1105.65080 |

[28] | R. K. Pandey, N. Kumar, A. Bhardwaj, and G. Dutta, “Solution of Lane-Emden type equations using Legendre operational matrix of differentiation,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7629-7637, 2012. · Zbl 1246.65115 |

[29] | F. Yin, J. Song, F. Lu, and H. Leng, “A coupled method of Laplace transform and legendre wavelets for Lane-Emden-type differential equations,” Journal of Applied Mathematics, vol. 2012, Article ID 163821, 16 pages, 2012. · Zbl 1264.65227 |

[30] | L. M. B. Assas, “Variational iteration method for solving coupled-KdV equations,” Chaos, Solitons and Fractals, vol. 38, no. 4, pp. 1225-1228, 2008. · Zbl 1152.35466 |

[31] | Z. M. Odibat, “Construction of solitary solutions for nonlinear dispersive equations by variational iteration method,” Physics Letters A, vol. 372, no. 22, pp. 4045-4052, 2008. · Zbl 1220.35143 |

[32] | E. M. Abulwafa, M. A. Abdou, and A. A. Mahmoud, “The solution of nonlinear coagulation problem with mass loss,” Chaos, Solitons & Fractals, vol. 29, no. 2, pp. 313-330, 2006. · Zbl 1101.82018 |

[33] | E. M. Abulwafa, M. A. Abdou, and A. A. Mahmoud, “Nonlinear fluid flows in pipe-like domain problem using variational-iteration method,” Chaos, Solitons & Fractals, vol. 32, no. 4, pp. 1384-1397, 2007. · Zbl 1128.76019 |

[34] | N. Bildik and A. Konuralp, “The use of variational iteration method, differential transform method and adomian decomposition method for solving different types of nonlinear partial differential equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 65-70, 2006. · Zbl 1115.65365 |

[35] | S. Momani and S. Abuasad, “Application of He’s variational iteration method to Helmholtz equation,” Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1119-1123, 2006. · Zbl 1086.65113 |

[36] | S. Momani and Z. Odibat, “Numerical comparison of methods for solving linear differential equations of fractional order,” Chaos, Solitons & Fractals, vol. 31, no. 5, pp. 1248-1255, 2007. · Zbl 1137.65450 |

[37] | S. T. Mohyud-Din, M. A. Noor, K. I. Noor, and M. M. Hosseini, “Solution of singular equation by He’s variational iteration method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 2, pp. 81-86, 2010. · Zbl 1401.65089 |

[38] | N. H. Sweilam and M. M. Khader, “Variational iteration method for one dimensional nonlinear thermoelasticity,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 145-149, 2007. · Zbl 1131.74018 |

[39] | D. D. Ganji, H. Tari, and M. Bakhshi Jooybari, “Variational iteration method and homotopy perturbation method for nonlinear evolution equations,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1018-1027, 2007. · Zbl 1141.65384 |

[40] | Z. Odibat and S. Momani, “Numerical methods for nonlinear partial differential equations of fractional order,” Applied Mathematical Modelling, vol. 32, no. 1, pp. 28-39, 2008. · Zbl 1133.65116 |

[41] | E. Hizel and S. K. Kü, “A numerical analysis of the Burgers-Poisson (BP) equation using variational iteration method,” in Proceedings of the 3rd WSEAS International Conference on Applied and Theoretical Mechanics, Tenerife, Spain, December 2007. |

[42] | S. T. Mohyud-Din and M. A. Noor, “Modified variational iteration method for solving Fisher’s equations,” Journal of Applied Mathematics and Computing, vol. 31, no. 1-2, pp. 295-308, 2009. · Zbl 1177.65158 |

[43] | M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for solving higher-order nonlinear boundary value problems using He’s polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 141-156, 2008. · Zbl 1151.65334 |

[44] | M. A. Noor and S. T. Mohyud-Din, “Modified variational iteration method for heat and wave-like equations,” Acta Applicandae Mathematicae, vol. 104, no. 3, pp. 257-269, 2008. · Zbl 1162.65397 |

[45] | M. A. Noor and S. T. Mohyud-Din, “Modified variational iteration method for solving Helmholtz equations,” Computational Mathematics and Modeling, vol. 20, no. 1, pp. 40-50, 2009. · Zbl 1177.65169 |

[46] | M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for fifth-order boundary value problems using He’s polynomials,” Mathematical Problems in Engineering, vol. 2008, Article ID 954794, 12 pages, 2008. · Zbl 1151.65334 |

[47] | M. A. Noor and S. T. Mohyud-Din, “Modified variational iteration method for solving fourth-order boundary value problems,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 81-94, 2009. · Zbl 1176.65083 |

[48] | M. A. Noor and S. T. Mohyud-Din, “Modified variational iteration method for Goursat and Laplace problems,” World Applied Sciences Journal, vol. 4, no. 4, pp. 487-498, 2008. |

[49] | S. Abbasbandy, “A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 59-63, 2007. · Zbl 1120.65083 |

[50] | S. Abbasbandy, “Numerical solution of non-linear Klein-Gordon equations by variational iteration method,” International Journal for Numerical Methods in Engineering, vol. 70, no. 7, pp. 876-881, 2007. · Zbl 1194.65120 |

[51] | S. T. Mohyud-Din and M. A. Noor, “Solving Schrödinger equations by modified variational iteration method,” World Applied Sciences Journal, vol. 5, no. 3, pp. 352-357, 2008. |

[52] | S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Modified variational iteration method for solving Sine Gordon equations,” World Applied Sciences Journal, vol. 5, no. 3, pp. 352-357, 2008. · Zbl 1162.65397 |

[53] | M. A. Noor and S. T. Mohyud-Din, “Solution of singular and nonsingular initial and boundary value problems by modified variational iteration method,” Mathematical Problems in Engineering, vol. 2008, Article ID 917407, 23 pages, 2008. · Zbl 1155.65083 |

[54] | M. A. Noor and S. T. Mohyud-Din, “Variational iteration decomposition method for solving eighth-order boundary value problems,” Differential Equations and Nonlinear Mechanics, vol. 2007, Article ID 19529, 16 pages, 2007. · Zbl 1143.49023 |

[55] | M. M. Hosseini, “Adomian decomposition method with Chebyshev polynomials,” Applied Mathematics and Computation, vol. 175, no. 2, pp. 1685-1693, 2006. · Zbl 1093.65073 |

[56] | W. C. Tien and C. K. Chen, “Adomian decomposition method by Legendre polynomials,” Chaos, Solitons and Fractals, vol. 39, no. 5, pp. 2093-2101, 2009. |

[57] | Z. Odibat, “On Legendre polynomial approximation with the VIM or HAM for numerical treatment of nonlinear fractional differential equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 9, pp. 2956-2968, 2011. · Zbl 1210.65132 |

[58] | M. Razzaghi and S. Yousefi, “The Legendre wavelets operational matrix of integration,” International Journal of Systems Science, vol. 32, no. 4, pp. 495-502, 2001. · Zbl 1006.65151 |

[59] | F. Mohammadi and M. M. Hosseini, “A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations,” Journal of the Franklin Institute, vol. 348, no. 8, pp. 1787-1796, 2011. · Zbl 1237.65079 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.