Asymptotic methods for solitary solutions and compactons

Abstract and Applied Analysis

Volumn 2012, Issue , 2012, Pages

  He, Ji Huan ^a  

^a SOOCHOW UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84869494625     PISSN: 10853375     EISSN: 16870409     Source Type: Journal    
DOI: 10.1155/2012/916793     Document Type: Review

References (143)

1. Russell J. S., Report on waves. 1844. in Proceedings of the14th Meeting of the British Association for the Advancement of Science
2. Zabusky N. J., Kruskal M. D., Interaction of "solitons" in a collisionless plasma and the recurrence of initial states. Physical Review Letters 1965 15 6 240 243 2-s2.0-33846361348 10.1103/PhysRevLett.15.240
3. Korteweg D. J., de Vires G., On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary wave. Philosophical Magazine 1895 539 422 443
4. Gardner C. S., Greene J. M., Kruskal M. D., Miura R. M., Method for solving the Korteweg-deVries equation. Physical Review Letters 1967 19 19 1095 1097 2-s2.0-36049057587 10.1103/PhysRevLett.19.1095
5. He J.-H., Meyers R., Soliton perturbation. Encyclopedia of Complexity and Systems Science 2009 9 New York, NY, USA Springer 8453 8457
6. He J.-H., Zhu S. D., Meyers R., Solitons and compactons. Encyclopedia of Complexity and Systems Science 2009 9 New York, NY, USA Springer 8457 8464
7. He J.-H., Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B 2006 20 10 1141 1199 10.1142/S0217979206033796 2220565 ZBL1102.34039 (Pubitemid 43599585)
8. He J. H., An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. International Journal of Modern Physics B 2008 22 21 3487 3578 2-s2.0-50949112266 10.1142/ S0217979208048668
9. He J.-H., New interpretation of homotopy perturbation method. Addendum: some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B 2006 20 18 2561 2568 10.1142/S0217979206034819 2251264 (Pubitemid 44154678)
10. Rosenau P., Hyman J. M., Compactons: solitons with finite wavelength. Physical Review Letters 1993 70 5 564 567 2-s2.0-12044253199 10.1103/PhysRevLett.70.564
11. 3 films. Advanced Functional Materials 2011 21 11 2072 2079 2-s2.0-79957967241 10.1002/adfm.201001979
12. Lin J. S., Hildemann L. M., A nonsteady-state analytical model to predict gaseous emissions of volatile organic compounds from landfills. Journal of Hazardous Materials 1995 40 3 271 295 2-s2.0-0029159554 10.1016/0304-3894(94) 00088-X
13. Okrasiński W., Płociniczak Ł., A nonlinear mathematical model of the corneal shape. Nonlinear Analysis: Real World Applications 2012 13 3 1498 1505 10.1016/j.nonrwa.2011.11.014
14. He J.-H., A remark on a nonlinear mathematical model of the corneal shape. Nonlinear Analysis: Real World Applications 2012 13 6 2863 2865 10.1016/j.nonrwa.2012.04.014
15. Herman R. L., Exploring the connection between quasistationary and squared eigenfunction expansion techniques in soliton perturbation theory. Nonlinear Analysis: Theory, Methods & Applications 2005 63 5-7 e2473 e2482 2-s2.0-28044436277 10.1016/j.na.2005.02.034 (Pubitemid 41691393)
16. He J.-H., Non-perturbative methods for strongly nonlinear problems [Ph.D. thesis] 2006 Berlin, Germany de-Verlag im Internet GmbH
17. He J.-H., Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos, Solitons & Fractals 2004 19 4 847 851 10.1016/S0960-0779(03)00265-0 2009670 ZBL1135.35303
18. He J.-H., Generalized Variational Principles in Fluids 2003 Science & Culture Publishing House of China x+237 2094422
19. Lighthill M. J., Whitham G. B., On kinematic waves. II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society London Series A 1955 229 317 345 0072606 10.1098/rspa.1955.0089 ZBL0064.20906
20. Whitham G. B., Linear and Nonlinear Waves 1974 New York, NY, USA John Wiley & Sons xvi+636 0483954
21. Zheng W., A fluid dynamics model for the low speed traffic systems Acta Mechanica Sinica 1994 26 2 149 157
22. He J.-H., Variational approach to foam drainage equation. Meccanica 2011 46 6 1265 1266
23. He J.-H., Variational approach to impulsive differential equations using the semi-inverse method. Zeitschrift für Naturforschung A 2011 66 632 634
24. He J.-H., Variational approach for nonlinear oscillators. Chaos, Solitons & Fractals 2007 34 5 1430 1439 2335394 10.1016/j.chaos.2006.10.026 ZBL1152.34327 (Pubitemid 46907588)
25. Liu H. M., He J.-H., Variational approach to chemical reaction. Computers and Chemical Engineering 2004 28 9 1549 2-s2.0-2442537034 10.1016/j. compchemeng.2004.01.006 (Pubitemid 38638279)
26. Öziş T., Yildirim A., Application of He's semi-inverse method to the nonlinear Schrödinger equation. Computers & Mathematics with Applications 2007 54 7-8 1039 1042 10.1016/j.camwa.2006.12.047 2395642 ZBL1157.65465 (Pubitemid 47488818)
27. Zhang J., Variational approach to solitary wave solution of the generalized Zakharov equation. Computers & Mathematics with Applications 2007 54 7-8 1043 1046 2395643 10.1016/j.camwa.2006.12.048 ZBL1141.65391 (Pubitemid 47488819)
28. Dittrich W., Reuter M., Classical and Quantum Dynamics 1994 2nd Berlin, Germany Springer x+385 Advanced Texts in Physics 10.1007/978-3-642-56430-7 1883279
29. Carini A., Genna F., Some variational formulations for continuum nonlinear dynamics. Journal of the Mechanics and Physics of Solids 1998 46 7 1253 1277 10.1016/S0022-5096(98)00016-7 1630697 ZBL1030.74006 (Pubitemid 128378430)
30. Liu G.-L., A vital innovation in Hamilton principle and its extension to initial-value problems. Proceedings of the 4th International Conference on Nonlinear Mechanics August 2002 Shanghai, China Shanghai University Press 90 97
31. He J.-H., Hamiltonian approach to nonlinear oscillators. Physics Letters A 2010 374 23 2312 2314 10.1016/j.physleta.2010.03.064 2629840 ZBL1237.70036
32. He J. H., Zhong T., Tang L., Hamiltonian approach to duffing-harmonic equation. The International Journal of Nonlinear Sciences and Numerical Simulation 2010 11 supplement 43 46
33. He J. H., Preliminary report on the energy balance for nonlinear oscillations. Mechanics Research Communications 2002 29 2-3 107 111 2-s2.0-0036526626 10.1016/S0093-6413(02)00237-9 (Pubitemid 34992264)
34. Afrouzi G. A., Ganji D. D., Talarposhti R. A., He's energy balance method for nonlinear oscillators with discontinuities. International Journal of Nonlinear Sciences and Numerical Simulation 2009 10 3 301 304 2-s2.0-65349177656
35. Zhang H. L., Xu Y. G., Chang J. R., Application of he's energy balance method to a nonlinear oscillator with discontinuity. International Journal of Nonlinear Sciences and Numerical Simulation 2009 10 2 207 214 2-s2.0-65349088127
36. Ganji S. S., Ganji D. D., Karimpour S., He's Energy balance and He's variational methods for nonlinear oscillations in engineering. International Journal of Modern Physics B 2009 23 3 461 471 2-s2.0-66349083019 10.1142/S0217979209049644
37. Yildirim A., Saadatnia Z., Askari H., Khan Y., KalamiYazdi M., Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach. Applied Mathematics Letters 2011 24 12 2042 2051 10.1016/j.aml.2011.05.040 2826123
38. Xu L., A Hamiltonian approach for a plasma physics problem. Computers and Mathematics with Applications 2011 61 8 1909 1911 2-s2.0-79953744737 10.1016/j.camwa.2010.06.028
39. Xu L., He J. H., Determination of limit cycle by Hamiltonian approach for strongly nonlinear oscillators. International Journal of Nonlinear Sciences and Numerical Simulation 2010 11 12 1097 1101 2-s2.0-79951700937
40. Durmaz S., Altay Demirba S., Kaya M. O., High order hamiltonian approach to nonlinear oscillators. International Journal of Nonlinear Sciences and Numerical Simulation 2010 11 8 565 570 2-s2.0-78649523624
41. Didenkulova I., Pelinovsky E., Sergeeva A., Statistical characteristics of long waves nearshore. Coastal Engineering 2011 58 1 94 102 2-s2.0-78349305819 10.1016/j.coastaleng.2010.08.005
42. Lin C. C., A new variational principle for isenergetic flows. Quarterly of Applied Mathematics 1952 9 421 423 0044978 ZBL0046.18301
43. Washizu K., Variational Methods in Elasticity and Plasticity 1982 Oxford, UK Pergamon Press xii+412 0391680
44. Lin C. C., Hydrodynamics of helium II. 21 Proceedings of the International School of Physics 1963 Academic Press 93 146
45. Tannehill J. C., Anderson D. A., Pletcher R. H., Computational Fluid Mechanics and Heat Transfer 1997 Taylor & Francis
46. Herivel J. W., The derivation of the equations of motion of an ideal fluid by Hamilton's principle. Mathematical Proceedings of the Cambridge Philosophical Society 1955 51 344 349 0071194 ZBL0068.18802
47. Bretherton F. P., A note on Hamilton's principle for perfect fluids. The Journal of Fluid Mechanics 1970 44 19 31
48. Ecer A., Akay H. U., Investigation of transonic flow in a cascade using the finite element method. Journal of American Institute of Aeronautics and Astronautics 1981 19 9 1174 1182 10.2514/3.60057 626387 ZBL0479.76070 (Pubitemid 12506472)
49. Ecer A., Akay H. U., A finite element formulation for steady transonic euler equations. AIAA Journal 1983 21 3 343 350 2-s2.0-0020716916 (Pubitemid 13502828)
50. Goncharov V., Pavlov V., On the Hamiltonian approach: applications to geophysical flows. Nonlinear Processes in Geophysics 1998 5 4 219 240 2-s2.0-0032322759 (Pubitemid 29429399)
51. Holm D. D., Zeitlin V., Hamilton's principle for quasigeostrophic motion. Physics of Fluids 1998 10 4 800 806 10.1063/1.869623 1613939 ZBL1185.76848 (Pubitemid 128613768)
52. Hiroki F., Youhei F., Clebsch potentials in the variational principle for a perfect fluid. Progress of Theoretical Physics 2010 124 3 517 531 2-s2.0-78650657014 10.1143/PTP.124.517
53. Larsson J., A practical form of Lagrange-Hamilton theory for ideal fluids and plasmas. Journal of Plasma Physics 2003 69 3 211 252 2-s2.0-4243091770 10.1017/S0022377803002290
54. Salmon R., Hamiltonian fluid mechanics. Annual Review of Fluid Mechanics 1988 20 225 256 2-s2.0-0023770258
55. Seliger R. L., Whitham G. B., Variational principles in continuum mechanics. Proceedings of the Royal Society A 1968 305 1 25
56. Wagner H. J., On the use of Clebsch potentials in the Lagrangian formulation of classical electrodynamics. Physics Letters A 2002 292 4-5 246 250 1889420 10.1016/S0375-9601(01)00795-2 ZBL0995.78002
57. Clebsch A., Uber die integration der hydrodynamischen gleichungen. Journal für die Reine und Angewandte Mathematik 1859 56 1 1 10
58. He J.-H., Wu X.-H., Variational iteration method: new development and applications. Computers & Mathematics with Applications 2007 54 7-8 881 894 2395625 10.1016/j.camwa.2006.12.083 ZBL1141.65372 (Pubitemid 47488841)
59. He J.-H., Variational iteration method-some recent results and new interpretations. Journal of Computational and Applied Mathematics 2007 207 1 3 17 2332941 10.1016/j.cam.2006.07.009 ZBL1119.65049 (Pubitemid 46935380)
60. He J.-H., Wu X.-H., Construction of solitary solution and compacton-like solution by variational iteration method. Chaos, Solitons & Fractals 2006 29 1 108 113 10.1016/j.chaos.2005.10.100 2213340 ZBL1147.35338 (Pubitemid 43063201)
61. He J.-H., A short remark on fractional variational iteration method. Physics Letters A 2011 375 38 3362 3364 10.1016/j.physleta.2011.07.033 2826245
62. He J. H., Wu G. C., Austin F., The variational iterational method which should be follow. Nonlinear Science Letters A 2010 1 1 30
63. Assas L. M. B., Variational iteration method for solving coupled-KdV equations. Chaos, Solitons & Fractals 2008 38 4 1225 1228 10.1016/j.chaos.2007.02.012 2437911 ZBL1152.35466
64. Odibat Z. M., Construction of solitary solutions for nonlinear dispersive equations by variational iteration method. Physics Letters A 2008 372 22 4045 4052 10.1016/j.physleta.2008.01.089 2418404 ZBL1220.35143
65. Hesameddini E., Latifizadeh H., Reconstruction of variational iteration algorithms using the laplace transform. International Journal of Nonlinear Sciences and Numerical Simulation 2009 10 11-12 1377 1382 2-s2.0-77950427865
66. Pande C. S., Cooper K. P., On the analytical solution for self-similar grain size distributions in two dimensions. Acta Materialia 2011 59 3 955 961 2-s2.0-78650706889 10.1016/j.actamat.2010.10.019
67. He J. H., First Course of Functional Analysis 2011 Asian Academic Publisher
68. Zhou L.-H., He J. H., The variational approach coupled with an ancient Chinese mathematical method to the relativistic oscillator. Mathematical & Computational Applications 2010 15 5 930 935 2777699
69. He J.-H., Recent development of the homotopy perturbation method. Topological Methods in Nonlinear Analysis 2008 31 2 205 209 2432078 ZBL1159.34333
70. He J. H., Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons & Fractals 2005 26 3 695 700 2-s2.0-18844426016 10.1016/j.chaos.2005.03.006 (Pubitemid 40682499)
71. He J. H., A note on the homotopy perturbation method. Thermal Science 2010 14 2 565 568 2-s2.0-77955858024
72. He J. H., Homotopy perturbation method with an auxiliary term. Abstract and Applied Analysis 2012 2012 7 857612 10.1155/2012/857612
73. Khan Y., Mohyud-Din S. T., Coupling of He's polynomials and Laplace transformation for MHD viscous flow over a stretching sheet. International Journal of Nonlinear Sciences and Numerical Simulation 2010 11 12 1103 1107 2-s2.0-79951702509
74. Zhang W., Li X., Approximate damped oscillatory solutions for generalized KdV-Burgers equation and their error estimates. Abstract and Applied Analysis 2011 2011 26 807860 10.1155/2011/807860 2835261 ZBL1228.35211
75. Khan H., Islam S., Ali J., Ali Shah I., Comparison of different analytic solutions to axisymmetric squeezing fluid flow between two infinite parallel plates with slip boundary conditions. Abstract and Applied Analysis 2012 2012 18 835268 10.1155/2012/835268 2879935 ZBL1235.76028
76. Jiang Y.-Q., Zhu J.-M., Solitary wave solutions for a coupled MKdV system using the homotopy perturbation method. Topological Methods in Nonlinear Analysis 2008 31 2 359 367 2432094 ZBL1153.65101
77. Zhang B.-G., Li S.-Y., Liu Z.-R., Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations. Physics Letters A 2008 372 11 1867 1872 10.1016/j.physleta.2007.10.072 2398391 ZBL1220.34010
78. Schmalholz S. M., A simple analytical solution for slab detachment. Earth and Planetary Science Letters 2011 304 1-2 45 54 2-s2.0-79952630107 10.1016/j.epsl.2011.01.011
79. He J.-H., Modified Lindstedt-Poincaré methods for some strongly non-linear oscillations. I. Expansion of a constant. International Journal of Non-Linear Mechanics 2002 37 2 309 314 1907214 10.1016/S0020-7462(00)00116-5 (Pubitemid 32943373)
80. He J.-H., Bookkeeping parameter in perturbation methods. International Journal of Nonlinear Sciences and Numerical Simulation 2001 2 3 257 264 10.1515/IJNSNS.2001.2.3.257 1857700 ZBL1072.34508 (Pubitemid 33679256)
81. Shou D. H., He J. H., Application of parameter-expanding method to strongly nonlinear oscillators. International Journal of Nonlinear Sciences and Numerical Simulation 2007 8 1 121 124 2-s2.0-34249996463
82. Xu L., Determination of limit cycle by He's parameter-expanding method for strongly nonlinear oscillators. Journal of Sound and Vibration 2007 302 1-2 178 184 10.1016/j.jsv.2006.11.011 2295336 (Pubitemid 46209206)
83. Wang S. Q., He J. H., Nonlinear oscillator with discontinuity by parameter-expansion method. Chaos, Solitons & Fractals 2008 35 4 688 691 2-s2.0-34748917561 10.1016/j.chaos.2007.07.055 (Pubitemid 47484447)
84. Xu L., Application of He's parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire. Physics Letters, Section A 2007 368 3-4 259 262 2-s2.0-34547661887 10.1016/j.physleta.2007.04.004
85. Darvishi M. T., Karami A., Shin B.-C., Application of He's parameter-expansion method for oscillators with smooth odd nonlinearities. Physics Letters A 2008 372 33 5381 5384 10.1016/j.physleta.2008.06.058 2438230 ZBL1223.70065
86. Zengin F. Ö., Kaya M. O., Demirba S. A., Application of parameter-expansion method to nonlinear oscillators with discontinuities. International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 3 267 270 2-s2.0-47049103543
87. He J.-H., Wu X.-H., Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals 2006 30 3 700 708 2238695 10.1016/j.chaos.2006. 03.020 ZBL1141.35448 (Pubitemid 43903182)
88. Wu X.-H., He J.-H., Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method. Computers & Mathematics with Applications 2007 54 7-8 966 986 2395634 10.1016/j.camwa.2006. 12.041 ZBL1143.35360 (Pubitemid 47488812)
89. Wu X.-H., He J.-H., EXP-function method and its application to nonlinear equations. Chaos, Solitons & Fractals 2008 38 3 903 910 10.1016/j.chaos. 2007.01.024 2423371 ZBL1153.35384 (Pubitemid 351633038)
90. Bekir A., Boz A., Exact solutions for a class of nonlinear partial differential equations using exp-function method. International Journal of Nonlinear Sciences and Numerical Simulation 2007 8 4 505 512 2-s2.0-36248983300
91. Mokhtari R., Variational iteration method for solving nonlinear differential-difference equations. International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 1 19 23 2-s2.0-42449156432
92. Yildirim A., Exact solutions of nonlinear differential-difference equations by He's homotopy perturbation method. International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 2 111 114 2-s2.0-47049085208
93. Zhang S., Application of Exp-function method to high-dimensional nonlinear evolution equation. Chaos, Solitons & Fractals 2008 38 1 270 276 2417662 10.1016/j.chaos.2006.11.014 ZBL1142.35593 (Pubitemid 351503992)
94. Zhou X.-W., Exp-function method for solving Huxley equation. Mathematical Problems in Engineering 2008 2008 7 538489 10.1155/2008/538489 2402979 ZBL1151.92007 (Pubitemid 351701666)
95. Zhou X. W., Wen Y. X., He J. H., Exp-function method to solve the nonlinear dispersive K(m, n) equations. The International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 301 306
96. Wazwaz A.-M., 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 2008 202 1 275 286 10.1016/j.amc.2008.02.013 2437158 ZBL1147.65109
97. Tao Z. L., A note on the variational approach to the Benjamin-Bona-Mahony equation using He's semi-inverse method. Int. J. Comput. Math. 2010 87 1752 1754
98. Wazwaz A.-M., Multiple-soliton solutions for the KP equation by Hirota's bilinear method and by the tanh-coth method. Applied Mathematics and Computation 2007 190 1 633 640 10.1016/j.amc.2007.01.056 2338741 (Pubitemid 46856693)
99. Wang M., Li X., Zhang J., The (G ′ / G) -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A 2008 372 4 417 423 10.1016/j.physleta.2007.07.051 2381823
100. Zhang S., A generalized auxiliary equation method and its application to (2 + 1) -dimensional Korteweg-de Vries equations. Computers & Mathematics with Applications 2007 54 7-8 1028 1038 10.1016/j.camwa.2006.12.046 2395641 (Pubitemid 47488817)
101. Geng L., Cai X. C., He's frequency formulation for nonlinear oscillators. European Journal of Physics 2007 28 5 923 931 2-s2.0-36048969106 10.1088/0143-0807/28/5/016 (Pubitemid 350092681)
102. Zhang H. L., Application of He's frequency-amplitude formulation to an x 1 / 3 force nonlinear oscillator. International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 3 297 300 2-s2.0-47049091120
103. Zhao L., Chinese mathematics for nonlinear oscillators. Topological Methods in Nonlinear Analysis 2008 31 2 383 387 2432097 ZBL1146.01303
104. Fan J., Application of He's frequency-amplitude formulation to the Duffing-harmonic oscillator. Topological Methods in Nonlinear Analysis 2008 31 2 389 394 2432098 ZBL1146.01302
105. He J. H., An improved amplitude-frequency formulation for nonlinear oscillators. The International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 211 212
106. He J. H., Comment on 'He's frequency formulation for nonlinear oscillators'. European Journal of Physics 2008 29 4 L19 L22 2-s2.0-46749142553 10.1088/0143-0807/29/4/L02
107. He J. H., Application of ancient chinese mathematics to optimal problems. Nonlinear Science Letters A 2011 2 2 81 84
108. Qin S. T., Ge Y., A novel approach to Markowitz Portfolio model without using lagrange multipliers. The International Journal of Nonlinear Sciences and Numerical Simulation 2010 11 supplement s331 s334
109. Cha X. S., Optimization and Optimal Control 1983 Beijing, China Tsinghua University Press
110. He J.-H., Yang Q., Solitary wavenumber-frequency formulation using an ancient Chinese arithmetic. International Journal of Modern Physics B 2010 24 24 4747 4751 10.1142/S0217979210054245 2756285 ZBL1218.34041
111. He J.-H., He Chengtian's inequality and its applications. Applied Mathematics and Computation 2004 151 3 887 891 10.1016/S0096-3003(03)00531-9 2052467 ZBL1043.01004
112. He J. H., Max-min approach to nonlinear oscillators. The International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 207 210
113. Kartashov Y. V., Vysloukh V. A., Malomed Boris A., Torner L., Solitons in nonlinear lattices. Reviews of Modern Physics 2011 83 1 247 305
114. Wu G. C., Zhao L., He J. H., Differential-difference model for textile engineering. Chaos, Solitons & Fractals 2009 42 1 352 354 2-s2.0-67649613021 10.1016/j.chaos.2008.12.011
115. Zhu S. D., Exp-function method for the Hybrid-Lattice system. The International Journal of Nonlinear Sciences and Numerical Simulation 2007 8 461 464
116. Zhu S. D., Exp-function method for the discrete mKdV lattice. International Journal of Nonlinear Sciences and Numerical Simulation 2007 8 3 465 468 2-s2.0-34548577316
117. Dai C.-Q., Wang Y.-Y., Exact travelling wave solutions of the discrete nonlinear Schrödinger equation and the hybrid lattice equation obtained via the exp-function method. Physica Scripta 2008 78 1, article 015013 6 10.1088/0031-8949/78/01/015013 2447535 ZBL1144.81450
118. Mokhtari R., Variational iteration method for solving nonlinear differential-difference equations. International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 1 19 23 2-s2.0-42449156432
119. Yildirim A., Exact solutions of nonlinear differential-difference equations by He's homotopy perturbation method. International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 2 111 114 2-s2.0-47049085208
120. He J., Some new approaches to Duffing equation with strongly and high order nonlinearity. II. Parametrized perturbation technique. Communications in Nonlinear Science & Numerical Simulation 1999 4 1 81 83 10.1016/S1007- 5704(99)90065-5 1703402 ZBL0932.34058
121. He J.-H., A review on some new recently developed nonlinear analytical techniques. International Journal of Nonlinear Sciences and Numerical Simulation 2000 1 1 51 70 10.1515/IJNSNS.2000.1.1.51 1743120 ZBL0966.65056
122. Ding X. H., Zhang L., Applying He's parameterized perturbation method for solving differential-difference equation. International Journal of Nonlinear Sciences and Numerical Simulation 2009 10 9 1249 1252 2-s2.0-72449145355
123. He J.-H., Elagan S. K., Wu G.-C., Solitary-solution formulation for differential-difference equations using an ancient chinese algorithm. Abstract and Applied Analysis 2012 2012 6 861438 2-s2.0-84859768501 10.1155/2012/861438
124. He J.-H., Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods in Applied Mechanics and Engineering 1998 167 1-2 57 68 10.1016/S0045-7825(98)00108-X 1665221 ZBL0942.76077
125. Drǎgǎnescu G. E., Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives. Journal of Mathematical Physics 2006 47 8, article 082902 9 10.1063/1.2234273 2258596 ZBL1112.74009 (Pubitemid 44327387)
126. Odibat Z. M., Momani S., Application of variational iteration method to nonlinear differential equations of fractional order. International Journal of Nonlinear Sciences and Numerical Simulation 2006 7 1 27 34 2-s2.0-30344464250 (Pubitemid 43246579)
127. Shawagfeh N. T., Analytical approximate solutions for nonlinear fractional differential equations. Applied Mathematics and Computation 2002 131 2-3 517 529 10.1016/S0096-3003(01)00167-9 1920243 ZBL1029.34003 (Pubitemid 34813503)
128. Bildik N., Konuralp A., 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 2006 7 1 65 70 2-s2.0-32644457439 (Pubitemid 43246582)
129. Ghorbani A., Saberi-Nadjafi J., He's homotopy perturbation method for calculating adomian polynomials. International Journal of Nonlinear Sciences and Numerical Simulation 2007 8 2 229 232 2-s2.0-34249865652
130. Ghorbani A., Beyond Adomian polynomials: He polynomials. Chaos, Solitons & Fractals 2009 39 3 1486 1492 10.1016/j.chaos.2007.06.034 2512946 ZBL1197.65061
131. Momani S., Odibat Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order. Physics Letters A 2007 365 5-6 345 350 10.1016/j.physleta.2007.01.046 2308776 ZBL1203.65212
132. He J. H., A generalized poincaré-invariant action with possible application in strings and E-infinity theory. Chaos, Solitons & Fractals 2009 39 4 1667 1670 2-s2.0-63449085451 10.1016/j.chaos.2007.06.047
133. Das S., Solution of fractional vibration equation by the variational iteration method and modified decomposition method. International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 4 361 366 2-s2.0-51749111612
134. Odibat Z., Momani S., Applications of variational iteration and homotopy perturbation methods to fractional evolution equations. Topological Methods in Nonlinear Analysis 2008 31 2 227 234 2432080 ZBL1172.26303
135. Momani S., Odibat Z., Hashim I., Algorithms for nonlinear fractional partial differential equations: a selection of numerical methods. Topological Methods in Nonlinear Analysis 2008 31 2 211 226 2432079
136. Odibat Z., Momani S., Applications of variational iteration and homotopy perturbation methods to fractional evolution equations. Topological Methods in Nonlinear Analysis 2008 31 2 227 234 2432080 ZBL1172.26303
137. Ganji Z. Z., Ganji D. D., Jafari H., Rostamian M., Application of the homotopy perturbation method to coupled system of partial differential equations with time fractional derivatives. Topological Methods in Nonlinear Analysis 2008 31 2 341 348 2432092
138. Li Z.-B., He J.-H., Fractional complex transform for fractional differential equations. Mathematical & Computational Applications 2010 15 5 970 973 2777702 ZBL1215.35164
139. He J. H., Elagan S. K., Li Z. B., Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Physics Letters A 2012 376 4 257 259
140. Jumarie G., Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution. Journal of Applied Mathematics & Computing 2007 24 1-2 31 48 10.1007/BF02832299 2311947 ZBL1145.26302 (Pubitemid 46850236)
141. Yang X., Local fractional integral transforms. Progress in Nonlinear Science 2011 4 1 225
142. Chen W., Zhang X. D., Korošak D., Investigation on fractional and fractal derivative relaxation-oscillation models. International Journal of Nonlinear Sciences and Numerical Simulation 2010 11 1 3 9 2-s2.0-78649351641
143. Wu G. C., Variational iteration method for q -difference equations of second order. Journal of Applied Mathematics 2012 2012. http://www.hindawi.com/ journals/jam/2012/102850/ 102850 10.1155/2012/102850