Variational iteration method for initial and boundary value problems using he's polynomials

International Journal of Differential Equations

Volumn 2010, Issue , 2010, Pages

  Mohyud Din, Syed Tauseef ^a   Yildirim, Ahmet ^b   Hosseini, M M ^c  

^a HITEC University Taxila Cantt   (Pakistan)

^b EGE UNIVERSITY   (Turkey)

^c YAZD UNIVERSITY   (Iran)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84863695619     PISSN: 16879643     EISSN: 16879651     Source Type: Journal    
DOI: 10.1155/2010/426213     Document Type: Review

References (67)

1. Abbasbandy S., 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 2007 207 1 59 63 10.1016/j.cam.2006. 07.012 MR2332947 ZBL1120.65083 (Pubitemid 46935386)
2. Abbasbandy S., Numerical solution of non-linear Klein-Gordon equations by variational iteration method. International Journal for Numerical Methods in Engineering 2007 70 7 876 881 10.1002/nme.1924 MR2317382 (Pubitemid 46779426)
3. Abdou M. A., Soliman A. A., Variational iteration method for solving Burger's and coupled Burger's equations. Journal of Computational and Applied Mathematics 2005 181 2 245 251 10.1016/j.cam.2004.11.032 MR2146836 (Pubitemid 40715094)
4. Abdou M. A., Soliman A. A., New applications of variational iteration method. Physica D 2005 211 1-2 1 8 10.1016/j.physd.2005.08.002 MR2200437 ZBL1084.35539 (Pubitemid 41491773)
5. Abassy T. A., El-Tawil M. A., El Zoheiry H., Solving nonlinear partial differential equations using the modified variational iteration Padé technique. Journal of Computational and Applied Mathematics 2007 207 1 73 91 10.1016/j.cam.2006.07.024 MR2332949 ZBL1119.65095 (Pubitemid 46935388)
6. Batiha B., Noorani M. S. M., Hashim I., Variational iteration method for solving multispecies Lotka-Volterra equations. Computers & Mathematics with Applications 2007 54 7-8 903 909 10.1016/j.camwa.2006.12.058 MR2395627 ZBL1141.65370 (Pubitemid 47488840)
7. Biazar J., Ghazvini H., He's variational iteration method for fourth-order parabolic equations. Computers & Mathematics with Applications 2007 54 7-8 1047 1054 10.1016/j.camwa.2006.12.049 MR2395644 (Pubitemid 47496210)
8. Barari A., Omidvar M., Ghotbi A. R., Ganji D. D., Application of homotopy perturbation method and variational iteration method to nonlinear oscillator differential equations. Acta Applicandae Mathematicae 2008 104 2 161 171 10.1007/s10440-008-9248-9 MR2443274
9. Adomian G., Solution of the Thomas-Fermi equation. Applied Mathematics Letters 1998 11 3 131 133 10.1016/S0893-9659(98)00046-9 MR1625415 ZBL0947.34501 (Pubitemid 128389570)
10. Chan C. Y., Hon Y. C., A constructive solution for a generalized Thomas-Fermi theory of ionized atoms. Quarterly of Applied Mathematics 1987 45 3 591 599 MR910465 ZBL0639.34021
11. Cedillo A., A perturbative approach to the Thomas-Fermi equation in terms of the density. Journal of Mathematical Physics 1993 34 7 2713 2717 10.1063/1.530090 MR1224172 ZBL0777.34057
12. Fooladi F., Hosseinzadeh E., Barari A., Domairry G., Highly nonlinear temperature dependent fin analysis by variational iteration method. Heat Transfer Research. In press
13. Ganji D. D., Tari H., Jooybari M. B., Variational iteration method and homotopy perturbation method for nonlinear evolution equations. Computers & Mathematics with Applications 2007 54 7-8 1018 1027 10.1016/j.camwa.2006.12.070 MR2395640 ZBL1141.65384 (Pubitemid 47488816)
14. Sadighi A., mirgang@nit.ac.ir Ganji D. D., ddg-davood@yahoo.com Solution of the generalized nonlinear Boussinesq equation using homotopy perturbation and variational iteration methods. International Journal of Nonlinear Sciences and Numerical Simulation 2007 8 3 435 443 2-s2.0-0021496906
15. Ganji D. D., ddg-davood@yahoo.com Sadighi A., Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations. International Journal of Nonlinear Sciences and Numerical Simulation 2006 7 4 411 418 2-s2.0-0000013352 (Pubitemid 44433692)
16. Ganji Z. Z., Ganji D. D., Rostamiyan Y., Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by an analytical technique. Applied Mathematical Modelling 2009 33 7 3107 3113 10.1016/j.apm.2008.10.034 MR2502734
17. Ghorbani A., as-gh56@vahoo.com 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-s2.0-27144469470
18. Ghorbani A., as-gh56@yahoo.com Beyond Adomian polynomials: He polynomials. Chaos, Solitons & Fractals 2009 39 3 1486 1492 2-s2.0-33745852188 10.1016/j.chaos.2007.06.034
19. Golbabai A., Javidi M., mo-javidi@yahoo.com A variational iteration method for solving parabolic partial differential equations. Computers and Mathematics with Applications 2007 54 7-8 987 992 2-s2.0-19144365030 10.1016/j.camwa.2006.12.042 ZBL1141.65385 (Pubitemid 47496208)
20. He J.-H., Variational approach to the Thomas-Fermi equation. Applied Mathematics and Computation 2003 143 2-3 533 535 10.1016/S0096-3003(02)00380-6 MR1981713 ZBL1022.65083
21. He J.-H., jhhe@dhu.edu.cn 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-s2.0-34748815591 10.1142/S0217979208048668
22. He J.-H., Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B 2006 20 10 1141 1199 10.1142/S0217979206033796 MR2220565 ZBL1102.34039 (Pubitemid 43599585)
23. He J.-H., Illiu@yc.shu.edu.cn Variational iteration method-a kind of non-linear analytical technique: some examples. International Journal of Non-Linear Mechanics 1999 34 4 699 708 2-s2.0-0002228705 (Pubitemid 129537033)
24. He J.-H., Variational iteration method for autonomous ordinary differential systems. Applied Mathematics and Computation 2000 114 2-3 115 123 10.1016/S0096-3003(99)00104-6 MR1778658 ZBL1027.34009
25. 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 MR2213340 ZBL1147.35338 (Pubitemid 43063201)
26. He J.-H., Variational iteration method-some recent results and new interpretations. Journal of Computational and Applied Mathematics 2007 207 1 3 17 10.1016/j.cam.2006.07.009 MR2332941 ZBL1119.65049 (Pubitemid 46935380)
27. He J.-H., Wu X.-H., Variational iteration method: new development and applications. Computers & Mathematics with Applications 2007 54 7-8 881 894 10.1016/j.camwa.2006.12.083 MR2395625 ZBL1141.65372 (Pubitemid 47488841)
28. He J.-H., The variational iteration method for eighth-order initial-boundary value problems. Physica Scripta 2007 76 6 680 682 10.1088/0031-8949/76/6/016 MR2393451 ZBL1134.34307 (Pubitemid 350115899)
29. He J.-H., Recent development of the homotopy perturbation method. Topological Methods in Nonlinear Analysis 2008 31 2 205 209 MR2432078 ZBL1159.34333
30. He J.-H., Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering 1999 178 3-4 257 262 10.1016/S0045-7825(99)00018-3 MR1711041 ZBL0956.70017
31. He J.-H., Homotopy perturbation method for solving boundary value problems. Physics Letters A 2006 350 1-2 87 88 10.1016/j.physleta.2005.10.005 MR2199322 (Pubitemid 43087673)
32. He J.-H., Comparison of homotopy perturbation method and homotopy analysis method. Applied Mathematics and Computation 2004 156 2 527 539 10.1016/j.amc.2003.08.008 MR2087530 ZBL1062.65074
33. He J.-H., jhhe@dhu.edu.cn Homotopy perturbation method for bifurcation of nonlinear problems. International Journal of Nonlinear Sciences and Numerical Simulation 2005 6 2 207 208 2-s2.0-4243080805 (Pubitemid 40588629)
34. He J.-H., The homotopy perturbation method nonlinear oscillators with discontinuities. Applied Mathematics and Computation 2004 151 1 287 292 10.1016/S0096-3003(03)00341-2 MR2037967 ZBL1039.65052
35. He J.-H., A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics 2000 35 1 37 43 10.1016/S0020-7462(98)00085-7 MR1723761
36. Hon Y. C., Adomian's decomposition method for Thomas-Fermi. Southeast Asian Bulletin of Mathematics 1996 20 3 55 58 MR1430609 ZBL0858.34017
37. Inokuti M., Sekine H., Mura T., Nemat-Naseer S., General use of the Lagrange multiplier in nonlinear mathematical physics. Variational Method in the Mechanics of Solids 1978 New York, NY, USA Pergamon Press 156 162
38. Kaya D., An explicit and numerical solutions of some fifth-order KdV equation by decomposition method. Applied Mathematics and Computation 2003 144 2-3 353 363 10.1016/S0096-3003(02)00412-5 MR1993436 ZBL1024.65096
39. Kaya D., On the solution of a Korteweg-de Vries like equation by the decomposition method. International Journal of Computer Mathematics 1999 72 4 531 539 10.1080/00207169908804874 MR1735443 ZBL0948.65104 (Pubitemid 129581084)
40. Konuralp A., The steady temperature distributions with different types of nonlinearities. Computers & Mathematics with Applications 2009 58 11-12 2152 2159 10.1016/j.camwa.2009.03.007 MR2557342
41. Lu J., Variational iteration method for solving two-point boundary value problems. Journal of Computational and Applied Mathematics 2007 207 1 92 95 10.1016/j.cam.2006.07.014 MR2332950 ZBL1119.65068 (Pubitemid 46935389)
42. Ma W. X., Zhou D. T., Explicit exact solution of a generalized KdV equation. Acta Mathematica Scientia 1997 17 168 174
43. Ma W.-X., You Y., Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions. Transactions of the American Mathematical Society 2005 357 5 1753 1778 10.1090/S0002-9947-04-03726-2 MR2115075 ZBL1062.37077 (Pubitemid 40622693)
44. Ma W. X., You Y., Rational solutions of the Toda lattice equation in Casoratian form. Chaos, Solitons & Fractals 2004 22 2 395 406 10.1016/j.chaos.2004.02.011 MR2060875 ZBL1090.37053
45. Ma W.-X., Wu H., He J., Partial differential equations possessing Frobenius integrable decompositions. Physics Letters A 2007 364 1 29 32 10.1016/j.physleta.2006.11.048 MR2288009
46. Momani S., Abuasad S., Application of He's variational iteration method to Helmholtz equation. Chaos, Solitons & Fractals 2006 27 5 1119 1123 10.1016/j.chaos.2005.04.113 MR2164845 ZBL1086.65113 (Pubitemid 41329806)
47. Mohyud-Din S. T., Noor M. A., Noor K. I., Travelling wave solutions of seventh-order generalized KdV equations using He's polynomials. International Journal of Nonlinear Sciences and Numerical Simulation 2009 10 2 223 229
48. Mohyud-Din S. T., Noor M. A., Homotopy perturbation method for solving fourth-order boundary value problems. Mathematical Problems in Engineering 2007 2007 15 98602 MR2278494 10.1155/2007/98602 ZBL1144.65311 (Pubitemid 46198135)
49. Mohyud-Din S. T., syedtauseefs@hotmail.com Noor M. A., noormaslam@hotmail.com Noor K. I., khalidanoor@hotmail.com Traveling wave solutions of seventh-order generalized KdV equations by variational iteration method using Adomian's polynomials. International Journal of Modern Physics B 2009 23 15 3265 3277 2-s2.0-0041791040 10.1142/S0217979209052467
50. Noor M. A., noormaslam at hotmail.com Mohyud-Din S. T., syedtauseef sat hotmal.Com Homotopy perturbation method for solving nonlinear higher-order boundary value problems. International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 4 395 408 2-s2.0-34748871953
51. Noor M. A., noormaslam@hotmail.com Mohyud-Din S. T., syedtauseefs@hotmail.com Variational iteration method for solving higher-order nonlinear boundary value problems using He's polynomials. International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 2 141 157 2-s2.0-0034276401
52. Noor M. A., Mohyud-Din S. T., Modified variational iteration method for heat and wave-like equations. Acta Applicandae Mathematicae 2008 104 3 257 269 10.1007/s10440-008-9255-x MR2453425 ZBL1162.65397
53. Noor M. A., Mohyud-Din S. T., Variational homotopy perturbation method for solving higher dimensional initial boundary value problems. Mathematical Problems in Engineering 2008 2008 11 696734 MR2417233 10.1155/2008/696734 ZBL1155.65082
54. Noor M. A., Mohyud-Din S. T., Solution of singular and nonsingular initial and boundary value problems by modified variational iteration method. Mathematical Problems in Engineering 2008 2008 23 917407 MR2430646 10.1155/2008/917407 ZBL1155.65083
55. Rafei M., Daniali H., Application of the variational iteration method to the Whitham-Broer-Kaup equations. Computers & Mathematics with Applications 2007 54 7-8 1079 1085 10.1016/j.camwa.2006.12.054 MR2398129 ZBL1138.76024 (Pubitemid 47488824)
56. Sweilam N. H., Harmonic wave generation in non linear thermoelasticity by variational iteration method and Adomian's method. Journal of Computational and Applied Mathematics 2007 207 1 64 72 10.1016/j.cam.2006.07.013 MR2332948 (Pubitemid 46935387)
57. Sweilam N. H., Fourth order integro-differential equations using variational iteration method. Computers & Mathematics with Applications 2007 54 7-8 1086 1091 10.1016/j.camwa.2006.12.055 MR2398130 ZBL1141.65399 (Pubitemid 47488826)
58. Tatari M., Dehghan M., On the convergence of He's variational iteration method. Journal of Computational and Applied Mathematics 2007 207 1 121 128 10.1016/j.cam.2006.07.017 MR2332953 ZBL1120.65112 (Pubitemid 46950910)
59. Wazwaz A.-M., The modified decomposition method and Padé approximants for solving the Thomas-Fermi equation. Applied Mathematics and Computation 1999 105 1 11 19 10.1016/S0096-3003(98)10090-5 MR1706059 ZBL0956.65064 (Pubitemid 129525433)
60. Wazwaz A.-M., Analytic treatment for variable coefficient fourth-order parabolic partial differential equations. Applied Mathematics and Computation 2001 123 2 219 227 10.1016/S0096-3003(00)00070-9 MR1847912 ZBL1027.35006 (Pubitemid 32716101)
61. Wazwaz A.-M., The decomposition method for approximate solution of the Goursat problem. Applied Mathematics and Computation 1995 69 2-3 299 311 10.1016/0096-3003(94)00137-S MR1326677 ZBL0826.65077
62. 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 10.1016/j.camwa.2006.12.041 MR2395634 ZBL1143.35360 (Pubitemid 47488812)
63. Xu L., xlanmail@21cn.com The variational iteration method for fourth order boundary value problems. Chaos, Solitons & Fractals 2009 39 3 1386 1394 2-s2.0-22944440739 10.1016/j.chaos.2007.06.013
64. Xu L., He's homotopy perturbation method for a boundary layer equation in unbounded domain. Computers & Mathematics with Applications 2007 54 7-8 1067 1070 10.1016/j.camwa.2006.12.052 MR2398127 (Pubitemid 47488822)
65. Xu L., Variational iteration method for solving integral equations. Computers & Mathematics with Applications 2007 54 7-8 1071 1078 10.1016/j.camwa.2006.12.053 MR2398128 ZBL1141.65400 (Pubitemid 47488823)
66. Yusufoǧlu E., eyusufoglu@dumlupinar.edu.tr New solitonary solutions for the MBBM equations using Exp-function method. Physics Letters A 2008 372 4 442 446 2-s2.0-33751205605 10.1016/j.physleta.2007.07.062
67. Zhou X.-W., Wen Y.-X., He J.-H., Exp-function method to solve the nonlinear dispersive K (m, n) equations. International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 3 301 306 2-s2.0-40549134152