Application of the homotopy perturbation method for the Fokker-Planck equation

International Journal for Numerical Methods in Biomedical Engineering

Volumn 26, Issue 9, 2010, Pages 1144-1154

  Yildirim, Ahmet ^a  

^a EGE UNIVERSITY   (Turkey)

Author keywords

Fokker planck equation; He's homotopy perturbation method; Kolmogorov equation; Nonlinear phenomena

Indexed keywords

ALTERNATIVE METHODS; APPROXIMATE SOLUTION; EXACT SOLUTION; FINITE DIFFERENCE; FINITE ELEMENT; HOMOTOPY PERTURBATION METHOD; KOLMOGOROV EQUATIONS; NON-LINEAR PHENOMENA; NONLINEAR DIFFERENTIAL EQUATION; SCIENCE AND ENGINEERING; SPECTRAL METHODS;

COMPUTATIONAL COMPLEXITY; DIFFERENTIAL EQUATIONS; FINITE ELEMENT METHOD; FOKKER PLANCK EQUATION; INITIAL VALUE PROBLEMS; PERTURBATION TECHNIQUES; SPECTROSCOPY; TIMING JITTER;

NONLINEAR EQUATIONS;

EID: 77956512231     PISSN: 20407939     EISSN: 20407947     Source Type: Journal    
DOI: 10.1002/cnm.1200     Document Type: Article

References (46)

1. Risken H. The Fokker-Planck Equation: Method of Solution and Applications. Springer: Berlin, 1989.
2. Rogers C, Shadwich WF. Backlund Transformation and their Applications. Academic Press: New York, 1982.
3. Gu CH. Darboux Transformations in Soliton Theory and its Geometric Applications. Shanghai Scientific and Technical Publishers: Shanghai, 1999.
4. Cherruault Y, Adomian G. Decomposition methods: a new proof of convergence. Mathematical and Computer Modelling 1993; 18:103-106.
5. Wazwaz AM. A reliable modification of Adomian decomposition method. Applied Mathematics and Computation 1997; 102:77-86.
6. Wazwaz AM. A new algorithm for calculating Adomian polynomials for nonlinear operators. Applied Mathematics and Computation 1997; 102:77-86.
7. Yusufoǧlu E, Bekir A. Numerical Solution of equal-width wave equation. Computers and Mathematics with Applications 2007; 54:1147-1153.
8. Wazwaz AM. Construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method. Chaos, Solitons and Fractals 2001; 12:2283-2293.
9. Dehghan M, Tatari M. Solution of a semilinear parabolic equation with an unknown control function using the decomposition procedure of Adomian. Numerical Methods for Partial Differential Equations 2007; 23:499-10
10. Dehghan M, Tatari M. The use of He's variational iteration method for solving a Fokker-Planck equation. Physica Scripta 2006; 74:310-316.
11. He JH, Wu XH. Exp-function method for nonlinear wave equations. Chaos, Solitons and Fractals 2006; 30:700-708.
12. Xu F. Application of Exp-function method to symmetric regularized long wave (SRLW) equation. Physics Letters A 2007; 372:252-257.
13. He JH. Variational iteration method-a kind of non-linear analytical technique: some examples. International Journal of Non-linear Mechanics 1999; 34:699-708.
14. He JH. Variational iteration method for autonomous ordinary differential system. Applied Mathematics and Computation 2000; 114:115-123.
15. He JH, Wu XH. Construction of solitary solution and compacton-like solution by variational iteration method. Chaos, Solitons and Fractals 2006; 29:108-113.
16. 1/3 force by He's variational iteration method. Journal of Sound and Vibration 2007; 306:372-376.
17. Wazwaz AM. The variational iteration method: a powerful scheme for handling linear and nonlinear diffusion equations. Computer Mathematics with Applications 2007; 54:933-939.
18. Sadighi A, Ganji DD. Exact solutions of nonlinear diffusion equations by variational iteration method. Computers and Mathematics with Applications 2007; 54:1112-1121.
19. Golbabai A, Javidi M. A variational iteration method for solving parabolic partial differential equations. Computers and Mathematics with Applications 2007; 54:987-992.
20. Momani S, Odibat Z, Alawneh A. Variational iteration method for solving the space- and time-fractional KdV equation. Numerical Methods for Partial Differential Equations 2008; 24:262-271.
21. Yi{dotless}ldi{dotless}ri{dotless}m A. Variational iteration method for modified Camassa-Holm and Degasperis-Procesi equations. Communications in Numerical Methods in Engineering 2008; DOI: 10.1002/cnm.1154.
22. Tatari M, Dehghan M, Razzaghi M. Application of the Adomian decomposition method for the Fokker-Planck equation. Mathematical and Computer Modelling 2007; 45:639-650.
23. He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Applied Mathematics and Computation 2004; 151:287-292.
24. He JH. Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons and Fractals 2005; 26:695-700.
25. He JH. Homotopy perturbation method for bifurcation of nonlinear problems. International Journal of Nonlinear Science and Numerical Simulation 2005; 6:207-208.
26. He JH. Homotopy perturbation method for solving boundary value problems. Physics Letters A 2006; 350:87-88
27. He JH. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering 1999; 178:257-262.
28. He JH. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-linear Mechanics 2000; 35:37-43.
29. He JH. Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation 2003; 135:73-79.
30. Öziş T, Yi{dotless}ldi{dotless}ri{dotless}m A. A note on He's homotopy perturbation method for van der Pol oscillator with very strong nonlinearity. Chaos, Solitons and Fractals 2007; 34:989-991.
31. Öziş T, Yi{dotless}ldi{dotless}ri{dotless}m A. A comparative study of He's homotopy perturbation method for determining frequency- amplitude relation of a nonlinear oscillator with discontinuities. International Journal of Nonlinear Science and Numerical Simulation 2007; 8:243-248.
32. Öziş T, Yi{dotless}ldi{dotless}ri{dotless}m A. Traveling wave solution of Korteweg-de Vries equation using He's homotopy perturbation method. International Journal of Nonlinear Science and Numerical Simulation 2007; 8:239-242.
33. 1/3) force by He' modified Lindstedt-Poincaré method. Journal of Sound and Vibration 2007; 301:415-419.
34. Yi{dotless}ldi{dotless}ri{dotless}m A, Öziş T. Solutions of singular IVPs of Lane-Emden type by homotopy perturbation method. Physics Letters A 2007; 369:70-76.
35. Yi{dotless}ldi{dotless}ri{dotless}m A. On the solution of the non-linear Korteweg-de Vries equation by the homotopy perturbation method. Communications in Numerical Methods in Engineering 2008; DOI: 10.1002/cnm.1146.
36. Shakeri F, Dehghan M. Solution of delay equations via a homotopy perturbation method. Mathematical and Computer Modelling 2008; 48:486-498.
37. Biazar J, Ghazvini H. Exact solutions for non-linear Schrödinger equations by He's homotopy perturbation method. Physics Letters A 2007; 366:79-84.
38. He JH. Recent development of the homotopy perturbation method. Topological Methods in Nonlinear Analysis 2008; 31:205-209.
39. He JH. Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B 2006; 20:1141-1199.
40. He JH. New interpretation of homotopy perturbation method. International Journal of Modern Physics B 2006; 20:2561-2568.
41. Yi{dotless}ldi{dotless}ri{dotless}m A. Exact solutions of nonlinear differential-difference equations by He's homotopy perturbation method. International Journal of Nonlinear Science and Numerical Simulation 2008; 9:111-114.
42. Dehghan M, Shakeri F. Use of He's homotpy perturbation method for solving a partial differential equation arising in modeling of flow in porous media. Journal of Porous Media 2008; 11:765-778.
43. Dehghan M, Shakeri F. Solution of an integro-differential equation arising in oscillating magnetic fields using He's homotopy perturbation method. Progress in Electromagnetic Research (PIER) 2008; 78:361-376.
44. Shakeri F, Dehghan M. Inverse problem of diffusion equation by He's homotopy perturbation method. Physica Scripta 2007; 75:551-556.
45. Dehghan M, Shakeri F. Solution of a partial differential equation subject to temperature overspecification by He's homotopy perturbation method. Physica Scripta 2007; 75:778-787.
46. Saadatmandi A, Dehghan M, Eftekhari A. Application of He's homotopy perturbation method for non-linear system of second-order boundary value problems. Nonlinear Analysis: Real World Applications 2008; DOI: 10.1016/j.nonrwa.2008.02.032.