×

zbMATH — the first resource for mathematics

Asymptotic methods for solitary solutions and compactons. (English) Zbl 1257.35158
Summary: This paper is an elementary introduction to some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations. Particular attention is paid throughout the paper to giving an intuitive grasp for the variational approach, the Hamiltonian approach, the variational iteration method, the homotopy perturbation method, the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform, and ancient Chinese mathematics. Hamilton principle and variational principles are also emphasized. The reviewed asymptotic methods are easy to be followed for various applications. Some ideas on this paper are first appeared.

MSC:
35Q51 Soliton equations
35C08 Soliton solutions
35R11 Fractional partial differential equations
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. S. Russell, “Report on waves,” Tech. Rep., 1844, in Proceedings of the14th Meeting of the British Association for the Advancement of Science.
[2] N. J. Zabusky and M. D. Kruskal, “Interaction of “solitons” in a collisionless plasma and the recurrence of initial states,” Physical Review Letters, vol. 15, no. 6, pp. 240-243, 1965. · Zbl 1201.35174 · doi:10.1103/PhysRevLett.15.240 · staff.ustc.edu.cn
[3] D. J. Korteweg and G. de Vires, “On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary wave,” Philosophical Magazine, vol. 539, pp. 422-443, 1895. · JFM 26.0881.02
[4] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Physical Review Letters, vol. 19, no. 19, pp. 1095-1097, 1967. · Zbl 1103.35360 · doi:10.1103/PhysRevLett.19.1095
[5] J.-H. He, “Soliton perturbation,” in Encyclopedia of Complexity and Systems Science, R. Meyers, Ed., vol. 9, pp. 8453-8457, Springer, New York, NY, USA, 2009.
[6] J.-H. He and S. D. Zhu, “Solitons and compactons,” in Encyclopedia of Complexity and Systems Science, R. Meyers, Ed., vol. 9, pp. 8457-8464, Springer, New York, NY, USA, 2009.
[7] 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 · doi:10.1142/S0217979206033796
[8] J. H. He, “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering,” International Journal of Modern Physics B, vol. 22, no. 21, pp. 3487-3578, 2008. · Zbl 1149.76607 · doi:10.1142/S0217979208048668
[9] J.-H. He, “New interpretation of homotopy perturbation method. Addendum: some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 18, pp. 2561-2568, 2006. · doi:10.1142/S0217979206034819
[10] P. Rosenau and J. M. Hyman, “Compactons: solitons with finite wavelength,” Physical Review Letters, vol. 70, no. 5, pp. 564-567, 1993. · Zbl 0952.35502 · doi:10.1103/PhysRevLett.70.564
[11] S. Walia, R. Weber, K. Latham et al., “Oscillatory thermopower waves based on Bi2Te3 films,” Advanced Functional Materials, vol. 21, no. 11, pp. 2072-2079, 2011. · doi:10.1002/adfm.201001979
[12] J. S. Lin and L. M. Hildemann, “A nonsteady-state analytical model to predict gaseous emissions of volatile organic compounds from landfills,” Journal of Hazardous Materials, vol. 40, no. 3, pp. 271-295, 1995. · doi:10.1016/0304-3894(94)00088-X
[13] W. Okrasiński and Ł. Płociniczak, “A nonlinear mathematical model of the corneal shape,” Nonlinear Analysis: Real World Applications, vol. 13, no. 3, pp. 1498-1505, 2012. · Zbl 1239.34004 · doi:10.1016/j.nonrwa.2011.11.014
[14] J.-H. He, “A remark on a nonlinear mathematical model of the corneal shape,” Nonlinear Analysis: Real World Applications, vol. 13, no. 6, pp. 2863-2865, 2012. · Zbl 1257.34010 · doi:10.1016/j.nonrwa.2012.04.014
[15] R. L. Herman, “Exploring the connection between quasistationary and squared eigenfunction expansion techniques in soliton perturbation theory,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5-7, pp. e2473-e2482, 2005. · Zbl 1224.35363 · doi:10.1016/j.na.2005.02.034
[16] J.-H. He, Non-perturbative methods for strongly nonlinear problems [Ph.D. thesis], de-Verlag im Internet GmbH, Berlin, Germany, 2006.
[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 · doi:10.1016/S0960-0779(03)00265-0
[18] J.-H. He, Generalized Variational Principles in Fluids, Science & Culture Publishing House of China, 2003. · Zbl 1054.76001
[19] M. J. Lighthill and G. B. Whitham, “On kinematic waves. II. A theory of traffic flow on long crowded roads,” Proceedings of the Royal Society London Series A, vol. 229, pp. 317-345, 1955. · Zbl 0064.20906 · doi:10.1098/rspa.1955.0089
[20] G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York, NY, USA, 1974. · Zbl 0373.76001
[21] W. Zheng, “A fluid dynamics model for the low speed traffic systems,” Acta Mechanica Sinica, vol. 26, no. 2, pp. 149-157, 1994 (Chinese).
[22] J.-H. He, “Variational approach to foam drainage equation,” Meccanica, vol. 46, no. 6, pp. 1265-1266, 2011. · Zbl 1271.76341
[23] J.-H. He, “Variational approach to impulsive differential equations using the semi-inverse method,” Zeitschrift für Naturforschung A, vol. 66, pp. 632-634, 2011.
[24] J.-H. He, “Variational approach for nonlinear oscillators,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1430-1439, 2007. · Zbl 1152.34327 · doi:10.1016/j.chaos.2006.10.026
[25] H. M. Liu and J.-H. He, “Variational approach to chemical reaction,” Computers and Chemical Engineering, vol. 28, no. 9, p. 1549, 2004. · doi:10.1016/j.compchemeng.2004.01.006
[26] T. Özi\cs and A. Yıldırım, “Application of He’s semi-inverse method to the nonlinear Schrödinger equation,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1039-1042, 2007. · Zbl 1157.65465 · doi:10.1016/j.camwa.2006.12.047
[27] J. Zhang, “Variational approach to solitary wave solution of the generalized Zakharov equation,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1043-1046, 2007. · Zbl 1141.65391 · doi:10.1016/j.camwa.2006.12.048
[28] W. Dittrich and M. Reuter, Classical and Quantum Dynamics, Advanced Texts in Physics, Springer, Berlin, Germany, 2nd edition, 1994. · Zbl 0793.70001 · doi:10.1007/978-3-642-56430-7
[29] A. Carini and F. Genna, “Some variational formulations for continuum nonlinear dynamics,” Journal of the Mechanics and Physics of Solids, vol. 46, no. 7, pp. 1253-1277, 1998. · Zbl 1030.74006 · doi:10.1016/S0022-5096(98)00016-7
[30] G.-L. Liu, “A vital innovation in Hamilton principle and its extension to initial-value problems,” in Proceedings of the 4th International Conference on Nonlinear Mechanics, pp. 90-97, Shanghai University Press, Shanghai, China, August 2002.
[31] J.-H. He, “Hamiltonian approach to nonlinear oscillators,” Physics Letters A, vol. 374, no. 23, pp. 2312-2314, 2010. · Zbl 1237.70036 · doi:10.1016/j.physleta.2010.03.064
[32] J. H. He, T. Zhong, and L. Tang, “Hamiltonian approach to duffing-harmonic equation,” The International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, supplement, pp. 43-46, 2010. · Zbl 06942545
[33] J. H. He, “Preliminary report on the energy balance for nonlinear oscillations,” Mechanics Research Communications, vol. 29, no. 2-3, pp. 107-111, 2002. · Zbl 1048.70011 · doi:10.1016/S0093-6413(02)00237-9
[34] G. A. Afrouzi, D. D. Ganji, and R. A. Talarposhti, “He’s energy balance method for nonlinear oscillators with discontinuities,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 3, pp. 301-304, 2009. · Zbl 06942402
[35] H. L. Zhang, Y. G. Xu, and J. R. Chang, “Application of he’s energy balance method to a nonlinear oscillator with discontinuity,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 207-214, 2009. · Zbl 06942391
[36] S. S. Ganji, D. D. Ganji, and S. Karimpour, “He’s Energy balance and He’s variational methods for nonlinear oscillations in engineering,” International Journal of Modern Physics B, vol. 23, no. 3, pp. 461-471, 2009. · Zbl 1165.74325 · doi:10.1142/S0217979209049644
[37] A. Yildirim, Z. Saadatnia, H. Askari, Y. Khan, and M. KalamiYazdi, “Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2042-2051, 2011. · Zbl 1272.70110 · doi:10.1016/j.aml.2011.05.040
[38] L. Xu, “A Hamiltonian approach for a plasma physics problem,” Computers and Mathematics with Applications, vol. 61, no. 8, pp. 1909-1911, 2011. · Zbl 1219.82034 · doi:10.1016/j.camwa.2010.06.028
[39] L. Xu and J. H. He, “Determination of limit cycle by Hamiltonian approach for strongly nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 12, pp. 1097-1101, 2010. · Zbl 06942719
[40] S. Durmaz, S. Altay Demirba\vg, and M. O. Kaya, “High order hamiltonian approach to nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 8, pp. 565-570, 2010. · Zbl 06942667
[41] I. Didenkulova, E. Pelinovsky, and A. Sergeeva, “Statistical characteristics of long waves nearshore,” Coastal Engineering, vol. 58, no. 1, pp. 94-102, 2011. · Zbl 1191.86006 · doi:10.1016/j.coastaleng.2010.08.005
[42] C. C. Lin, “A new variational principle for isenergetic flows,” Quarterly of Applied Mathematics, vol. 9, pp. 421-423, 1952. · Zbl 0046.18301
[43] K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press, Oxford, UK, 1982. · Zbl 0498.73014
[44] C. C. Lin, “Hydrodynamics of helium II,” in Proceedings of the International School of Physics, vol. 21, pp. 93-146, Academic Press, 1963.
[45] J. C. Tannehill, D. A. Anderson, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Taylor & Francis, 1997. · Zbl 0569.76001
[46] J. W. Herivel, “The derivation of the equations of motion of an ideal fluid by Hamilton’s principle,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 51, pp. 344-349, 1955. · Zbl 0068.18802
[47] F. P. Bretherton, “A note on Hamilton’s principle for perfect fluids,” The Journal of Fluid Mechanics, vol. 44, pp. 19-31, 1970. · Zbl 0198.58901 · doi:10.1017/S0022112070001660
[48] A. Ecer and H. U. Akay, “Investigation of transonic flow in a cascade using the finite element method,” Journal of American Institute of Aeronautics and Astronautics, vol. 19, no. 9, pp. 1174-1182, 1981. · Zbl 0479.76070 · doi:10.2514/3.60057
[49] A. Ecer and H. U. Akay, “A finite element formulation for steady transonic euler equations,” AIAA Journal, vol. 21, no. 3, pp. 343-350, 1983. · Zbl 0515.76064 · doi:10.2514/3.8078
[50] V. Goncharov and V. Pavlov, “On the Hamiltonian approach: applications to geophysical flows,” Nonlinear Processes in Geophysics, vol. 5, no. 4, pp. 219-240, 1998.
[51] D. D. Holm and V. Zeitlin, “Hamilton’s principle for quasigeostrophic motion,” Physics of Fluids, vol. 10, no. 4, pp. 800-806, 1998. · Zbl 1185.76848 · doi:10.1063/1.869623
[52] F. Hiroki and F. Youhei, “Clebsch potentials in the variational principle for a perfect fluid,” Progress of Theoretical Physics, vol. 124, no. 3, pp. 517-531, 2010. · Zbl 1377.76037 · doi:10.1143/PTP.124.517
[53] J. Larsson, “A practical form of Lagrange-Hamilton theory for ideal fluids and plasmas,” Journal of Plasma Physics, vol. 69, no. 3, pp. 211-252, 2003. · doi:10.1017/S0022377803002290
[54] R. Salmon, “Hamiltonian fluid mechanics,” Annual Review of Fluid Mechanics, vol. 20, pp. 225-256, 1988.
[55] R. L. Seliger and G. B. Whitham, “Variational principles in continuum mechanics,” Proceedings of the Royal Society A, vol. 305, pp. 1-25, 1968. · Zbl 0198.57601 · doi:10.1098/rspa.1968.0103
[56] H. J. Wagner, “On the use of Clebsch potentials in the Lagrangian formulation of classical electrodynamics,” Physics Letters A, vol. 292, no. 4-5, pp. 246-250, 2002. · Zbl 0995.78002 · doi:10.1016/S0375-9601(01)00795-2
[57] A. Clebsch, “Uber die integration der hydrodynamischen gleichungen,” Journal für die Reine und Angewandte Mathematik, vol. 56, no. 1, pp. 1-10, 1859. · ERAM 056.1468cj
[58] 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 · doi:10.1016/j.camwa.2006.12.083
[59] 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 · doi:10.1016/j.cam.2006.07.009
[60] 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 · doi:10.1016/j.chaos.2005.10.100
[61] J.-H. He, “A short remark on fractional variational iteration method,” Physics Letters A, vol. 375, no. 38, pp. 3362-3364, 2011. · Zbl 1252.49027 · doi:10.1016/j.physleta.2011.07.033
[62] J. H. He, G. C. Wu, and F. Austin, “The variational iterational method which should be follow,” Nonlinear Science Letters A, vol. 1, pp. 1-30, 2010.
[63] L. M. B. Assas, “Variational iteration method for solving coupled-KdV equations,” Chaos, Solitons & Fractals, vol. 38, no. 4, pp. 1225-1228, 2008. · Zbl 1152.35466 · doi:10.1016/j.chaos.2007.02.012
[64] 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 · doi:10.1016/j.physleta.2008.01.089
[65] E. Hesameddini and H. Latifizadeh, “Reconstruction of variational iteration algorithms using the laplace transform,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 11-12, pp. 1377-1382, 2009. · Zbl 1191.65165
[66] C. S. Pande and K. P. Cooper, “On the analytical solution for self-similar grain size distributions in two dimensions,” Acta Materialia, vol. 59, no. 3, pp. 955-961, 2011. · doi:10.1016/j.actamat.2010.10.019
[67] J. H. He, et al., First Course of Functional Analysis, Asian Academic Publisher, 2011.
[68] L.-H. Zhou and J. H. He, “The variational approach coupled with an ancient Chinese mathematical method to the relativistic oscillator,” Mathematical & Computational Applications, vol. 15, no. 5, pp. 930-935, 2010. · Zbl 1371.34053
[69] J.-H. He, “Recent development of the homotopy perturbation method,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 205-209, 2008. · Zbl 1159.34333
[70] J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 695-700, 2005. · Zbl 1072.35502 · doi:10.1016/j.chaos.2005.03.006
[71] J. H. He, “A note on the homotopy perturbation method,” Thermal Science, vol. 14, no. 2, pp. 565-568, 2010.
[72] J. H. He, “Homotopy perturbation method with an auxiliary term,” Abstract and Applied Analysis, vol. 2012, Article ID 857612, 7 pages, 2012. · Zbl 1235.65096 · doi:10.1155/2012/857612
[73] Y. Khan and S. T. Mohyud-Din, “Coupling of He’s polynomials and Laplace transformation for MHD viscous flow over a stretching sheet,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 12, pp. 1103-1107, 2010. · Zbl 06942720
[74] W. Zhang and X. Li, “Approximate damped oscillatory solutions for generalized KdV-Burgers equation and their error estimates,” Abstract and Applied Analysis, vol. 2011, Article ID 807860, 26 pages, 2011. · Zbl 1228.35211 · doi:10.1155/2011/807860
[75] H. Khan, S. Islam, J. Ali, and I. Ali Shah, “Comparison of different analytic solutions to axisymmetric squeezing fluid flow between two infinite parallel plates with slip boundary conditions,” Abstract and Applied Analysis, vol. 2012, Article ID 835268, 18 pages, 2012. · Zbl 1235.76028 · doi:10.1155/2012/835268
[76] Y.-Q. Jiang and J.-M. Zhu, “Solitary wave solutions for a coupled MKdV system using the homotopy perturbation method,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 359-367, 2008. · Zbl 1153.65101
[77] B.-G. Zhang, S.-Y. Li, and Z.-R. Liu, “Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations,” Physics Letters A, vol. 372, no. 11, pp. 1867-1872, 2008. · Zbl 1220.34010 · doi:10.1016/j.physleta.2007.10.072
[78] S. M. Schmalholz, “A simple analytical solution for slab detachment,” Earth and Planetary Science Letters, vol. 304, no. 1-2, pp. 45-54, 2011. · doi:10.1016/j.epsl.2011.01.011
[79] J.-H. He, “Modified Lindstedt-Poincaré methods for some strongly non-linear oscillations. I. Expansion of a constant,” International Journal of Non-Linear Mechanics, vol. 37, no. 2, pp. 309-314, 2002. · Zbl 1116.34320 · doi:10.1016/S0020-7462(00)00116-5
[80] J.-H. He, “Bookkeeping parameter in perturbation methods,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 2, no. 3, pp. 257-264, 2001. · Zbl 1072.34508 · doi:10.1515/IJNSNS.2001.2.3.257
[81] D. H. Shou and J. H. He, “Application of parameter-expanding method to strongly nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 1, pp. 121-124, 2007. · Zbl 06942250
[82] L. Xu, “Determination of limit cycle by He’s parameter-expanding method for strongly nonlinear oscillators,” Journal of Sound and Vibration, vol. 302, no. 1-2, pp. 178-184, 2007. · Zbl 1242.70038 · doi:10.1016/j.jsv.2006.11.011
[83] S. Q. Wang and J. H. He, “Nonlinear oscillator with discontinuity by parameter-expansion method,” Chaos, Solitons & Fractals, vol. 35, no. 4, pp. 688-691, 2008. · Zbl 1210.70023 · doi:10.1016/j.chaos.2007.07.055
[84] L. Xu, “Application of He’s parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire,” Physics Letters, Section A, vol. 368, no. 3-4, pp. 259-262, 2007. · doi:10.1016/j.physleta.2007.04.004
[85] M. T. Darvishi, A. Karami, and B.-C. Shin, “Application of He’s parameter-expansion method for oscillators with smooth odd nonlinearities,” Physics Letters A, vol. 372, no. 33, pp. 5381-5384, 2008. · Zbl 1223.70065 · doi:10.1016/j.physleta.2008.06.058
[86] F. Ö. Zengin, M. O. Kaya, and S. A. Demirba\vg, “Application of parameter-expansion method to nonlinear oscillators with discontinuities,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 3, pp. 267-270, 2008. · Zbl 06942345
[87] J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700-708, 2006. · Zbl 1141.35448 · doi:10.1016/j.chaos.2006.03.020
[88] X.-H. Wu and J.-H. He, “Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 966-986, 2007. · Zbl 1143.35360 · doi:10.1016/j.camwa.2006.12.041
[89] X.-H. Wu and J.-H. He, “EXP-function method and its application to nonlinear equations,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 903-910, 2008. · Zbl 1153.35384 · doi:10.1016/j.chaos.2007.01.024
[90] A. Bekir and A. Boz, “Exact solutions for a class of nonlinear partial differential equations using exp-function method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 4, pp. 505-512, 2007. · Zbl 06942300
[91] R. Mokhtari, “Variational iteration method for solving nonlinear differential-difference equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 1, pp. 19-23, 2008. · Zbl 1401.65152
[92] A. Yildirim, “Exact solutions of nonlinear differential-difference equations by He’s homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 111-114, 2008. · Zbl 06942330
[93] S. Zhang, “Application of Exp-function method to high-dimensional nonlinear evolution equation,” Chaos, Solitons & Fractals, vol. 38, no. 1, pp. 270-276, 2008. · Zbl 1142.35593 · doi:10.1016/j.chaos.2006.11.014
[94] X.-W. Zhou, “Exp-function method for solving Huxley equation,” Mathematical Problems in Engineering, vol. 2008, Article ID 538489, 7 pages, 2008. · Zbl 1151.92007 · doi:10.1155/2008/538489 · eudml:54841
[95] X. W. Zhou, Y. X. Wen, and J. H. He, “Exp-function method to solve the nonlinear dispersive K(m, n) equations,” The International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, pp. 301-306, 2008. · Zbl 06942352
[96] A.-M. Wazwaz, “Solitary wave solutions of the generalized shallow water wave (GSWW) equation by Hirota’s method, tanh-coth method and Exp-function method,” Applied Mathematics and Computation, vol. 202, no. 1, pp. 275-286, 2008. · Zbl 1147.65109 · doi:10.1016/j.amc.2008.02.013
[97] Z. L. Tao, “A note on the variational approach to the Benjamin-Bona-Mahony equation using He’s semi-inverse method,” Int. J. Comput. Math., vol. 87, pp. 1752-1754, 2010. · Zbl 1197.65159 · doi:10.1080/00207160802464571
[98] A.-M. Wazwaz, “Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh-coth method,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 633-640, 2007. · Zbl 1243.35148 · doi:10.1016/j.amc.2007.01.056
[99] M. Wang, X. Li, and J. Zhang, “The (G\(^{\prime}\)/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417-423, 2008. · Zbl 1217.76023 · doi:10.1016/j.physleta.2007.07.051
[100] S. Zhang, “A generalized auxiliary equation method and its application to (2+1)-dimensional Korteweg-de Vries equations,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1028-1038, 2007. · Zbl 1143.35363 · doi:10.1016/j.camwa.2006.12.046
[101] L. Geng and X. C. Cai, “He’s frequency formulation for nonlinear oscillators,” European Journal of Physics, vol. 28, no. 5, pp. 923-931, 2007. · Zbl 1162.70019 · doi:10.1088/0143-0807/28/5/016
[102] H. L. Zhang, “Application of He’s frequency-amplitude formulation to an x1/3 force nonlinear oscillator,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 3, pp. 297-300, 2008.
[103] L. Zhao, “Chinese mathematics for nonlinear oscillators,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 383-387, 2008. · Zbl 1146.01303
[104] J. Fan, “Application of He’s frequency-amplitude formulation to the Duffing-harmonic oscillator,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 389-394, 2008. · Zbl 1146.01302
[105] J. H. He, “An improved amplitude-frequency formulation for nonlinear oscillators,” The International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, pp. 211-212, 2008. · Zbl 06942340
[106] J. H. He, “Comment on ‘He’s frequency formulation for nonlinear oscillators’,” European Journal of Physics, vol. 29, no. 4, pp. L19-L22, 2008. · doi:10.1088/0143-0807/29/4/L02
[107] J. H. He, “Application of ancient chinese mathematics to optimal problems,” Nonlinear Science Letters A, vol. 2, no. 2, pp. 81-84, 2011.
[108] S. T. Qin and Y. Ge, “A novel approach to Markowitz Portfolio model without using lagrange multipliers,” The International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, supplement, pp. s331-s334, 2010. · Zbl 1401.91594
[109] X. S. Cha, Optimization and Optimal Control, Tsinghua University Press, Beijing, China, 1983. · Zbl 0535.76109
[110] J.-H. He and Q. Yang, “Solitary wavenumber-frequency formulation using an ancient Chinese arithmetic,” International Journal of Modern Physics B, vol. 24, no. 24, pp. 4747-4751, 2010. · Zbl 1218.34041 · doi:10.1142/S0217979210054245
[111] J.-H. He, “He Chengtian’s inequality and its applications,” Applied Mathematics and Computation, vol. 151, no. 3, pp. 887-891, 2004. · Zbl 1043.01004 · doi:10.1016/S0096-3003(03)00531-9
[112] J. H. He, “Max-min approach to nonlinear oscillators,” The International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, pp. 207-210, 2008. · Zbl 06942339
[113] Y. V. Kartashov, V. A. Vysloukh, A. Malomed Boris, and L. Torner, “Solitons in nonlinear lattices,” Reviews of Modern Physics, vol. 83, no. 1, pp. 247-305, 2011.
[114] G. C. Wu, L. Zhao, and J. H. He, “Differential-difference model for textile engineering,” Chaos, Solitons & Fractals, vol. 42, no. 1, pp. 352-354, 2009. · doi:10.1016/j.chaos.2008.12.011
[115] S. D. Zhu, “Exp-function method for the Hybrid-Lattice system,” The International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, pp. 461-464, 2007. · Zbl 06942293
[116] S. D. Zhu, “Exp-function method for the discrete mKdV lattice,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 465-468, 2007. · Zbl 06942294
[117] C.-Q. Dai and Y.-Y. Wang, “Exact travelling wave solutions of the discrete nonlinear Schrödinger equation and the hybrid lattice equation obtained via the exp-function method,” Physica Scripta, vol. 78, no. 1, article 015013, 6 pages, 2008. · Zbl 1144.81450 · doi:10.1088/0031-8949/78/01/015013
[118] R. Mokhtari, “Variational iteration method for solving nonlinear differential-difference equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 1, pp. 19-23, 2008. · Zbl 1401.65152
[119] A. Yildirim, “Exact solutions of nonlinear differential-difference equations by He’s homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 111-114, 2008. · Zbl 06942330
[120] J. He, “Some new approaches to Duffing equation with strongly and high order nonlinearity. II. Parametrized perturbation technique,” Communications in Nonlinear Science & Numerical Simulation, vol. 4, no. 1, pp. 81-83, 1999. · Zbl 0932.34058 · doi:10.1016/S1007-5704(99)90065-5
[121] J.-H. He, “A review on some new recently developed nonlinear analytical techniques,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 1, no. 1, pp. 51-70, 2000. · Zbl 0966.65056 · doi:10.1515/IJNSNS.2000.1.1.51
[122] X. H. Ding and L. Zhang, “Applying He’s parameterized perturbation method for solving differential-difference equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 9, pp. 1249-1252, 2009. · Zbl 06942499
[123] J.-H. He, S. K. Elagan, and G.-C. Wu, “Solitary-solution formulation for differential-difference equations using an ancient chinese algorithm,” Abstract and Applied Analysis, vol. 2012, Article ID 861438, 6 pages, 2012. · Zbl 1247.34018 · doi:10.1155/2012/861438
[124] J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57-68, 1998. · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X
[125] G. E. Dr\uag\uanescu, “Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives,” Journal of Mathematical Physics, vol. 47, no. 8, article 082902, 9 pages, 2006. · Zbl 1112.74009 · doi:10.1063/1.2234273
[126] Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential equations of fractional order,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 27-34, 2006. · Zbl 1378.76084
[127] N. T. Shawagfeh, “Analytical approximate solutions for nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 517-529, 2002. · Zbl 1029.34003 · doi:10.1016/S0096-3003(01)00167-9
[128] 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
[129] A. Ghorbani and J. Saberi-Nadjafi, “He’s homotopy perturbation method for calculating adomian polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 229-232, 2007. · Zbl 1401.65056
[130] A. Ghorbani, “Beyond Adomian polynomials: He polynomials,” Chaos, Solitons & Fractals, vol. 39, no. 3, pp. 1486-1492, 2009. · Zbl 1197.65061 · doi:10.1016/j.chaos.2007.06.034
[131] S. Momani and Z. Odibat, “Homotopy perturbation method for nonlinear partial differential equations of fractional order,” Physics Letters A, vol. 365, no. 5-6, pp. 345-350, 2007. · Zbl 1203.65212 · doi:10.1016/j.physleta.2007.01.046
[132] J. H. He, “A generalized poincaré-invariant action with possible application in strings and E-infinity theory,” Chaos, Solitons & Fractals, vol. 39, no. 4, pp. 1667-1670, 2009. · doi:10.1016/j.chaos.2007.06.047
[133] S. Das, “Solution of fractional vibration equation by the variational iteration method and modified decomposition method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 4, pp. 361-366, 2008. · Zbl 06942361
[134] Z. Odibat and S. Momani, “Applications of variational iteration and homotopy perturbation methods to fractional evolution equations,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 227-234, 2008. · Zbl 1172.26303
[135] S. Momani, Z. Odibat, and I. Hashim, “Algorithms for nonlinear fractional partial differential equations: a selection of numerical methods,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 211-226, 2008. · Zbl 1172.26302
[136] Z. Odibat and S. Momani, “Applications of variational iteration and homotopy perturbation methods to fractional evolution equations,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 227-234, 2008. · Zbl 1172.26303
[137] Z. Z. Ganji, D. D. Ganji, H. Jafari, and M. Rostamian, “Application of the homotopy perturbation method to coupled system of partial differential equations with time fractional derivatives,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 341-348, 2008. · Zbl 1163.35312
[138] Z.-B. Li and J.-H. He, “Fractional complex transform for fractional differential equations,” Mathematical & Computational Applications, vol. 15, no. 5, pp. 970-973, 2010. · Zbl 1215.35164
[139] J. H. He, S. K. Elagan, and Z. B. Li, “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus,” Physics Letters A, vol. 376, no. 4, pp. 257-259, 2012. · Zbl 1255.26002
[140] G. Jumarie, “Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution,” Journal of Applied Mathematics & Computing, vol. 24, no. 1-2, pp. 31-48, 2007. · Zbl 1145.26302 · doi:10.1007/BF02832299
[141] X. Yang, “Local fractional integral transforms,” Progress in Nonlinear Science, vol. 4, pp. 1-225, 2011.
[142] W. Chen, X. D. Zhang, and D. Koro\vsak, “Investigation on fractional and fractal derivative relaxation-oscillation models,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 1, pp. 3-9, 2010.
[143] G. C. Wu, “Variational iteration method for q-difference equations of second order,” Journal of Applied Mathematics, vol. 2012, Article ID 102850, 2012, http://www.hindawi.com/journals/jam/2012/102850/. · Zbl 1251.65170 · doi:10.1155/2012/102850
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.