The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method

Journal of Mathematical Analysis and Applications

Volumn 345, Issue 1, 2008, Pages 476-484

  Inc, Mustafa ^a  

^a FIRAT UNIVERSITY   (Turkey)

Author keywords

Fractional derivative; Lagrange multiplier; Time and space fractional Burgers equations; Variational iteration method

Indexed keywords

EID: 43949121726     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2008.04.007     Document Type: Article

References (72)

1. Ablowitz M.J., and Clarkson P.A. Solitons: Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press, Cambridge
2. Miura M.R. Backlund Transformation (1978), Springer-Verlag, Berlin
3. Gu C.H., et al. Darboux Transformation in Solitons Theory and Geometry Applications (1999), Shangai Science Technology Press, Shanghai
4. Gardner C.S., et al. Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19 (1967) 1095
5. Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27 (1971) 1192
6. Malfliet W. Solitary wave solutions of nonlinear wave equations. Amer. J. Phys. 60 (1992) 659
7. Inc M., and Fan E.G. Extended tanh-function method for finding travelling wave solutions of some nonlinear partial differential equations. Z. Naturforsch. A 60 (2005) 7
8. Yan Z., and Zhang H.Q. New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics. Phys. Lett. A 252 (1999) 291
9. Inc M., and Evans D.J. A study for obtaining more solitary pattern solutions of fifth-order KdV-like equations. Int. J. Comput. Math. 81 (2004) 473
10. Wang M.L. Exact solutions for a compound KdV-Burgers equation. Phys. Lett. A 215 (1996) 279
11. Yan Z., and Zhang H.Q. New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water. Phys. Lett. A 285 (2001) 355
12. Fu Z., et al. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett. A 290 (2001) 72
13. Yan Z. The extended Jacobian elliptic function expansion method and its application in the generalized Hirota-Satsuma coupled KdV system. Chaos Solitons Fractals 15 (2003) 575
14. Ray S.S., and Bera R.K. An approximate solutions of a nonlinear fractional differential equation by Adomian decomposition method. Appl. Math. Comput. 167 2 (2005) 561-571
15. Bagley R.L., and Torvik P.J. A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27 (1983) 201-210
16. Bagley R.L., and Torvik P.J. Fractional calculus-A different approach to the analysis of viscoelastically damped structures. AIAA J. 21 (1983) 741-748
17. Bagley R.L., and Torvik P.J. Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J. 23 (1985) 918-925
18. Ichise M., Nagayanagi Y., and Kojima T. An analog simulation of non integer order transfer functions for analysis of electrode processes. J. Electron. Chem. Interf. Electrochem. 33 (1971) 253-265
19. Sun H.H., Onaral B., and Tsao Y. Application of positive reality princible to metal electrode linear polarization phenomena. IEEE Trans. Bimed. Eng. 31 (1984) 664-674
20. Sun H.H., Abdelvahab A.A., and Onaral B. Linear approximation of transfer function with a pole of fractional order. IEEE Trans. Automat. Control 29 (1984) 441-444
21. Mandelbrot B. Some noises with 1 / f spectrum, a bridge between direct current and white noise. IEEE Trans. Inform. Theory 13 (1967) 289-298
22. Hartley T.T. Chaos in a fractional order Chua system. IEEE Trans. Circuits Syst. 42 (1995) 485-490
23. Mainardi F. Fractional calculus: Some basic problem in continuum and statistical mechanics. In: Carpinteri A., and Mainardi F. (Eds). Fractals and Fractional Calculus in Continuum and Mechanics (1997), Springer, New York 291-348
24. Rossikhin Y.A., and Shitikova M.V. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50 (1997) 15-67
25. Magin R.L. Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32 (2004) 1-104
26. Podlubny I. Fractional Differential Equations (1999), Academic Press, San Diego
27. A. Luchko, R. Groneflo, The initial value problem for some fractional differential equations with the Caputo derivative, preprint series, A08-98, Fachbereich Mathematik und Informatik, Freie Universität Berlin, 1998
28. Miller K.S., and Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons, New York
29. Oldham K.B., and Spainer J. The Fractional Calculus (1974), Academic Press, New York
30. Caputo M. Linear model of dissipation whose q is almost frequency independent. J. Roy. Astr. Soc. 13 (1967) 529-539
31. Atanackovic T.M., and Stankovic B. On a system of differential equations with fractional derivatives arising in rod theory. J. Phys. A 37 (2004) 1241-1250
32. Shawagfeh N.T. Analytical approximate solutions for nonlinear fractional differential equations. Appl. Math. Comput. 131 (2002) 517-529
33. Daftardar-Gejji V., and Babakhani A. Analysis of a system of fractional differential equations. J. Math. Anal. Appl. 293 (2004) 511-522
34. Daftardar-Gejji V., and Babakhani A. Adomian decomposition: A tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301 (2005) 508-518
35. Daftardar-Gejji V., and Babakhani A. An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl. 316 (2006) 753-763
36. Momani S. Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos Solitons Fractals 28 (2006) 930-937
37. Biler P., Funaki T., and Woyczynski W.A. Fractal Burger equations. J. Differential Equations 148 (1998) 9-46
38. Biler P., Karch G., and Woyczynski W.A. Critical nonlinearity exponent and self-similar asyptotics for Lévy conservation laws. Ann. Henri Poincaré 5 (2001) 613-637
39. Mann J.A., and Woyczynski W.A. Growing fractal interfaces in the presence of self-similar hopping surface diffusion. Phys. A 291 (2001) 159-183
40. Stanescu D., Kim D., and Woyczynski W.A. Numerical study of interacting particles approximation for integro-differential equations. J. Comput. Phys. 206 (2005) 706-726
41. He J.H. A new approach to nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 2 4 (1997) 230-235
42. He J.H. Commun. Nonlinear Sci. Numer. Simul. 2 4 (1997) 235-236
43. He J.H. Approximate anlytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Engrg. 167 1-2 (1998) 57-68
44. He J.H. Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput. Methods Appl. Mech. Engrg. 167 1-2 (1998) 69-73
45. He J.H. Variational iteration method a kind of non-linear analytical technique; some examples. Internat. J. Non-Linear Mech. 34 (1999) 699-708
46. Ganji D.D., Jannatabadi M., and Mohseni E. Application of He's variational iteration method to nonlinear Jaulent-Miodek equations and comparing it with ADM. J. Comput. Appl. Math. 207 (2007) 35-45
47. Tian L., and Yin J. Shock-peakon and shock-compacton solutions for K (p, q) equation by variational iteration method. J. Comput. Appl. Math. 207 (2007) 46-52
48. He J.H., and Wu X.H. Construction of solitary solution and compacton-like solution by variational iteration method. Chaos Solitons Fractals 29 (2006) 108-113
49. Inc M. Numerical simulation of KdV and mKdV equations with initial conditions by the variational iteration method. Chaos Solitons Fractals 34 (2007) 1071-1081
50. Inc M. Numerical doubly-periodic solution of the (2 + 1)-dimensional Boussinesq equation with initial conditions by the variational iteration method. Phys. Lett. A 366 (2007) 20-24
51. Inc M. Exact and numerical solitons with compact support for nonlinear dispersive K (m, p) equations by the variational iteration method. Phys. A 375 (2007) 447-456
52. Dehghan M., and Tatari M. The use of He's variational iterational method for solving a Fokker-Planck equation. Phys. Scr. 74 (2006) 310-316
53. Draganescu G., Cofan N., and Rujan D. Nonlinear vibrations of a nano sized sensor with fractional damping. J. Optoelectron. Adv. Mater. 7 (2005) 877-884
54. Draganescu G. Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives. J. Math. Phys. 47 (2006) Art. No. 082902
55. Odibat Z., and Momani S. Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 7 (2006) 27-34
56. He J.H. Some asymptotic methods for strongly nonlinear equations. Internat. J. Modern Phys. B 20 (2006) 1141-1199
57. Dehghan M., and Shakeri F. Application of He's variational iteration method for solving the Cauchy reaction-diffusion problem. J. Comput. Appl. Math. 214 (2008) 435-446
58. Dehghan M., and Shakeri F. Approximate solution of a differential equation arising in astrophysics using the variational iteration method. New Astron. 13 (2008) 53-59
59. Shakeri F., and Dehghan M. Numerical solution of a biological population model using He's variational iteration method. Comput. Math. Appl. 54 (2007) 1197-1209
60. Tatari M., and Dehghan M. On the convergence of He's variational iteration method. J. Comput. Appl. Math. 207 (2007) 121-128
61. Abbasbandy S. A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials. J. Comput. Appl. Math. 207 (2007) 59-63
62. Abassy T.A., El-Tawil M.A., and El Zoheiry H. Solving nonlinear partial differential equations using the modified variational iteration Padé technique. J. Comput. Appl. Math. 207 (2007) 73-91
63. Inokvti M., Sekine H., and Mura T. General use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Nasser S. (Ed). Variational Method in the Mechanics of Solids (1978), Pergamon Press, Oxford
64. He J.H. Generalized Variational Principles in Fluids (2003), Science and Culture Publishing House of China (in Chinese)
65. J.H. He, Non-perturbative methods for strongly nonlinear problems, Disertation, de-Verlag in GmbH, Berlin, 2006
66. Kaya D., and Yokuş A. A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations. Math. Comput. Simulation 60 (2002) 507-512
67. Kaya D. An application of the decomposition method for the KdVB equation. Appl. Math. Comput. 152 (2004) 279-288
68. Momani S., and Al-Khaled K. Numerical solutions for systems of fractional differential equations by the decomposition method. Appl. Math. Comput. 162 (2005) 1351-1365
69. Momani S. Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method. Appl. Math. Comput. 165 (2005) 459-472
70. Momani S. An explicit and numerical solutions of the fractional KdV equations. Math. Comput. Simulation 70 (2005) 110-118
71. Momani S. Analytic and approximate solutions of the space- and time- fractional telegraph equations. Appl. Math. Comput. 170 (2005) 1126-1134
72. Momani S., and Odibat Z. Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Phys. Lett. A 355 (2006) 271-279