He's homotopy perturbation method for solving the shock wave equation

Applicable Analysis

Volumn 88, Issue 7, 2009, Pages 997-1004

  Berberler, Murat Erşen ^a   Yildirim, Ahmet ^a  

^a EGE UNIVERSITY   (Turkey)

Author keywords

Conservation law; Shock wave equations; Soliton solutions; The homotopy perturbation method

Indexed keywords

EID: 70349618139     PISSN: 00036811     EISSN: 1563504X     Source Type: Journal    
DOI: 10.1080/00036810903114767     Document Type: Article

References (30)

1. J. Behrens, Atmospheric and ocean modeling with an adaptive finite element solver for the shallow-water equations, Appl. Numer. Math. 26(1-2) (1998), pp. 217-226.
2. A. Bermudez and M.E. Vazquez, Upwind methods for hyperbolic conservation laws with course terms, Comput. Fluids 23 (1994), pp. 1349-1071.
3. R. Bruno, Convergence of approximate solutions of the Cauchy problem for a 2 2 nonstrictly hyperbolic system of conservation laws, in Nonlinear Hyperbolic Problems, M. Shearer, ed., Vieweg Braunschweig, Taormina, 1993, pp. 487-494.
4. J. Smaller, Shockwaves and Reaction-Diffusion Equations, Springer, New York, 1983.
5. F.M. Allan and K. Al-Khaled, An approximation of the analytic solution of the shock wave equation, J. Comput. Appl. Math. 192 (2006), pp. 301-309.
6. J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput. 151 (2004), pp. 287-292.
7. J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simulation 6 (2005), pp. 207-208.
8. J.H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A 350 (2006), pp. 87-88.
9. J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178 (1999), pp. 257-262.
10. J.H. He, A coupling method of a homotopy technique and a perturbation technique for nonlinear problems, Int. J. Nonlinear Mech. 35 (2000), pp. 37-43.
11. J.H. He, Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput. 135 (2003), pp. 73-79.
12. J.H. He, Recent development of the homotopy perturbation method, Topol. Methods Nonlinear Anal. 31 (2008), pp. 205-209.
13. J.H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Phys. B 20 (2006), pp. 1141-1199.
14. J.H. He, New interpretation of homotopy perturbation method, Int. J. Modern Phys. B 20 (2006), pp. 2561-2568.
15. T. Öziş and A. Yildirim, Determination of periodic solution for a u(1/3) force by He' modified Lindstedt-Poincaré method, J. Sound Vibrat. 301 (2007), pp. 415-419.
16. A Yildirim and T. Öziş, Solutions of singular IVPs of Lane-Emden type by homotopy perturbation method, Phys. Lett. A 369 (2007), pp. 70-76.
17. F. Shakeri and M. Dehghan, Inverse problem of diffusion equation by He's homotopy perturbation method, Phys. Scr. 75 (2007), p. 551.
18. M. Dehghan and F. Shakeri, Solution of a partial differential equation subject to temperature overspecification by He's homotopy perturbation method, Phys. Scr. 75 (2007), p. 778.
19. E. Yusufoǧlu, Homotopy perturbation method for solving a nonlinear system of second order boundary value problems, Int. J. Nonlinear Sci. Numer. Simulation 8 (2007), p. 353.
20. T. Öziş and A. Yildirim, A note on He's homotopy perturbation method for van der Pol oscillator with very strong nonlinearity, Chaos Solitons Fractals 34 (2007), pp. 989-991.
21. T. Öziş and A. Yildirim, A comparative study of He's homotopy perturbation method for determining frequency-Amplitude relation of a nonlinear oscillator with discontinuities, Int. J. Nonlinear Sci. Numer. Simulation 8 (2007), pp. 243-248.
22. T. Öziş and A. Yildirim, Traveling wave solution of Korteweg-de Vries equation using He's homotopy perturbation method, Int. J. Nonlinear Sci. Numer. Simulation 8 (2007), pp. 239-242.
23. A. Yildirim, Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method, Comput. Math. Appl. 56 (2008), pp. 3175-3180.
24. A. Yildirim, He's homotopy perturbation method for nonlinear differential-difference equations, Int. J. Comput. Math. (2008), pp. 1-5. Available at www.informaworld.com/ smpp/content db=all content=a904886860.
25. A. Yildirim, The Homotopy perturbation method for approximate solution of the modified KdV equation, Zeitschr. Naturforsch. A J. Phys. Sci. 63a (2008), pp. 621-626.
26. A. Yildirim, Application of the homotopy perturbation method for the Fokker-Planck Equation, Commun. Numer. Methods Eng. (2008). Available at www3.interscience.wiley. com/journal/121538859/abstract?CRETRY=1&SRETRY=0.
27. K. Al-Khaled, Theory and computations in hyperbolic model problems, Ph.D. thesis, University of Nebraska-Lincoln, USA, 1996.
28. C.Y. Chow, An Introduction to Computational Fluid Mechanics, Wiley, New York, 1979.
29. J. Kevorkian, Partial Differential Equations, Analytical Solution Techniques, Wadsworth and Brooks, New York, 1990.
30. E.D. Banta, Lossless propagation of one-dimensional, finite amplitude sound waves, J. Math. Anal. Appl. 10 (1965), pp. 166-173.